Université Montpellier 2 Sciences et Techniques du Languedoc
Thèse présentée en vu d’obtenir le grade de
Docteur de l’Université Montpellier 2 Discipline : Informatique Ecole Doctorale : Informations, Structures, Systemes (I2S) par
Da SILVA David
Caractérisation de la nature multi-échelles des plantes par des outils de géométrie fractale, application à la modélisation de l’interception de la lumière. Sous la direction de :
Christophe Godin Hervé Sinoquet Thèse soutenue le 24 Novembre 2008 devant le jury composé de :
Mme Marie-Catherine Vilarem M. Jacques Levy Vehel M. Michael Chelle Mme Annick Lesne M. Philippe Balandier M. Christophe Godin M. Christophe Fiorio
Université Montpellier 2 INRIA INRA LHES CEMAGREF INRIA Université Montpellier 2
Présidente Rapporteur Rapporteur Jury Jury Directeur de thèse Invité
Université Montpellier 2 Sciences et Techniques du Languedoc
Thesis submitted for the degree of
Doctor of Philosophy in Computer Science by
Da SILVA David
Characterizing multiscale nature of plants using fractal geometry descriptors, application on light-interception modeling. Advisors:
Christophe Godin Hervé Sinoquet Thesis defended on November 24th 2008 before the jury composed of:
Mme Marie-Catherine Vilarem M. Jacques Levy Vehel M. Michael Chelle Mme Annick Lesne M. Philippe Balandier M. Christophe Godin M. Christophe Fiorio
Université Montpellier 2 INRIA INRA LHES CEMAGREF INRIA Université Montpellier 2
Présidente Rapporteur Rapporteur Jury Jury Directeur de thèse Invité
à MaMie, elle sait ce qui compte. . .
Remerciements
J
e voudrais tout d’abord exprimer mes plus profonds remerciements aux deux géants qui ont bien voulu me préter leurs épaules, merci Christophe, merci Hervé. Cette thèse est le résultat d’une aventure tant scientifique qu’humaine et elle n’aurait été possible sans la participation d’un grand nombre de camarades de voyage, un grand merci à tous d’avoir partagé cette aventure, et mes plus plates excuses à ceux que j’aurais oublié en écrivant ces lignes à la hâte. Je profite de ce moment pour rendre hommage à Hervé, sa disparition lors de ma thèse m’a fait réaliser combien il m’avait apporté. Son enthousiasme et son courage resteront une grande source d’inspiration. Merci Hervé, et salut. Je remercie les différents instituts, UM2, CIRAD, INRA, INRIA et leurs directeurs(trices)/président(e)s qui ont permis la réalisation de cette thèse par un accueil, toujours chaleureux, au seins des différentes UMR, merci au LIRMM, à AMAP, au PIAF et à DAP. J’adresse de sincères remerciements aux membres de mon jury de thèse. Merci à Marie-Catherine Vilarem pour l’avoir présidé, pour sa gentillesse et pour sa grande disponibilité. Merci à Michaël Chelle et Jacques Levy-Vehel d’en avoir été les rapporteurs minutieux. Merci à Annick Lesne, Philippe Balandier et Chistophe Fiorio pour leurs remarques constructives et les échanges que nous avons eu lors de la soutenance. Cette thèse s’est principalement déroulé entre Clermont-Ferrand, au PIAF, et Montpellier dans l’équipe VirtualPlants. Merci à tous pour votre accueil. Merci à Nicolas, Bénédicte et Florent pour leur accueil chaleureux dans ce pays froid. Merci à Philippe et Mister André pour m’avoir initier aux joies des mesures de terrains. Merci à Gabriela pour sa présence et son courage. Merci professeur Pradal, pour les nombreuses discussion, et surtout pour ta vision du monde, je pense sincèrement que tu es plus proche du vrai que la plupart d’entre nous. Merci Yann pour ta patience envers mon inaptitude naturelle aux stats, merci surtout pour le reste, nos discussion, ta disponibilité et ta franchise, tu ne te caches pas derrière Markov ! Pour continuer dans la branche, celle des stats, merci à ma voisine de bureau, Florence, a.k.a Angie, ma mission s’achève là, mais je suis sur qu’il y en aura d’autres, on a un labo à monter. . . Merci Sam pour ton efficacité, y’a des bouts de toi dans fractalysis. Merci aussi à l’homme des ages farouches, Jérome, merci pour la chasse aux mammouths et pour la grimpette, je ne vais pas m’étendre, à quoi ça sert ? Szymon, my polish brother, we shared this great period and I’m very glad for that. For sure, more than few words on a card or a thesis page will remain. Merci à toi Moïse, PlantGL-man, pour ton aide et ton amitié, merci aussi à Céline pour sa bonne cuisine et sa compréhension de mon humour.
vii
Merci à Nico, LaFlèche, Pierre, Patrick, Romain, Mikaël, Damien, Evelyne, Yolande, Michel, Rodolphe, Colin, Pascal, Jean-Baptiste, Vincent, Etienne, Thomas, Chakkrit, Yassin, d’avoir croisé mon chemin et de l’avoir rendu plus agréable. Evidemment je remercie ma famille, au sens étendu du terme, qui pour des raisons encore mystérieuses, m’ont toujours soutenu et continue de le faire, merci. Cette thèse n’aurait pas pu voir le jour sans la générosité des habitants du Canta Aoussel. Fina, Michel, il faudrait plus que trois ans de thèse pour faire le tour de votre grand cœur, je vous dois beaucoup plus que vous je ne pourrais vous rendre, merci. Jack, le frère que je n’ai jamais eu. . . tu sais déjà tous ce que je pourrais te dire, simplement merci d’être toi et d’être là. Finalement, Maud n’a pas qu’un air d’ange. . . et c’est le mien. Merci de me supporter, de me soutenir, de m’avoir choisi. . .
Avignon, January 14, 2009.
viii
Contents Contents
ix
Introduction
1
1 Multiscale nature of plants . . . . .
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Plant architecture . . . . . . . . . . . Plant architecture representations 1.2.1 Spatial representations . . . . . . . 1.2.2 Geometric representations . . . . . 1.2.3 Topological representations . . . .
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2 Generation of computational models of plants 2.1 2.2 2.3
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29 31 33 33 33 34 36 37 38 40 40 40 47
Allain & Cloitre’s lacunarity . . . . . . . . . . . . . . . . . . . . . . . Centered lacunarity : Λ+ . . . . . . . . . . . . . . . . . . . . . . . . . . Assessing plant lacunarity . . . . . . . . . . . . . . . . . . . . . . . . .
51 53 57 64
Iterated Function Systems . . . . . . . L-Systems . . . . . . . . . . . . . . . . . . Three-dimensional digitizing systems 2.3.1 Contact digitizing . . . . . . . . . . . 2.3.2 Non contact digitizing . . . . . . . . Data base . . . . . . . . . . . . . . . . . . 2.4.1 Virtual plants . . . . . . . . . . . . . 2.4.2 Digitized plants . . . . . . . . . . . .
3 Fractal dimension of plants 3.1 3.2
3.3
Measuring objects . . . . . . . . . . . . . Notions of dimensions . . . . . . . . . . 3.2.1 Topological dimension . . . . . . . . 3.2.2 Hausdorff Besicovitch dimension . . 3.2.3 Self-similarity dimension . . . . . . . 3.2.4 Compass dimension . . . . . . . . . 3.2.5 Box counting method . . . . . . . . . 3.2.6 Two-surface dimension . . . . . . . . 3.2.7 Effective dimension . . . . . . . . . . Fractal dimension of plants . . . . . . 3.3.1 Box counting dimension . . . . . . . 3.3.2 Two-surface dimension . . . . . . . .
4 Lacunarity of plants 4.1 4.2 4.3
ix
5 Modelling light interception 5.1
5.2
69
Models representing plants with explicit description of the geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Ray tracing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Radiosity method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Models representing plants as a volume and statistical functions 5.2.1 Gap frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Departure from random canopy . . . . . . . . . . . . . . . . . . . . . 5.2.3 Canopy structure as parameter . . . . . . . . . . . . . . . . . . . . . .
6 MµSLI M 6.1 6.2
6.3 6.4 6.5
Introduction . . . . . . . . . . . . . . . . . . Modeling framework . . . . . . . . . . . . . 6.2.1 Multiscale representation of plants . . . . 6.2.2 Multiscale model of light interception . . 6.2.3 Assessing light interception . . . . . . . Plant database . . . . . . . . . . . . . . . . . 6.3.1 Plant material . . . . . . . . . . . . . . . Results and clumping analysis . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . 6.5.1 6.5.2 6.5.3 6.5.4
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Requirements in canopy structure description for an tion of light interception . . . . . . . . . . . . . . . Sensitivity analysis . . . . . . . . . . . . . . . . . . A unifying approach . . . . . . . . . . . . . . . . . Implementation issues and complexity . . . . . . .
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7 Modeling of light transmission under heterogeneous forest canopy 113 7.1
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Stand reconstruction . . . . . . . 7.1.1 Experimental unit . . . . . . . . 7.1.2 Crown reconstruction . . . . . . 7.1.3 Opacity evaluation . . . . . . . Estimation of light transmission 7.2.1 Beam path length . . . . . . . .
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Conclusion and prospects
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Bibliography
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A Papers and communication
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B Reference Manual for OpenAlea Fractalysis and MµSLIM modules 191 x
B.1 F R A C T A L Y S I S . . . . B.1.1 BCM . . . . . . B.1.2 MatrixLac . . B.2 MµSLIM . . . . . . . . B.2.1
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xi
Introduction
P
lants have a major role in several societal challenges such as, sustainable agronomy and development, environmental changes, and food security. To address these questions, one key approach is to better control and understand Plant development. To reach this goal, researchers are developing computer models of plant to understand the various biological processes that drive the plants development and their interactions. In the past years, models that take the structure into account have become increasingly popular with the evolution of dedicated programming tools and with the escalation of computer power. These Functional-Structural Plant Models (FSPMs) describe the development of the three-dimensional plant structure as it interacts with the environment, as an emerging property of their components development. The purpose of FSPMs is to better understand the link between the ecophysiological processes and the plant architecture, at several spatial and temporal scales [Godin et al. 2005]. Plant structure defines the main interface with the environment, and is the result of both the topological organization of the plant components, i.e. their adjacency, and their geometry, i.e. their shape, position and orientation. Each of these two aspects of plants structure have been intensively studied in the recent years and greatly differ according to species, environmental conditions or time period. They also differ by their role in FSPMs, the geometry is a key factor of the interaction between the plant and the environment (ressource capture, heat dissipation, disease propagation), while topology is critical for the modeling of transport fluxes (water, nutrients, phytosynthates, signals) within the plant. The high numbers of plant components, possibly defined at different scales, and their replications create a complex structure difficult to represent in models. In this thesis I study the complexity of plants structures in modelling approaches, with a focus on the geometrical aspect. In particular, we analyze two major properties found in plants: • the (strong) irregularity of plant shapes, • the degree of similarity resulting from the repetition of many levels of substructure with the same general pattern. These two notions can be described and analyzed as a function of the scale at which they are observed.
In the context of geometry, one of the most widely used framework to perform such multiscale analysis is the fractal geometry introduced by Mandelbrot [1983]. In the first part of this thesis, I characterize the complexity of plant geometry using two descriptors from fractal geometry. First, I evaluate the fractal dimension of plants, which describes the way they physically penetrate space at different scales. 1
2
Introduction
Second, I characterize the gap and cluster sizes distribution within plant, using threedimensional lacunarity estimation. In the second part, I investigate the influence of the multiscale organization of plant geometry on one of the the principal ecophysiological process: light interception. Light interception models usually use either a very detailed description of plant geometry or an abstraction of it using strong hypotheses like homogeneous spatial distribution of leaves. Detailed descriptions address the issue of plant structure but does not allow to grasp the main structural determinants of the process due to their over-parameterization. On the other hand, global descriptions are attractively simple because of the few parameters they contain, but do not capture the irregular nature of plant shapes which severely limits the generalization capacity of the approach. The development of an intermediate model allowing to assess the influence of the multiscale structure of plants while describing its irregularity with few parameters is a challenging problem. I address this problem by introducing a new model of light interception that bridges the gap between detailed and simple approaches.
Document organization The first chapter briefly describes the architecture of plants and their multiscale organization. The second chapter is an introduction to simulation methods that allow us to generate virtual plants. As an alternative way to create virtual plants, this chapter also presents various methods used for field measurement to obtain computer representation of real plants. A database of 3D plant mock-ups is constituted. The third chapter introduces the notion of fractal dimension. The relation between measure and dimension is first emphasized. Then, different ways of estimating the fractal dimension of an object are presented. Two particular methods are used to assess the fractal dimension of the 3D plants of our database: the box-counting and the two-surface methods. The fourth chapter introduces the notion of lacunarity. Classical definitions and properties are first presented and their limitation assessed. For better gap and cluster description, I introduce a variant of the notion of lacunarity. I then characterize some of its structural properties. Finally, these different notions are applied to our 3D plant database, and compared. The fifth chapter starts the investigation of the relation between light interception and plant structure. It recalls the basics of radiative transfers and how they are applied to plants. An emphasis is made on models that put the plant multiscale organization to contribution. The sixth chapter presents our multiscale light interception model (MµSLIM) and the results we obtained. This chapter is in the form of a published paper in SIAM journal Multiscale Modeling and Simulations. The seventh chapter illustrates how the MµSLIM model can be used as a component of a different research topics, and describe its integration in the OpenAlea plant modeling platform.
1
Multiscale nature of plants
“Natural abilities are like natural plants; they need pruning by study.” Sir F. Bacon
T
hree-dimensional structure of a plant is the result of both the topological organization of the plant components, i.e. their adjacency, and their geometry, i.e. their shape, position and orientation. This organization is the result of organogenesis that occurs throughout plant life. Understanding the growth process of plants is, thus, the key to understanding how the complex geometry of plants is created. According to Godin [2000], the term “plant architecture”was introduced in different contexts. For Hallé et al. [1978], it denotes the architectural model of a tree species, i.e. a set of rules that describes the growth patterns of an average individual of this species [Barthélémy et al. 1989; Robinson 1996; Barthélémy and Caraglio 2007]. For Ross [1981], plant architecture is “a set of features delineating the shape, size, geometry and external structure of a plant”. Here we shall use the more general definition proposed by Godin [2000], for whom plant architecture is “any individual description based on decomposition of the plant into components, specifying their biological type and/or their shape, and/or their location/orientation in space and/or the way these components are physically related one with another”. This definition better suits the purpose of this work for it puts emphasis on the geometry of individuals [Takenaka 1994]. Depending on the model’s aim, plant representation and its underlying formalism can strongly vary. For instance, plant transpiration can be assessed using only information about total leaf area, disregarding the woody structure while it would be mandatory for water fluxes modeling. We will start this chapter with the biological and botanical basics that underlie plant architecture. We will then present the most used representations in nowadays functional-structural plant modeling. The size of the basic units used in these representations, defining the scale at which the plant will be studied, can also be discussed.
3
1.1. Plant architecture
1.1
5
Plant architecture A plant results from the functioning of undifferentiated cells constituting the apical meristem. The term “meristem”is derived from the Greek word merizein, meaning to divide reflecting its functioning. Located at the tip of a stem, the shoot apical meristem generates groups of cells with different potentialities [Lyndon 1998] (Figure 1.1 a.). The apical growth process of the stem gives the plant the ability to develop in one direction and is the result of two processes: organogenesis and elongation. During the organogenesis, the insertion zone of a leaf on the stem is called a node and the region which separates two successive nodes is called the internode. According to the plant phyllotaxy, i.e. the arrangement of the leaves on the stem of a plant [Adler et al. 1997], more than one leaf can be inserted at one node and buds embedding lateral meristems may develop at the axil of each leaf (Figure 1.1 b.). The formation of these axillary meristems defines the branching process, and branching structures are created if these meristems themselves enter an apical growth process (Figure 1.1 b.). The unit formed by a node together with its leaf (or leaves) and lateral bud(s) plus the subtending internode represents the basic structural unit of the plant commonly called the metamer [White 1979].
Figure 1.1 – a. Shoot apex and b. stem organization. Each axis ends in an apical meristem usually protected in a bud. A stem is composed of a succession of metamers (dark grey) comprising the node (i.e the insertion point of the leaf on a stem), the corresponding one or several leaves and associated lateral buds and the preceding internode.
Plant development is achieved by the repetition of elementary botanical entities whose morphological, dimensional, functional and anatomical features change during ontogeny and according to several processes. A plant is thus a modular organism that can be viewed as a sequence of metamers [Harper and White 1986; Room et al. 1994]. The growth process can be continuous or rhythmic, but rhythmic extension of leafy axes is the more frequent growth pattern in plants and growth units are defined as the sequence of metamers developed during uninterrupted period of extension [Barthélémy and Caraglio 2007]. The apical growth and the branching processes are responsible for two basic types of modularity [Godin and Caraglio 1998]:
Chapter 1. Multiscale nature of plants
6
a.
b.
c.
Figure 1.2 – Different levels of modularity expression in plants. a. Kleinia, Asteraceae (Photo Hallé F.) where the modules are generated by the linear structure of its definite axes. b. Arocaria (Photo Grosfeld J.) with modules as the branching growth units. c. Parinari (Photo Caraglio Y.) where a modular representation can be given by its leafy shoots.
• Nodal modularity: the iterative nature of the apical growth process results in stems being constituted of sequence of metamers. • Axial modularity: every plant can be decomposed into a set of axes and can thus be considered as a modular structure made from the repetition of a module of type “axis”. Because of their origin in the fundamental growth processes, these basic modularities can be found in every plants. However, other types of modularities can exist in a plant, each one corresponding to an interpretation of the plant at a particular scale. A scale of description is defined by the choice of a unit to decompose the plant, the finer the unit, the higher the scale of description. The unit choice can be based on either botanical or artificial criteria (e.g. fixed time period). There is thus the theoretical possibility of finding numerous types of modularity for a single plant [Barthélémy 1991]. Furthermore, the study of a plant often requires the simultaneous description of its organization at different scales, i.e. a multiscale description [Godin and Caraglio 1998]. The relation between two scales can be of two types: if one modularity is a refinement of the other (e.g. growth units refined into internodes), they are nested. If the two modularities are not a refinement of each other, they are overlapping. In what follows, we will only consider nested modularities.
1.2
Plant architecture representations The modular nature of plants inspired several types of representations, all based on plant architecture decomposition. A single scale decomposition can be of two types: spatial or organ-based. The spatial decomposition divides space in voxels and considers the ones occupied by plant, whereas organ-based decomposition makes use of
1.2. Plant architecture representations
7
plant modules and can be divided in two classes: geometric or topological decomposition. The following classification is from Godin [2000].
a.
b.
c.
Figure 1.3 – Modular representation of an apple tree [Costes et al. 2003]. a. Spatial decomposition of the tree, cell colors depends on enclosed plant surface. b. Geometric decomposition including spatial and geometrical informations of leaves. c. Complete representation of the tree including topological, spatial and geometrical informations of the components.
1.2.1
Spatial representations In such representations plant components are not directly considered. Instead the plant is represented by the set of 3D voxels containing them. The simplest technique consists of using voxels with identical size, δ, leading to uniform subdivision of the space occupied by the plant, e.g. its bounding box. An illustration of the spatial representation of an apple tree [Costes et al. 2003] can be seen in Figure 1.3 a. Biological parameters characterizing the components (leaf area density, optical properties, etc. . . ) can be attached to each voxel as attributes. A voxel representation, Pδ , can thus be summarized as the list of occupied voxels, I, and optional attributes, ρ:
Pδ = {(i, ρi )}i∈ I . Each voxel is identified by its indices ( xi , yi , zi ) in the grid, with its spatial coordinates being ( xi .dx, yi .dy, zi .dz), where dx, dy, and dz are constants and δ = max (dx, dy, dz). This type of voxel based representations have been used in the context of light interception modeling [Sinoquet and Bonhomme 1992] and plant growth simulation [Greene 1989]. If required, a multiscale geometric representation can be achieved by using hierarchical voxel refinement, a voxel is decomposed into a set of voxels with smaller size. The size of the voxels varies according to the local irregularity of the object; the more irregular the shape, the finer the voxels (Figure 1.4). A multiscale geometric representation with m scales, P , can be formalized as a set of m nested voxel representations where the different voxel sizes are multiples of one another: dxs = k.dxs+1 P = {Pδs }06s6m | ∀s dys = k.dys+1 , k ∈ N+∗ . dzs = k.dzs+1
8
Chapter 1. Multiscale nature of plants
The spatial coordinate of a voxel A well-known version of multiscale geometric representation is the octree. An octree is obtained by dividing voxel size by 2 at each scale. They have been broadly used, specifically in computer graphics when dealing with light rendering [Sillion 1995]. This kind of representation needs only spatial positioning of plant components,
Figure 1.4 – Octree representation of an apple tree. The object geometry is approximated by voxels whose sizes are locally adapted to its irregularity. Representation generated using PlantGL library.
therefore it can be implemented from easy-to-obtain data like photographs [Phattaralerphong and Sinoquet 2005].
1.2.2
Geometric representations This type of representation takes the architecture of plants into account. The plant is decomposed into organs such as leaves, fruits, metamers or different type of growth units, and their geometry and spatial position and orientation are considered [Godin 2000]. The connections between the organs are disregarded and not all types of organs need to be used (Figure 1.3 c.). For instance, when dealing with light interception, one may be only interested in leaves positioning for they are the main organs responsible for photosynthesis. These types of modular representations are frequently used to obtain accurate descriptions of the plant interface with the environment [Chelle and Andrieu 1998; Sinoquet et al. 1998]. Similarly to spatial representation a multiscale approach can be defined by considering different comparable modularities. The shapes of coarser scale components are successively decomposed into smaller shapes corresponding to the geometries of their components in the finer scale decomposition. The decomposition is repeated until the desired level of accuracy is obtained. In practice this is generally done in the reverse way, by gathering the components of the finest scale, usually the one associated with metamer modularity, into coarser components until the scale of the whole plant is reached as illustrated in Figure 1.5. This type of multiscale representation has been used to characterize the complex geometry of canopy and isolated trees [Boudon et al. 2006]. A new approach using multiscale geometric representation to model radiative transfer in canopies will be presented in chapter 6 of this document.
1.2. Plant architecture representations
a.
9
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c.
Figure 1.5 – Multiscale geometric representation of an apple tree. The decomposition is based on architectural modules: a. Leaves ; b. Branches ; c. Crown.
1.2.3
Topological representations These organ-based representations put the emphasis on the connections between plant components. An increasing number of structure-function models use topological representations to address various processes such as water fluxes within plants [Früh 1997; Dauzat et al. 2001], carbon partitioning [Valentine 1985; Nikinmaa 1992; Allen et al. 2006] or signals [Janssen and Lindenmayer 1987; Prusinkiewicz 1998] through plant components. The use of topological representation of plant architecture was initiated by Honda [1971]. He demonstrated that complex crown shapes could be obtained using limited number of geometric parameters and that plant architecture is very sensitive to change in these parameters [Godin 2000]. Topological representations to obtain realistic rendering of 3D plants have become increasingly widespread as computers have become more powerful [Fitter 1986; de Reffye et al. 1988; Prusinkiewicz and Lindenmayer 1990; Ford et al. 1990; Prusinkiewicz et al. 1994; Weber and Penn 1995]. The above list is not exhaustive, however all these applications using a topological representation of plant architecture have a common underlying structure, namely that of a tree graph. Let us consider the set of components resulting from decomposition of a plant into modules. The network made by these connected components can be represented by a binary relation defined over the set of plant components, i.e. a tree graph [Godin and Caraglio 1998]. Definition 1.1
A graph G is a pair G = (V , E ) where
V denotes a finite set of vertices, E denotes a finite set of edges, each edge being represented by a pair of vertices, E ⊆ V × V . Because of the special nature of plant growth, graphs representing plant topology are of a particular type, known as rooted tree graphs. In graph theory, a tree is a graph in which any two vertices are connected by exactly one path. Alternatively, any connected graph with no cycles is a tree.
Chapter 1. Multiscale nature of plants
10
A tree is called a rooted tree if one vertex, called the root, has been tagged. In which case the edges have a natural orientation, towards or away from the root.
Definition 1.2
In this document we will restrict to the case where the edges are all directed away from the root. Any vertex with no successor, i.e. no outgoing edge, is called a leaf of the rooted tree. Godin and Caraglio [1998] introduced a global formalism to represent plant organization using rooted trees. They distinguished two types of connections to mark the hierarchical organization of components in plants. When two components have been created by the same apical meristem, their connection is noted “<” (precedes). When two components have been created by different apical meristems, their connection is noted “+” (bears). Additional information can be associated with plant organs in topological representations by adding features to the corresponding vertices in the tree graph. This information may correspond to the spatial position of an organ in space, its geometry, or any other characteristic of the organ. The resulting representation is called an augmented tree graph. An illustration is given Figure 1.6 a.
a.
b.
Figure 1.6 – Equivalence between an augmented tree graph, a., and an axial tree, b. [Godin and Caraglio 1998]. Vertices in one representation are edges in the second and vice versa. Letter A denotes terminal components called apex. Tagging edges with connection information and grouping components into axis are equivalent.
A dual way of representing plant modularity using rooted tree graph, called axial trees, has been proposed by Prusinkiewicz and Lindenmayer [1990] in the context of plant growth simulation with L-systems. In axial trees, plants are described as rooted tree graphs where vertices represent connecting points between plant components and edges represent the components themselves, referred to as branch segment (Figure 1.6 b.). A similar distinction is made for the connection types motivated by the botanical notion of “branch axis ”. At each node of the axial tree, at most one outgoing straight segment is distinguished. All remaining edges are called lateral or side segments. The following definition is from [Prusinkiewicz and Lindenmayer 1990]. Definition 1.3
A sequence of segments is called an axis if:
• the first segment in the sequence originates at the root of the plant or as a lateral segment at some node,
1.2. Plant architecture representations
11
• each subsequent segment is a straight segment and, • the last segment is not followed by any straight segment in the plant. An axis and all its descendants define a axial sub-tree. Axes have order, the one originating at the root has order 0. Any axis born by a n-order parent axis has order n + 1. The order of an axial sub-tree is equal to the lowest-order of its constituting axes. As we discussed previously, several modularities exist at the same time on a plant (at least the nodal and axial modularities). For each of this modularities a particular topological representation can be defined. The set of these topological representations along with their relations define the multiscale topological structure of the plant [Godin and Caraglio 1998]. Let us consider the topological structure of a plant for a given modularity. Consider now a comparable modularity, coarser than the previous one: every constituent of the coarse modularity can be decomposed into a set of constituents of the fine modularity and reciprocally, every constituent of the fine modularity is a component of some coarse constituent. This decomposition relation between constituents of both modularities can be modeled by an onto mapping from the set of constituents of the fine modularity to the set of constituents of the coarse modularity. Two comparable modularities can thus be represented by a quotiented graph defined as follows: Definition 1.4
A quotiented graph G is a triple (h, V , π) where:
• h is a simple graph, called the support of G , • V is a set of vertices, • π is an onto mapping from V|h to V , V|h being the restriction of V to h. The graph h represents the topological structure associated with the finest modularity, V represents the set of constituents of the coarse modularity and the onto mapping π defines how every constituent v of the coarse modularity is decomposed into a set of finer constituents. π only maps vertices of h, corresponding to the fine modularity, onto vertices of V , corresponding to the coarse modularity, see illustration between scale 3 and scale 2 in Figure 1.7. The vertex π ( x ) is called the complex of x and reciprocally, x is a component of π ( x ). The onto mapping π induces a partition ΠG over the set of vertices V|h : Π G = { π −1 ( v ), v ∈ V } . We shall denote Π( x ) the block of ΠG containing the vertex x of V|h . The partition ΠG is thus a representation of the coarse modularity in terms of the elements of the fine modularity. To ensure coherence between the different scales representations of a plant, the quotient operation must ensure that, if there exists an edge between two components at scale n and if the complexes of this two components are different, then there must be an edge between the two complexes at scale n − 1 [Godin et al. 2005]. Connection between two modules results from the connection between two of their components. Intuitively, quotiented graphs can be generalized by recurrently applying new quotient operations on the successive quotient graphs obtained from an initial graph.
12
Chapter 1. Multiscale nature of plants
Applying several consecutive quotient operations on an initial object rapidly leads us to collapse the entire object into one single vertex. The object representations at the different scales form a pyramidal structure whose bottom is made of vertices at the highest scale and whose top is made of the vertex at the most macroscopic scale representing the entire plant. This entire structure defines a multiscale graph. However, there is no guarantee that a quotiented graph whose support is a tree graph is also a tree graph. Graphs that verify such property are called quotiented tree graph and multiscale tree graph if more than two scales are considered. A multiscale tree graph (MTG) is a multiscale graph whose support graph is a tree graph and where all quotiented graphs are quotiented tree graphs.
Definition 1.5
Precise definitions and properties defining a multiscale tree graph are discussed in [Godin and Caraglio 1998].
Figure 1.7 – Illustration of a multiscale tree graph. The decomposition of all vertices is not shown for readability reasons [Ferraro 2000].
Conclusion In this chapter we gave botanical and biological notions that underlie plant’s development (apical growth, branching process,. . . ) and showed that it organizes plant structure at different scales. Different ways of representing this complex organization have been shown, their use depending on the objective of the plant study. Emphasis was put on the topological representation because of its importance regarding the understanding of plant organization and development as underlined by several authors [Hallé et al. 1978; Room et al. 1994]. Godin and Caraglio [1998] defined a general mathematical formalism for describing a plant multiscale topological structure and its growth. Moreover, all presented representations can be extracted from a MTG with appropriate informations associated with it, namely position and geometry of components.
Generation of computational models of plants
“L’imagination se lassera plutôt de concevoir que la nature de fournir.” B. Pascal
O
ur understanding of natural processes mainly depends on our ability to create models that simplifies reality while producing results with acceptable reliability. In the last decades, the increasing computer power has generated considerable development of new models focusing on the functioning of plants [Godin and Sinoquet 2005]. These models have, in turn, triggered the need of 3D computational models of plants, called virtual plants [Room et al. 1996; Prusinkiewicz 2004a], to test or validate the models and for visualization purposes. The ability to generate virtual plants with adapted geometric representation allows to make virtual experiments and measurements impossible to set in the real world due to time, cost or feasibility constraints [Godin et al. 2005]. The modular structure of plants at different scales, leading to different levels of selfsimilarity, inspired some authors to use tools from a particular field of mathematics: fractal geometry. This new geometric framework allowed modelers to simulate the complex geometry of plants [Mandelbrot 1983; Oppenheimer 1986] and was successfully used to illustrate how models with few parameters can generate intricate structures [Smith 1984; Barnsley 1988; Prusinkiewicz and Hanan 1989], and artificial plants in modeling applications [Prusinkiewicz and Lindenmayer 1990; Chen et al. 1994]. To provide a formal description of the development of simple multicellular organisms Lindenmayer [1968] introduced an innovative paradigm based on rewriting rules. This formalism, called L-systems, initiated the modeling of dynamical structure that evolve throughout time, now known as Dynamical Systems with a Dynamical Structure (DS2 ) Giavitto and Michel [2003]. Plants with their growth process can be considered DS2 . The extension of the L-systems paradigm to the complex branching structures that are plants [Prusinkiewicz and Hanan 1989; Prusinkiewicz and Lindenmayer 1990], marked an important milestone in the development of computational models of plants. Regardless of the generation process, the quality of these artificial plants should be assessed by comparison with real measurement. The level of precision required to perform detailed comparison can only be provided by three-dimensional (3D) digitizing systems, which are frequently tedious and necessitate human operator.
13
2
14
Chapter 2. Generation of computational models of plants
In this chapter we will start by presenting two widespread simulation systems that are able to synthesize plant-like structure: Iterated Function Systems and L-systems. We then briefly review different types of 3D digitizing methods that can be used to get 3D mockups of real plants. Finally, the database of 3D mock-ups that will be used throughout this document, obtained by either methods, will be presented.
2.1. Iterated Function Systems
2.1
15
Iterated Function Systems Modularities present in plants tends to reproduce similar structures at different locations, but also at different scales [Godin 2003]. This produces self-similar structures that can be more or less obvious depending on plants. This property associated with their strong irregular geometry is a reminder to particular fractal objects: self-similar objects. Intuitively, a part of a self-similar object is exactly or approximately similar to its entire self, i.e. the parts have the same shapes as the whole. Many objects in the real world, such as coastlines, are statistically self-similar: parts of them show the same statistical properties at many scales [Mandelbrot 1967]. The self-similarity property can actually be used to characterize such complex and multiscale objects [Falconer 1990]. Different techniques have been proposed to generate self-similar objects. One of the
a.
b.
Figure 2.1 – Plants with high degree of self-similarity: a. Romanesco cauliflower and b. Fern.
most widespread is called Iterated Function Systems1 (IFS) and provides a very convenient way of generating self-similar fractal sets [Barnsley 1988]. This kind of objects have well defined mathematical properties and specific multiscale structure where any component at any scale is a smaller replication of the entire object. Let us consider a non-empty compact subset of Rn , A, with the usual Euclidian metric on Rn , d. The distance between points x, y ∈ Rn is noted d( x, y). A mapping ω : A → A is called a contraction on A if there is a number c with 1 < c such that 1 ∀ x, y ∈ Rn d (ω ( x ), ω (y)) 6 d( x, y) c
Definition 2.1
If equality holds, i.e. d (ω ( x ), ω (y)) = 1c d( x, y), then ω transforms sets into geometrically similar ones, and ω is called a similarity. Let F be a set of A. The transformation of F by a contraction is the set formed by all transformed points of F : ω (F ) = {ω ( x ) | x ∈ F }. An Iterated Function System, Ω, consists of a family of contractions {ω1 , . . . , ωn } on A: Ω = {ωi }i6n , with {c1 , . . . , cn } the contraction factors , and where
Definition 2.2
1 or
schemes, but always abbreviated IFS.
16
Chapter 2. Generation of computational models of plants
n > 2. We define the transformation of F by Ω as Ω(F ) = ω1 (F ) ∪ ω2 (F ) ∪ · · · ∪ ωn (F ) . Definition 2.3
We note
Let F0 be an initial set, and F1 = Ω(F0 ), hence Fk = Ω(Fk−1 ) = Ωk (F0 ).
F∞ = lim Ω(Fk ) = lim Ωk (F0 ) . k→∞
k→∞
A fundamental property of IFSs was proven by Hutchinson [1981] who showed, using a metric defined on A that IFS can always be described in a Banach space. Therefore the Banach fixed point theorem can be applied, and the procedure of iterating an IFS always yields a fixed point: its attractor. This property is expressed by theorem2.1, whose proof can be found in [Falconer 1990, 1997]. Let Ω = {ωi }i6n be an IFS on A ∈ Rn . Then there exists a unique, non-empty set F ∈ A that satisfies F = Ω(F ) .
Theorem 2.1
Moreover, F = F∞ and do not depend on the initial set F0 . This property can be illustrated by the von Koch curve. Let us consider a segment as shown in Figure 2.2 a., and an IFS, Ω, composed of n = 4 similarities (ω1 , ω2 , ω3 , ω4 ), with identical contraction factors c = 3. The application of Ω on the initial segment yields the new object illustrated in Figure 2.2 b. By iteratively applying Ω on the obtained object (Figure 2.2 c.) we obtain the von Koch curve (Figure 2.2 d.) which is the attractor of this IFS.
Figure 2.2 – Left: construction procedure of the von Koch curve using an IFS composed of n = 4 similarities (ω1 , ω2 , ω3 , ω4 ), with identical contraction factors c = 3. Right: self-similarity of the von Koch curve, a side of the curve is both symmetrical and scale-invariant, it can be continuously magnified by 3 without changing shape.
2.1. Iterated Function Systems
17
This curve is perfectly self-similar: any proper part of the curve, magnified by a 3-fold scaling yields the total initial shape, as shown in Figure 2.2. Note that although the von Koch curve lies in a finite space (it is included in a disc of diameter 1 for example) the length between any two of its points is infinite.
Remark 2.1
Various well-known 2D and 3D self-similar sets can be produced using IFS: the middle-third Cantor set, the Sierpinski triangle and carpet, Menger sponge, etc. . . , see Figure 2.6. One of the advantage of IFS is that, under certain conditions, the self-similarity of the attractor can be characterized by a simple number: its fractal dimension (see chapter 3 for detailed discussion). The fractal dimension , Da , is estimated in terms of the IFS contractions: n Da X 1 =1 ci i =1
For this relation to be valid, we need a condition that ensure that the components ωi (F ) of F do not overlap too much [Falconer 1990]. This condition is expressed by the open set condition (OSC), which Falconer [1997] defines as: An IFS Ω = {ωi }16i6n satisfies the OSC if there exists a non-empty bounded open set V ⊂ A such that
Definition 2.4
n [
ωi (V ) ⊂ V
and
ωi (V ) ∩ ω j (V ) = ∅ for i 6= j .
i =1
The OSC is easily verified for sets such as the von Koch curve, see Figure 2.3.
Figure 2.3 – The open set condition for the von Koch curve. The open set V is the interior of the bounding triangle with ω1 , . . . , ω4 the similarities.
As illustrated in Figure 2.1, self-similarity is indeed present in nature. Hence it is possible to produce good geometric representation of natural occurring objects using IFS formalism. One of the most famous example is the fern Barnsley [1988] generated using IFS (Figure 2.4). In a similar way it is possible to generate 3D objects using IFS that look like plants. These self-similar objects have well defined mathematical properties that allow one to test and validate new analysis method as it will be presented in chapter 3 and used in chapter 4. An example of the generation of an artificial crown is shown in Figure 2.5. A major interest in the use of IFS is that the construction process directly generates a multiscale representation of the object [Godin 2003]. Let us consider the von Koch curve, and more specifically its construction process illustrated in Figure 2.2. Each of these curves2 is build from the previous one by applying the IFS similarities. The 2 except
for the first one
Chapter 2. Generation of computational models of plants
18
Step 0
Step 1
Step 5
Step 10
Step ∞
Figure 2.4 – Fractal fern generated using Iterated Function Systems [Barnsley 1988].
true von Koch curve is the limit of this series of curves, but each step toward the limit adds details on the curve. One can then consider that going through the IFS steps is equivalent to going through the scales of the real curve. At each step, the obtained representation is a refinement of the previous, coarser, representation. Therefore, it is possible to associate a MTG to the von Koch curve [Godin 2003]. The generation starts with a segment which would be the geometry associated with the root of the MTG. This segment is then decomposed into four parts of size one third of the previous segment. This is equivalent in the MTG to the decomposition of the root into four new vertices. These nodes being connected to the root by the decomposition relation and to each others by their adjacency relationship. In this particular case, the graph at each scale is a chain graph since it describes a curve. Each one of these new nodes can also be decomposed into 4 nodes of the finer scale, and so on, until the desired level of precision is reached. This decomposition along the scales is consistant with the geometric representation. Indeed the adjacency between two nodes, n1 and n2 at a coarse scale is transposed into the adjacency between the last component of n1 and the first of n2 . An illustration of this association can be seen in Figure 2.12.
Figure 2.5 – Construction of an artificial crown. The initial object was a tapered ellipsoid and the IFS transformation was made of n = 5 duplications of a contracted object by a factor c = 3. This IFS verifies the OSC, its fractal dimension is given by equation 2.1, Da = 1.4649.
2.1. Iterated Function Systems
19
a.
b.
c.
d. Figure 2.6 – Well-known self-similar sets. a. 5 first steps of middle-third Cantor dust generation ; b. 6 first steps of the Sierpinski triangle ; c. the Sierpinski carpet and d. 3D Cantor dust.
Chapter 2. Generation of computational models of plants
20
2.2
L-Systems A L-system is a parallel rewriting system, namely a variant of a formal grammar (a set of rules and symbols). Originally L-systems were introduced and developed by Aristid Lindenmayer (the L of L-system stands for Lindenmayer) to provide a formal description of the development of simple multicellular organisms [Lindenmayer 1968]. They have thus the ability to model the development of structures. Subsequently this system was extended to describe plants and their complex branching structures described as bracketed string [Prusinkiewicz and Kari 1996]. Definition 2.5
A L-system is a tuple G = (V, ω, P), where
V (the alphabet) is a set of symbols (representing different types of components that compose the biological object). The set of all words (strings) that can be generated from V is noted V∗, ω (start, axiom or initiator) is a string of symbols from V defining the initial state of the system, P is a set of production rules or productions defining the way variables can be throughout time, all symbols can be rewritten (as default as themselves). A production consists of two strings - the predecessor and the successor strings. If there is exactly one production for each symbol, then the L-system is said to be deterministic. The rewriting system is defined by the application of as many production rules as possible on a string. This operation is called a derivation and the nth derivated word is obtained after n recursive derivations, starting from the axiom. The words generated with such L-systems represent linear structures.
F[+F]F[-F[-F]F]F Figure 2.7 – Bracketed string representation of an axial tree. + and symbols stands for left orientation and right orientation, respectively.
To represent branching structures, two new symbols were introduced to delimit a branch: ‘[ ’and ‘] ’. These extended L-systems are called bracketed L-systems, and the set of words they generate are well formed expressions, i.e. as many opening symbols ‘[’as closing symbols, ‘]’. The bracketed L-systems can generate strings that can represent axial trees as introduced by Prusinkiewicz and Lindenmayer [1990] and presented in chapter 1. Let us define the words x1 , x2 , . . . , xn+1 ∈ V ∗ that do not include brackets, and the words α1 , α2 , . . . , αn ∈ {V ∪ {[, ]}}∗ that are fell formed. An axial tree can be represented by the string m m = x 1 [ α 1 ] x 2 [ α 2 ] · · · x n [ α n ] x n +1 ,
2.2. L-Systems
21
where the sequence x1 , x2 , . . . , xn+1 defines the 0-order axis of the axial tree. The (sub)words α1 , α2 , . . . , αn represent its axes that have order 1, and can be similarly decomposed in axial sub-trees that have order 2, and so on recursively. An example of an axial tree and its string representation are shown in Figure 2.7. The string representation also allows to attach additional informations to each represented plant module. This is usually done by using parameters attached to modules that encapsulate one or several numerical parameters, to the desired string symbol. Such a symbol is called a parametric symbol, and a string composed of parametric symbols is a parametric word. Subsequently, the L-system formalism has been widely extended to deal with context sensitive rules [Prusinkiewicz and Lindenmayer 1990; Hanan 1992; Prusinkiewicz and Lindenmayer 1990], to interact with the environment [Prusinkiewicz et al. 1994; Mech 1997], or to convey signals through plant modules efficiently [Karwowski 2002]. To obtain a geometric interpretation of the plant structure described by an axial tree, a geometry must be associated with the string symbols. This is usually performed by defining rules that bound symbols to geometric command that are interpreted by a LOGO-like turtle. The turtle interpretation relies on the relative positioning of one component with respect to its parent. The geometry can be viewed as the result of the path walked by the turtle. This technique is well-adapted to represent bracketed strings, the geometry is positioned as the string is parsed, Figure 2.8.
Figure 2.8 – The turtle (tip of the red arrow) draws a straight line for each F encountered, + and symbols cause rotation to the left and right, respectively. [ pushes the current state of the turtle onto a pushdown stack were position and orientation are saved. ] pops the state from the stack, the previous stored state in the stack becomes the current state.
An example of tree-like structure generated with L-systems and drawn with VPlants/Lpy is shown in Figure 2.9. For more complex geometric representation, homomorphism rules can be defined [Hanan 1992]. These rules link the string symbols to a geometric interpretation allowing the separation of the plant structure and development from its geometric interpretation. These rules are applied just before the generation of the model geometry and allow the rendering of complex structures as illustrated in Figure 2.10.
22
Chapter 2. Generation of computational models of plants
a. n = 7 , δ = 20◦ ω: X P1 : X → F[+X]F[-X]+FX P2 : F → FF
b. n = 7 , δ = 35◦ ω: X P1 : X → F[+FX][-FX]FX P2 : F → FX
Figure 2.9 – Example of plant-like structures generated by bracketed L-systems. n is the number of iteration used to obtain the representation, and δ is the branching angle.
Figure 2.10 – Consecutive steps of a simulation illustrating the development of a flower. The geometrical properties of each symbol are generated using homomorphisms. They represent continuous change of flower organs throughout time, which creates the impression of continuous growth of the entire flower. Simulation created by Chaubert F. and Stoma S.
The recursive nature of the L-system rules can lead to self-similar structures Prusinkiewicz [2004b]. In particular, L-systems can be used to generate the attractor of iterated function systems. For example, the topology of the von Koch curve can be obtained with a L-system. Using the parametric possibilities of the L-systems, we can add geometric information to precise the organization of the four new components. We just need to indicate the contraction factor of the line as a parameter of the symbol representing it, here F, and the directions of the rotation: left ‘+’or right ‘-’with their angle values as parameters. Hence the L-system version of von Koch
2.2. L-Systems
23
curve: ω : F(x) P1 : F(x) → F( 3x )+(60)F( 3x )-(120)F( 3x )+ (60)F( 3x ) . the fractal fern in Figure 2.4 generated using IFS can be reproduced using Lsystems as illustrated in Figure 2.11. A formal link between L-systems and IFS can be found in [Prusinkiewicz and Lindenmayer 1990; Prusinkiewicz 2004b].
Step 0
Step 1
Step 5
Step 10
Figure 2.11 – Fractal fern generated using L-systems.
Unified representation Both IFS and L-systems methods produce complex object that are close to the mathematical definition of attractor, these objects are called prefractal. In this work we will use the geometry of these pre-fractals into models. The natural representation of an IFS is a subset of Rn while it is a string for the L-systems. This two representation being incompatible, we need a unifying way of representing the geometry produced by either methods. We previously explained how an IFS could be represented with an MTG. Similarly to the IFS, each iteration of the L-system can be seen as the definition of a new, refined, scale of the object. The same MTG can be associated with this L-system representation of the von Koch curve. In this case the adjacency relationship at each scale is explicit and additional information about geometry, particularly regarding the decreasing size of components, is needed to obtain a geometric representation of the curve. This illustrates the fact that both the IFS and L-systems generation methods yield a multiscale organization that can be represented using the MTG formalism. The basic association of a MTG with the L-system is shown in Figure 2.12.
24
Chapter 2. Generation of computational models of plants
Figure 2.12 – Multiscale tree graph as a unifying structure embedding topological and geometric informations of von Koch curve. a. the L-system string representing each step of the von Koch curve generation. For readability purpose, orientation symbols ’+’ and ’-’ are assigned default parameters (60) and (120), respectively. b.The MTG associated to the von Koch curve [Godin 2003]. The black arrows represent the decomposition relation between nodes of successive scales, while the grey arrows express the inner-scale adjacency relation. Red segments within each nodes show the geometry associated from generation step of the von Koch curve depicted in c.
2.3
Three-dimensional digitizing systems Three-dimensional digitizing is the process of collecting position, orientation and size of plant organs. According to Moulia and Sinoquet [1993], digitizing systems can be separated into two major classes: contact digitizing and non contact digitizing.
2.3.1
Contact digitizing Three kinds of contact digitizers have been used in the literature: mechanical, (ultra-)sonic and magnetic. The first mechanical device was build as an articulated arms by Lang [1973]. The number of degrees of freedom is determined by the number of arms composing the device. One potentiometer is placed at each joint for angle measurement, allowing the computation of the tip of the final segment position. More recently, Takenaka et al. [1998] built a device called a Pocometer where a string is stretched between a fixed point and the point to be recorded. The length and angle of the string determine the point 3D coordinates. The main problem with mechanical devices is that some points may not be reached due to insufficient degrees of freedom or intricate vegetation. The limited active volume, specially in the case of articulated arms, may also be an issue. The sonic digitizers use the principle of triangulation. A probe, used as the pointer, emits a sound that is heard by a set of four microphones placed within the same plane. Probe position is computed using the time shift between sound reception by the microphones. The time-distance conversion is made by using the speed of sound in still air. Consequently, the use of this device is almost restricted to laboratory usage since wind fluctuation or air temperature in the field easily modify sound speed. Digitizing dense canopies with this method may raise some difficul- Figure 2.13 – Magnetic field ties, vegetation between emitter and receptors disturbing generator.
2.4. Data base
25
sound propagation as well. The magnetic digitizers include a magnetic field generator (Figure 2.13) and a pointer to be positioned by an operator at each point to be measured. The pointer includes coils where electrical currents are induced when located in the magnetic fields. The spatial coordinates and orientation angles in 3D can be derived from the values of induced currents [Polhemus 1993]. This method is likely to be the most suitable for plant digitizing. First, it allows the measurement of orientation angle of components which is a valuable feature, for instance, leaf orientation is of major importance when dealing with light interception. Second, the active volume can be large, up to 80 m3 , and third, the magnetic field is not disturbed by vegetation between the magnetic field generator and the pointer, however, it is sensitive to the presence of metal within the active volume. Thus, it can be used to digitize dense canopies, and has been intensively used in the past decade [Sinoquet and Rivet 1997; Sinoquet et al. 1997; Godin et al. 1999; Costes et al. 2003; Sonohat et al. 2006]. Contact methods are more tedious, as they need an operator moving the pointer onto the vegetative surfaces. However, the operator can simultaneously record additional information about the measured organs, including the identification of the plant components and topology [Godin et al. 2005]. This has fostered the development of helper-software like 3A [Adam et al. 1999], that drives the magnetic digitizer while allowing the recording of plant topology using the MTG notation developed by Godin et al. [1997].
2.3.2
Non contact digitizing These systems compute spatial coordinates using mainly the principle of triangulation or the parallaxe method. Triangulation methods need different point of view of the same object and comprise stereovision [Ivanov et al. 1995], photogrammetry [Boissard 1985] and laser triangulation [Walklate 1989]. Parallaxe techniques use optical telemeters which are, for the most part, based on laser beam techniques. The reflection of a sent beam in the canopy is picked up by a dedicated camera, the position is then computed using trigonometry relations. The main drawback of these techniques comes from occlusion, these methods getting much more complicated for in situ characterization when for instance, the canopy gets denser [Moulia and Sinoquet 1993]. In the context of root-system investigation, non destructive methods were proposed. Southon and Jones [1992] utilized nuclear magnetic resonance (NMR) while X-ray tomography was used by Heeraman et al. [1997], allowing them to reconstruct the three-dimensional structure from stack of images.
2.4
Data base
2.4.1
Virtual plants Three fractal plants were generated from the 3D IFS generation process illustrated in Figure 2.5, AC1 (n = 5, c = 3), AC2 (n = 7, c = 3) and AC3 (n = 9, c = 3). A classical 3D cantor dust [Mandelbrot 1983] was also generated using an IFS (n = 8, c = 3). Each IFS was developed over 5 iterations and are represented in Figure 2.15. In addition to these self-similar plants a stochastic 3D cantor dust was generated using a recursive algorithm derived from the method known as curdling trema genera-
Chapter 2. Generation of computational models of plants
26
tion [Mandelbrot 1983; Plotnick et al. 1993]. This method is a recursive procedure that starts with an initiator, frequently a square, which is called trema generators by Mandelbrot [1983]. The size of the initiator is divided by a subdivision factor, s, generating a set of subcomponents, from which only a subset of n elements are considered for the next step. This process is applied recursively and converges at the limit to a set called a curd, which elements are considered to be “leaves”. The construction of the two-dimensional triadic Cantor set is shown in Figure 2.14. A trema generator can be compared to an IFS and its associated curd to the attractor of the IFS, likewise, curds log(n) have self-similarity dimension: Da = log(s) .
a.
b.
c.
d.
Figure 2.14 – Generation of a two-dimensional Cantor dust with the curdling trema generation method. The trema generator is a square (a.) which is subdivided into subcomponents with s = 3 (b.). Only the corner elements are considered (c.) for the next step (d.).
A stochastic method, called curdling and random trema generation [Mandelbrot 1983], can be derived from the curdling trema generation method. In this case, the subcomponents considered for the next step are defined as a proportion of the total number of subcomponents. This ratio can be seen as a probability, p, for a given subcomponents to participate in the next step. The classical three-dimensional cantor dust is generated using the 8 corner subcomponents among the 27 possible. We generated our stochastic 3D cantor dust using the curdling and random trema generation method 8 over 5 iteration levels. with s = 3 and p = 27
Figure 2.15 – From left to right, the three artificial canopies : AC1 (n = 5, c = 3), AC2 (n = 7, c = 3), AC3 (n = 9, c = 3), on the top, the cantor dust and on the bottom a stochastic cantor dust.
2.4. Data base
2.4.2
27
Digitized plants Four 4-year old peach trees (variety August Red) were digitized one month after bud break in May 2001, in CTIFL-Balandran in South-East of France (43◦ 830 N, 4◦ 350 E). Two of them were trained as tight goblet (TG) system, and the two other ones as wide double-Y (WDY). TG is an open center (goblet-shaped) structure employing several primary scaffolds. Planting distances in TG system were 6 × 3m. WDY derives from goblet, with larger planting distances, 7 × 4m, and four primary scaffolds arranged by pairs [Giauque 2003]. Tree height was about 2.5m. Four 3-year old mango trees grown in a commercial farm were digitized in March
Figure 2.16 – Four four-year old peach trees (cv. August Red) were digitized in May 2001 in CTIFL Center, Nîmes, South of France, at current-year shoot scale, one month after bud break.
2001 in Saint-Paul, La Réunion Island (20◦ 530 S, 55◦ 320 E). Two of them belonged to variety Lirfa, and the two other ones to variety José. A square planting system was used with a 6-meter distance between trees and a north-west–south-east row orientation. Tree height was about 1.5m. Full information on peach and mango trees can be found in [Sonohat et al. 2006] and [Urban et al. 2003], respectively. All trees were 3D-digitized with a 3Space Fastrak electro-magnetic device (Polhemus
Figure 2.17 – Four 3-year old mango trees grown in a commercial farm were digitized in March 2001 in Saint-Paul, La Réunion Island.
inc., Colchester, VT, USA). Plant digitizing was driven by software Pol95 [Adam 1999]. Mango trees were digitized at leaf scale according to Sinoquet et al. [1998]’s method. For each leaf, the pointer was located at the junction between lamina and petiole, with pointer X − axis parallel to the midrib and pointer X − Y plane parallel to lamina. With this configuration, measured angles were midrib azimuth, midrib elevation and rolling angle of lamina around the midrib. In addition, leaf length was measured with a ruler. Fifty leaves were harvested for leaf area (A cm2 ) measurement with a planimeter, and leaf length (L cm) was measured with a ruler. This sample was used to set a relationship between A and L2 : A = k.L2 .
(2.1)
28
Chapter 2. Generation of computational models of plants
Coefficient k was constant for both varieties, with a value of 0.1826. Peach trees were digitized at the leafy shoot scale according to Sinoquet and Rivet [1997]’s method. For each leafy shoot, the spatial bottom and top coordinates of the shoot were recorded with the digitizer. For each shoot type in each cultivar, namely short and long (> 5cm) shoots, 15 to 30 shoots were digitized at leaf scale, as described above. These data were used to establish foliage reconstruction rules for all digitized shoots. Reconstruction rules included allometric relationships at shoot and leaf scale, random sampling in leaf angle distributions and additional hypotheses. The reconstruction method has been fully presented and assessed in [Sonohat et al. 2006]. At the same time as digitizing, the position of the digitized organ in the multiscale tree organization was recorded using Godin et al. [1999]’s method. For mango trees 4 scales were used : plant, scaffold, current-year-shoot (CYS) i.e. leafy shoot, and leaf. One additional scale was used for peach trees, i.e. one-year-old shoot (OYOS) between scaffold and CYS scales. Of course reconstructed peach leaves were assigned to the corresponding digitized shoot. Each tree is encoded as a MTG where the geometries of the last scale nodes are the leaves and the geometry of their complexes are convex hulls containing them.
Conclusion In this chapter, we described two efficient ways of generating multiscale plants. The L-systems formalism emphasizes the topological representation while the IFS one is only a geometric representation. We showed the unifying capacity of MTG to describe multiscale structures generated with either L-systems or IFS. We will use this representation for the geometry of our database. Finally, we presented different digitizing techniques that allow a precise reconstruction of plants as illustrated in the database of 3D object and plant mock-ups that will be used throughout this work. We designed artificial plant crowns with IFS, which made it possible to set up 3D plant mockups with controlled fractal characteristics (fractal dimension, lacunarity: these notions are further presented in chapter 3 & 4). These artificial crowns will make it possible to evaluate the accuracy of the analysis tools and models we will develop.
3
Fractal dimension of plants
“Man’s mind, once stretched by a new idea, never regains its original dimensions.” O. W. Holmes
P
lant geometry is a key factor for modeling ecophysiological interaction of plant and the environment, and modeling these interactions requires measures of this geometry. Light interception, for instance, requires the measurement of crown surface. This surface intercepts sunlight and provides the plant with the energy to realize photosynthesis. But what should we measure? Should we measure the surface of each individual leaf? Should we measure the surface of a convex envelope of the crown, or should we consider its volume, which encompasses mostly empty space? [Zeide 1998]. In fact measuring intricate structures with cuts and gaps, like tree crowns, is not done as easily as for classical Euclidian objects. This raises the question of the possibility of characterizing the geometry of such irregular and complex objects using few parameters as a sphere is characterized by its radius or a box by its width, length and height. As explained by Godin [2003], classical geometry has its root in the regular Euclidian geometry and most of nowadays scientific education is based on Leibniz’s and Newton’s calculus. Therefore it is natural for one to try to describe Nature within this framework, but one will frequently fail. A simple reason to that is given by Mandelbrot [1983] when he says: “clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in straight line.” As a consequence of such an “irregularity”more, measurement of the extend of many natural objects increases when smaller yardsticks are used [Richardson 1961]. These objects are highly irregular and, within certain limits, behave likewise the self-similar objects presented in the previous chapter. In the late 19th century the efforts of mathematicians to measure such objects and characterize how they fill space led them to new ways of defining the concept of measure, which in turn make it possible to generalize the notion of dimension. Dimension was, and still is, associated with the number of degrees of freedom of a mobile point inside an object. A point is of dimension 0, a line is of dimension 1, a plane is of dimension 2, etc. . . For instance, according to the previous definition, the von Koch curve has dimension one, but it does not behave like a curve: the length between any two of its points is infinite. No small piece of it is line-like, but neither is it like a piece of the plane. In some sense, we could say that it is too big to be thought of as a one-dimensional object, but too thin to be a two-dimensional object, leading to the question of whether its dimension might best be described in some sense by number between one and two. It would be better described by a fractional number between 1 and 2, i.e. a fractal dimension. Fractal geometry was introduced as a new conceptual framework that is an attempts 29
30
Chapter 3. Fractal dimension of plants
to summarize the messy complexity of shapes we see around us [Mandelbrot 1983]. The fractal dimension describes the complexity of an object geometry based on the quantity of substance they deploy in space as one zooms in. In this chapter we will show how the measure of an object is intrinsically related to its dimension, whose estimation depends on its formal definition. We will present some of the many definitions that exist before describing more carefully two of them well adapted to plant study: the box dimension and the two-surfaces dimension. Usage limitations of these descriptors and how they should be adapted for plant geometry characterization will then be discussed.
3.1. Measuring objects
3.1
31
Measuring objects The measurement of an object is usually achieved through a tiling process. First, a measurement unit, or tile, of size δ is chosen. We define the size of any set F as
|F | = sup {d( x, y), ∀ x, y ∈ F } ,
(3.1)
with d the usual distance or metric. Then the object to be measured is entirely paved with these tiles. Finally, by multiplying the number of tiles necessary to cover the object by their size, we obtain a estimation of the object measurement. Depending on the yardsticks size, δ, the measurement of an object can be made with an arbitrary precision, we will call it a δ-measure. A δ-measure ignores irregularities of size less than δ and thus can be considered as the measurement at scale δ. Figure 3.1 illustrates the tiling process underlying the measurement of objects and the increase of necessary yardsticks as their size decreases.
Figure 3.1 – Increase of δ-sized unit necessary to cover a cube (a.), a square (b.), and a segment (c.) as 1 1 δ decreases. Value of δ is 1 (first column), (second column), and (third column). Illustration from 3 9 [Godin 2003].
One can see in this trivial example that the measurement does not change with the scale. Let us consider the measurement, Mδ (F ), of the square F (Figure 3.1 b.) at scale δ: 1 1 Mδ (F ) = 1 × (1)2 = 9 × ( )2 = 81 × ( )2 = 1 (3.2) | {z } | {z3 } | {z 9 } δ =1
δ= 31
δ= 19
This measurement approach is so well-established that we usually forget that it is based on two important premises. First yardsticks have to be of the same dimension as the object to be measured. Indeed, it would be extremely difficult measure a volume using a surface yardstick. Second, the object must be regular enough, i.e. it must be rectifiable as defined by Tricot [1993]. Intuitively, the concept of rectifiability can be
Chapter 3. Fractal dimension of plants
32
illustrated by the fact that when zooming in on an object, after some time, there are no new details appearing. Figure 3.2 illustrates this idea on a curve. Therefore the object can be assimilated to a set of small parts where each of them has simple geometry with known measure. The measure of the object is thus the sum of the measures of the parts. This means that there is convergence of the δ-measures of a rectifiable object when δ decreases. In fact, the rectifiability hypothesis is the major paradigm underlying Leibniz’s and Newton’s calculus. Although rectifiability paradigm is pervasive in
Figure 3.2 – Successive zooming on the curve have a segment appeared. At this point the curve is similar to a segment. Illustration from [Godin 2003].
modern science, with significant validated applications, it is not adapted to deal with strongly irregular object. For instance, how do we measure objects with new details appearing at each zoom as the curve represented Figure 3.3 or fractal objects?
Figure 3.3 – Non-rectifiable curve, with each successive zoom, new details appear. This is a fractal curve in Mandelbrot’s meaning from Latin fractus meaning broken. Illustration from [Godin 2003].
As a motivative prelude to trying to answer this question, let us examine again the von Koch curve presented in section 2.1, and try to measure its length with the tiling procedure. Starting with the initial segment of the IFS construction procedure as the yardstick, the first measure of the von Koch curve is 1. We will then use a series of yardsticks with decreasing size, δk , each size being one third of the previous size: δ δk = k3−1 . Obtained δk -measurements, Mδk , of the curve are presented Table 3.1:
δk
1
1 3
1 9
1 27
···
1 3k
Nδk
1
4
16
64
···
4k
1
4 3
16 9
64 27
···
k 4 3
Mδk
Table 3.1 – Measurement of von Koch curve. δk is the size of the yardsticks, Nδk is the number of δk -sized yardsticks needed to tile the curve and Mδk is the obtained δk measure.
3.2. Notions of dimensions
33
Unlike the example of the square measurement given in equation 3.2, the length of the von Koch curve varies with the size of the measurement unit. Furthermore, the length increases as the yardstick size decreases leading to an infinite length of a curve however contained in finite 2D space: the bounding triangle of the von Koch curve, see Figure 2.3. In what follows we will show how measurement and dimension are bounded to each other and how we can obtain a measure of von Koch curve length.
3.2
Notions of dimensions There are many definitions of fractal dimension and none of them should be considered as the universal one Mandelbrot [1983]. From the theoretical point of view the most important is probably the Hausdorff Besicovitch dimension for almost all other definitions derive from it. Moreover the term fractal was coined by Mandelbrot [1983] to define “a set for which the Hausdorff Besicovitch dimension strictly exceeds the topological dimension.”
3.2.1
Topological dimension The topological dimension, also known as the Euclidian - or Cartesian - dimension, is the one that everybody knows. It is associated with the number of degrees of freedom, i.e. the number of coordinates needed to travel through the object. Poincaré gave an alternative inductive definition that can be summarized as follows: the dimension of a space is defined as one plus the dimension of the lowest dimensional object with the capacity to separate any neighborhood of the space into two parts. As an axiom, the dimension of a point is 0. The dimension of a line which can be split in two disjoint parts by a point (of dimension 0) is 1. The dimension of a plane which can be split in two disjoint parts by a line (of dimension 1) is 2. The dimension of a volume which can be split in two disjoint parts by a line (of dimension 2) is 3, etc. . . For any object F , its topological dimension will be noted Dt (F ).
3.2.2
Hausdorff Besicovitch dimension Among the wide variety of fractal dimension that can be found in literature [Falconer 1990], the definition of Hausdorff Besicovitch is probably one of the more important because it is defined on any set and is defined in terms of measures [Falconer 1997]. In this part we will remind the definition and the main properties given in [Falconer 1990] as done by Godin [2003]. Suppose that F is a subset of Rn . If {Ui } is a countable collection of sets of diameter, at most δ that covers F , we say that {Ui } is a δ-cover of F [Falconer 1990]. The diameter of Ui is defined as: |Ui | = sup (| x − y|). x,y∈Ui
For any non negative number d and δ, we define ( ∞ ) X hdδ (F ) = inf |Ui |d | {Ui } δ-cover of F i =1
(3.3)
Chapter 3. Fractal dimension of plants
34
This definition looks at all δ-covers of F and seeks to minimize the sum of the diameters to the power of d. As δ decreases, the class of δ-covers of F is reduced, so the infimum increases and approaches a limit [Falconer 1997]. Furthermore the infimum en∞ X sures that there is no overlapping between Ui sets, hence for classical objects, |Ui | Dt i =1
estimate the size of F at scale δ. We define the d-dimensional Hausdorff measure of F as: hd (F ) = lim hdδ (F ) . δ →0
(3.4)
This limit exists, perhaps as 0 or ∞, for all F ∈ Rn [Falconer 1997]. Here are the most interesting properties of hd : • hd is a measure: let F and G be two disjoint sets hd (F ∪ G) = hd (F ) + hd (G) • hd isometry invariant: let t be an isometry, hd (t(F )) = hd (F ) • scaling property: let λF = {λx | x ∈ F } then hd (λF ) = λd hd (F ) • critical value: for a given set F , there is, at most, one value for d at which hd (F ) is finite and not null. This last property defines a number that characterize the dimension of any set F . This value is called the Hausdorff dimension 1 and will be noted Dh (F ). A major drawback of Hausdorff dimension is that in many cases it is hard to estimate by computational methods. The estimation generates the optimization problem over the set of all δ-covers of finding the one with fewer number of elements, for which we do not have any construction method.
3.2.3
Self-similarity dimension Another way of characterizing the dimension of an object, is through the parameter that controls the relationship between the paving-measurement of an object and the size of the tile. As illustrated in Figure 3.1 and equation 3.2, this definition concurs with the topological one for Euclidean objects, i.e. objects from the standard geometry, as distinguished by Mandelbrot [1983]. This agreement is due to the fact that, for Euclidean objects, at some point there are no more details smaller than the yardstick size, δ. But this is no longer the case for more complex objects where new details appears 1 sometimes
referred as Hausdorff-Besicovitch dimension
3.2. Notions of dimensions
35
Figure 3.4 – Graph of hd against d for a set F . The Hausdorff dimension is the value of d at which the jump from 0 to ∞ occurs.
for any value of δ. We saw previously that using the classical measurement procedure described in section 3.1, the von Koch curve has an infinite length. This is mainly because we were using measurement unit with inadequate dimension. In what follows we will show that if we use the “correct”dimension we can obtain the measure of IFS attractors. In fact we will show that the dimension of an IFS lies in its definition. We will use a set of tiles of decreasing sizes {δk } defined by the k th step of the IFS construction process. Let Ω = {ωi }06i6n be an IFS of contraction factor ci = c, ∀i. Let also Nδk be the number of tiles required to cover the object. The measurement of the object at scale δk is noted Mδk : Mδk = Nδk .δkd , (3.5) where d is the dimension of the tiles. At step k, size of tiles and number of tiles are 1 δk = ( )k s
and
Nδk = nk ,
where n is the number of similarities of the IFS. One can see that in the general case, this last equality is only verified for sets verifying the Open Set Condition, see section 2.1. When the OSC is not verified, overlapping of elements could occur inducing a less important number of tiles to cover the set. By taking k out of the two previous relations, we obtain log( Nδk ) = Let us denote Da =
log(n) , log(s)
log(n). log(δk ) log(n) 1 = . log( ) . 1 log(s) δk log( s )
then Nδk = (
1 Da ) , δk
(3.6)
therefore we can link the measurement of an object to the size of tiles:
Mδk = (
1 Da d 1 ) δk = ( ) Da −d . δk δk
(3.7)
This relation shows that the measurement is independent of the scale is when d = Da . Da is the self-similarity dimension and for attractors obtained from IFS it is often
Chapter 3. Fractal dimension of plants
36
relatively easy to calculate, see [Falconer 1990]. In the case of IFS with unique contraction factor, for instance the one generating the von Koch curve, the self-similarity dimension is log n (3.8) Da = log c At each step, the von Koch curve is obtained with 4 duplications of the previous step log 4 reduced by a factor 3, thus its self-similar dimension is Da = log 3 = 1.2619. By using yardsticks of this dimension, from equation 3.7, we can see that the measure of the von Koch curve becomes independent of the scale and equal to 1. This dimension characterizes irregularity of theoretical self-similar sets that we can generate using IFS, but can not be used for any other set, we don’t know the similarities, n, that can be used to generate the set, nor their contraction factors, c.
3.2.4
Compass dimension This method was used by Richardson [1961] to estimate frontiers or coastlines length of countries on a map . The procedure consists in covering the curve using yardsticks of size δ beginning from an arbitrary point. The sequence of determined points defines a polygonal approximation of the curve where all segments have the same size, δ.The estimated length of the frontier, Mδ , is the product of the number of yardsticks required to cover the curve, Nδ , and the size of the yardsticks, δ:
Mδ = Nδ × δ .
(3.9)
Figure 3.5 – Polygonal approximation of a curve with decreasing yardstick size δ
By using decreasing sizes of yardstick, Richardson exhibited a power-law relationship2 between the length and the measuring scale similar to the existing for selfsimilar sets 1 (3.10) Mδ ∝ ( )d . δ The compass (or dividers) dimension, Dc , is defined as: Dc = 1 + d
(3.11)
The compass dimension can be easily extended to work in higher dimension, the yardsticks need only to be replaced by n-dimensional balls of radius δ to cover the object. In this case, Mδ = Nδ × δn , hence the generalized compass dimension Dc = n + d , 2 f (δ)
∝ g(δ) means that, when δ → 0, f (δ) → g(δ)
(3.12)
3.2. Notions of dimensions
37
can be seen as one of the definition of the box-counting dimension detailed in section 3.2.5 [Falconer 1990]. As noticed by Tricot [1993], this method is not wellfounded theoretically, and furthermore is exact only for statistically self-similar object, see [Tricot 1993] for detailed exposition on the analysis of nonrectifiable curves. Indeed, the estimated length may vary depending on starting position, thus a faithful estimation of the compass dimension should require reiterations with different starting positions [Sugihara and May 1990]. Also while the point of slope change may indicate the operational scale of a different generative process [Kent and Wong 1982; Wiens 1989], it may simply reflect the limited spatial resolution of the data being analyzed [Walker and Kenkel 1993; Gautestad and Mysterud 1994].
3.2.5
Box counting method The box-counting dimension is one of the most widely used fractal dimension, mostly because of its simplicity of usage and implementation. It consists of building a sequence of grids dividing space in homogeneous unit elements of decreasing size δ and counting the number Nδ of grid elements intercepted by the studied object. The estimator of the fractal dimension of the object is defined as log Nδ . δ→0 log 1 δ
Db = lim
(3.13)
If the limit does not exists then we should talk about the lower and upper boxcounting dimension [Falconer 1990]. In fact a formal definition of the box-counting dimension is not restricted to grid sequence: The lower and upper box-counting dimensions of a subset F of Rn Falconer [1990], are given by
Definition 3.1
log( Nδ (F )) δ →0 log( 1δ ) log( Nδ (F )) Db (F ) = lim δ →0 log( 1δ ) Db (F ) = lim
(3.14) (3.15)
and the box-counting dimension of F by log( Nδ (F )) , δ →0 log( 1δ )
Db = lim
(if this limit exists), where Nδ (F ) is any of the following: i the smallest number of closed balls of radius δ that cover F ; ii the smallest number of cubes of side δ that cover F ; iii the number of δ-mesh cubes that intersect F ; iv the smallest number of sets of diameter at most δ that cover F ; v the largest number of disjoint balls of radius δ with centers in F .
(3.16)
Chapter 3. Fractal dimension of plants
38
a.
b.
c.
Figure 3.6 – Box counting method applied on the von Koch curve. The grids are obtained by subdividing the an initial box a. with increasing factor: b. 3, and c. 9. These grids correspond to the iii. of definition 3.1. The boxes covering the curve are colored in grey.
A cube is an interval in R1 , a square in R2 , a cube in R3 . δ-mesh cubes are therefore the elements of a regular grid.
Remark 3.1
According to Falconer [1990], the box-counting method (BCM) is one of the most used because of its ease of mathematical calculation and empirical estimation. In practice the counting of elements of a regular grid, does not involve any optimization process to find the smallest or largest number of any kind of δ-cover of F , hence a relatively simple computer implementation. The box dimension, Db , is estimated as the slope of the regression line between
log ( Nδ ) and log 1δ . It is important to note that several factors may influence the accuracy of this method, e.g. the choice of a proper range of scales and the orientation and alignment of the grid [Foroutan-Pour et al. 1999; Halley et al. 2004]. Regarding its usage for estimating the fractal dimension of plants, Da Silva et al. [2006] realized an appraisal that will be presented in section 3.3.1.
3.2.6
Two-surface dimension Originally introduced in Zeide and Pfeifer [1991], this method estimates the fractal dimension of tree crown populations. The authors suggest that a power law relates obtained measures, the total leaf area, L, and the surface area of a convex hull enveloping the tree crown, E . This empirical procedure is based on the fact that the (simple) surface of the convex hull is related to the measurement unit (or scale) by the power 2 whereas leaf area is supposed to be related to it by the power of its dimension, d : E ∝ δ2 and L ∝ δd . Therefore we obtain the relationship d
L = k.E 2 ,
(3.17)
where k is a constant. A similar relationship can be exhibited using the volume, V , of the convex hull d L = k.V 3 , (3.18)
3.2. Notions of dimensions
39
in fact it was the relation 3.18 that was used in [Zeide and Pfeifer 1991] to estimate the fractal dimension of tree crowns. The comparison was made for a high (enough) number of individuals with varying sizes, and an estimate of d2 (or d3 ), was given by the slope of a log-log regression between the two variables. The resulting fractal dimension is the two-surface dimension and will be noted Ds . Boudon et al. [2006] extended this method for computing dimension of individual plants using a multiscale approach of plant structure (see section 2.4.2). The possibility to use the two-surface method on a single tree stems from two hypotheses: 1. measurement of leaf area follows the scaling property, i.e., the measure at scale δ of an object dilated by a factor λ is identical to the measure of the original (non-dilated) object at scale λδ , 2. at a given scale, or level of organization, a plant can be decomposed into a set of modules of the same type [Boudon et al. 2006]. This set of modules, called plant modularity Godin and Caraglio [1998], forms a collection of branching systems with varying sizes.
a.
b.
c.
d.
Figure 3.7 – a. Artificial canopy generated using IFS and b., its convex hull ; c. branching system that can be used as decomposition module with d., its envelope. Illustration from [Boudon et al. 2006].
The authors were able to estimate the fractal dimension at a particular scale of trees by considering a particular plant modularity. This “local dimension”reflects the self-similarity of leafy modules around the considered scale. Moreover, if the plant is self similar, all these “local dimensions”should tend to be identical. To verify this, an individual fractal dimension can be assessed for each tree by merging the relations of all modularities, i.e. at all scales. An interesting property of this approach is that it uses plant organization to setup a biological partitioning of plant leaves that optimizes their tiling with convex envelopes. The box-counting dimension could be estimated using these envelopes as boxes, defining a variant of the grid-based box-counting method [Boudon et al. 2006].
40
Chapter 3. Fractal dimension of plants
Results obtained with the two-surface method and the box-counting variant are presented in section 3.3.2.
3.2.7
Effective dimension After defining rigorous fractal dimension definitions, the notion of “effective dimension ”introduced by Mandelbrot [1983] should be reminded because it concerns the relation between mathematical sets and natural objects. Moreover it deals with the relation between an object and the scale at which it is being observed. Let us remind the thread ball example given by Mandelbrot [1983]: “. . . , a ball of 10 cm diameter made of a thick thread of 1 mm diameter possesses (in latent fashion) several distinct effective dimension. To an observer placed far away, the ball appears as a zero-dimensional figure: a point. [. . . ] As seen from a distance of 10 cm resolution, the ball of thread is a three-dimensional figure. At 10 mm, it is a mess of one-dimensional threads. At 0.1 mm, each thread becomes a column and the whole becomes a three dimensional figure again. At 0.01 mm, each column dissolves into fibers, and the ball again becomes one-dimensional, and so on, with the dimension crossing over repeatedly from one value to another. ” Natural objects can not be truly fractal, they may exhibit fractal properties across a range of scales, but not at all or completely different ones outside this cutoffs. Therefore it is important, specially when dealing with biological problems, to precisely define the context of study and the range of scales within which it will be done. The relevance of using fractal theory is dependent on objectives, a forester interested in forest wood quantity will settle for a representation of tree trunks by cylinder whereas an ecologist interested in modelling habitat availability on tree trunks will need more accurate estimation of bark surface than the coarse estimation given by the cylinder diameter [Kenkel and Walker 1996].
3.3
Fractal dimension of plants
3.3.1
Box counting dimension In this section, we study the application of the box counting method (BCM) to estimate the fractal dimension of 3D plant foliage. Most of the previous studies using the BCM were carried out on woody structures, and especially on root systems [Fitter 1987; Eshel 1998; Oppelt et al. 2000]. For practical reasons, in most works, box counting dimension, Db , was estimated from 2D photographs [Morse et al. 1985; Critten 1997; Castrejon-Pita et al. 2002]. Unfortunately, such a technique always under-estimates the actual fractal dimension [Falconer 1990], and so is not very accurate. Recently the BCM was used on 3D digitized root systems [Oppelt et al. 2000], however, the accuracy of the estimated values could not be evaluated. In what follows, we study the application of the BCM to both artificial and real 3D plant foliage. We use artificial crowns with known theoretical fractal dimensions to characterize the accuracy of BCM and we extend the approach to 3D digitized plants. The limits of BCM is then analyzed and discussed in this context. In particular, errors are experimentally characterized for the estimated values of the fractal dimension. Most of this work is from [Da Silva et al. 2006], the original paper is included in appendix A.1.
3.3. Fractal dimension of plants
41
In the original study, nine 3D plants from the database (section 2.4) were included. The four digitized peachtrees and the five artificial plants generated from theoretical models, namely AC1, AC2, AC3, Cantor dust and Stochastic Cantor Dust. The geometric scenes representing the plant crowns were designed using the PlantGL library [Pradal et al. 2008a]. In the present section, we added the analysis of a similar subsequent study made with the four digitized mango trees presented in section 2.4. 3.3.1.1 BCM implementation The BCM has been extensively used to estimate fractal dimension of objects embedded in the plane. Its adaptation to 3D consists of building a sequence of 3D grids dividing space in homogeneous voxels of decreasing size δ and counting the number Nδ of grid voxels intercepted by the studied object. To implement this method, we approximated all the geometric objects by triangular meshes. The intersection of each triangle with the grid voxels can then be computed analytically in time proportional to the number of triangles in the mesh and the number of voxels [Andres et al. 1997]. However, to simplify the voxelisation process, we represent each triangle by a set of points [Pfister et al. 2000]. The number of points used is chosen such as the distance between two points is small compared to the minimal voxel size. The intersection algorithm is thus reduced to checking whether a voxel contains at least one point. The grid sequence is obtained by dividing the original bounding-box size, δ0 , by a range of consecutive integers acting as subdivision factors. Thus the series of δn is a decreasing series formed by { δS0 }0≤i
a.
b.
c.
Figure 3.8 – Box counting method applied on an apple tree. The grids are obtained by subdividing the bounding-box of the apple tree mockup with increasing factor: a. 5, b. 15, and c. 25. Colors reflect the density of enclosed tree surface surface but have no influence in this context except for visual confort.
The simplicity of usage of the BCM has a cost, indeed many pitfalls may arise when estimating fractal dimension with this method. In the following section, we identified the most important and proposed solutions to tackle them.
Chapter 3. Fractal dimension of plants
42 3.3.1.2 Critical appraisal
Data correlation The major problem of the BCM estimator, as pointed out by Reeve [1992], is that the numbers of intercepted voxels at each scale are correlated positively, and the correlation structure is completely ignored in the estimation procedure. This violates the assumption of data independence used in regression analysis. Although the linear regression may produce a decent estimate of the slope, thus of the boxdimension, the associated statistics, p-values and confidence intervals, do not apply [Halley et al. 2004]. The consequence is a direct underestimation of confidence interval associated with the estimated fractal dimension. To eliminate the correlation, we introduce a new estimator, namely local scale variation estimator (LSV), based on the relative increase of intercepted voxels against the relative decrease in scale. This estimator can be derived from the BCM estimator as follows. Assuming the power law is verified for each scale δ 1 Nδ ∝ ( ) Db , δ
(3.19)
the differential form of this equation leads to 1 d log( Nδ ) ∝ d( Db log( )) , δ dδ dNδ ∝ − Db Nδ δ
(3.20)
which gives a variational interpretation of the fractal dimension. Db thus expresses the linear coefficient that corresponds to the ratio of new details due to a certain ratio of zoom in the structure. However, in this equation it is assumed that both dNδ and dδ are infinitely small. This is not usually the case for the scales used in BCM, except at very small scales. It is possible to generalize this variational principle to non-infinitely small quantities. Let Nδ be the number of intercepted voxels at scale δ. We define ∆Nδ,∆δ as ∆Nδ,∆δ = Nδ+∆δ − Nδ . (3.21) e = ∆Nδ,∆δ . Similarly, The relative increase at scale δ in the number of boxes is denoted N Nδ the relative increase of zoom when passing from cell size δ to δ + ∆δ. we denote δe = ∆δ δ Thus, assuming equation 3.19 is still satisfied, we have − Db (δ + ∆δ)− Db − δ− Db e e N∝ = 1+δ −1, δ − Db
(3.22)
which leads to a generalized form of equation 3.20, where variations of Nδ and δ need not be infinitely small, e ) ∝ − Db log(1 + δe) log(1 + N (3.23) e ) and Db can thus be estimated by performing a linear regression between log(1 + N log(1 + δe). Both BCM and LSV estimator will be used to estimate the box-counting dimension of our data-base objects.
3.3. Fractal dimension of plants
43
Number of voxels as a function of scale In general, we may expect that the number of intercepted voxels is a monotonously increasing function of scale. However this is not always the case due to a quantization effect which results from discrepancy between discretization with the 3D grid and space occupation of the plant at some scales. This can be illustrated with the 2D Cantor dust represented in Figure 3.9, a grid with smaller cell size that better correspond to the object positioning have less intercepted voxels than a coarser grid.
a.
b.
c.
Figure 3.9 – The number of intercepted voxels between the 2D Cantor dust (a.) and a grid obtained by subdividing its bounding box by a factor 7 (b.) and 9 (c.) are 36 and 16, respectively.
Figure 3.10 contains plots of the number of voxels intercepted at the different scales for each object. The local variation of the curves comes from the fact that the number of intercepted voxels at one scale depends of the adjustment of the grid. Some shiftings, up to a factor δ in each direction, and reorientations of the grid may lead to overestimating the number of voxels at one scale, causing local variation of the curve. Thus, the discrete quantization of the 3D shape of the object into voxels introduces some fuzziness in its representation, depending on scale. It can be seen in Figure 3.10 that the quantization effect is far more pronounced with the artificial crowns and Cantor dusts than the digitized peach trees. This difference is attributed to the less deterministically distributed foliage of the digitized trees as the less quantized curve of Stochastic Cantor, compared to the classical Cantor, illustrates. Scale Range When the grid voxel size is smaller than the leaf size, the evaluation of the dimension is modified by the dimension of the leaves surfaces. To avoid this √ effect, a minimum voxel size, δmin , is determined such as δmin ≥ Al , where Al is the mean leaf area. Since every voxel size δi is obtained from the bounding box size δ0 as δ0 . Let Vbb be the bounding box volume. δi = δS0 , the minimum size must be δmin = Smax i An uni-dimensional proportionality factor is defined by √ 3 V Smax = √ bb . (3.24) Al Setting Smax as the upper bound for the subdivision factors {Si }0≤i
44
Chapter 3. Fractal dimension of plants
Figure 3.10 – The number of intercepted voxels as a function of the scale.
a factor Smax 3 instead of Smax was considered as the contraction limit. This factor can be explained as follows. Let us consider a grid with voxels equal in size to the mean leaf size. Optimally a leaf will be included into a single voxel (Figure 3.11 a.). All the possible shifting configurations of the grid may cause the leaf also be included in any of the twenty-six neighboring voxels (Figure 3.11 b., c. and d.). Considering voxels of bigger sizes with a factor 3 can be seen as including the twenty-seven possible small voxels into the same large one and so limits the errors found in finer grids (Figure 3.11 e. to h.). Of course, the optimal grid for one leaf will not be the optimal grid for all leaves; therefore, artifact effects of grid adjustment may persist. We experimentally observed that this persistence is limited, see Figure 3.12. Orientation of the Grid Optimal voxel coverage of the plant depends on the orientation of the grid relative to the plant. For this, we made a sensitivity analysis to evaluate how the estimated fractal dimension is affected by changes in the grid’s orientation. A set of random grid orientations were selected and fractal dimension was estimated for each orientation. Table 3.2 gives the mean and variance of the estimated fractal dimension across orientations for all the considered plants. We can observe a low variability in the absolute values of the results: the standard deviations are inferior to one per cent of the mean values. From this, we conclude that the orientation of the grid has only limited effect on the BCM evaluation method.
3.3. Fractal dimension of plants
45
a.
b.
c.
d.
e.
f.
g.
h.
Figure 3.11 – a. An optimal leaf inclusion into a grid. The more common situations encountered: b., c. and d. Using bigger voxels ( e. to h.) reduces voxel counting errors due to leaves positioned at voxel frontiers.
Figure 3.12 – Evolution of AC2 slope during BCM estimation. Its self similar dimension is Da = 1.77. Theh number i of voxels intercepted at various scales for AC2 with the slopes highlighted. In
the range 0, Smax , the slope is primarily influenced by the structure of AC2 and the box-dimension 3 i h Db = 1.765. In the range Smax , S , the slope is also partially influenced by the fractal dimension of max 3 individual leaves and is sensitive to local variation due to grid adjustment. When this range is taken into account for the fractal dimension estimation, Db drops from 1.765 to 1.584. Finally for grids with voxel sizes smaller than Smax , the slope is directly related to individual leaf fractal dimension (0 in our representation since we use points). With a naive range of evaluation including all points, the fractal dimension drops to 1.172.
3.3.1.3 Error Characterization To characterize the error made during the estimation, a comparison with theoretical fractal dimension can be used. In the case of plants corresponding to IFS attractors, the theoretical fractal dimension, D, is known. But there is no such dimension for real plants; however, it has been shown that, when plant’s topology is known, a faithful
Chapter 3. Fractal dimension of plants
46
estimate of the plant fractal dimension can be obtained using the two-surface method [Boudon et al. 2006]. This value will be used as reference value for the peach trees. A Table 3.2 – Fractal dimension results for studied canopies and their properties. Da is the reference (theoretical) value of the fractal dimension for the artificial crowns, the values found with the twosurface method are taken as reference for the digitized trees. For Db estimation, Db gives the mean estimated value and σ the standard deviation over all considered rotations. The minimum standard error r2 over all rotations is shown. All results are obtained with Smax 3 as the upper limit. BCM Db
LSV Db
Relative Bias
Db
σ
r2
√ 3
Vbb
√
Al
Canopy
Da
Db
σ
r2
AC1
1.47
1.4889
0.0056
0.97
0.0128
1.8761
0.0457
0.33
1.83
0.0143
128
AC2
1.77
1.7305
0.0053
0.99
0.0223
1.9409
0.06
0.58
2.29
0.0143
160
AC3
2
1.97
0.0074
0.99
0.015
2.0705
0.0534
0.74
1.85
0.0143
129
Cantor
1.89
1.8835
0.0174
0.94
0.0034
2.2286
0.0852
0.09
0.99
0.0041
243
Stoc. Cantor
1.89
1.8896
0.0105
0.97
0.0002
2.1218
0.0933
0.17
2.43
0.01
243
Peach 1
2.33
2.3221
0.0043
0.99
0.0033
2.2832
0.0115
0.97
2.97
0.0439
67
Peach 2
2.36
2.3516
0.0056
0.99
0.0035
2.3416
0.0117
0.97
2.97
0.0459
64
Peach 3
2.38
2.307
0.0064
0.99
0.0306
2.3022
0.0195
0.97
3.04
0.0463
65
Peach 4
2.33
2.3218
0.0076
0.99
0.0035
2.3147
0.0175
0.98
3.27
0.0449
72
Mango 1
2.42
2.4616
0.0143
0.99
0.0172
2.5278
0.0093
0.99
1.44
0.0654
22
Mango 2
2.46
2.5751
0.0097
0.99
0.0468
2.7202
0.008
0.99
1.66
0.0663
25
Mango 3
2.02
2.4664
0.0088
0.99
0.2209
2.5691
0.0141
0.99
1.45
0.0610
24
Mango 4
2.48
2.6777
0.0099
0.99
0.0797
2.7237
0.014
0.99
1.21
0.0659
18
Smax
classical Student’s t-test on the computed Db distributions shows that a significant bias in the BCM estimation exists. However, results reported in Table 3.2 (cols 3-6) show that this bias is less than 3.1% of the theoretical value for the studied canopies. Estimating Fractal Dimension from the LSV Method We use the LSV estimator of the box counting method, presented on section 3.2.5, on the theoretical and digitized plants. The δe values were defined using couple of successive scales 1 δ − δi = , δe = i+1 δi δi
(3.25)
e values from the corresponding N values. and N Since it is based on a local estimation, it is sensible to the local variation of the number of box as a function of scales introduced by the quantization effect. The local variations in this estimation are reflected in the variance and standard error of the computed fractal dimensions, giving a better estimation of the reliability of the results compared to the classical box counting method. Experimentally, we observe that results on theoretical plants are very sensitive to quantization effect as shown by dispersion of the data in Figure 3.13 and the minimum
3.3. Fractal dimension of plants
47
Figure 3.13 – Estimated fractal dimension with the LSV method for AC2, Cantor Dust Peach 2, and Mango 2. This new estimator is very sensitive to quantization effect leading to a dispersion of the measurements in AC2 and Cantor Dust. On the contrary the method gives an estimation of d close to that obtained with the two-surface method (i.e d = 2.36) for Peach tree 2.
standard error in Table 3.2 (cols 7-9). The minimum r2 for the estimated dimensions on these objects are between 0.09 to 0.74. This effect is much less important on real plants; the minimum r2 values are between 0.97 and 0.98. In this case, the results seems more relevant. The difference with theoretical values is small (less than 3.2%).
3.3.2
Two-surface dimension In this section, we used the method developed by Boudon et al. [2006] and we extended the analysis to all digitized trees in our database. In particular we analyzed a set of 4 mango trees with the two-surface method and results are shown Table 3.3. In addition, the outcome of the relation (3.18) that uses the volume of the envelopes was computed for both species and are shown Table 3.4. For the peach trees, we observed close values at CYS and OYOS scales and for the entire tree (Table 3.3), thus confirming the self-similar structure of their foliage. However, because of the reconstruction method (see section 2.4.2), the envelope surface of small branching systems, mainly CYS, are strongly correlated with their leaf area. This influences the bottom part of the cloud of points, leading to linear point sets. On the opposite, mango trees were digitized at leaf scale and a greater variability in the cloud of points is clearly visible in mango plots of Figure 3.14. One can also notice that scaffold and crown points are not as well aligned with CYS points for mango
Chapter 3. Fractal dimension of plants
48
Table 3.3 – Two-surface dimension of the peach and mango trees computed using relation (3.17). For each tree an individual dimension is computed as well as “local dimensions”for different scales. Dimension not available (N.A.) for individual trees due to low number of instances can be computed when all trees instances are merged. CYS
OYOS
Scaffolds
All Scales
CYS
Scaffolds
All Scales
Peach1
2.44
2.47
N.A.
2.33
Mango1
2.13
N.A.
2.42
Peach2
2.44
2.30
N.A.
2.36
Mango2
2.19
N.A.
2.46
Peach3
2.42
2.44
N.A.
2.38
Mango3
1.76
N.A.
2.02
Peach4
2.42
2.29
N.A.
2.33
Mango4
2.15
N.A.
2.48
All
2.44
2.35
2.77
2.35
All
1.94
3.056
2.24
trees than for peach trees. This, along with discrepancies between fractal dimension of CYS and entire tree, see Table 3.3, points up the fact that mango trees do not exhibit a self-similar organization despite a well marked fractal nature of its small leafy modules, namely CYS. The value above 3 found for the scaffold scale for the mango trees can be explained by high leaf density cumulated with the fact that the value is estimated using a linear regression on few points (16), which makes it very sensitive to extremal points. Finally, the variant of the box-counting method yields fractal dimensions in accordance with the values previously found for the peach trees while, for mango trees, the values were the largest, see Table 3.4. This can be partially explained by the low number of defined scales (3) that does not allow to compute reliable regression coefficients. But most importantly, the high density of foliage and the highly intricate structure of mango trees induces important intersection of the envelopes, especially at CYS scale. Therefore these sets of convex envelopes are not valid covers in the sense of definition 3.1 and do not allow reasonable estimation of the fractal dimension using the variant of the box-counting method and explain the high values found, especially the one above 3.
Conclusion In this chapter we presented fractal geometry as a framework allowing to measure and characterize objects with very complex geometry. In the purpose of characterizing plant geometry we detailed two methods to estimate the fractal dimension, the box-counting and the two-surface methods. This parameter is of major importance in the study of irregularity: it characterizes the way plants physically penetrate space as one zooms in. The accuracy of the BCM for evaluating the fractal dimension of 3D crowns was studied. Several factors that may influence this accuracy were examined and practical solutions proposed. In particular a proper voxel size limit is determined dependent on leaf sizes and the BCM bias was quantified. The problem of data dependency used during the regression analysis was discussed and a new estimator, LSV, that does not violate the independence assumption was described. The LSV estimator appears to be an interesting indicator to determine whether the quantization effect disturbs the
3.3. Fractal dimension of plants
49
Table 3.4 – Recapitulation of the two-surface dimension and the box dimension for digitized trees. The two-surface dimension was assessed with the relation between the surface of the convex envelopes and the leaf area, column (3.17), and with the relation between the volume of the convex envelopes and the leaf area, column (3.18). The box dimension was estimated with the classical 3D grids dividing space into voxels, column “classical ”, and with the convex envelopes from the biological partitioning as boxes, column “variant ”. The last column is the box dimension estimated from the new estimator, LSV, presented in section 3.2.5.
Ds
Trees
Db
(3.17)
(3.18)
classical
variant
LSV
Peach1
2.33
2.27
2.32
2.36
2.28
Peach2
2.36
2.32
2.35
2.37
2.34
Peach3
2.38
2.34
2.31
2.40
2.30
Peach4
2.33
2.29
2.32
2.34
2.31
Mango1
2.42
2.15
2.46
2.87
2.53
Mango2
2.46
2.17
2.58
3.10
2.72
Mango3
2.02
1.54
2.47
2.70
2.57
Mango4
2.48
2.26
2.68
2.96
2.72
fractal dimension estimation. Eventually it has to be improved to support more robust evaluations. The two-surface method was also assessed for estimating the fractal dimension of 3D canopies. It appears more reliable than the BCM because its accuracy is not influenced by external factors others than the ones related to the generation of the multiscale representation and the 3D mockups. However, this method requires an explicit specification of the organization of the plant and cannot be used with raw digitized data. The use of different methods to estimate the fractal dimension of plants allows one to determine more precisely if the plant tends to be self-similar and also to better apprehend if such a number has a real meaning regarding the characterization of tree foliage. Nevertheless, fractal dimension only describes the multiscale behavior of plants in a quantitative way, i.e. how many details appear when one zooms in, and as illustrated by Stochastic Cantor dust, two objects can have the same fractal dimension but look completely different. Although fractal dimension is a meaningful number that generalizes the classical notion of Euclidian dimension, it does not suffice to entirely characterize the multiscale nature of objects. In particular, fractal dimension does not say much about where the details appear when one zooms in. However, the spatial location of plant elements, at a given scale, has evident effects on ecophysiological process like light interception, and should also be characterized. In the next chapter we will try to complement fractal dimension with such a multiscale descriptor.
50
Chapter 3. Fractal dimension of plants
Figure 3.14 – Linear regression between the log of the convex envelope surfaces and the log of the leaf surfaces, for all branching systems at all scales of peach and mango trees.
4
Lacunarity of plants
“A complex system that works is invariably found to have evolved from a simple system that works.” J. Gaule
T
he modeling of ecophysiological interactions of plants with the environment such as light or rain interception, heat dissipation or disease propagation are related to the quantitative aspect of plant geometry. As we showed in chapter 3, plant geometry is often intricate and can partially be characterized by fractal dimension. However, plant geometry also strongly depends on the spatial distribution of components. In particular, the amount and size of the gaps are of major interest for such ecophysiological processes, allowing, for instance the penetration of light within a tree crown. Mandelbrot [1983] introduced the notion of lacunarity in the specific purpose of characterizing the gap distribution within an object body. A precise definition of lacunarity was proposed by Allain and Cloitre [1991] along with the gliding-box algorithm to compute it. Although this lacunarity was successfully used by Plotnick et al. [1993] to identify different textures, the gap sizes are difficult to identify. In this chapter we will present Allain and Cloitre [1991]’s lacunarity before introducing a variant based on the same principles. The properties of this variant will be investigated before assessing the lacunarity of virtual and digitized plants, with both lacunarity definitions.
51
4.1. Allain & Cloitre’s lacunarity
4.1
53
Allain & Cloitre’s lacunarity The fractal dimension can be viewed as a measure of the irregularity of an object but it does not deliver precise information on the spatial location of its components. In fact, objects can have identical fractal dimension but very different appearance, or texture [Mandelbrot 1983], as illustrated in Figure 4.1. These sets (curds) are obtained using the recursive curdling and trema generation method presented in section 2.4. All these curds have the same self-similar dimension while showing different textures.
Curd 1
Curd 2
Curd 3
Figure 4.1 – Objects on the first row are the generators (s = 5, n = 8) of the curds of the second row obtained with the curdling and trema generation method. All this curds have the same self-similarity log 8 dimension: Da = log 5 = 1.292.
One way of distinguishing one texture from another is through their gaps, more precisely, their size and frequency. The notion of lacunarity, from Lacuna which is Latin for gap, was originally introduced for this purpose by Mandelbrot [1983]. Several methods were given for estimating lacunarity [Mandelbrot 1983; Gefen et al. 1983; Lin and Yang 1986; Allain and Cloitre 1991; Obert 1993; Tolle et al. 2003], and are usually based on the variation of a mass-radius relation. Objects with wide range of gap sizes will have high lacunarity, on the opposite objects where all gap sizes are similar will have low lacunarity and will tend to be homogeneous. At a given scale, if you look at different points of an object with a lot of different gap sizes, there is a good chance that the neighborhood of this different points will look different, the object will appear heterogeneous. On the contrary, an object with same sizes of gap will almost look the same everywhere, almost homogeneous. In fact Gefen et al. [1983] defines lacunarity of an object as a measure of its deviation from translational invariance at a particular scale. An object can be heterogeneous at small scales while being fairly homogeneous at larger scale, or vice versa. Lacunarity can thus be considered a scale dependent measure of heterogeneity or texture [Plotnick et al. 1993].
Chapter 4. Lacunarity of plants
54
In this section we will focus on the definition of lacunarity, introduced by Allain and Cloitre [1991], denoted AC-lacunarity, based on the analysis of the variations of a mass distribution function. To computed this lacunarity, the authors described a gliding-box algorithm, which is, to date, the most used method to estimate lacunarity in ecological sciences and biology [Halley et al. 2004]. Gliding-box algorithm To compute the AC-lacunarity, the gliding-box algorithm uses a discretization of the object by a regular grid of size N. A box of radius δ is then centered on one grid element so that the entire box fits into the grid, and the box mass is defined as the number of embedded grid elements occupied by the object. The box is then moved over all possible centers on the grid and its mass is recorded. This procedure generates a frequency distribution of box masses where n(η, δ) is the number of boxes of radius δ having a mass η. Box sizes are odd multiples of the cells size so they can be centered on grid cells. The probability Q(η, δ) is defined as Q(η, δ) =
n(η, δ) , Bδ
(4.1)
where Bδ = ( N − δ + 1) Dt , is the total number of boxes of size δ strictly contained in the grid, and Dt is the topological, or Euclidian, dimension of the grid. Analyzing the properties of probability distributions is typically done by studying its moments. Let Xδ be the random variable representing the mass of boxes of radius δ, and having Q(. , δ) as probability distribution, i.e. Q(η, δ) = P( Xδ = η ) where P( Xδ = η ) is the probability for a box to have a mass η. The qth moment of the probability distribution is q
m q = E ( Xδ ) =
X
η q Q(η, δ) =
X
η q P ( Xδ = η ) .
(4.2)
η
η
The expected value, or mean, of the random variable Xδ denoted E( Xδ ) is the first order moment: X m1 = E ( Xδ ) = η q P ( Xδ = η ) , η
and its variance , noted Var ( Xδ ) is related to the second order moment as follow: h i m2 − (m1 )2 = Var ( Xδ ) = E ( Xδ − E( Xδ ))2 The AC-lacunarity at scale δ is defined as the relative variance (variance compared to the mean) plus 1: Λδ =
m2 Var ( Xδ ) + (m1 )2 Var ( Xδ ) = = +1. ( m1 )2 ( m1 )2 E ( Xδ )2
(4.3)
Lacunarity is equal to 1 when the variance is equal to 0, i.e. when there is translational invariance at scale δ. In statistics, Λ = ζ 2 + 1, where ζ is the coefficient of variation. ζ is defined as the ratio of the standard deviation to the mean. It is a normalized measure of dispersion of a probability distribution [Johnson and Leone 1977].
Remark 4.1
4.1. Allain & Cloitre’s lacunarity
55
The coefficient of variation is a dimensionless number allowing the comparison between sets with different units or dimensions, therefore, so is lacunarity. The lacunarity is expected to have the following properties • Objects with more homogeneous gap sizes should have smaller lacunarity due to smaller variance of mass distribution. • Lacunarity should decrease toward 1 when δ gets big enough. Indeed, at some point the gliding-boxes will enclose almost the entire object leading to almost identical masses and therefore to a variance close to 0. Plotnick et al. [1993] successfully used the lacunarity to characterize hierarchical random maps obtained using the curdling and random trema generation explained in section 2.4. The resulting maps can be viewed as a N × N array where each element is set to “one”with a probability P. By changing the probability pi at each iteration, the authors were able to generate maps with very different textures but with identical values of P, where Y P= pi , (4.4) i
and where pi is the probability at iteration i. An example of maps obtained after three iterations with a subdivision factor of 6 and a overall value of P = 0.5 is shown in Figure 4.2.
Top
Middle
Bottom
Same
Figure 4.2 – Hierarchical random maps with N = 216 and P = 0.5 over 3 iterations. P = p1 × p2 × p3 . ‘Top’map, p1 = 0.5 and p2 = p3 = 1. ‘Middle’map, p1 = p3 = 1 and p2 = 0.5. ‘Bottom’map, p1 = p2 = 1 and p3 = 0.5. ‘Same’map, p1 = p2 = p3 = 29 36 , each iteration of the curdling and random trema process offers 6 × 6 = 36 possible pieces, and setting n to 29 gives the best approximation to 0.5, 29 3 i.e. ( 36 ) = 0.523.
The authors stressed that the use of a single-valued lacunarity estimate based on a single gliding-box size is of limited value. In fact the major interest of lacunarity measurements lies in the use of a wide range of gliding-box sizes, yielding a multiscale view of the object heterogeneity. Lacunarity curves for the maps of Figure 4.2 are shown in Figure 4.3. The highest value of lacunarity is found for δ = 1, the value being only a function of elements set to “one”. Thus, all maps having the same P value have their initial point on the same location. One should note the slight down shift for 3 the ‘Same’ map due to the fact that P = ( 29 36 ) = 0.523, illustrating the fact that for the same gliding-box size, denser maps have lower lacunarity [Plotnick et al. 1995]. The minimum value would be reached when the gliding-box size is N, the unique glidingbox contains the entire map, hence a variance of 0 and a lacunarity value of 1. Away from the endpoints, the shapes of curves yield good characterization of the maps. The
56
Chapter 4. Lacunarity of plants
Figure 4.3 – Log-log plot of lacunarity against gliding-box size for hierarchical random maps shown in Figure 4.2. The lacunarity was calculated using Allain and Cloitre [1991]’s glidingbox algorithm. The sizes of the gliding-boxes ranged from δ = 1 to 99 by step of 2.
curve for the ‘Top’ map shows high value of lacunarity until the size of the gliding-box reaches the size of the clusters (36), then lacunarity values rapidly decreases. Similarly the ‘Middle’ and ‘Bottom’ maps that have cluster sizes of 6 and 1, respectively, exhibits an inflexion point as the gliding-box size reaches these values. As explained by Allain and Cloitre [1991], the lacunarity curve for self-similar sets should be a straight with a slope equal to Da − Dt . We estimated of the slope of the almost linear curve for the ‘Same’ map and found a value of −0.134 ± 0.007 with r2 = 0.98. Thus, the estimated value for fractal dimension was −0.134 + 2 = 1.866 which was in very log(291 )
good accordance with the self-similar dimension of the set: Da = log(6) = 1.879. Plotnick et al. [1993] also demonstrated, using different values for P, that the shapes of the lacunarity curves depend on the pattern of aggregation and are independent of the density, changes are only reflected in the positions of the curves. In a subsequent study, Plotnick et al. [1995] used lacunarity as a general technique for the analysis of spatial patterns and showed how it can be applied for the description of nonfractal and multifractal patterns.
Figure 4.4 – Log-log plot of lacunarity against gliding-box size for curds shown in Figure 4.1. The lacunarity was calculated using Allain and Cloitre [1991]’s gliding-box algorithm.
Applying the gliding-box algorithm to compute the lacunarity of the curds presented Figure 4.1 produce almost linear curves as expected for self similar sets, see Figure 4.4. The slopes of these curves yield, for Curd 1 to 3, respective fractal dimensions of 1.25, 1.34 and 1.30, which are close to these sets self-similar dimension value: Da = 1.292. As previously, lacunarity curves allow a comparison between the
4.2. Centered lacunarity : Λ+
57
curd textures, Curd 2 having the bigger gaps have the higher lacunarity while the lowest values are obtained for Curd 1 which is, visually the most homogeneous. This characterization is only possible for the portion of the curve where the gliding boxes have reached the size of intermediate clusters and before they are too big to have a significant mass variability. This limitation does not allow to easily characterize gap or cluster sizes at different scales, which, in the context of characterizing complex objects like plants, would be interesting. To tackle this, we propose a variant form of lacunarity based on the gliding-box algorithm ofAllain and Cloitre [1991].
4.2
Centered lacunarity : Λ+ Let T be the cells of the regular grid embedding the object, and T + the subset of T occupied by the object. The mass of the gliding-box is defined as previously but its shiftings are now restricted to elements of T + . For all δ values, the gliding-box is centered on every T + elements whether or not it is entirely contained in the grid. The frequency distribution of the box masses, statistical moments and lacunarity are defined as previously. The total number of boxes, Bδ , is now independent of δ and has a constant value equal to the number of occupied cells, | T + |. We call this new form of lacunarity, centered lacunarity and denote it by Λ+ : Λ+ δ =
Var ( Xδ+ ) E( Xδ+ )2
+1,
(4.5)
where Xδ+ is the random variable representing the mass of boxes centered on T + elements.
Figure 4.5 – Log-log plot of centered lacunarity against gliding-box size for curds shown in Figure 4.1.
The centered lacunarity value when δ = 0 (boxes have the size of a voxel) is obviously 1, all boxes having a mass of 1 hence a variance equals 0. At the other end of the curve, for large values of δ, the variance value goes back to 0 when the entire object is included in all gliding-boxes. With no restrictions on the gliding-boxes positioning, this happens when the radius of the gliding-box is equal to the size of the object, i.e. the gliding-box size is twice the size of the object, δ = N. The centered lacunarities for the curds of Figure 4.1 are shown in Figure 4.5. The curves now clearly differ reflecting more accurately the differences between the curd textures. The three curves show two clearly marked local minima with almost identical lacunarity values.
Chapter 4. Lacunarity of plants
58
Characterization of the centered lacunarity Our aim is to understand what voxel configurations lead to null or low centered lacunarity. Let us first introduce some definitions. We will work in Euclidean space Rn , with the usual metric. If x, y are two points of Rn , the distance between them is noted d( x, y). We will denote A the subset of Rn corresponding to the points of the studied object, x ∈ A means that the point x belongs to the set A and P ⊂ A means that P is a subset of set A. Definition 4.1
A closed ball of center x ∈ A and radius δ is defined by
Bδ ( x ) = {y ∈ A | d( x, y) 6 δ} , and the subset BδA ( x ) ⊂ A is
BδA ( x ) = Bδ ( x ) ∩ A
A set A is δ-connected if ∀ x, y ∈ A there exists a finite sequence of points {zi } ∈ A where i ∈ N such that BδA ( x ) ∩ BδA (z1 ) 6= ∅, . . . , BδA (zi ) ∩ BδA (zi+1 ) 6= ∅, . . . , BδA (zn ) ∩ BδA (y) 6= ∅ .
Definition 4.2
Intuitively, we think of a set A as δ-connected if it consists in just one piece at scale δ (Figure 4.6). Definition 4.3
The δ-parallel body, Aδ , of set A is the set of points within distance δ of A: Aδ =
[
Bδ ( x ) .
x ∈A
Let kK be the number of connected kcomponents of Aδ . Aδ induces a partition of A: Aδ = Aδ ∈ A k∈[0,K] , such that ∀k, Aδ is included in a single connected component of Aδ . The δ-parallel body of set A is shown for δ = δ1 and δ = δ2 in Figure 4.6. Aδ1 is composed of 4 components while Aδ2 has a unique component. Property 4.1
A is δ-connected iff its δ-parallel body has a unique connected component.
δ-connection defines a grouping notion between elements, the complementary idea of separation is given by the concept of δ-isolation. Definition 4.4
A subset P ∈ A is δ-isolated in A if
∀ x ∈ A \ P,
d( x, P) > δ ,
(4.6)
with d( x, P) = min {d( x, y), ∀y ∈ P}. Similarly to the δ-connection, there is a strong relation between the δ-parallel body of A and δ-isolation. More precisely, the relation lies between the induced partion components: Property 4.2
If Aδ has more than one element, then they are δ-isolated.
This property follows directly from definition 4.3 of δ-parallel body. From properties 4.1 and 4.2, we have the important following corollary Corollary 4.1
The δ-connected components of Aδ are δ-isolated.
4.2. Centered lacunarity : Λ+
59
Figure 4.6 – Thenδ-body o Aδ1 of set A for δ = δ1 is composed of 4 components (A is not δ1 -connected). Its components
Akδ1
k ∈[0,3]
are δ-isolated and δ-connected. Aδ2 has a unique component (A is δ2 -
connected), and Aδ2 is δ-connected.
We have defined notions and properties that characterize relations between the elements of a set. The notions of δ-connection and δ-isolation are related to the lacunarity through the positioning of the gliding boxes. The movements of the gliding-box at scale δ for the centered lacunarity are restricted to the object elements, therefore the subsets that are δ-isolated will not have influence on each others. Let x, y ∈ A and M be a mass function (measure) defined on A. Definition 4.5
The differential mass from x to y at scale δ is defined as: h i A A A ∆Mδ ( x, y) = M Bδ ( x ) \ Bδ ( x ) ∩ Bδ (y) , = M BδA ( x ) − M BδA ( x ) ∩ BδA (y) .
Now, we define a property that characterizes local mass invariance. Definition 4.6
∀y ∈
Theorem 4.1
BδA ( x ):
A is said to be Locally Mass Invariant (LMI) at scale δ, if ∀ x ∈ A, and ∆Mδ ( x, y) = ∆Mδ (y, x )
The two following propositions are equivalent:
i. A is LMI at scale δ
ii. There exists a partition of A into δ-isolated subsets Aiδ i , where ∀i, ∀ x, y ∈ Ai : the mass is constant, i.e. : M BδA ( x ) = M BδA (y) Proof. Let us consider the partition Aδ = Akδ ∈ A k induced by the connected components of Aδ . i. ⇒ ii. ∀k
∀ x, y ∈ Akδ , we have either: BδA ( x ) ∩ BδA (y) 6= ∅ , or
BδA ( x ) ∩ BδA (y) = ∅ .
Chapter 4. Lacunarity of plants
60
• If BδA ( x ) ∩ BδA (y) 6= ∅, M BδA ( x ) = M BδA ( x ) ∩ BδA (y) + ∆Mδ ( x, y) A A = M Bδ ( x ) ∩ Bδ (y) + ∆Mδ (y, x ) = M BδA (y) • If BδA ( x ) ∩ BδA (y) = ∅, since Akδ is δ-connected, there exists a finite sequence of points {zi }i∈[0,n] ∈ Akδ such that Bδ ( x ) ∩ Bδ (z1 ) 6= ∅, . . . , Bδ (zi ) ∩ Bδ (zi+1 ) 6= ∅, . . . , Bδ (zn ) ∩ Bδ (y) 6= ∅. Using the yields: LMI hypothesis A A A A M B δ ( x ) = M B δ ( z1 ) = . . . = M B δ ( z n ) = M B δ ( y ) . Therefore the partition induced by Aδ verifies ii. i. ⇐ ii. Ai is δ-isolated, thus ∀ x ∈ Ai BδA ( x ) ∈ Ai and ∀i, ∀ x, y ∈ Ai M BδA ( x ) = M BδA (y) , therefore ∀ x ∈ Ai , ∀y ∈ BδA ( x ): ∆Mδ ( x, y) = M BδA ( x ) − M BδA ( x ) ∩ BδA (y) = M BδA (y) − M BδA ( x ) ∩ BδA (y)
= ∆Mδ (y, x ) . A is LMI at scale δ. Each δ-isolated Aiδ has a characteristic value, mi . The particular case when all of these subsets have the same characteric mass leads to the centered lacunarity theorem: Theorem 4.2
The two following propositions are equivalent:
i. A is LMI at scale δ, and all the subsets induced by its δ-parallel body Aδ have the same characteristic mass m. ii. Λ+ δ (A) = 0. Proof. From the centered lacunarity definition and from previous theorem 4.1: j
i. ⇒ ii. ∀i, j and ∀ x ∈ Aiδ , ∀y ∈ Aδ , we have M BδA ( x ) = M BδA (y) = m ⇒ Λ+ (A) = 0 i. ⇐ ii. ∀ x, y ∈ A, M BδA ( x ) = M BδA (y) , directly yields, from theorem 4.1, that A is LMI at scale δ
From theorem 4.2, we can assert that a null value for the centered lacunarity at scale δ expresses a mass invariance of the δ-connected subsets. This theorem defines a subset of all the possible sets (of the considered space) for which we do not have characterization, yet. Figure 4.7 show some simple and some less simple sets that have a null centered lacunarity. An infinite straight line has obviously a null centered
4.2. Centered lacunarity : Λ+
61
lacunarity at all scale δ. A set of parallel straight lines has a null lacunarity at all scale δ smaller than the minimum distance between any two lines (Figure 4.7 a.). The repetition, finite or infinite, of identical δ-isolated components LMI at scale δ has also a null centered lacunarity at this scale (Figure 4.7 b.). Regular circle-like line has a null centered lacunarity for all scales smaller than their radius (Figure 4.7 c.), but more complex closed lines can also have null centered lacunarity for some specific scales (Figure 4.7 d.). In the case of δ-isolated union of LMI sets, if they all have the same mass at scale δ, the centered lacunarity of this set at scale δ is null as well (Figure 4.7 e.), otherwise the centered lacunarity at this scale is not null (Figure 4.7 f.). Similarly, LMI sets at different scales do not have null centered lacunarity at neither of these scales(Figure 4.7 g.). Finally, non δ-isolated union of LMI sets at scale δ do not have null centered lacunarity at this scale.
a.
b.
c.
d.
e.
f.
g.
h.
Figure 4.7 – a., b. and c.: Simple examples of sets having null centered lacunarity at scale δ. d.: More complex closed line also having null centered lacunarity at scale δ, or e.: any δ-isolated union of the previous sets. f.: The centered lacunarity is not null if the components do not have the same mass, or g.: if they are not LMI at the same scale δ, or h.: if the components are not δ-isolated.
The generator of Curd 2 clearly shows that this set has identical δ-isolated subsets, and allows to predict the scale at which it will be LMI. The subdivision factor is 5, so due to the iteration process, the different sizes of clusters will be 5 and 25. Since the gliding boxes are centered on a cell, the set will exhibit LMI at scale δ = 4 and δ = 24, and consequently the centered lacunarity should be null at these scales. This previsions were confirmed as illustrated in Figure 4.8. One could also notice that in the particular case of Curd 2, the lacunarity values only changes every two steps of gliding-box radius increase. This is due to the particular structure of this curd generator where the distance between any two points is 2, thus the mass of a gliding box centered on a particular element changes only if its radius is increased by 2. For the two other curds presented Figure 4.1, the centered lacunarity curves also exhibits two minima, but without reaching 0 (Figure 4.9). The minima are the scale
62
Chapter 4. Lacunarity of plants
Figure 4.8 – Centered lacunarity for a highly-clustered set at scale δ = 4 and 24. When the radius of the gliding-box is equal to δ, lacunarity value reaches 0.
Figure 4.9 – Centered lacunarity curves for Curd 1 (left) and Curd 3 (right). The minimum values of lacunarity are reached for δ = 4 and 20 and δ = 2 and 14 for Curd 1 and 3, respectively.
at which we are the closer to LMI, and we can see that the scales are not the same as for Curd 2. For instance, in Curd 1, the pattern of the corners are not δ-isolated in the center. Consequently, for δ = 24, the characteristic mass of the δ-connected subset of the middle is higher than the ones of the corner subsets. As we can see in Figure 4.9 the minimum is reached for a slightly smaller value of δ, for which the LMI condition is better approached. By removing two opposed element in the center of Curd 1, we can generate a alternate pattern that is LMI for the same scale as Curd 2. The shift between the scale where minima are reached and the difference of lacunarity values should represent the perturbation effect of the two removed elements from LMI, see Figure 4.10. For non LMI sets, the values of gilding-box radius for which a minimum is reached seems strongly related to the periodicity induced by the connexity, in the
4.2. Centered lacunarity : Λ+
63
Figure 4.10 – Compared centered lacunarity for Curd 1 and the a curd obtained from the same generator modified so that it exhibits LMI at scale δ = 4 and 24.
discrete geometry sense, of the generator. More thorough analysis is required before being able to confirm such a relation. In practice lacunarity values can be computed using matrix convolutions. The grid embedding the object is converted into a matrix of size N filled with 0, then occupied sites are assigned with the value 1. The gliding-box of size δ acts as a δ-sized kernel filled with 1. The convolution produces a new matrix of size N where the value in each cell represents the mass of the gliding-box of size δ centered on that location. A naive gliding-box algorithm would have to explore the entire matrix ( N 2 ) for each of its element ( N 2 ) in order to assess the mass, thus having a complexity in O( N 4 ), while the convolution method is in O( N 3 ). An example of such a convolution for a gliding-box of size δ = 3 is given Figure 4.11. To obtain the mass distribution used for the centered lacunarity, we only consider the values from the originally occupied cells. In the case of AC-lacunarity, only cells with coordinates within [ δ−2 1 , M − δ−2 1 ] are considered, i.e. the centers of the gliding-boxes entirely included in the grid. The main benefit of this technique is that from one convolution, different forms of lacunarity can be calculated by changing the subset of considered cells.
Figure 4.11 – Mass distribution for Curd 1 and a gliding-box of size δ = 3 computed with matrix convolution.
64
4.3
Chapter 4. Lacunarity of plants
Assessing plant lacunarity
Figure 4.12 – Allain and Cloitre [1991]’s lacunarity (left) and centered lacunarity (right) for, from top to bottom, artificial crowns, peach trees and mango trees.
We computed both AC-lacunarity and the centered lacunarity to the artificial crowns and the digitized trees of our database. Results are shown in Figure 4.12. AClacunarity yielded almost straight lines for the artificial crowns, as expected because of their self-similar nature, but also for the peach trees which tends to confirm their self-similar behavior as noted in chapter 3. In comparison to the peach tree results, the mango tree curves exhibited more variability reflecting the higher diversity of their 3D shape as shown in Figure 2.17. The slightly more pronounced concavity is due to the higher leaf area density (LAD) of the mango trees, more rapidly leading to homogeneity of the gilding-box masses. This difference of LAD is clearly visible when comparing lacunarity at the same scale, i.e same x-axis value. The results from
4.3. Assessing plant lacunarity
65
the centered lacunarity yielded the same informations regarding the LAD differences, but with more characteristic curve profiles. As explained, a local minimum seems defining a box radius, having the property of being the closest LMI, and therefore being a compromise between the size of δconnection, clusters size, and δ-isolation, gap size. This box has low mass variation so when gliding through a cluster it contains the main part of it, so box centered in the middle of a cluster weight almost the same as the boxes centered on its fringe. The mass of the frontier boxes being possibly compensated by the border of surrounding clusters, hence the compromise between gap and cluster size. As a matter of fact, the value of the minimum found for each artificial crown define a δ for which the subsets induced by the δ-parallel body are the closest to be δ-isolated and LMI, thus the closest to verify conditions of theorem 4.2. However, a sensitivity analysis conducted on a broader set of artificial crowns, with larger and well identified variability of gaps and cluster sizes, is required to confirm it. The curves of peach trees do not show this minimum and the one found for mango tree is not clearly related to any defined botanical entities (current year shoot or branches) or gap size. The local maximum that appears on all curves we suspect is due to an edge effect. When the gliding box-size reaches the size allowing it to contain the major part of the object, the box centered on the border of the object will have a mass significantly smaller than those centered within the object, thus leading to a local maximum of lacunarity. The simple example in Figure 4.13 seems to confirm this hypothesis. An image is generated by positioning a set of full discs at random, leaving only few gaps. The centered lacunarity curve obtained showed a unique maximum when the radius of the gliding-boxes reached the third of the set’s width, i.e. a gliding-box surface almost the half of the set’s one. This size has no relation with discs size nor with their organization, thus leaning in favor of the edge hypothesis and raising, yet, another discrete geometry question: can we analytically estimate, for a given shape (possibly simple), the gliding-box size leading to the higher mass variability and hence, higher lacunarity?
Figure 4.13 – Centered lacunarity for set composed of 16 discs positioned at random inside a 150x150 pixels image. A disc radius is 15 pixels. The maximum lacunarity is obtained for δ = 50.
The centered lacunarity profiles were quite similar for the trees of a same species, and different from one species to another. This property was also verified for the artificial crowns profiles. In fact their generation processes, which are fairly similar, can be seen as phenotype expression, suggesting that the centered lacunarity profiles
66
Chapter 4. Lacunarity of plants
could be used to characterize species. This is only rough conjecture that need to be examined more carefully. The original purpose of lacunarity is to describe the distribution of gaps, which, as explained in the introduction, can have an important effect on ecophysiological processes such as light interception. Light interception of a plant can be roughly described as the intersection of plant organs with beams coming from the sun direction.It can be estimated as the projection of the plants along this direction, the 3D plant being shortened to its 2D projection. Studies of light interception are usually realized for a set of direction that discretize the sky vault, for example, the 46 directions of the turtle sky introduced by Den Dulk [1989]. We tried to establish a link between multiscale analysis and ecophysiological processes by comparing the 3D centered lacunarity of trees to the 2D centered lacunarity their projections along these 46 directions. The directional lacunarities were grouped according to the elevation of the light direction, Figure 4.14 – Projection images of Peach2 and low elevation corresponds to light com- Mango2 for different elevations. ing from direction close to the horizon while zenith corresponds to an elevation of 90◦ . Results are shown in Figure 4.15, vertical red lines were added to represent the average size of the different botanical entities (scales) defined for the trees, namely leaf, current year shoot (CYS), one year old shoot (OYOS) and crown (see section 2.4.2). First, 3D lacunarity is markedly superior to the 2D one. The organization of plants is such that inside gaps are much less visible when looking at the structure from a projection point of view. Second, there is no obvious correlation between elevation and lacunarity. Mango1 and 3 exhibiting, however, a denser structure when viewed from higher elevations, expressing better interception for light coming from zenithal directions. Third, the fairly dense projection images, as illustrated Figure 7.5, yielding the shapes of lacunarity curves of Figure 4.15 tend to confirm the edge effect hypothesis concerning the end curve bump.
Conclusion In this chapter we presented a second descriptor that enriches the multiscale analysis. Fractal dimension measures how much space is filled as a function of scale, and lacunarity complements it by measuring how space is filled [Halley et al. 2004]. We presented a widely used method to assess lacunarity and showed that it is not perfectly suited for our purposes, characterizing the gap and cluster sizes of isolated objects such as tree crowns. We therefore proposed an alternative definition of lacunarity, more adapted to isolated object. It is independent of a bounding box since it is centered on the studied object, and was called: centered lacunarity. The edge effect was identified, and seems to be similar in both 3D and 2D. Centered lacunarity has interesting properties related to cluster and gap sizes. We initiated their analysis and
4.3. Assessing plant lacunarity
67
Figure 4.15 – 3D and 2D centered lacunarity for all peach and mango trees. 2D lacunarities are grouped according to the elevation of the projection direction. Red lines represent the average size of the trees botanical entities.
68
Chapter 4. Lacunarity of plants
demonstrated some properties, but further work in that direction is required. We also made a first step toward the main goal of this work which is the estimation of the impact of multiscale organization of plants regarding ecophysiological process, specially light interception. In fact, from an ecophysiological point of view, using biological values in voxel cells, like leaf area, for computing lacunarity may be more meaningful than the binary presence/absence type of data currently used. This next step would be easily achieved, as the matrix convolution method we use, is fully compliant with this kind of changes. As Deering and West [1992] stated: “it is important to recognize that while Euclidean geometry is not realized in nature, neither is strict mathematical fractal geometry. Specifically, there is a lower limit to self-similarity in most biological systems, and nature adds an element of randomness to its fractal structures. Nonetheless, fractal geometry is far closer to nature than is Euclidean geometry” In the last two chapters we saw that fractal geometry provides novel insights where Euclidian tools were found to be unadapted for describing the complex geometry that can be found in nature and especially in plants. In the next part of the document we will present how the multiscale nature of plants can used in the modeling of light interception.
5
Modelling light interception
“You can’t have a light without a dark to stick it in.” A. Guthrie
L
ight is a major factor impacting plant development. First, it provides plants with their main source of energy through photosynthesis. Second, plants are sensitive to light conditions (direction, duration, wavelengths) and react to it. Green plants only require carbon dioxide and a simple inorganic nitrogen compound for metabolic synthesis of organic molecules (such as glucose) using light energy, they are photoautotrophic. This process, called photosynthesis, can be summarized as: photosynthesis splits water to liberate O2 and fixes CO2 into sugar. In plants, the primary sites of photosynthesis are the leaves because of their high chloroplast concentration. Chloroplasts are organelles that contain chlorophyll pigments, the necessary proteins for photosynthesis. Chlorophyll absorbs the blue and red but almost nothing in the green, hence the color of plants. In fact interaction between plants and light results in important modifications of its electromagnetic spectrum, and more specifically of the Morphogenetically Active Radiations (MAR). Plants have photoreceptors, namely phytocromes, able to sense changes in the spectral composition of light, thus allowing it to perceive its light environment [Smith and Whitelam 1997]. Phytocromes along with other MAR photoreceptors act as a signals system [Aphalo et al. 1999] being able to trigger morphogenesis changes that will modify the plant architecture, which, in turn, will modify light interception. Therefore, this photomorphogenesis process allows light to modify plant structure by influencing creation, size and organization of its organs [Varlet-Grancher et al. 1993]. Moreover, photomorphogenesis mechanism creates a feed-back loop regulating plant architecture in order to optimize light-interception. Photomorphogenesis is thus an essential element of functional-structural plant modeling, but its understanding requires to quantify the relation between plant architecture and light interception. Therefore, as a first step, we need to model radiative transfer within plants. The light interception process can be seen as the intersection between a photon trajectory and a plant organ. Non intercepted photons go through plant space and constitute the transmitted radiation. The intercepted photons are either absorbed and used in processes like photosynthesis or scattered. Scattered photons can, in turn, be intercepted or not, and if interception occurs, be absorbed or scattered again, and so on. The scattering mechanism is composed of two processes, reflection and transmission. Reflection, that bounces back the photon in an other direction, and transmission, that allows the photon to go through the intercepting element, depend on 69
70
Chapter 5. Modelling light interception
the element properties and therefore can vary with wavelengths whereas interception does not. Since the pioneer work of Monsi and Saeki [1953], many models have been developed and reviews can be found in [Ross 1981; Myneni et al. 1989] and in [Sillion and Puech 1994; Glassner 1995] (computer graphics oriented). The models allow the computation of radiative transfers with or without taking scattering into account, but they all need, as an input, information about the plants to be studied and the light environment. Models can be classified according to the plant description they require, ranging from detailed description where every component is represented as a surface, to global description where canopy is considered as a volume with its structural parameters described using statistical functions. In this chapter we will review the major approaches that are used to assess radiative transfers in these two types of model with an emphasis on contributions using plant structure and particularly their multiscale organization as an improvement factor.
5.1. Models representing plants with explicit description of the geometry
5.1
71
Models representing plants with explicit description of the geometry In this section we will consider the explicit representation of plants as a set of surfaces that we will call primitives. A primitive is a polygon, usually a triangle, that describes the shape, size and orientation of a plant part. This type of description can be obtained with methods described in chapter 2. The set of primitives that are considered for illumination computing is generally called a scene. Models using this type of description tentatively estimate the distribution of light in a scene with restriction to geometric optics [Glassner 1995]. To obtain a global illumination solution, the radiative transfers between primitives need to be solved. These transfers are expressed by the radiance equation initially introduced by Kajiya [1986], which expresses the radiance scattered at a point of a surface as a function of radiances reflected to this point by other surfaces [Chelle and Andrieu 1998]. The relevant quantity to describe radiant energy transfer is radiance. It describes the amount of light emitted from a location in space, x, to a specific direction (θ, φ) expressed in polar coordinates, per unit time, per unit area orthogonal to the light direction and per unit solid angle [Sillion and Puech 1994]. The radiance is noted L( x, θ, φ), and its SI unit is watts per steradian per square meter . A corresponding quantity, E, called irradiance and expressed in (W.m−2 ), represents the total incident flux density at a point.
Definition 5.1
The notion of solid angle (see Figure 5.1) is the 3D generalization of the 2D angle, it measures the projected area of an object on a unit sphere centered at the view point. Two-dimensional angle is defined as the ratio of the arc to the circle radius, similarly, a solid angle, Ω, is defined as the ratio of a portion of the sphere surface, S, to the square of its radius, r: S Ω= 2. r A solid angle Ω is a measure of how big an object appears to an observer. For instance, a small object nearby could subtend the same solid angle as a large object far away. In the following we adopted the nomenclature from [Chelle and Andrieu
Figure 5.1 – The solid angle is proportional to the projected area, S, of an object onto a sphere surface, of radius r, centered at the view point, divided by the square of the sphere’s radius, r: Ω = rS2 . The plane containing a oriented primitive A j separate its solid angle domain in two, Ω j+ and Ω j− . The former being the one containing the normal to A j .
1998]. Let us consider a primitive A j , the domain Ω j of solid angles starting from → the center of A j is divided in two hemispheres. The sets of directions − ω such that − → − → − → − → − → ω . n j > 0 and ω . n j < 0, where n j is the normal to A j , define the Ω j+ and Ω j−
Chapter 5. Modelling light interception
72
hemispheres, respectively. An illustration of this hemispheres distinction can be seen in Figure 5.1 When an incident light beam of radiance L( x, θi , φi ) arrives at a point x of A j , it is reflected along all directions of Ω j+ according to the bidirectional reflectance distribution function (BRDF) at this point and it is transmitted along all directions of Ω j− according to the bidirectional transmittance distribution function (BTDF).
Figure 5.2 – A BRDF can be defined as a reflectance function ρ(θr , φr , θi , φi ), where (θi , φi ) define the incident direction and (θr , φr ) the re→ flected direction. − n is the normal direction of the primitive.
Reflection and refraction, i.e. transmission, depends on the object phase function which can be decomposed in two distributions: BRDF and BTDF which relates light incident in a given direction to light reflected, refracted, along a second direction for a given object, see illustration in Figure 5.2. Hence L( x, θr , φr ) = ρ j ( x, θr , φr , θi , φi ) E( x, θi , φi ) , L( x, θt , φt ) = τj ( x, θt , φt , θi , φi ) E( x, θi , φi ) ,
(5.1) (5.2)
where ρ j is the reflectance function of A j in x and τj the transmittance function. And, where the subscripts i, r and t stands for incident, reflected and transmitted, respectively. The incident irradiance E( x, θi , φi ) can be expressed in terms of incident radiance [Nicodemus et al. 1977]: E( x, θi , φi ) = L( x, θi , φi ) cos θi dωi . In the case of a light source, the object itself emits radiance that we will denote Le ( x, θ, φ). Therefore, the energy equilibrium in a scene is expressed by the following integral equation : Z L( x, θ0 , φ0 ) = ρ j ( x, θ0 , φ0 , θ, φ) L( x, θ, φ) cos θdω Z
+
Ω j+
Ω j−
τj ( x, θ0 , φ0 , θ, φ) L( x, θ, φ) cos θdω + Le ( x, θ0 , φ0 ) .
(5.3)
5.1. Models representing plants with explicit description of the geometry
73
The radiance leaving A j at point x in direction (θ0 , φ0 ) is the sum of reflected radiance coming from domain Ω j+ , transmitted radiance coming from domain Ω j− and emissivity of the surface which is non-zero only for light sources. This type of integral equation where the unknown term appears on both sides can rarely be solved analytically, thus numerical methods are usually used to compute approximate solutions [Sillion and Puech 1994]. The finite element method can be used and the collocation method or Galerkin approach can be applied to obtain solutions [Glassner 1995]. As noted by Chelle and Andrieu [1998], this particular equation is frequently either solved stochastically with Monte Carlo based methods [Ross and Marshak 1988] or deterministically with the radiosity method [Borel et al. 1991]. In what follows, we will only present these two most widely used approaches.
5.1.1
Ray tracing As stated by Glassner [1995], “ray tracing method is nothing more than an application of Monte Carlo methods to the full radiance equation”. The principle of ray tracing is to mimic photons trajectories, assuming light travels in straight line. The idea is to cast a large number of photons from the light sources and follow them into the scene, this method is called photon tracing. An alternative and dual way, called visibility tracing is to consider the reverse path of photons from a trajectory end point on a primitive. Although visibility tracing is widely used in computer graphics, we will describe the ray tracing approach using photon tracing, for it better corresponds with the modeled phenomenon. A first approach is to compute light interception from one direction, considering the light source as a collimated one. For direct radiation, the sun is considered a point source situated at an infinite distance while for diffuse radiation, the sky vault is discretized in a set of directions, e.g. Den Dulk [1989]’s turtle sky, from which light rays are parallel. Direct plant illumination can be obtained by projecting the geometry of its organs along the light direction to an orthogonal plan above the scene. The projection is usually obtained using a z-buffer. A z-buffer is a twodimensional array managing the depth of plant primitives to be represented. Each element of the buffer accounts for a particular light beam, and for each beam the z-buffer array stores which primitive has the smaller depth value along the projection direction. Each primitive stored in the z-buffer receives light proportionally to its occurrence number. The technique called ray casting based on the idea of shooting rays and finding the first element blocking the path is exactly equivalent. As ray casting does not account for scattered light, it would be fully correct only in the case of black leaves [Planchais and Sinoquet 1998]. However it has been successfully used for estimating direct illumination of vegetation in several occasions [Chen et al. 1993; Pearcy and Figure 5.3 – Classical ray Yang 1996; Planchais and Sinoquet 1998; Fournier and Antracing, [Glassner 1995]. drieu 1999]. It is for exampla the underlying process for computing light-interception in VegeSTAR software [Adam et al. 2004]. In fact ray
74
Chapter 5. Modelling light interception
casting can be thought of as a simplified version of the ray tracing technique capable of simulating light scattering. Ray tracing is based on the same idea of casting light beams towards the scene and testing for intersection with objects within the scene. But, when the beam hits an object, it can be either absorbed, reflected, refracted, or any combination of these processes, depending on the normal direction of the surface. These processes decrease the energy of the beam, and when it reaches zero it is totally absorbed. Reflected and refracted beams continue their path within the scene until being totally absorbed or reaching a predefined maximum bounce number. This classical ray tracing leads us to a structure introduced by Whitted [1980], and shown schematically in Figure 5.3. In this case, all primitives reached by at least one beam will receive light. Ray tracing is, in fact, a recursive ray casting where increased accuracy is achieved with recursivity. The major drawback of this method is the possible high number of computed beams. Indeed, if m is the maximum recursivity allowed, each original beam can generate up to 2m new beams. Computation time being proportional to the total number of beams, n, it dramatically increases with desired accuracy, its complexity is in O(n.2m ). Instead of using a single direction for reflected and refracted beams, Cook [1986] proposed to sample the directions of reflection and refraction. This approach may be viewed as a direct use of Monte Carlo methods using the BRDF and the BTDF. This technique called distributed ray casting or stochastic ray tracing is illustrated schematically in Figure 5.4. Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. As the number of √ samples increases, convergence toward a solution is achieved at a rate of η for η samples [Disney et al. 2000].
Figure 5.4 – Distributed ray tracing, [Glassner 1995]. The length of arrows illustrate the probability value of this particular direction in the BRDF or BTDF.
Another Monte Carlo approach called path tracing, which creates a path history for a given beam was proposed by Kajiya [1986]. That is, instead of spawning a set of new beams at an intersection, one direction is chosen, and one beam is cast to follow it, until absorption. See [Sillion and Puech 1994; Glassner 1995] for a more detailed presentation of path tracing. As ray tracing requires many beam-surface intersection tests, several approaches have been designed to speed up the process. The most common way of reducing the
5.1. Models representing plants with explicit description of the geometry
75
number of intersection tests is space subdivision into many non-overlapping subregions which is frequently done using an octree. But there are several other procedures like directional subdivision or bounding volumes hierarchies [Glassner 1988]. A review of acceleration techniques for ray tracing can be found in Arvo and Kirk [1989]. It is worth noting that, in the context of plants, the idea of using hierarchies of models introduced by Kajiya [1985] was implemented by Meyer and Neyret [2000] to render complex natural scenes. The authors used a multiscale decomposition of pine-tree into boughs, cones and needles to develop an efficient rendering algorithm that intensively uses the similarities of the components at different scales.
5.1.2
Radiosity method The classical radiosity method does not solve the global illumination equation 5.3. Instead it solves a simplified problem where all surfaces are supposed to be Lambertian and small enough so that radiosity and irradiance across a surface is constant [Glassner 1995]. A Lambertian surface scatters light so that the reflected and transmitted radiance are the same for any direction of the respective hemispheres. This simplification hypothesis has been successfully used for plant leaves in dense crop canopies [Chelle 2006]. Depending on the object material, this hypothesis tackles the possible complexity of its BRDF and BTRF. Reflectance and transmittance of A j will therefore be simply noted ρ j and τj , respectively. ([Sillion and Puech 1994]) Radiosity is the total power leaving a point on a surface, per unit area on the surface. Radiosity, usually denoted B, is a function of position on the surface and is defined for a point x of the primitive A j as : Z B( x ) = L( x, θ, φ) cos θdω . (5.4)
Definition 5.2
Ωj
Because of classical radiosity hypotheses, radiance is a function of primitive only, thus Z Bj = L j cos θdω Ωj
Z
= Lj = πL j
π
Z
2π
cos θ sin θdθdφ , 0
0
In the following, we will show how, starting from equation 5.3 we obtain the radiosity equation as expressed in [Chelle and Andrieu 1998]. The Lambertian assumption also simplifies the general global illumination equation 5.3, but incoming radiance still depends on direction. Moreover the incident radiance depends on the radiance of all other primitives. Without loss of generality and because plants are rarely light sources, we will consider the emissivity term equal to zero and decompose the radiosity into two terms. The radiosity of uncollided light from light sources, typically sun and sky, and the radiosity resulting of scattered light from other primitives. Let us note B0 the radiosity due to direct illumination, hence equation 5.4 can be written Z Z 0 (5.5) Bj = Bj + ρ j ( x ) L j (θ, φ) cos θdω + τj ( x ) L j (θ, φ) cos θdω , |{z} Ω j+ Ω j− {z } | direct light scattered light
Chapter 5. Modelling light interception
76
where the two integrals terms only account for radiance from other primitives. Let us consider two points x and y from primitives A j and Ak , respectively. If y is the point visible from x in the direction (θ jk , φjk ), then it is unique and x is also visible from y in the direction (θkj , φkj ). The property of invariance of the radiance along a view line, shown in [Sillion and Puech 1994] states that L( x, θ jk , φjk ) = L(y, θkj , φkj ) .
(5.6)
Since Ak is a Lambertian surface, B(y) = πL(y, θkj , φkj ) .
(5.7)
By expending the differential solid angle dω =
cos θkj dAk →k , k− xy
(5.8)
and restricting the integration domain to the points of Ak visible from x, we can express the radiance from Ak as a surface integral. The simplest way to achieve the domain restriction is to introduce a visibility function V ( x, y) such that ( 1 if x and y are mutually visible V ( x, y) = 0 otherwise . Using equations (5.6), (5.7) (5.8) and the visibility function, the radiance from primitive Ak , Z L( x, θ jk , φjk ) cos θ jk dω , Ak
can be rewritten as:
1 π
Z Ak
B(y)
cos θ jk cos θkj dAk V ( x, y) . →k k− xy
(5.9)
For the purpose of illuminating other primitives, we consider the constant radiosity as an average of the pointwise radiosities as suggested by Sillion and Puech [1994]: Z 1 Bj = B( x )dx , (5.10) a j x∈ A j where a j is the area of primitive A j . Therefore the equation 5.9 becomes Z cos θ jk cos θkj 1 Bk . dAk V ( x, y) , →k π Ak k− xy
(5.11)
which, if disregarding Bk , is a purely geometric term accounting for the proportion of scattered light from Ak that reaches A j . This term is called the form factor and is denoted Fjk : Z cos θ jk cos θkj 1 Fjk = dAk V ( x, y) . (5.12) →k π Ak k− xy Introducing the form factors in equation 5.5 yields the radiosity equation as expressed by Chelle and Andrieu [1998]: X X Bj = B0j + ρi Bk Fjk + τi Bk Fjk . (5.13) k ∈Ω j+
k ∈Ω j−
5.1. Models representing plants with explicit description of the geometry
77
This equation is written for one primitive, therefore there exist similar equations for each of the N primitives of the scene. Theses equations constitute a linear system with N unknown radiosities that can be grouped to form the following matrix equation: 0 B1 1 − χ11 F11 −χ12 F12 · · · −χ1N F1N B1 0 −χ21 F21 1 − χ22 F22 · · · − χ F B 2N 2N 2 B2 . .. = .. .. .. .. .. . . . . . .
−χ N1 FN1
−χ N2 FN2
or simply put
· · · 1 − χ NN FNN
BN
B0N
M · B = B0 ,
(5.14)
where B and B0 are vector containing the Bj and B0j , respectively, and where ( χ jk =
ρj τj
if if
− → → n j .− nk < 0 − → − → n .n > 0 . j
k
Once the matrix has been built, it could be inverted to find the solution B = M −1 · B 0 .
(5.15)
Figure 5.5 – A taxonomy of form factor algorithm from [Cohen et al. 1993].
To build the matrix M, we first need to compute the form factors. A plethora of methods to compute form factor exists and are summarized in Figure 5.5. The complexity of algorithms to compute form factors, hence to compute the M matrix is in O(n2 ) [Glassner 1995]. Beside this complexity, related to the size of the matrix, its invertion becomes rapidly very costly. If n is the size of the matrix, inversion
Chapter 5. Modelling light interception
78
using classical Gaussian elimination is in O(n3 ) [Cormen et al. 2001]. So, rather than explicitly invert the matrix, iterative methods such as Jacobi or Gauss-Seidel are used. Each iteration cost of such methods is in O(n2 ). These methods yield approximate solutions that are acceptable considering other simplifications hypotheses already made. Consequently, computation of the form factor matrix, M, constitute the bulk of the work, and reducing the number of form factors will impact both the computation time of M, and a solution. Here again, many solutions were proposed, Borel et al. [1991] proposed to set form factors of distant primitives to 0, yielding sparse matrix, which makes a full-matrix solution possible [Chelle et al. 1998]. This technique introduces a bias in the solution because the amount of small neglected transfers could result in significant lost energy in the system. Goel et al. [1991] made periodicity assumptions allowing to reduce the number of form factors to compute while Chelle and Andrieu [1998] explicitly restrict the radiosity method to a neighboring sphere around each primitive. But the most used technique is the hierarchical radiosity that makes use of clustering. The idea behind hierarchical radiosity is that the computing time of a form factor should be proportional to its importance in regard of the global energy balance [Sillion and Puech 1994]. For instance when two primitives are close to each other, it may be necessary to subdivide the primitives to precisely compute the interaction. On the contrary, if the distance between two sets of primitives is important, each of these sets may be considered as a group, and only one form factor is computed between these two groups. At the heart of hierarchical radiosity is the idea of a hierarchy of subdivided primitives and an iterative algorithm including three particular phases: Refine, Gather, and Push/Pull [Soler 1998]. The Refine phase determines, for each pair of primitives, which should be subdivided. The Gather phase propagates the energy along the links (computed form factors) defined by the Refine phase. The Push/Pull phase maintains the radiosity coherence within the hierarchical structure. So, although the algorithm starts by computing O(n2 ) interactions, this value of n is much lower than for nonhierarchical radiosity [Glassner 1995]. It is considered that with a well balanced hierarchy, the radiosity of a scene containing n primitives can be obtained in O(n. log n) [Soler et al. 2003]. Soler et al. [2003] developed a specific method to use hierarchical radiosity with plants. Instead of a hierarchy based on proximity, the hierarchy of this model is based on the plant multiscale organization. It allowed the authors to use the self-similarity of plants to decrease both computation time and memory cost. To achieve this, instantiation was used on top of hierarchical radiosity and clustering. At each scale of the plant, entities considered similar were bound to a unique instance that allow the sharing of the geometry and pre-computed BRDF and BTDF for a predetermined set of directions. The recursive hierarchy of the multiscale organization of plants, induces a nested instanciation procedure that drastically decreases the computation time. As an example, let us consider a well-balanced hierarchy of instances over k levels as done in [Soler et al. 2003]. Let us also consider that each level is composed of N elements among which p are (sub)instances. N − p is thus the number of primitives in each instance. The total number of primitives, n, is therefore n = ( N − p)(1 + p + · · · + pk−1 ) + pk N = O( pk N ) . {z } |{z} | Level 0 to k − 1
level k
(5.16)
5.2. Models representing plants as a volume and statistical functions
79
As stated before, a hierarchical radiosity solution for this kind of scene can be performed in O(n. log n) time. The key idea of Soler et al. [2003]’s algorithm is to compute local radiosity solution recursively in each instance. Let C (i ) denote the cost for solving level i and its sublevels, i.e. the cost of the local solution plus the cost of the (sub)instances of level i: C (i ) = N. log N + pC (i − 1) . (5.17) At level k the cost is C (k) = N. log N, and therefore for the entire scene C (0) = N. log N (1 + p + · · · + pk ) = O( pk N. log N ) = O(n. log N ) .
(5.18)
When the self-similarity of a plant is strongly expressed at each scale (N n), this method drastically improves the gain in computation time since O(n. log N ) O(n. log n) . Chelle et al. [1998]’s nested radiosity introduces a mixed model that uses a geometric model for computing radiative transfer in a sphere-based neighborhood of each primitive and a statistical approach to account for influence of distant primitives. The radiosity equation 5.13 is modify as: Bj = B0j + ρi
X k ∈Ω j+
Bk Fjk + τi
X k ∈Ω j−
f ar
Bk Fjk + Bj
,
(5.19)
f ar
where Bj is the radiosity due to far primitives computed with a statistical model. In this case, the complexity of radiosity solution is directly bounded to the choice of the neighborhood sphere.
5.2
Models representing plants as a volume and statistical functions In this section plants are described using the turbid medium analogy in which plant organs are considered as a set of infinitely small primitives homogeneously distributed in a given volume v. This analogy was introduced by Monsi and Saeki [1953] to conduct photosynthesis calculations. We will now describe the principles of simulating radiative transfer in statistical models as done in [Sinoquet et al. 1993; Combes 2002]. The features of these small primitives having an effect on the canopy radiation are described with two statistical functions: the leaf area density function and the leaf orientation function. Leaf area density function It characterizes the quantity of leaf area in a small volume dv around the point ( x, y, z) and is usually noted u( x, y, z). This density function only describes the mean display of foliage elements, but does not provide any information about the relative spatial location of the leaves, i.e about the mutual shading
Chapter 5. Modelling light interception
80
[Sinoquet and Andrieu 1993]. This quantity is frequently described by the Leaf Area Index which corresponds to the leaf area above a given unit ground area: Z T u( x, y, z)dz , (5.20) LAI = 0
where T is the top of the canopy. In the case of homogeneous canopy, the LAI is constant. Another used quantity is the Leaf Area Density (LAD), defined as the ratio of total leaf area of v to its volume V (v): RRR u( x, y, z)dxdydz . (5.21) LAD = V (v) Leaf orientation function This function defines the quantity of leaves having their normal direction within a small solid angle dω, and is noted g(θ, φ). Simplifying assumptions are generally made, first leaf inclination and azimuth distribution are supposed independent, so g(θ, φ) may be split as follow [Sinoquet and Andrieu 1993]: g(θ, φ) = g(θ ).g(φ) . Second, azimuth distribution is supposed to be uniform, so the orientation distribution is only a function of φ which obey: Z π 2 g(φ)dφ = 1 . 0
This function can also be analytically defined. Several standardized distributions
Figure 5.6 – Cumulative distribution for standardized distribution functions defined by the principal orientation of leaves. Planophile: Horizontal. Erectophile: Vertical. Plagiophile: Inclined at 45◦ . Extremophile: Horizontal and Vertical. Spherical: g(φ) = sin(φ). Uniform: None.
describing the global trends in the foliage orientation were defined by [de Wit 1965] and are illustrated Figure 5.6. In statistical models, the location of primitives is not know, hence light interception can not be estimated as their projection along the light direction. Instead, it can be expressed as a probability. Let us call p the probability of a beam b to be intercepted at a definite level L of the canopy, its complement p0 = 1 − p ,
5.2. Models representing plants as a volume and statistical functions
81
corresponds to the probability of the beam b to reach level L without being intercepted. It defines the gap-frequency. This quantity expresses the radiation attenuation in a turbid medium.
5.2.1
Gap frequency In this part we will describe the link between the gap-frequency concept and the Beer-Lambert’s law. This law, introduced by Monsi and Saeki [1953] to express the radiation attenuation, depends on the elevation h of the beams direction, Ω, and is usually displayed as: GΩ .LAI , (5.22) p0 = exp − sin(h) where GΩ is the extinction coefficient, also referred to as e.g., the mean projection of unit foliage area, the G-function or the mean projection coefficient [Ross 1981; Chen and Black 1992; Stenberg 2006]. Let C be the canopy and pi (C) its set of leaves, `. We will denote S(`) the area of leaf `, and SΩ (`) the area of its projected surface along the direction Ω. If N is the total number of leaves in C , the G-function can be written X SΩ (`) 1 `∈π (C) GΩ = , N X S(`) `∈π (C)
and we consider
∀ ` ∈ π (C),
SΩ (`) = GΩ .S(`) .
The gap frequency at the bottom of a canopy containing a unique leaf is p0 = 1 −
SΩ (`) G .S(`) = 1− Ω , SΩ (C) SΩ (C)
(5.23)
which corresponds to the fraction of projected canopy surface that receives light. For N similar leaves distributed at random (Poisson) in the canopy, GΩ .S(`) N p0 = 1 − . (5.24) SΩ (C) Because of the turbid medium hypothesis concerning the size of primitives, we have S(`) S(C) and thus S(`) → 0, S(C) allowing us to use the Taylor series in equation 5.24, switching from the binomial form to the exponential one GΩ .S(`) . (5.25) p0 = exp − N SΩ (C) If we recall that the projected surface of the canopy on the ground is definition of LAI, we can write LAI =
N.S(`). sin(h) , SΩ (C)
SΩ (C) , from the sin(h)
Chapter 5. Modelling light interception
82
and with replacement in equation 5.25, we obtain the Beer-Lambert expression for the gap-frequency: GΩ .LAI . (5.26) p0 = exp − sin(h) The computation of gap-frequency expresses the radiation attenuation, i.e. the propability of light interception (1 − p0 ). If I0 is the incident radiation flux above the canopy, I0 .p0 is the available flux at level L, and I0 .(1 − p0 ) is the intercepted flux at this level. The scattered flux is a fraction of the intercepted flux according to the scattering phase function. This function accounts for the angular distribution of scattered radiation for incident radiation of a given direction [Sinoquet et al. 1993]. Let us denote ρ L the scattering coefficient at level L and PL→K the radiative attenuation between level L and K computed using Beer-Lambert expression. The flux intercepted at level L, IL , can be written using a formalism close to the radiosity method: IL = I0 .(1 − p0 ) +
X
IK .ρK .PK → L ,
(5.27)
K
where the two terms of the right member of the equation, respectively, account for interception of the incident and scattered fluxes, see [Norman and Jarvis 1975; Ross 1981; Sinoquet et al. 2001] for detailed discussion.
5.2.2
Departure from random canopy This approach was developed for homogeneous canopy, but this hypothesis is rarely verified. A frequently used solution to solve this problem is to divide the canopy into different layers [Nilson 1971; Ross 1981; Norman and Welles 1983; Verhoef 1984] or voxels [Kimes and Kirchner 1982; Myneni 1991; Sinoquet and Bonhomme 1992; Gastellu-Etchegorry et al. 1996; Knyazikhin et al. 1997; de Castro and Fetcher 1998]. In layers models, leaves distribution is considered horizontally homogeneous, and the scattering can be estimated by the phase functions [Ross and Marshak 1988] whose estimation can be eased by simplifying assumption as like considering leaves as bi-lambertian diffusers [Sinoquet et al. 1993]. When scattering is taken into account in models using voxels [Sinoquet and Bonhomme 1992; Gastellu-Etchegorry et al. 1996], the scattered radiations are estimated within each voxel and then propagated to their adjacent neighbors. However, numerous canopies depart from the random (Poisson) leaf dispersion, even within a layer or a voxel [Myneni et al. 1989]. Some of them tend to avoid leaf overlapping, while others show more marked mutual shading. Distributions where leaves tend to avoid overlapping result from a more regular dispersion. They are called regular distributions. On the contrary, mutual shading proceeds from more aggregative distribution. These types of distribution are called clumped distributions. Nilson [1971] suggested two approaches to take non-random leaf dispersion into account: binomial and Markov models. We will rapidly present them as done in [Sinoquet et al. 1993]. In these two approaches, the canopy is subdivided in N layers whose thickness is measured in term of leaf area index, dL, with LAI = N.dL.
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83
Binomial models In binomial models, the N sublayers are statistically independent. For representing regular leaf dispersion, the possible events, within a sublayer, are “one contact”or “zero contact”with the probabilities: p1 =
GΩ .dL sin(h)
p0 = 1 −
and
GΩ .dL . sin(h)
Since pn = 0, ∀n, there is no mutual shading within a sublayer. The probability law Pn of n contact for the N sublayers is a positive binomial law with parameters N and GΩ .dL . From the independence of sublayers, the gap-frequency, P0 , of the canopy is: sin(h) GΩ .dL N N P0 = p0 = 1 − sin(h) LAI GΩ .dL = exp · log . (5.28) dL sin(h) For representing clumped leaf dispersion, the possible events, within a sublayer, may be “n contact”, with n > 0 with the probability: n pn =
GΩ .dL sin(h)
GΩ .dL sin(h)
n +1 .
The gap-frequency (n = 0) of a sublayer is: p0 =
1 GΩ .dL sin(h)
.
The physical meaning of this distribution is not clear, but it indicates that leaf area element overlap within a sublayer Sinoquet et al. [1993]. The probability distribution Pn of n contacts for the N sublayers is a negative binomial law with parameters N and GΩ .dL . From the independence of sublayers, the gap-frequency P0 , of the canopy is: sin(h) GΩ .dL − N N P0 = p0 = 1 + sin(h) LAI GΩ .dL = exp − · log . (5.29) dL sin(h) Relations 5.28 and 5.29 are identical except for the sign before dL, and both tend to relation 5.26 if dL goes to 0. The binomial models enable to express the gap-frequency in a single expression LAI GΩ .dL P0 = exp − · log , (5.30) dL sin(h) where the sublayer thickness dL becomes a leaf dispersion parameter, also denoted dL, with a positive or negative value according to the kind of leaf dispersion:
Chapter 5. Modelling light interception
84 dL = dL = dL =
+|dL| > 0 for regular leaf dispersion, 0 for random leaf dispersion, +|dL| < 0 for clumped leaf dispersion.
Markov models Markov model involves dependent sublayers, and consequently conditional probability. We assume that the layers are sufficiently thin so that the only possible events within a sublayer are “one contact”or “zero contact”. The underlying Markov chain illustrating this model is shown in Figure 5.7. Let us denote Xi the random variable such that Xi = 1 if contact occurs in layer i, and Xi = 0 otherwise. The conditional probabilities of contact are expressed as: Ω .dL p ( Xi = 1 | Xi − 1 = 0 ) = µ G sin(h)
if no interception occured within the previous layer,
Ω .dL p ( Xi = 1 | Xi − 1 = 1 ) = µ 0 G sin(h)
if interception occured within the previous layer,
where parameters µ and µ0 account for the degree of dependence between foliage elements location of two successive sublayers. The gap-frequency, P0 , of the canopy is: P0 = p( X N = 0, . . . , X1 = 0) = p( X N = 0| X N −1 = 0, . . . , X1 = 0) p( X N −1 = 0, . . . , X1 = 0) = p( X N = 0| X N −1 ) p( X N −1 = 0, . . . , X1 = 0)
=
N Y
p ( X i = 1 | X i − 1 = 0 ) p ( X1 = 0 )
i =2
with the probability of zero contact for the first layer, p ( X1 = 0) = 1 −
GΩ .dL . sin(h)
P0
GΩ .dL GΩ .dL N −1 = 1− · 1−µ sin(h) sin(h) GΩ .dL LAI − dL GΩ .dL = 1− · exp · log 1 − µ . sin(h) dL sin(h)
Thus,
(5.31)
In theory, combination of parameters dL and µ allows the description of a large range of leaf dispersions. In practice, the determination of the parameters from field measurement is almost impossible, hence the use of a simplified version with dL = 0. In that case, equation 5.31 leads to: µ.GΩ .LAI P0 = exp − . (5.32) sin(h) This relation can be used as a simple alternative to account for leaf dispersion by varying µ around 1, as dL around 0 in the case of binomial models. µ > 1 for regular leaf dispersion, µ = 1 for random leaf dispersion, µ < 1 for clumped leaf dispersion.
5.2. Models representing plants as a volume and statistical functions
85
This parameter possibly changes with direction [Nilson 1971; Stenberg 1996] and botanical parameters [Casella and Sinoquet 2003; Niinemets et al. 2004]. However, there is no direct way of inferring µ from the structural parameters, although Sinoquet et al. [2005] have shown that it is related to the spatial variations of LAI. Indeed the product µ.LAI can be regarded as the LAI of an equivalent random canopy showing the same gap-frequency. Thus a clumped canopy shows a higher gap-frequency and therefore a lower equivalent LAI, i.e. µ < 1. Up to now, this parameter is usually estimated by inversion of Equation 5.32 from field measurement of canopy transmittance. As stated by Cescatti [1997b], when using this approach, µ becomes an empirical correction factor and it no longer provides useful information on the canopy architecture. Figure 5.7 – Markov chain underlying Nilson [1971]’s sublayer approach for accounting the leaf distribution departure from random. p is the interception conditional propability when no interception occurred in the previous state. q is the interception conditional propability when interception occurred in the previous state.
5.2.3
Canopy structure as parameter To include canopy structure in the calculation of radiation attenuation, models using a voxel representation usually use different values of Leaf Area Density (LAD), the ratio of the leaf area contained in that voxel to its volume. This variation of LAD is a way of representing the effect of clumping that causes higher gap-frequency than a homogeneous random leaf distribution, voxels with low LAD simulate gaps, while voxels with high LAD mimic the clumping. The radiation attenuation also depends on the path length of a beam into a voxel, the longer the path, the higher the probability to be intercepted. Hence the modified version of relation 5.32 for a voxel k, and a beam b: GΩ .LAD.L b , (5.33) p0,k = exp − sin(h) where L is the path length of the beam in the voxel k, and the global P0b for that beam b along the trajectory of b. is the product of all p0,k Based on the fact that an important factor of departure from homogeneity in canopies is the crown morphology, another type of model has been developed: the geometric-optical models [Mottus et al. 2006]. In these models, plants are described using pre-determined shapes (cylinders, cone, ellipsoids, etc.). The volume inside is usually considered as a turbid medium, possibly divided into layers. Manifold shape forms are taken into account by the lengths of beam paths in Equation 5.33. The computation of beam path lengths within arrays of crowns in these models is consequently a crucial procedure [Norman and Welles 1983]. Models of this type are frequently used to predict canopy reflectance and obtain canopy structural parameters from remote sensing images. A review of these models can be found in [Pinty et al.
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Chapter 5. Modelling light interception
2004]. They have also been used to estimate transmitted radiation through canopy [Norman and Jarvis 1975; Wang and Jarvis 1990; Cescatti 1997a; Brunner 1998; Nilson 1999; Courbaud 2003]. However, in this case they are usually calibrated with the µ parameter to account of within crown clumping. The only models doing so without the addition of an empirical parameter, were introduced by Norman and Jarvis [1975] and applied to conifer species. The authors assumed that spruce crowns were made of whorls regularly distributed along the trunk, with shoots randomly distributed in whorls and needles randomly dispersed in shoots. Oker-Blom and Kellomaki [1983] proposed a simplification of this approach, where shoots were directly distributed at random in the tree crown of Scots pines. Using these models, it was shown that (i ) grouping of needles at any level of the canopy decrease the light absorption by comparison with a horizontally homogeneous canopy, (ii ) the average light conditions for the lowest part of the canopy were improved by the clumping effect since it enables more light ot pass through the upper levels of canopy. This kind of model, called grouping model, better takes into account the foliage distribution according to the plant organization at several scales. It also authorizes a better understanding of the influence of plant structure on its light interception capabilities. Moreover it allows to estimate the influence of structure on light interception. For instance, Stenberg [1996] successfully modeled the transmittance for conifers using the mean shoot silhouette area to needle surface area ratio as the extinction coefficient, replacing a complex function by a unique parameter value.
Conclusion In this chapter, I presented the wide variety of light models for plants. Two main categories were defined according to the type of plant representation. We saw that models using detailed descriptions address the issue of plant structure, but they do not allow to grasp the main structural determinants of the process due to their over-parameterization [Sinoquet et al. 2007]. On the other hand, statistical models are attractively simple because of the few parameters they contain. They allow readily estimation of light attenuation with the Beer-Lambert law when the canopy is homogeneous. In reality, forest canopies are far from homogeneous. This leads us to the use of more detailed light extinction models that account for inhomogeneity in foliage within trees. Because such models require more elaborate parameterization, they are less suited for large-scale applications. Thus, there is interest in simplified models that can be readily parameterized [Duursma and Mäkelä 2007]. The grouping model approach is the only one that provides a simple way to establish the relation between the structure of the plant and its light interception capabilities, which is an objective of this work. Unfortunately, the existing models of that type were developed for conifers, with predetermined number of scales and predefined shapes, therefore of limited usage. To be able to assess the relation between plant structure and light interception, we need a general model that bridges the gap between the two groups we presented. It should be able to estimate light-interception using a range of plant descriptions, from detailed to statistical, without any limitation of shapes. Because it is difficult to determine the clumping index (µ) of the Markov model, the following important question in modeling the architecture of canopies remains: at which structural level (canopy,
5.2. Models representing plants as a volume and statistical functions
87
crown, branch, shoot) may the phytoelements be treated as random distributed in space [Oker-Blom et al. 1991]? To address all these issues, we developed a new light interception model able to take into account the multiscale nature of plant canopies. It is a general grouping model, i.e. with no restriction concerning the number of scales, that can combine the estimation of light interception with both geometric and statistical plant description.
Multiscale framework for modeling and analyzing light interception by trees
“You will find something more in woods than in books. Trees and stones will teach you that which you can never learn from masters.” St Bernard
T
he main objective that fostered the developement of this new model, was to assess the influence of multiscale organization of real plants on light interception. Norman and Jarvis [1975] and Oker-Blom and Kellomaki [1983] proposed models based on the multiscale structure of conifers. These models allow them to study the effect of grouping of needles on light interception, but were specifically designed for one species, with adapted shapes, and a fixed number of scales. We generalize this approach with a model that can take into account any number of scales, and without restriction on the shapes used. To assess the influence of the multiscale organization, the model is able to compute light interception with detailed description of the plant and compare the results with the ones obtained for a statistical description of the same plant. The choice of description can be made for each scale separately. Therefore it can estimate the scale by scale influence of structure and have the ability to switch from detailed to statistical description seamlessly. This chapter is a sligthly modified transcription of the paper: Multiscale Framework for Modeling and Analyzing Light Interception by Trees. Da SILVA David, BOUDON Frédéric, GODIN Christophe and SINOQUET Hervé Published in the SIAM journal: Multiscale Modeling and Simulations, 7 (2) (2008), pp 910-933. The original version of the paper is included in appendix A.3.
Abstract This paper presents a new framework for modeling light interception by isolated trees which makes it possible to analyze the influence of structural tree organization on light capture. The framework is based on a multiscale representation of the plant organization. Tree architecture is decomposed into a collection of components representing clusters of leaves at different scales in the tree crown. The components are represented by porous envelopes automatically generated as convex hulls 89
6
90
Chapter 6. MµSLI M
containing components at a finer scale. The component opacity is defined as the interception probability of a light beam going through its envelope. The role of tree organization on light capture was assessed by running different scenarii where the components at any scale were either randomly distributed or localized to their actual three-dimensional (3D) position. The modeling framework was used with 3D digitized fruit trees, namely peach and mango trees. A sensitivity analysis was carried out to assess the effect of the spatial organization in each scale on light interception. This modeling framework makes it possible to identify a level of tree description that achieves a good compromise between the amount of measurement required to describe the tree architecture and the quality of the resulting light interception model.
6.1. Introduction
6.1
91
Introduction Light capture by plants is an essential process for plant growth and survival. Indeed light provides plants with energy which can be used for carbon fixation through foliage photosynthesis and for transpiration which allows water and nutriment transport within the plant [Jones 1992]. Light interception by plant foliage is governed by simple basic principles: Photons coming from the sun direction (direct radiation) and the whole sky hemisphere (diffuse radiation) may be intercepted by the plant elements or transmitted onto the soil surface if they pass in the foliage gaps. Intercepted photons may then be absorbed or scattered in any direction. Scattered photons may then either be intercepted again or leave the canopy. The interception process can be seen as the intersection between the photon trajectory—a line—and the plant organ. It thus depends only on the canopy structure, i.e., the spatial distribution of the plant organs, and organ geometry, namely shape, size, and orientation [Ross 1981]. If the detailed three-dimensional (3D) geometry is known, light interception can be easily and accurately computed by using 3D computer plant mock-up. Ray-tracing methods [Foley et al. 1995], the Z-buffer approach [Foley et al. 1995], or plant image processing [Sinoquet et al. 1998] can be used. Although accurate, this class of computation methods shows several shortcomings. First, it does not allow one an understanding of which structural features are the main determinants of light interception by plants. Second, although methods exist to exhaustively measure the detailed 3D geometry—in particular 3D digitizing [Sinoquet and Rivet 1997]—these techniques are very tedious and do not allow describing large sets of large trees. Third, if scattering is computed, the algorithm complexity dramatically increases due to multiple interception-scattering events and the high number of traced photons needed for convergence. Fourth, even though the basic processes are computed without any assumption, the simulation results are sensitive to measurement errors in the detailed canopy structure and in the optical properties of plant organs. For more than 50 years, more simple approaches have been proposed to estimate light interception by plants [Monsi and Saeki 1953]. The most common approach abstracts the plant canopy as a turbid medium [Ross 1981], i.e., a medium made of infinitely small foliage particles randomly dispersed in the vegetation volume and thus having a uniform optical density, i.e., transparency. In such a medium, light penetration can be expressed by the Beer–Lambert law; i.e., the probability that a photon crosses the vegetation volume without any interception can be written as p0 = exp(− G.LAD.L).
(6.1)
Hence, p0 is the probability of zero interception and corresponds to the canopy porosity. LAD is the leaf area density, and G is the extinction coefficient, namely the projection coefficient of plant elements on a plane perpendicular to the direction Ω [Ross 1981]. G depends on the angle between Ω and the normal of the plant’s elements. The product G.LAD can be regarded as the optical density of the vegetation. Note that leaves are usually the only elements taken into account because they represent the solar collector of the plant. Finally L is the distance travelled by the photons in the canopy. If scattering is disregarded, which is the case in this study where we focus on the effect of canopy structure on light interception, the intercepted light is proportional to p = 1 − p0 , i.e., the probability of light interception that defines the canopy opacity.
Chapter 6. MµSLI M
92
For a photon direction Ω, the distance L is constant for horizontally homogeneous vegetation canopies, e.g., grasslands. However, for tree crowns, L depends on the point where the photons enter the canopy. Usually, tree crowns are abstracted by envelopes, and beams are sent from a grid of points above the tree. The contribution of each beam to light interception is then summed up to compute total light interception probability in this direction. Finally the contribution of each direction Ω is summed up by weighting each directional probability with the fraction of incident radiation coming from direction Ω. Several light interception models for tree crowns are based on these principles [Charles-Edwards and Thornley 1973; Mann et al. 1979]. However, the assumption of uniform random distribution of leaf elements is rarely verified in actual tree crowns [Whitehead et al. 1990; Cohen et al. 1995; Sinoquet and Rivet 1997]. Indeed leaves are grouped around shoots, with higher density at the crown periphery. This leads to an overall clumped dispersion of the foliage, nonuniform LAD distribution, and lesser interception by crowns showing foliage clumping. The simplest way to deal with the nonrandom location of leaf elements is to introduce a leaf dispersion parameter µ in the Beer–Lambert equation p0 = exp(− G.µ.LAD.L).
(6.2)
Parameter µ equals 1 for random distribution. It is less than 1 for clumped foliage, i.e., crown porosity, p0 , is higher, and µ could be greater than 1 if foliage would show regular dispersion. Indeed the product µ.LAD can be regarded as the LAD of an equivalent random canopy showing the same porosity. Thus a clumped canopy shows a higher porosity and therefore a lower equivalent LAD, i.e., µ < 1. The parameter µ possibly changes with direction Ω [Nilson 1971] and botanical parameters [Niinemets et al. 2004; Casella and Sinoquet 2007]. However, there is not yet a clear knowledge about the structural parameters determining the degree of foliage clumping, although Sinoquet et al. [2007] have shown that µ is related to the spatial variations of LAD. Two other approaches have been proposed to deal with nonrandom and nonuniform foliage. In the first one, the crown volume is divided into voxels, and a value of LAD is assigned to each voxel [Kimes and Kirchner 1982; Cohen et al. 1987; Myneni 1991]. This approach shows two shortcomings. On the one hand, computed light interception depends on voxel size [Knyazikhin et al. 1997; Sinoquet et al. 2005]. On the other hand, assigning LAD values to each voxel needs a huge number of field measurements [Cohen et al. 1987]. The second approach is based on the botanical multiscale structure of trees and was applied to conifer species. Norman and Jarvis [1975] assumed that spruce crowns were made of whorls regularly distributed along the trunk, with shoots randomly distributed in whorls and needles randomly dispersed in shoots. Oker-Blom and Kellomaki [1983] proposed a simplification of the Norman and Jarvis approach, where shoots were directly distributed at random in the tree crown of Scots pines. This kind of model, called grouping model, better takes into account the foliage distribution according to the plant organization at several scales. It allows better rendering of foliage clumping without introducing a calibration parameter µ. The objective of this study was to develop a general modeling framework for computing light interception by single tree crowns. This framework includes most of the previously proposed methods in a unifying formalism: 3D plant mock-ups vs. turbid medium, mono- vs. multiscale approaches. This modeling framework is aimed
6.2. Modeling framework
93
at better understanding the effects of the crown organization on light capture at the whole tree scale, i.e., giving meaning to µ. The expected outcome of this study is to define ways of describing canopy structure as simple as possible and allowing accurate estimation of light interception, without the need of introducing any empirical dispersion parameter µ. In this paper, this approach was applied to a collection of fruit trees, namely four peach and four mango trees.
6.2
Modeling framework At a macroscopic level, the problem consists of estimating the amount of direct radiation intercepted by a vegetal component x (representing either the entire plant crown or a subbranching system) for each direction Ω of incident light. Light interception is computed in terms of STAR, silhouette to total area ratio: 4
STAR =
PLA , TLA
(6.3)
where PLA (m2 ) is the projected leaf area on a plane perpendicular to the incident direction Ω (i.e., the leaf area which intercepts light in direction Ω) and TLA (m2 ) is the total leaf area [Carter and Smith 1985; Oker-Blom and Smolander 1988]. The STAR is thus the relative irradiance of the leaf area. To take into account the clumping of leaves in plant crowns, this definition can be extended to the case where a canopy is decomposed into groups of leaves rather than directly into leaves. Groups of leaves can be regarded as macroscopic plant components, corresponding, for instance, to particular branching systems in the plant. In this case, the notion of PLA must be redefined since it now refers to the projected area of these coarser components, which are not entirely opaque. For this, we assume that the shape of a component x can be globally characterized by its convex envelope. According to Jackson and Palmer [1979], the PLA of x, denoted PLA x , can then be defined from the projected surface of the component envelope by introducing its opacity p x in direction Ω [Sinoquet et al. 2007]: 4
PLA x = PEA x .p x ,
(6.4)
where PEA x is the projected envelope area of x in the direction Ω and p x can be regarded as the probability of photon interception in the projected envelope. Reciprocally, 1 − p x is the envelope porosity. In the case of such multiscale plant structures, our original question thus amounts to estimating the opacity of the coarse components that are identified at different scales. Intuitively, the opacity in a particular direction Ω of such components, themselves composed of subcomponents (such as leaves or smaller branching systems) with defined shapes, is controlled by two independent factors: 1. On the one hand, it depends on the opacity of the subcomponents themselves. 2. On the other hand, it depends on the spatial distribution of the subcomponents and, more precisely, on how much the subcomponent silhouettes overlap when observed from direction Ω. In the case of opaque subcomponents (e.g., opaque leaves in a tree crown) the opacity of the composed object in a given direction Ω is solely a function of the directional
94
Chapter 6. MµSLI M
Figure 6.1 – Four-scale decomposition of an artificial tree. Leaves are represented by a set of geometric models (a). Leafy modules are defined here by botanical branching order 2 (b), 1 (c), and 0 (d). Illustration from [Boudon et al. 2006].
overlapping. In the case of porous subcomponents, the opacity is the result of these two factors applied to the smaller components. Possibly, the subcomponents themselves can be decomposed into smaller components with their own opacity. In this section, we first briefly recall how the structural organization of a plant can be formalized within a multiscale framework [Godin and Caraglio 1998; Boudon et al. 2006]. We then show how to compute the porosity factors of these elements and use the resulting hierarchical structure to compute light interception. We then show how sensitivity analysis of the model can be carried out to determine the influence of each scale of the hierarchy in the light interception.
6.2.1
Multiscale representation of plants Plant 3D mock-ups are represented by sets of geometric components for which the shape, size, spatial coordinates, and orientation of each component are well defined (Figure 6.1 a.). This information can be obtained either from direct measurements [Sinoquet and Rivet 1997; Godin et al. 1999] or from simulation models of plant architecture [Prusinkiewicz 2004a; Ferraro et al. 2005]. In both cases, the multiscale structural information is described as a multiscale tree graph (MTG) [Godin and Caraglio 1998]. At each scale i (i = 1 . . . n), the plant is regarded as a set of botanical components (e.g., branches, shoots, leaves) arranged as a rooted tree graph. Components at a scale i are made of components at scale i + 1 and together define a partition of the set of components at scale i + 1. Scale 1 corresponds to the whole tree and scale n to the set of leaves (an MTG includes at least these two scales) (Figure 6.1 d.). Each component is associated with a shape. At the leaf scale (i = n), components (leaves) are represented with a set of polygons. At the other scales (i 6= n), component shapes are convex hulls containing the shapes of their subcomponents (Figure 6.1).
6.2. Modeling framework
6.2.2
95
Multiscale model of light interception Using this framework, for each component j at any scale i, both STARi,j and PLAi,j values can be derived from (6.3) and (6.4): PLAi,j , TLAi,j
(6.5)
PLAi,j = PEAi,j .pi,j ,
(6.6)
STARi,j =
where PEAi,j is the projected envelope area of the component j at scale i and pi,j is its envelope opacity. For leaves (i.e., components at scale n), PLAn,j and STARn,j are simply the projected area of the leaf and the extinction coefficient of the leaf, respectively [Ross 1981]. At other scales i (i 6= n), components j are porous objects containing subcomponents at scale i + 1. In what follows, we shall show how such a multiscale organization of plants can be used to compute recursively the opacity of a plant crown and to interpret the light interception properties of the plant at different scales. At each scale i and for each component j at this scale, the envelope opacity pi,j can be estimated by casting a set of regularly spaced beams in the envelope. Let Bi,j be the set of beams intersecting with component j at scale i. The origin of beams b (b = 1 . . . Bi,j ) can be obtained from the cell centers of a regular grid perpendicular to the direction Ω and positioned above the tree. Each beam is affected with a cross-section area, Ab , corresponding to the surface area of a grid unit element; see Figure 6.2. b with each beam b: Consequently, we can associate a volume Vi,j b b Vi,j = Li,j .Ab ,
(6.7)
b is the travelling distance of beam b in component j. Based on the beam where Li,j bi,j and PEA [ i,j of the volume and projected area sampling we can define estimators V of the envelope of component j at scale i:
4
bi,j = V
B
i,j X
b Vi,j
and
4
[ i,j = PEA
b =1
B
i,j X
b PEAi,j = Bi,j .Ab .
(6.8)
b =1
bi,j and PEA [ i,j provide good The beam sampling must be dense enough to verify that V estimates of Vi,j and PEAi,j . With such a beam sampling, an estimator of the envelope b : opacity, pi,j , can be defined as the mean of beam opacities pi,j B
i,j 1 X b pi,j . pbi,j = Bi,j
4
(6.9)
b =1
At any scale (i < n), the model includes two options for computing opacity of a component. The first one (called option A) takes into account the Actual geometry of subcomponents in the component envelope. The second one (option R) uses the turbid medium analogy, i.e., assumes subcomponents to be Randomly distributed in the envelope volume with uniform density.
96
Chapter 6. MµSLI M
Figure 6.2 – Beam sampling illustration using a three-scale component representation. A component j at scale i is discretized using an Ω-oriented beam sampling where each beam b has the same crossb , and therefore a specific volume, V b . PEA section area, Ab , but a specific length in its envelope, Li,b i,j i,j is the projected envelope area onto a plane orthogonal to the direction Ω.
Figure 6.3 – Recursive pattern for computing the PLA of an element through its envelope opacity. The combined options of interscale distribution determine the path to follow in this equation scheme. The recursion ends when the last scale is reached; this scale must have a known or fixed opacity. For instance in this study leaves are considered opaque, i.e., opacity = 1. Hence for the all-A scenario the b is either 0 or 1 for all i and j. value of every pi,j
6.2. Modeling framework
97
Opacity computation in option A The actual geometry of each subcomponent k in any envelope j at scale i is taken into account. For each beam of the grid positioned above the tree crown, the opacity pib+1,k of subcomponent k, defined as its light interception probability, can be computed from the intersection between the beam and the subcomponent. If the subcomponent is intersected by the beam, i.e., pib+1,k > 0, the value of pib+1,k is 1 for a solid object, e.g., a leaf, and less than 1 for a porous object. If the subcomponent k is not intersected b of beam b to the opacity of by the beam pib+1,k = 0, therefore the contribution pi,j envelope j is computed by taking into account all subcomponents k, of scale i + 1, included in j: nj h i Y b b 1 − pi+1,k . (6.10) pi,j = 1 − k =1
Note that the product is due to the sequence of subcomponents intersected by the beam b, but the restriction to these particular subcomponents is made through the b equals 0 if value of their beam opacities. Indeed, the above equation shows that pi,j b equals 1 as soon as a subcomponent k intersected by all pib+1,k are equal to 0, and pi,j
beam b is opaque, pib+1,k = 1. For instance, the latter happens when subcomponents are leaves, i.e., i + 1 = n. When i + 1 6= n the value of pib+1,k is in [0, 1], depending on the options of finer scales; see Figure 6.3. Opacity computation in option R In option R, subcomponents are randomly and uniformly distributed. The probability that the beam b is not intercepted by the component j at scale i is computed from the product of gap fractions produced by each subcomponent k, of scale i + 1, included in j [Sinoquet et al. 2005]. The gap fraction for a beam b due to the subcomponent k at scale i + 1 is defined as the fraction of Ab free of the subcomponent projection, i.e., Ab − PLAib+1,k Ab
,
where PLAib+1,k is the portion of component k area projected onto the beam crosssection area, i.e., the restriction of PLAi+1,k to the beam b. The probability of no interception, assuming independence between events, is n
b p0 i,j =
j Y
" 1−
PLAib+1,k Ab
k =1
# .
(6.11)
In (6.11), the product expresses the effect of the uniform random distribution of leaves on the beam opacity in envelope j. Moreover, one can see that probability of beam interception by component k: pib+1,k
=
PLAib+1,k Ab
;
PLAib+1,k Ab
is exactly the
(6.12)
Chapter 6. MµSLI M
98 this leads to
j h i Y 1 − pib+1,k . = 1−
n
b b pi,j = 1 − p0 i,j
(6.13)
k =1
Equation (6.13) is exactly the same as (6.10) found in the case of actual distributions of components k in the envelope j. However, they differ by the interpretation of the product: In option A, the product expresses the sequence of components k intersected by the beam b, while in option R, it expresses the random position of components. This analogy between both equations is further investigated in section 6.5.3. In option R, we thus need to compute PLAib+1,k for each beam. To carry out this computation, we use the assumption of uniform density of foliage which makes it possible to write PLAib+1,k PLAi+1,k = (6.14) b Vi,j Vi,j b = A Lb ; (6.12) and (6.14) can be combined to express opacity pb as Vi,j b i,j i +1,k
pib+1,k
=
b PLAi+1,k Vi,j
Ab Vi,j
=
b PEAi+1,k pi+1,k Li,j
Vi,j
;
(6.15)
and (6.13) can then be written n
b pi,j = 1−
j Y
k =1
" 1−
b PEAi+1,k pi+1,k Li,j
Vi,j
# .
(6.16)
To summarize, we showed that the opacity of component j at scale i can be expressed by a unique set of recursive equations that enables us to express PLAi,j for each (i, j) as B
PLAi,j
i,j 1 X b = PEAi,j pi,j . Bi,j
(6.17)
b =1
This makes it possible to evaluate STARi,j from (6.5) based on an estimate of the plant total leaf area [Villalobos et al. 1995; Phattaralerphong et al. 2006]. The scheme described in Figure 6.3 illustrates the recursive procedure that uses (6.10) or (6.16), depending on the scenario. It can be used with any number of scales, and either option A or R can be independently used at each scale.
6.2.3
Assessing light interception Using this modeling framework, any particular scale can be added or removed or its spatial distribution switched from one option to another, e.g., changing the actual position of leaves (option A) to the hypothesis of random distribution of leaf area density (option R). This will be used to analyze the influence of one specific scale in light interception. In case of a crown filled with leaves, i.e., the number of scales is two, option A corresponds to the actual leaf distribution within the crown, e.g., as digitized in the field. Otherwise, option R leads to the basic random distribution of leaf area in the tree crown, as used in many turbid medium models [Charles-Edwards and Thornley 1973; Mann et al. 1979]. In case of more scales, the scenario corresponding
6.2. Modeling framework
AA
99
AR
RA
RR
-R
Figure 6.4 – Example of distribution options for an object with three scales, namely crown, shoot, and leaves. Hence a 2-character string represents interscale distribution options. AA represents the actual distribution of leaves in the crown. With AR the actual position of shoots are used, but the opacity of their envelopes is computed with the turbid medium analogy, whereas RA uses the turbid medium analogy for shoots but with their real opacity taken into account. The RR distribution corresponds to the model where leaves and shoots are randomly distributed in shoots and in the crown, respectively, i.e., grouping model. Finally, -R uses the turbid medium analogy and supposes a uniform distribution of leaves in the crown.
to the selected combination of options A and R is encoded as a 6 string of characters, containing the number of scales minus 1 character. The first and last characters of the string refer to scale 2 and scale n, respectively. The option for scale 1 can only be A, i.e., the actual tree crown. This is why scale 1 is not included in the string. For example, let us consider a tree organized in three scales, e.g., crown, shoots, and leaves. A scenario AA would correspond to actual arrangement of shoots within the crown and to actual arrangement of leaves within shoots. This corresponds to the real plant structure and is assumed to be the true value. Another example illustrated in Figure 6.4 is the scenario RR corresponding to the Oker-Blom and Kellomaki [1983] model, where leaves and shoots are randomly distributed in shoots and in the crown, respectively. In addition, any inner scale (i ∈ [2, n − 1]) can be discarded to distribute components at scale i directly in envelopes at scale i − 2. Discarded scales are denoted “-” in the scenario string. For instance, in the example above, the scenario -R means that leaves are directly randomly distributed in the tree crown. Finally, note that discarding a scale and then using the actual position is equivalent to using the actual distribution for the discarded scale, i.e., -A ≡ AA. As presented in section 6.2.2, our goal is to estimate each plant’s global light interception efficiency in terms of STAR. This efficiency is evaluated at the plant scale, i.e., i = 1 and j = 1, by integrating directional STAR values over the sky directions Ω. In the following i and j indexed notations will thus be omitted and the Ω indexed notation will indicate directional quantities, whereas Ω-free notation will stand for integrated values. Ω-integrated values for STAR and PEA are obtained by summing up the directional values weighted by coefficients, ωΩ , derived from the standard
Chapter 6. MµSLI M
100 overcast sky radiance distribution [Sinoquet et al. 2004]: 4
STAR =
X
STARΩ .ωΩ
and
4
PEA =
Ω
X
PEAΩ .ωΩ .
(6.18)
Ω
STARΩ values are computed according to (6.3) from PLAΩ whose evaluation is described in Figure 6.3. Originally the leaf dispersion parameter, µ, introduced in the Beer–Lambert equation as a LAD modifier, expresses the departure of the actual crown gap fraction from the gap fraction of a crown with random distribution of leaves and equivalent leaf area density. Indeed, µ is 1 for random distribution of leaves, whereas leaf clumping leading to higher crown porosity yields a µ strictly less than 1. It can be extended to compare two scenarii, expressing a relative dispersion coefficient, µ, of a test porosity, p0 , against a reference one, p0 hre f i . By analogy with the Beer–Lambert formalism, where p0 hre f i would be expressed using (6.1) and p0 using (6.2), the following relationship is defined [Sinoquet et al. 2007]: p0 = p0 hre f i µ .
(6.19)
Using the complementary relationship between opacity and porosity and (6.3) and (6.4), an Ω-integrated µ value can be expressed as a function of STAR and PEA: TLA log 1 − STAR. PEA . (6.20) µ= STARhre f i . TLA log 1 − PEA It allows us to study the global or scale-by-scale spatial organization through the computation of µ from a specific scenario. For instance, in the example illustrated in Figure 6.4, the definition of µ as introduced by Nilson [1971] corresponds to comparing the situation AA to the reference situation -R and is therefore noted µ(AA/-R). The STAR of these scenarii are noted STARhAAi and STARh-Ri , respectively. The scale-byscale organization is given by scenarii which differ by only one letter corresponding to the studied scale. For instance the shoot distribution within the crown and the leaf distribution in the shoots are given by the ratio of STARhAAi to STARhRAi and STARhAAi to STARhARi , respectively.
6.3 6.3.1
Plant database Plant material The database is constituted of the four 4-year old peach trees (variety August Red) and the four 3-year old mango trees (two of them belonged to variety Lirfa and the two others to variety José) that are described in section 2.4.2. Illustrations of the trees are reminded in Figures 6.5 and 6.6, and multiscale components informations are available in Table 6.1.
6.4. Results and clumping analysis
101
Figure 6.5 – Peach tree scale decomposition. Top: top view; bottom: side view and from left to right: leaves, CYS, OYOS, scaffold, crown.
Figure 6.6 – Mango tree scale decomposition. Top: top view; bottom: side view and from left to right: leaves, CYS, scaffold, crown.
6.4
Results and clumping analysis Ω-integrated values for STAR and µ according to several hypotheses are shown in Tables 6.2 and 6.3 for peach and mango trees, respectively. Actual STAR values (line 0) for peach trees did not show a large range, i.e., between 0.245 and 0.275. This means that foliage irradiance was about 25% of the available light above the tree. For mango trees the range of STAR values was greater, i.e., between 0.243 and 0.364. These values are in the interval commonly reported for fruit trees [Willaume et al. 2004]. For both species, the classical assumption of random distribution of leaves in crowns (row 1) led to large overestimation of the STAR values. Therefore light interception models based on this assumption are biased, as already reported by Whitehead et al. [1990]; Chen et al. [1994]; Cohen et al. [1995]. This confirms that the actual foliage distribution shows significant departure from randomness. The values of the dispersion parameter µ were between 0.55 and 0.61 and between 0.70 and 0.81 for peach and mango trees,
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102
Nb elmt 1 4 86 1966 14405 1 5 88 1522 13709 1 5 93 1583 15118 1 6 101 1990 15589
Volume (dm3 ) 11811.50 2832.70 ± 1214.52 14.02 ± 29.27 0.97 ± 1.54 12200.89 1915.22 ± 507.09 13.98 ± 22.88 1.40 ± 2.06 12442.29 1934.32 ± 935.55 15.51 ± 24.89 1.52 ± 2.30 15911.77 2060.11 ± 773.34 12.48 ± 20.19 1.17 ± 1.91 -
Leaf area (dm2 ) 2773.85 693.45 ± 232.08 9.10 ± 14.85 1.41 ± 1.95 0.19 ± 0.05 2889.69 577.74 ± 190.48 9.99 ± 12.51 1.90 ± 2.32 0.21 ± 0.55 3242.96 648.58 ± 266.30 11.30 ± 15.24 2.05 ± 2.78 0.21 ± 0.55 3144.29 524.04 ± 186.99 8.30 ± 9.99 1.58 ± 2.01 0.20 ± 0.06
LAD (dm−1 ) 0.24 0.253 ± 0.04 1.31 ± 0.83 1.90 ± 0.58 0.24 0.30 ± 0.04 1.22 ± 0.72 1.77 ± 0.53 0.26 0.34 ± 0.05 1.35 ± 1.00 1.77 ± 0.52 0.20 0.26 ± 0.04 1.39 ± 1.10 1.84 ± 0.57 -
Mango a19
Crown Scaffold CYS Leaf
1 3 139 1475
1146.66 417.03 ± 68.42 6.07 ± 4.81 -
631.64 210.54 ± 11.95 4.54 ± 2.64 0.43 ± 0.17
0.55 0.51 ± 0.07 1.06 ± 0.72 -
Mango b7
Crown Scaffold CYS Leaf
1 3 252 2759
2073.24 793.35 ± 229.78 9.00 ± 6.47 -
1221.87 407.29 ± 126.59 4.85 ± 2.70 0.44 ± 0.19
0.59 0.51 ± 0.02 0.73 ± 0.52 -
Mango f21
Crown Scaffold CYS Leaf
1 2 147 1367
1188.56 614.09 ± 93.89 4.80 ± 4.93 -
509.19 254.60 ± 55.84 3.46 ± 2.40 0.37 ± 0.18
0.43 0.41 ± 0.03 0.77 ± 0.48 -
Crown Scaffold CYS Leaf
1 7 99 1135
869.39 146.10 ± 68.07 8.27 ± 6.92 -
492.87 70.41 ± 32.48 4.98 ± 3.19 0.43 ± 0.19
0.57 0.48 ± 0.05 0.83 ± 0.50 -
Peach 4
Peach 3
Peach 2
Peach 1
Scale Crown Scaffold OYOS CYS Leaf Crown Scaffold OYOS CYS Leaf Crown Scaffold OYOS CYS Leaf Crown Scaffold OYOS CYS Leaf
Mango g5
Table 6.1 – Scale-by-scale component demographies and their envelope volume, contained leaf area, and LAD. Abbreviations: OYOS = one-year-old shoot; CYS = current-year shoot; LAD = leaf area density.
respectively, expressing a clumped organization of the foliage and higher clumping for peach trees than for mango trees. The full grouping model (all-R, row 2) underestimated light interception (for all trees but Peach1). Indeed µ values above 1 means that the leaves in the actual canopy were less overlapping than they would have been in the canopy generated from the full grouping model. Consequently, actual STAR values were comprised between these two most usual modeling approaches. However, the two species showed different behavior: Actual STAR of peach trees was overestimated by 30% with scenario - --R, whereas scenario all-R underestimated actual STAR by only 10%. In the case
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103
Table 6.2 – Ω-integrated STAR for peach trees. 0 1 2 3 4 5 6 7 8 9 10 11 12 13
AAAA - - -R µ (AAAA/- - -R) RRRR µ (AAAA/RRRR) AAAR µ (AAAA/AAAR) AARA µ (AAAA/AARA) ARAA µ (AAAA/ARAA) RAAA µ (AAAA/RAAA) RRRA µ (RRRA/RRRR) RRAR µ (RRAR/RRRR) RARR µ (RARR/RRRR) ARRR µ (ARRR/RRRR) -RRR µ (AAAA/-RRR) - -RR µ (AAAA/- -RR) AA-R µ (AAAA/AA-R)
Peach1 0.2477 0.3389 0.5482 0.2422 1.0368 0.2372 1.071 0.2480 0.9987 0.2594 0.9281 0.2349 1.0868 0.2518 1.0643 0.2421 0.9995 0.2261 0.8999 0.2487 1.0432 0.2803 0.8117 0.3061 0.6871 0.2522 0.9717
Peach2 0.2715 0.3483 0.6127 0.2347 1.2582 0.2497 1.1445 0.2697 1.01 0.2890 0.8971 0.2412 1.2071 0.2519 1.1145 0.2352 1.003 0.2235 0.9304 0.2650 1.2087 0.2789 0.9554 0.3035 0.8202 0.2578 1.0882
Peach3 0.2456 0.3239 0.5734 0.2177 1.2110 0.2312 1.1036 0.2442 1.0096 0.2586 0.9154 0.2226 1.1782 0.2300 1.0884 0.2184 1.0048 0.2100 0.9466 0.2416 1.1782 0.2639 0.8835 0.2898 0.7386 0.2443 1.0094
Peach4 0.2754 0.3591 0.6082 0.2504 1.1599 0.2542 1.1331 0.2719 1.021 0.2927 0.904 0.2585 1.1257 0.2676 1.1082 0.2514 1.0066 0.2379 0.9274 0.2703 1.1257 0.3078 0.8280 0.3221 0.7615 0.2651 1.0627
of mango trees, the overestimation given by scenario --R was around 12% and equal to the underestimation given by scenario all-R. A deeper analysis was made by comparing the actual STAR values to those obtained by switching one scale option from A to R. This allows studying the dispersion pattern at the switched scale. For instance, the leaf dispersion within CYS in peach trees was assessed by comparing scenario AAAA to scenario AAAR. For peach trees (Table 6.2), only CYS in OYOS (row 4) showed a clear random dispersion (µ ' 1). STARhAAARi and STARhRAAAi (rows 3 and 6, respectively) were slightly less than the actual value; thus the distribution of leaves in CYS and scaffolds in crowns was slightly regular (µ ' 1.11 and µ ' 1.15, respectively). Finally STARhARAAi (row 5) was a bit greater than the actual value, showing a small clumping trend of OYOS in scaffolds (µ ' 0.91). For mango trees (Table 6.3), scenarii AAR and ARA (rows 3 and 4) led to small under- and overestimation of actual STAR, expressing slight regularity and clumpiness of leaves in CYS and CYS in scaffolds, respectively. By contrast, scenario RAA (row 5) markedly underestimated actual STAR, meaning that the actual distribution of scaffolds in the crown was regular (µ ' 1.26). The dispersion pattern for each scale was also assessed by comparing the STAR behavior values of the full grouping scenario (all-R) to those obtained by switching one scale option from R to A (rows 7 to 10 in Table 6.2 and rows 6 to 8 in Table 6.3 for peach and mango trees, respectively). This analysis confirmed the dispersion previously found at each scale. As the scaffold scale was shown to be regular, we tested a scenario where this scale
Chapter 6. MµSLI M
104 Table 6.3 – Ω-integrated STAR for mango trees. 0 1 2 3 4 5 6 7 8 9 10
AAA - -R µ (AAA/- -R) RRR µ (AAA/RRR) AAR µ (AAA/AAR) ARA µ (AAA/ARA) RAA µ (AAA/RAA) RRA µ (RRA/RRR) RAR µ (RAR/RRR) ARR µ (ARR/RRR) -RR µ (AAA/-RR) A-R µ (AAA/A-R)
Mango a19 0.2824 0.3282 0.7039 0.2544 1.2157 0.2736 1.064 0.2915 0.9366 0.2507 1.2465 0.2608 1.0458 0.2444 0.9335 0.2828 1.2195 0.3018 0.8683 0.3068 0.8363
Mango b7 0.2434 0.2717 0.7315 0.2193 1.2552 0.2347 1.0886 0.2497 0.9391 0.2124 1.3346 0.2222 1.0267 0.2065 0.8922 0.2456 1.2830 0.2582 0.8577 0.2587 0.8537
Mango f21 0.3637 0.4025 0.8112 0.3159 1.2809 0.3478 1.0859 0.3643 0.9964 0.3191 1.2596 0.3252 1.0495 0.3076 0.9576 0.3519 1.2047 0.3670 0.9829 0.3846 0.8949
Mango g5 0.3235 0.3564 0.7982 0.2811 1.3077 0.3084 1.1019 0.3053 1.1240 0.2927 1.2163 0.2892 1.0519 0.2820 1.0058 0.2956 1.0949 0.3262 0.9825 0.3190 1.0298
was discarded (rows 11 and 9 for peach and mango trees, respectively). Theoretically this should lead to STAR values closer to the actual ones because the scaffold subcomponents are no longer confined within the scaffold envelopes but distributed in the full crown volume. It is expected to be a way to simulate the regularity of scaffold distribution in the crown. Indeed STAR values in scenarios -RRR and -RR were greater than values given by the full grouping model for peach and mango trees, respectively. They were also greater than STAR values given by scenarii ARRR and ARR, meaning that disregarding the scaffold scale led to overestimating the actual regular dispersion of scaffolds. For all trees the actual STAR value was in between the values given by the full grouping scenario and that disregarding the scaffold scale. For peach trees scenario AA-R (row 13) gave STAR values very close to the actual ones. The profit of this scenario will be discussed in the next section.
6.5
Discussion
6.5.1
Requirements in canopy structure description for an accurate estimation of light interception Exhaustive measurement of the canopy structure, e.g., to get data for scenario all-A, is extremely tedious, especially for the peach trees where the number of leaves was about 15,000 per tree. Conversely, the simplest scenario --R uses only a few data to describe the tree structure, namely crown shape and volume and total leaf area. However, the computed light interception was shown to be largely overestimated. This confirms that the actual tree foliage distribution in crowns is not uniform and shows high clumping [Cohen et al. 1995]. Consequently, the simulation models using
6.5. Discussion
105
(a)
(b)
Figure 6.7 – Influence of measurement errors on the STAR estimation. (a) Effect of increasing measurement error. Random error generated by a uniform distribution (dark grey) or Student’s distribution (light grey) is introduced on leaf spatial coordinates so that the new position is in a sphere of radius 0.1, 0.2, 0.5, 1, 2, and 5 cm centered on its original position. (b) 1 cm positioning error of leaves on several scenarii using the R option. Errors are generated using a uniform distribution (dark grey) or Student’s distribution (light grey). The STAR values from the original tree are taken as the reference.
this scenario [Norman and Welles 1983; Cescatti 1997b] should include a calibration parameter µ. By contrast, the full grouping scenario underestimated light interception for both species. Moreover, the Oker-Blom and Kellomaki [1983] model, i.e., --RR and -RR for peach and mango trees, respectively, yielded STAR values close to the actual ones but only for mango trees. As the Oker-Blom and Kellomaki model can be regarded as a partial grouping model (i.e., leaves randomly distributed in shoots and shoots directly distributed at random in the tree), this questions the number of botanical scales that must be used in grouping models. However, the Oker-Blom and Kellomaki [1983] model was reported to be successful for conifer species, and our results show that this approach can also be used for some, but not all, fruit tree species. In the case of peach trees, the grouping scenarii did not work satisfactorily; however, the scenario AA-R gave nice results. This means that good estimations of light interception can be obtained by measuring the actual position of OYOS in the crown. In practice, this kind of measurement is really tractable [Sonohat et al. 2006], as a peach tree usually includes only about 100 OYOS (see Table 6.1). Moreover, the OYOS scale corresponds to the management unit used to train the tree (i.e., the fruit grower selects a given number of OYOS in the sunlit zones of the tree and prunes the other ones). Therefore an interesting use of the proposed modeling framework is to find out scenarii of canopy structure description as simple as possible that enables an accurate estimation of light interception without artificially using leaf dispersion parameters µ (especially because there is almost no means to estimate µ).
6.5.2
Sensitivity analysis Measurement errors effect. Magnetic digitizing is prone to measurement errors both on the location and orientation of organs. The error on spatial coordinates is typically
106
Chapter 6. MµSLI M
less than 1 mm in a controlled environment [Moulia and Sinoquet 1993] and is less than 1 cm for field measurements [Thanisawanyangkura et al. 1997]. The impact of such errors on STAR computation has already been assessed on peach trees by Sonohat et al. [2006]. In this case, measurements were conducted at the shoot scale, and leaf position and shape were reconstructed using allometric rules. It was shown that the STAR estimation at the shoot scale was inaccurate while it was satisfactory at the plant scale. In the case of mango trees, which were digitized at the leaf scale, we quantified the error in STAR estimation by generating mock-ups where random errors were introduced in leaf spatial coordinates and orientation angles. We designed two procedures for this. We first used a uniform distribution in the range [−ε, ε] to modify the coordinates and the orientation of each leaf. Alternatively, we used Student’s distribution, which yields a greater proportion of small errors. The impact of these two distributions on STAR computation was evaluated on a set of 100 mockups with ε values defined so that the new position of each leaf is in a sphere of radius 0.1, 0.2, 0.5, 1, 2, and 5 cm centered on its original position. Orientation angles were modified similarly with ε ranging from 5◦ to 45◦ . Results showed that positioning errors less than 1 cm had little effect on light interception capacities at the plant scale (Figure 6.7 a.), and for 1 cm, corresponding to the field measurement error, the STAR value error using scenario all-A was less than 5% regardless of the distribution used. For measurement errors greater than 1 cm the foliage rapidly tended to occupy a more important volume, leading to a lower leaf area density and thus a greater STAR. This effect was much more marked when errors were generated using a uniform distribution (Figure 6.7 a.). Note, however, that errors greater than 1 cm should be avoided if measurements are carefully conducted. Orientation errors had only a neglectable effect on STAR values integrated over the sky vault (less than 0.5% for all ε values; data are not shown). Positioning errors in plant organs may also affect the convex envelope of plant components. To assess the impact of this phenomenon on the STAR computation using scenarii with option R, we created a set of five mango mock-ups by introducing a 1 cm error on leaf position with the previously described distributions. For each of these trees, integrated STAR values were computed for scenarii AAA, AAR, ARR, RRR, and --R to assess the cumulative impact of option R. Results are shown in Figure 6.7 b.6.7, with the STAR value of the all-A scenario of the original tree taken as the reference. An increase in the STAR estimation error with the use of the R option at different scales is clearly visible (the effect is slightly reduced for Student’s distribution) and is due to the fact that the porosity of a component results from the product of its subcomponents’ porosities, (6.10). Using other types of bounding envelopes such as boxes, spheres, or ellipsoids induces a larger envelope volume [Boudon 2004] and thus leads to similar error effects. If the all-A scenario STAR value of the mock-up with leaf positioning errors is taken as the reference for the multiscale organization analysis, then we still find the same result trends. Beam sampling effect. As mentioned in section 6.2.2, the density of the beam sampling may influence STAR estimation. The plant is included in a bounding box such that its upper face is orthogonal to the light direction, Ω. This face is subdivided into cells of a regular grid whose centers define the locations of the beams. This defines a beam sampling density as BS , B being the total number of beams and S the area of the upper face. In the case of our regular grid we control the sampling density by the
6.5. Discussion
107
Figure 6.8 – Influence of beam sampling lineic density on the STAR estimation. The bounding box upper face surface S is considered as the unitary area, and the STAR value obtained with a density of 300 is taken as the reference for each scenario.
lineic density of the beams, d, defined as r d=
B , S
(6.21)
representing the number of beams per unit length along the grid axes. We assessed the effect of the sampling density by computing STAR values for several scenarii using different lineic densities: 20, 50, 100, 200, and 300, the latest being the one used to conduct the multiscale analysis of section 6.4. For each value of d, integrated STAR values were computed for scenarii AAA, AAR, ARR, RRR, and --R. Results show that convergence was rapidly obtained with increasing density (the error was less than 0.5% for densities over 100); see Figure 6.8. Using a lineic density over 100 guarantees a good STAR estimation regardless of the scenario used. Moreover, since the complexity of our algorithm is quadratic in the beam sampling lineic density (see section 6.5.4 for details), large gain in computation time is achieved by decreasing d from 300 to 100 with almost no loss of quality in the results.
6.5.3
A unifying approach In section 6.2.2, we observed that, surprisingly, the recursive equations (6.10) and (6.13) relating component porosities at different scales were actually identical despite the fact that they were derived from two different hypotheses. This raises the question as to whether these equations can be derived from a single unifying framework. To answer this question, let us call { x1 . . . xn } the set of subcomponents of a component c, and Ib (c) indicates whether the beam b intersects with the envelope of c (1 = yes, 0 = no). Let us also denote Xb (c) the random variable such that Xb (c) = 1 if the beam b is intercepted by an elementary component of c, i.e., it interacts with c, and
Chapter 6. MµSLI M
108
Xb (c) = 0 otherwise. With these definitions the porosity of a component c for the beam b is defined as p0b (c) = P( Xb (c) = 0| Ib (c) = 1); (6.22) reciprocally, its opacity is thus pb (c) = 1 − p0 b (c) = P( Xb (c) = 1| Ib (c) = 1).
(6.23)
Let us now consider a beam b that intersects a component c, i.e., P( Ib (c) = 0) = 0; thus the decomposition P( Xb (c) = 0) = P( Xb (c) = 0| Ib (c) = 1) P( Ib (c) = 1) + P( Xb (c) = 0| Ib (c) = 0) P( Ib (c) = 0) yields p0b (c) = P( Xb (c) = 0| Ib (c) = 1) = P( Xb (c) = 0). A beam b does not interact with a component c if and only if b does not interact with any subcomponent of c. Since going through a subcomponent without being intercepted does not modify the beam, all P( Xb ( xi ) = 0) are independent, and thus we have P( Xb (c) = 0) = P( Xb ( x1 ) = 0, . . . , Xb ( xn ) = 0) n n Y Y P ( Xb ( x k ) = 0) = = [1 − P( Xb ( xk ) = 1)] .
(6.24)
k =1
k =1
Using the fact that the probability for a beam b to interact with a subcomponent that it is not intersecting, P( Xb ( xk ) = 1| Ib ( xk ) = 0), is obviously 0, the decomposition of P( Xb ( xk ) = 1) yields P( Xb ( xk ) = 1) = P( Xb ( xk ) = 1| Ib ( xk ) = 1) P( Ib ( xk ) = 1).
(6.25)
Using expression (6.25) to replace P( Xb ( xk ) = 1) in the component porosity equation (6.24), and given the complementary relationship between opacity and porosity equation (6.23), the expression of opacity becomes P( Xb (c) = 1| Ib (c) = 1) = 1 −
n Y
[1 − P( Xb ( xk ) = 1| Ib ( xk ) = 1) P( Ib ( xk ) = 1)] , (6.26)
k =1
which illustrates our previous remark stating that opacity is controlled by two factors. The first is the opacity of the subcomponents: P( Xb ( xk ) = 1| Ib ( xk ) = 1). The second is their spatial distribution: P( Ib ( xk ) = 1). Indeed, when using option A, P( Ib ( xk ) = 1) is equal either to 0 (the beam does not intersect the component) or 1 (the beam intersects the component). Therefore this term acts as a filter to disregard all subcomponents not intersected by the beam b. This is equivalent to (6.10), where the opacity of subcomponents not intercepted is equal to 0. When using option R the spatial distribution of subcomponents can only be estimated. The probability of intersecting a subcomponent can be expressed as P( Ib ( xk ) = 1) =
PEAb ( xk ) , PEAb (c)
6.5. Discussion
109
whereas the expression of the porosity of the subcomponent for the beam b is P( Xb ( xk ) = 1| Ib ( xk ) = 1) =
PLAb ( xk ) . PEAb ( xk )
Since PEAb (c) is the beam cross-section, Ab , the product of these two quantities in (6.26) yields the expression of the beam interception probability: P( Xb ( xk ) = 1| Ib ( xk ) = 1) P( Ib ( xk ) = 1) =
=
PLAb ( xk ) PEAb ( xk ) PEAb ( xk ) PEAb (c) PLAb ( xk ) , Ab
which leads us to the second equation, (6.12), that was derived in case R. Consequently, we showed that it was possible to derive equations for both cases A and R from a unique expression, (6.26), that unify both situations.
6.5.4
Implementation issues and complexity This software has been written in Python and C++. It is a stand-alone module part of the VPlants software project (successor of AMAPmod [Godin et al. 1997]). In a first step a 3D shape is associated with each component of the MTG at each scale. The leaf geometry is defined using the PlantGL library [Pradal et al. 2008a], and convex hulls are computed with the QuickHull algorithm [Bradford-Barber et al. 1996] available in this library. The multiscale computation of opacity is carried out for each direction Ω as follows. It starts with a double Z-buffer approach: Two opposite orthographic cameras oriented along the Ω direction are used to generate two images of each geometric component of the plant from the same distance. Each pair of facing pixels from the two orthographic views represents the same beam b, and its cross-section area, Ab , is the area represented by a pixel expressed in metric units. The Z-values of each pixel yield the beam in- and out-points in the component. The travel b , is deduced from distance of a beam within the envelope of component j at scale i, Li,j these two values. This length is null for components associated with planar shape, e.g., leaves. Moreover, this double Z-buffer approach allows us to identify the beams intercepted by each component which is required for the use of option A, i.e., when actual component positions are used. In this approach convex envelopes are used since multiple in- and out-points are not taken into account; hence nonconvex envelopes will be treated as convex ones. This step also provides the projected area of the component shape by multiplying Ab with the number of pixels within the orthographic image. All of these quantities are scenario independent and thus are computed only once for each direction Ω. The volume of a convex hull, Vi,j , is computed separately using routines implemented in the PlantGL library [Pradal et al. 2008a]. Next the recursive scheme described in Figure 6.3 is applied to compute the opacity according to the scenario used. The two recursive procedures used to compute the b , are described opacity of a component, pi,j , and the beam opacity of a component, pi,j by Algorithms 1 and 2, respectively. The recursion starts from macroscopic components towards microscopic ones and ends when leaves are reached. It is important to note that in practice each value is computed once—the first time needed—and then stored for future use to save computation time.
Chapter 6. MµSLI M
110
Algorithm 1: Opacity(c, Ω, s) Input: component c, direction Ω, scenario s /* s = string in {A,R} */ Output: mean opacity of c if c == leaf then return 1 else sum = 0 ; foreach intercepted beam b do sum+ = OpacityBeam(c, Ω, s, b) return
sum size(intercepted beams)
Algorithm 2: OpacityBeam(c, Ω, s, b) Input: component c, direction Ω, scenario s, beam b Output: beam opacity of c if c == leaf then return 1 else option = s[0] ; /* get the current scale option */ s = s[1..n] ; /* and remove it from scenario */ porosity = 1 ; if option == A then foreach subcomponent x do if b ∩ x 6= ∅ then porosity ∗ = OpacityBeam(x, Ω, s, b) ; if option == R then foreach subcomponent x do p x = Opacity( x, Ω, s ) ; porosity
∗ = 1−
PEA x p x Lbc Vc
return 1 − porosity
Leaves are considered as opaque components; thus their opacity does not need to be computed. The test of whether or not a beam intersects a leaf is considered as the atomic operation. Therefore, the computation cost of one component opacity depends on the number of intersected beams, i.e., the number of pixels of its envelope projection in direction Ω, and the number of its subcomponents. In the worst-case scenario, every component is intersected by all beams. A very simple model allows us to evaluate the computation cost of our algorithm. Let us denote the number of scales by k and assume that the number of subcomponents, N, is identical for all components at every scale. The total number of components is consequently 1 + N + N 2 + · · · + N k−1 = O( N k−1 ) = O(n),
(6.27)
where n = N k−1 is the number of leaves, i.e., components of the last scale. Let us define a size ratio, δ, by comparing the size of a component to the (smaller) size of its subcomponents. We also assume that the size ratio between two scales is constant
6.5. Discussion
111
and less than 1, that is,
∀i, j, l
Vi+1,j = δ3 Vi,l
and
PEAi+1,j = δ2 PEAi,l .
(6.28)
Let Bi be the number of beams intercepted by a component at scale i with B0 = B. As a consequence of a constant size ratio, the number of intercepted beams at one scale is related to the number of intercepted beams at the previous scale: Bi+1 = δ2 Bi = δ2i B.
(6.29)
Consequently, the total cost γ is the sum of each component cost at every scale except the leaf scale: γ = 1 × NB + N × NBδ2 + N 2 × NBδ4 + · · · + N k−2 × NBδ2(k−2) = O N k−1 Bδ2(k−2) = O nBδ2(k−2) .
(6.30)
The worst case is reached if k = 2; thus the total cost is in O (nB). However, when k > 2, since δ < 1 the gain in complexity due to the hierarchical structure is proportional to δ2(k−2) . For example, using the plant illustrated in Figure 6.1, adding 1 scale to the 2 basic scales will reduce the complexity by 19 (δ = 13 ), and for 2 additional scales, the 1 . complexity will be reduced by 81
Conclusion This paper presented a new framework for modeling efficiently light interception by isolated trees. The turbid medium approach, usually limited to large canopies because of its statistical description of plants, has been adapted to isolated trees. The modeling was based on a multiscale representation of plants and on a porous envelope hypothesis. We defined recursive expressions to compute the opacity of components with two types of spatial distribution hypotheses that can be chosen at each scale independently. The combination of these options defined scenarii that allowed us to analyze the influence of the plant architecture on light interception through the generalization of a dispersion parameter µ which expresses the departure of a plant foliage from randomness. This model was then assessed on 3D digitized peach and mango trees. Peach trees were markedly more clumped than mango trees. The two species showed different clumping behaviors but with the same trend for scaffold branches toward regular positioning. We showed that Oker-Blom and Kellomaki’s partial grouping model can be used for mango trees but not for peach trees which shows a different type of clumping that does not fit their original assumption. Our model alleviates this problem by making it possible to use a variable number of scales and can thus be applied to both situations. Moreover, the true STAR values were always comprised between the values from the full grouping scenario and the partial grouping one disregarding the scaffold scale. This suggests that the multiscale organization is not the only factor involved in the light interception strategy of trees. The proposed multiscale framework may be used to optimize plant architecture measurement in the context of modeling light interception by plants. It also defines a versatile and incremental procedure to compute light interception up to a desired level of accuracy, ranging from coarse descriptions, i.e., using the turbid medium
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hypothesis at the plant scale, to detailed descriptions, e.g., obtained by 3D digitizing, with a series of intermediate options defined by the number of scales taken into account and their relative positioning. The multiscale organization of components represented by porous envelopes provides another advantage in terms of model flexibility; depending on the availability of field measurements the envelopes and their positions can either correspond to the actual field values or be inferred from statistical assumptions. MµSLI M bridges the gap between the two types of models described in chapter 5 and can be seen as a mixed model. It is mixed to a sense different ot that of Chelle and Andrieu [1998]’s. The light interception estimation procedure is changed as a function of scale, not of distance. It allows to assess the scale by scale influence of the structure on light interception. In particular, we can assess how the structure of each scale behaves from the light interception point of view. The number of scales can be defined by the user, and for each of them the type of resolution for light interception can be chosen. Therefore MµSLI M allows to make different compromises between accuracy and speed. The model accounts for crown clumping using the multiscale organization of the tree when available, but can also use the classical parameter µ as a modifier of LAD. MµSLI M only considers direct interception (radiations scattering is disregarded), which is usually sufficient for ecophysiological processes like photosynthesis where the active wavelengths are the ones that are the more absorbed, i.e. the less scattered. The output of the models are manyfold: STAR values, grey level images accounting for the transmission values that can be grouped into classes. But it can also be used for estimating the light quantity received by each component. The total attenuation of a beam is equal to the product of the attenuations due to each component on its trajectory. Therefore the light available to the jth component is equal to the incident light reduced by the product of the j − 1 preceding components, and can be computed at any scale. That property allows to define several level of estimation of light interception, from coarse to detailed.
7
Modeling of light transmission under heterogeneous forest canopy
“To live; alone and free like a tree, and in a brotherhood like a forest.” N. Hikmet
G
rowth and survival of regeneration saplings and understorey vegetation development is closely related to light available below the forest trees. Manipulating the forest structure by thinning adult trees is a major tool to control light transmission to the understorey. The transmission is related to the attenuation of light which is usually estimated with the Beer-Lambert law assuming homogeneous foliage within the canopy. However forest canopies are far from homogeneous, which requires models that can take into account the effect of clumping between and within trees. In this chapter we present an ongoing work in the context of ECOGER project that uses MµSLI M. This project has interest in evaluating the light transmission under heterogeneous forest canopy in regard of forest management processes like regeneration or biomass management. In this context the objective is twofold. First we want to estimate the light available under the canopy as transmission classes, not only as a mean value. The distribution of transmission classes is an important factor for regeneration and biodiversity of the understorey. Second, it has been shown that, for homogeneous canopy, basal area, mean age of trees and their species are good indicators to estimate the light transmission [Sonohat et al. 2004]. We want to establish a similar set of parameters to be measured so that light transmission under heterogeneous canopies can be accurately estimated. This will be achieved with a sensitivity analysis over several factors: the boundary effect, the space occupation by the crowns, their shapes and the spatial positioning of the trees.
In this chapter we illustrate the scalability of MµSLI M presented in chapter 6. This model originally developed for isolated trees can be readily used at a larger scale to model stands. It computes the distribution of light transmitted below a canopy that can be described at different levels, from coarse to multiscale detailed representation. The intermediate levels of description allow a continuous tradeoff between speed of computation and accuracy. To evaluate the accuracy of the model, we compare the model results with field measurement from several stands of Pinus sylvestris L. in the French Massif Central. We will first present the reconstruction of the stand mockup that will be used by MµSLI M. We will then explain how the light interception model will compute results that can be compared with field measurement of light transmission. Finally, the principles of the sensitivity analysis will be exposed before introducing the OpenAlea 113
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platform that provides a framework to conduct the analyzes. This chapter originates from a presentation at the 5th International Workshop on Functional-Structural Plant Models, Napier, New Zealand, November 2007 [Da Silva et al. 2007] (see appendix A.2).
7.1. Stand reconstruction
7.1
115
Stand reconstruction The reconstruction procedure is the first step of the process. From the dendrometric measurements on the field, a mockup of the stand is generated. This generation involves the construction of a crown(and trunk) for each tree of the stand and its positioning. Then, the light attenuation within the crown must be defined so that the light model can compute the transmittance under the canopy.
7.1.1
Experimental unit The pine stands are located in the Chaîne des Puys, a mid-elevation volcanic mountain range (45◦ 42’ N, 2◦ 58’E) at a place named Fontfreyde. The elevation is 900 m a.s.l., mean annual rainfall is about 820 mm, and mean annual temperature is about 7◦ C. The soil is a volcanic brown soil at pH 6.0 with no mineral deficiency. The pines were 30-year-old at time of measurement, with a density ranging from 500 to 4000 stem ha−1 . All trees in a square area, of size approximately three times their height, were located by their x, y coordinates, and measured for their total height (14.1 ± 2 m mean ± SD) and DBH (Diameter at Breast Height) (16.3 ± 5 cm mean ± SD). Top and bottom crown heights were also measured. Crown extents was assessed by visually projecting to the soil its characteristic points (i.e. the points that better describe the crown irregularities) in, at least, four directions. The azimuth and distance of those points from trunk were then measured (Figure 7.1 b.). The light measurements were conducted in the central zone of the experimental unit. The surrounding zone where dendrometric measures were done acts as a buffer zone accounting for the attenuation of the encompassing forest (Figure 7.1 a.). The light measurements were achieved using a grid composed of 64 PAR1 sensors, namely Solems PAR/CBE 80 sensors. One additional sensor, Sunshine sensor BF2, was positioned outside of the experimental unit in full light to measure global (total) and diffuse radiation. This extra sensor uses an array of photodiodes with a D computer-generated shading pattern to measure incident solar radiation and the G ratio, i.e. the diffuse to global radiation ratio.
7.1.2
Crown reconstruction To reconstruct the 3D envelopes of the trees from the field measurements, we used the PlantGL library [Pradal et al. 2008a]. This library contains several geometric models, including different types of envelopes and algorithms to reconstruct the geometry of plants at different scales. For this particular case, we used the skinned surface implemented in PlantGL which is a generalization of surface of revolution with varying profiles being interpolated. We rapidly remind the formal definition as presented in [Pradal et al. 2008a]. The envelope of a skinned hull is a closed skinned surface which interpolates a set of profiles { Pk (u), k = 0, . . . , K } positioned at angle {αk , k = 0, ..., K } around the z axis. Similarly to an extruded hull, all vertical profiles are split into two open profile curves 1 Photosynthetic
Active Radiations
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Chapter 7. Modeling of light transmission under heterogeneous forest canopy
Figure 7.1 – a. Experimental unit. b. Field data: each dot locates a tree (its trunk), and each arrow defines a specific azimuth and distance from the trunk, characterizing the crown extend. c. Zoom in on the interest zone where light measurement are conducted.
homogeneously parameterized. We assume thus that all these profiles Pk (u) are nonrational B-spline curves with common degree p and number n of control points Pi,k . From these profiles, a variational profile Q can be defined that gives a section for each angle α around the rotation axis (see Figure 7.2 b.). It is defined as Q(u, α) =
n X
Ni,p (u) Qi (α)
(7.1)
i =0
where the Ni,p (u) are the pth -degree B-splines basis functions and the Qi (α) are a variational form of control points. Q interpolates all the profiles Pk . Therefore, Qi (α) are computed using a global interpolation method on the control points Pi,k at αk with k ∈ [0, K ]. For this, let q be the chosen degree of the interpolation such as q < K. Qi (α) are defined as K X Qi ( α ) = Nj,q (α) Ri,j (7.2) j =0
where the control points Ri,j are computed by solving interpolation constraints that results in a system of linear equations:
∀i ∈ [0, n], ∀k ∈ [0, K ]
Pi,k = Qi (αk ) =
K X
Nj,q (αk ) Ri,j .
(7.3)
j =0
Geometrically, the surface of the skinned hull is obtained by rotating Q(u, α) around the z axis, α being the rotation angle. It is thus defined as S(u, α) = cos(α) Q x (u, α), sin(α) Q x (u, α), Qy (u, α) . (7.4) For the particular case of a single profile, the surface is a surface of revolution. This surface is thus built from any number of profiles with associated direction. A profile is supposed to pass through top and bottom points and at an intermediate point of maximum radius, i.e. the height of the crown maximum width. Two shape factors, CT and CB , are used to describe the shape of the profiles above and below the
7.1. Stand reconstruction
a.
117
b.
c.
Figure 7.2 – Skinned Hull reconstruction: a. The user defines a set of planar profiles in different planes around the z axis (in black). b. Profiles are interpolated to compute different sections (in grey). c. The surface is computed. Input profiles are iso-parametric curves of the resulting surface. Illustration from [Pradal et al. 2008a].
maximum width. Mathematically, two quarters of super-ellipse of degree CT and CB are used to define the top and bottom part of the profiles. Isopoints of the profiles are interpolated with B-Spline curves of given degree. Note that our envelopes can be viewed as extension of Cescatti [1997a]’s asymmetric hull with profiles in any direction instead of restricted directions (cardinal directions). Flexibility of our model enables us to measure the most adequate profiles in case of irregular crowns. For each azimuthal direction describing a tree crown, a profile is generated by connecting the top of the crown and the maximum radius point with the quarters of super-ellipse of degree CT and the maximum radius point and the bottom of the crown with the quarters of super-ellipse of degree CB . The maximum radius height varies according to the tree species. The 3D envelope of a tree is obtained using a skinned surface generated from all its profiles. The trunk of the tree is represented as a cylinder of DBH diameter and with the height equal to the crown bottom height. The 3D representation of the experimental unit presented in Figure 7.1, is shown in Figure 7.3.
7.1.3
Opacity evaluation Computing light transmission of a reconstructed canopy requires information about the light attenuation within a crown. This estimation can be readily obtained using Beer-Lambert law, as explained in chapter 6. It expresses the probability, p0 , of a photon to cross the vegetation without being intercepted (equation 6.1) p0 = exp (− G.LAD.L) . This quantity can be considered as the porosity of a crown, and its complement, p = 1 − p0 , as its opacity. An opaque crown would intercepted all beams, p0 = 0, and thus has an opacity p = 1.
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a.
b.
Figure 7.3 – 3D reconstruction of an experimental unit. a. Top view. b. 3D view. Colors are used to visually differentiate between tree species: green for pine (Pinus sylvestris L.), orange for birch (Betula L.), and yellow for poplar (Populus L.).
The opacity of a crown can be estimated using the Beer-Lambert if leaf area density and the G-function can be estimated for the crown. Or, it can directly be estimated from field photographs using PiafPhotem software (UMR PIAF, INRA, France). The image is digitized, thresholded and then the pixels are counted. The ratio of black pixels to the total number of pixels yields the opacity of the crown while the ratio of white pixels to the total produces its porosity. However, the opacity value is a mean value of the crown surface in the image and is estimated for each species over a sample of pictures. The thresholding procedure still requires human intervention, an automated procedure would allow an opacity estimation for each tree. An example of opacity evaluation is shown in Figure 7.4. Figure 7.4 – Thresholding of a crown photograph using PiafPhotem software (UMR PIAF, INRA, France). The black pixel count yields an opacity of 93.5%.
7.2
Estimation of light transmission The light transmission of such reconstructed stands can be computed using MµSLI M presented in chapter 6. The finer scale considered will be the crown, and the coarser
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scale, the stand. The model is a simple multiscale model with two scales where the finest scale is consituted of porous envelopes. An opacity value is associated with each envelope, the opacity value from the thresholded pictures for the crowns, and 1 for the trunks (considered as opaque organs). MµSLI M allows then to compute the light transmission of the canopy considered as a turbid medium of porous crowns (option R) or the transmission of the actual canopy (option A). In the case of option R, the envelope of the canopy is a convex hull containing all crowns envelopes, and its opacity is computed from the crown ones. In the A case, the attenuation of each beam is computed according to the opacities of the crowns encountered along its trajectory and according to the path length within each of these crowns. From the construction procedure, the crowns have a fixed opacity value according to the tree species. This value is used as a global opacity for the envelope and consequently for every beam regardless of the entry point into the envelope. Therefore the travelling distance of the beams into the envelope is not taken into account. A beam crossing a crown envelope on its border (small path length into the crown) will be attenuated as much as a beam going through the middle of this crown (long path length into the crown).
7.2.1
Beam path length To take this distance into account, the opacity must be assessed for each beam, and we have to estimate it using available data. In the sequel we shall describe a way to ˆ obtained from thresholded photos. do so using the global opacity value, p, The porosity expression that takes the beam travelling distance into account is related to G, the extinction coefficient that depends on light direction, and LAD, the leaf area density (from equations 6.1 and 6.9) 1 X b , (7.5) 1 − exp − G.LAD.L p= β b∈B
where B is the set of beams b, β is its cardinality, and Lb is the path length of the beam into the crown. ˜ can be We thus need to estimate the quantity G.LAD. A global opacity value, p, regarded as the result of opacity estimation using Beer-Lambert law2 in a infinite horizontally homogeneous layer: p˜ = 1 − exp (− G.LAI ) .
(7.6)
This relation yields an expression for G.LAI, where LAI is the leaf area index. In the case of crowns with a finite volume, V , the usual definition of LAI can be extended to be expressed as a function of the total leaf area, TLA, and the projected envelope area, PEA TLA LAI = , (7.7) PEA and since LAD is the ratio of total leaf area to crown volume, LAD = 2 The
tion
TLA . V
(7.8)
sin(h) term is discarded by considering the projection plane orthogonal to incident light direc-
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Chapter 7. Modeling of light transmission under heterogeneous forest canopy
We can express G.LAD as G.LAD =
G.LAI.PEA − log(1 − p˜ ).PEA = . V V
(7.9)
Using this G.LAD value with equation 7.5 naturally leads to a smaller envelope opacity value than the global one. This is due to the fact that the negative exponential of a mean value is less than the mean of negative exponential values. Therefore a numerical approximation is carried out using the value from equation 7.9 as the starting value to speed-up convergence. The process stops when the opacity computed using equation 7.5 is equal to the global opacity with an acceptable predefined error, e. This G.LAD approximation is done once for each crown and for each direction , and stored for further usage. The crown opacity for each beam, pb , is then computed using the expression pb = 1 − exp − G.LAD.Lb .
(7.10)
We can now compute the light transmission of the canopy along one direction. This is achieved by computing, for each beam, its total opacity resulting from its travelling through the canopy, i.e. possibly going through multiple crowns. The opacity is directly the ratio of transmitted light to available light, a value of 0 means that all light goes through while a value of 1 means that no light goes through. The gray-level image constituted by the beam porosity values is therefore a shadow map of the canopy transmittance. For each defined direction, MµSLI M, computes the opacity values of the beams and produces such an image in a plane orthogonal to the light direction as illustrated in Figure 7.5.
Figure 7.5 – For each direction of incident light the total opacity of beams yields a gray-level image traducing the canopy transmittance. Yellow points represent the discretization of the trajectory of the sun above the canopy.
7.2. Estimation of light transmission
7.2.2
121
Integration of directional transmissions To evaluate the accuracy of the model, we want to compare the results with field measurements. The field measures are realized with PAR sensors which measure radiation over a time period, usually from sunrise to sunset. The measured radiations result from both the direct and the diffuse light. Therefore we model the experimental conditions by using two sets of directions. The first set, for diffuse light, discretizes the sky hemisphere in 46 solid angle sectors of equal area, according to the Turtle sky proposed by Den Dulk [1989]. The directions used are the central direction of each solid angle sector. The second set is used to simulate the trajectory of the sun, and the directions are dependant of the location (latitude, longitude), the day of the year, and on the step of time we use to discretize the sun course. Each of these directions is associated with a weighting coefficients derived from the Standard OverCast (SOC) distribution of sky radiance [Moon and Spencer 1942]. For each direction of a set, the image generated on a plane orthogonal to the light direction, is projected on the ground. The lower the elevation, the more the image will be stretched by the projection. Each projected image is rotated according to the azimuth. Finally images are merged by using the weighting coefficients. Figure 7.6 illustrates this procedure for two directions.
Figure 7.6 – The fusion of two stretched and rotated images: a. high elevation, close to zenith, and b. low elevation, close to horizon.
We obtain two images, one for the integrated transmittance of diffuse light and one for the direct light. These two images are then merged using the diffuse to total D ratio, , from the extra sensor. The total transmittance of a sunny day will be closer G to the transmittance of direct light than the total transmittance of a cloudy one. The histogram of the gray-levels of the central zone can then be compared to the field measurements. The gray-level values of pixels, from 0 (black) to 1 (white), yield the transmittance classes simulated by MµSLI M. Results from field measurement for the presented experimental unit, denoted Fonfreyde2, are shown in Figure 7.7 a. We can see that the two classes between 10% and 20% of transmittance represent more than 80% of the surface of the stand. The classes of 5 − 10% and 20 − 25% represent around 5% of the surface, each. Less than 3% of the surface receives less than 5% of light. There is no transmittance above 25%.
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Chapter 7. Modeling of light transmission under heterogeneous forest canopy
The transmission classes from the model are shown in Figure 7.7 b., and the distribution of transmission classes resembles a normal distribution with a mean around 22%. The discrepancies may stem from several reasons that we will illustrate in next section.
7.3
Sensitivity analysis In this section we will illustrate the many factors which may be analyzed during the sensitivity analysis of the model. This is ongoing work that will be published, the results are given for illustrative purposes.
Crown porosity A first parameter which has influence on light transmission is the crown porosity. The value is estimated for each species on a small range of individuals and the thresholding procedure is subject to errors. However the usual error is in the range of 0% to 10%, and changes of porosity of that magnitude has a very little visible effect. Boundary effect The experimental unit includes a buffer zone dedicated to model the boundary effect. Despite of this, the light directions with little elevation have direct access to the zone of interest, only blocked by the tree trunks of the buffer zone. Although they have low weighting coefficients during the merging procedure, these directions still have non negligible influence. To tackle this problem, two options were considered: the replication and the opaque wall. The replication process consists in duplicating the reconstructed stand all around himself to simulate the non measured neighborhood. Whereas this option may seem a realistic approach of the problem, it raises several complications. First the problem of the position and orientation of the duplicates, yields too many parameters to calibrate. Second, the increase in the number of simulated trees has obvious impact on the computing time of the algorithm, and if not specifically handled, could become problematic in terms of memory management. On the contrary, surrounding the scene by an opaque wall, which is a more ‘artificial’approach, presents the advantage of simplicity. The opaque wall is a cylinder, associated with an opacity, positioned in the scene so that it surrounds the zone of interest. The cost in time and memory is unchanged and the only two additional parameters are the opacity affected to the wall and the ratio of the cylinder radius to its height. This ratio has to be set so that low elevation radiations are intercepted without adding further opacity to the beams going through the crowns of the buffer zone. We chose the opaque wall because of the simple parameterization an the fact that the computation time is unchanged. An illustration of the stand and an opaque wall is shown in Figure 7.7 f., and the results obtained with a wall opacity of 0.9 are shown in Figure 7.7 c. The opaque wall reduces the number of transmittance classes and the global transmittance under the canopy. The latter effect is traduced by a marked
7.3. Sensitivity analysis
123
shift toward lower classes of transmittance. The importance of this effect is due to the value of wall opacity but also to the height of the cylinder. It was arbitrarily set to the height of the bounding-box, but should be reduced so that it influences only the low elevation light directions. However, under different conditions, denser canopy or different light condition (different D G values), the effect of the opaque wall is not as marked as illustrated with this stand. The discrepancies are specially marked when results are compared with measurements made on sunny days, i.e. low D G . This raises the issue of validation of the reconstruction. Space occupation by crowns If gaps in the canopy are too large or misplaced, it can have an important influence on the transmission estimated for direct light. Diffuse light transmission is estimated with a large range of directions that can compensate for reconstruction errors. Direct light, on the contrary, is estimated with a reduced range of directions, thus being more sensitive to this type of errors. This sensibility will be estimated in future work by simulating direct transmission for a large range of days with different sun courses. Another way to improve the reconstruction procedure would be to enhance the ability of crowns to occupy space. The naive increase of the measured radii would fill the gaps, but also increase the interpenetration of crowns which trees tend to avoid. An illustration of the stand where all crown radii were augmented by 10% with the same opaque wall as previously is shown in Figure 7.7 g., with its results in Figure 7.7 d. When compared with the previous example with the same opaque wall, we can see that the radius augmentation has similar additional effects: reduced number of transmission classes and shift toward lower classes. A method of crown reconstruction by overlap minimization as developed by Piboule et al. [2005]; Boudon and Le Moguedec [2007] will also be investigated. This method adds a reconstruction phase were the initial shape of crowns are allowed to be modified, within a user defined range, to increase space occupation as long as overlapping stays minimal. This optimization procedure could be modified to take into account other factors than space occupation or overlapping. For instance, the individual light interception as optimization factor could be an interesting approach, for it mimics an important aspect of plant development. Crown shapes An important property of MµSLI M is its versatility. It can be readily used with simple or complex parameterization. The crown shapes can be smoothly and continuously degraded from skinned surface to revolution surface, and to simple shapes like cones or spheres, with no impact at all on the model functioning. Slowly changing from a complex crown described with many parameters to a simpler one, with few parameters, defines a series of steps. Estimating light transmission at each step will allow to characterize the influence of the shape simplification on light interception, and thus to determine the minimal set of field measurements necessary to obtain a prediction of light transmission with an acceptable accuracy. An example of crown shape changes is shown in Figure 7.7 h., where the crowns of pines were replaced by cones and others crowns by spheres. The light transmission (Figure 7.7 e.) was computed using the same opaque wall as previously. The results show that
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the shapes have an important influence on light interception. This is specially true on heterogeneous and discontinuous canopy as noted by Cescatti [1997b].
a.
b.
c.
d.
e.
f.
g.
h.
Figure 7.7 – a. Transmission classes obtained from field measurement over one day. b. Simulated transmission classes obtained for the initial reconstructed stand using MµSLI M. Simulated transmission classes for c.: the reconstructed stand with an opaque wall (opacity = 0.9) ; d.: the reconstructed stand with augmented radii and the same opaque wall ; e.: the reconstructed stand with simple shape and the same opaque wall. f., g. and h., their respective 3D representations.
Spatial distribution Finally an important characteristic of canopies is the spatial positioning of individuals and of group species. In the example shown in this chapter, the positioning of birch and poplar trees is clearly not random, in fact they tend to be organized in clusters. The taxonomy of heterogeneous forest stands has been successfully achieved by Goreaud et al. [2004]; Ngo et al. [2006]. The influence of spatial positioning on light transmission will determine wether or not the description defined by such a taxonomy could replace the explicit spatial positioning of trees. Being able to summarize
7.3. Sensitivity analysis
125
the spatial distribution of trees by the few parameters defined by the taxonomy will severely reduce the amount of field measurement. Figure 7.8 shows two examples of such modifications, one where all trees are positioned at random, and one where pines are positioned at random while birch and poplar trees are distributed in clusters using a Neyman-Scott process [Cressie 1993].
Figure 7.8 – Stands reconstruction modified by spatial point processes that modify the tree positions. Top row: the spatial point processes. Bottom: the associated 3D stand reconstruction. Left: all trees are positioned at random (Poisson). Right: pine trees are positioned at random while birch and poplar trees are positioned with a Neyman-Scott process with 2 clusters.
This sensitivity analysis will require the joint work of, at least, three different applications/models: • The stand reconstruction procedure that generates the 3D scene. This procedure generates the opaque wall and modifies the crowns shapes. • MµSLI M, to compute light transmission under the canopy previously generated. • The spatial point process that modifies the spatial locations of the trees in the reconstructed scene. Combining different procedures or models can be a difficult task, even for computer scientists. This kind of issue has fostered the development of the OpenAlea platform.
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7.4
Chapter 7. Modeling of light transmission under heterogeneous forest canopy
Model integration in OpenAlea The OpenAlea project was designed as an open source and operating-system independent platform to facilitate the integration and interoperability of heterogeneous models [Pradal et al. 2008b]. OpenAlea sets-up a component-based framework based on a Python-language centric approach, and allows the easy creation of models using a visual programming interface. OpenAlea implements the principles of a component framework, which allows users to combine dynamically existing and independent pieces of software into customized workflows. Each piece of software is defined as a node. It is a function object which provides a certain type of service. It reads data on its input ports and provides new data on its output ports. The MµSLI M node represented in Figure 7.9 needs 5 inputs: a multiscale scene Figure 7.9 – A graphical node is (MSS), the light direction, the multiscale list of options a visual representation of a function. Input ports at the top rep(A/R), the image size defining the numbers of beams to resent the input arguments and use, and a path to a saving directory. The outputs are the output ports at the bottom, the STAR defined by the option list, the turbid STAR (last resulting values. scale directly in first scale with option R, see chapter 6) and the grey-level of the light transmission. The output of the reconstruction procedure can be directly connected to the input port of MµSLI M node if their data types are compatible. Otherwise, an adaptor has to be inserted. Connections between nodes represent the flow of data. Nodes and their connections form a graph that defines a high level functional process called a dataflow. To easily access the different components, they are associated with a graphical node that visually represents the function defined by the node. Graphical nodes are used and connected to each others through a graphical user interface (GUI). OpenAlea provides a GUI, VisuAlea, which makes it possible to combine graphically different processing nodes provided by OpenAlea libraries and run the dataflow. Communications between nodes are graphically represented as edges between them. The graphical models show clearly the dependencies between the processes as a graphical network and ease the understanding of the structure of the model. Users can interactively edit, save and compose nodes. The Figure 7.10 show the dataflow of the stand reconstruction design for the above application. The graph is evaluated from top to bottom. It starts by reading the data from files containing field measurements for each species. Then, a module generates the shapes accordingly and merges the representations for all species into a 3D scene. The scene can be displayed by the plot3D node. Here, the node calls directly the PlantGL viewer [Pradal et al. 2008a]. Computation of light transmission of this stand is simply achieved by connecting the MµSLI M nodes at the end of the reconstruction dataflow, hence extending it. The resulting dataflow is shown in Figure 7.11, where with few additional nodes, the computation can be executed over a range of directions and the resulting image displayed. This new dataflow represents a new model which reconstructs stands and
7.4. Model integration in OpenAlea
127
Figure 7.10 – Snapshot of the OpenAlea visual modeling environment. The dataflow represents the reconstruction of a stand from field measurements. Note that the output of node for opaque wall is not connected to the dataflow and thus is not evaluated as the display of stand shows.
compute their light transmission.
Figure 7.11 – The dataflow for stand reconstruction is augmented with MµSLI M nodes and additional nodes allowing to loop over a range of light directions and display the resulting images.
Analyzing the effect of spatial positioning will require the generation of stands with modified positions of the trees. This can be readily done by adding nodes that modify the position of trees. The modification can take place either after the merging of species to affect all positions similarly, or before the merging if spatial organization modifications are to be made according to the species. MµSLI M was developed as an open source project and was designed to be used in the scientific community. One major drawback in the open source community is
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Chapter 7. Modeling of light transmission under heterogeneous forest canopy
the lack of visibility due to the profusion of projects. The diffusion of software is also problematic, in particular in the scientific community where each team has its specificity. To tackle these issues, OpenAlea provides a set of software tools to build, package, install, and distribute the modules in a uniform way on multiple operating systems. It decreases development and maintenance costs whilst increasing software quality and providing a larger diffusion. In particular, some compilation and distribution tools make it possible, with high level commands for users, to avoid most of the problems due to operating systems specificities. The OpenAlea platform is distributed under an open source licence to foster collaborative development and diffusion. All informations and software are available through the OpenAlea website: http://openalea.gforge.inria.fr
Conclusion In this chapter, we showed the scalability and versatility of MµSLI M. Due to its multiscale nature, this model developed for isolated trees with multiscale description can be easily used at stand scale with detailed or simple description of the crown shapes. If data were available, MµSLI M could compute light interception of a stand of trees with multiscale description. We also illustrated how its integration in the OpenAlea platform facilitates its reusability in other projects involving light interception. Concerning this project, the calibration of MµSLI M and the validation are ongoing tasks that need to be realized before moving on the next phase: the evaluation of relevant parameters that will realize the best trade-off between field measurement and accurate estimation of light transmission.
Conclusion and prospects
During this work we have studied the multiscale nature of plants and its use in modeling approaches. This work was motivated by the idea that the few parameters used in ecophysiological models with global descriptions, could be extracted from the structure of plants. The complexity and multiscale nature of plants led us to consider two approaches: a direct analysis and characterization of the plant geometry using fractal descriptors, and the introduction of the multiscale notion in ecophysiological models, radiative models in this case. The characterization of geometry was done using two types of descriptors. First we used the fractal dimension that generalizes the notion of Euclidian dimension to complex object, i.e. objects with a non integer dimension. We showed that the fractal dimension of plants gives a meaningful insight into the complexity of their geometry and expresses the way plant physically penetrate into space as a function of scale. In chapter 3, we studied two specific methods that we adapted to threedimensional representation of plant canopies: the box-counting method and the two surface method. The first one is very general and only requires spatial information about plant components. To obtain meaningfull results, this method must be used in specific conditions that we described. However, the reliability of this method is an issue. To address this question we defined a new estimator of the fractal dimension. This estimator has proven to be very sensitive to the quantization effect but can be used as a complementary descriptor confirming the results of the classical box-counting method. The second method (two-surface) is more robust and yields more reliable results but requires topological and botanical information to generate plant modularities. These additional information on plants can be tedious to obtain, explaining why the box counting method is easier to implement. To complement the characterization of plant geometry, in chapter 4 we studied how to quantify the spatial distribution of plant components, in our case, leaves. In particular we tried to assess the gaps and clusters sizes that can be found in the spatial organization of plants. We used the notion of lacunarity and proposed a modified definition adapted to our goal, the centered lacunarity. We showed that this variant of lacunarity has interesting properties. This descriptors are usually estimated for objects embedded in one or two-dimensional Euclidian space. To evaluate them on our three-dimensional representations of plants, we developed general tools and algorithm that are able to manage 3D objects. This work showed that fractal geometry provide a multiscale framework and descriptors that characterize the different levels of organization in plants. To our knowledge, this is the first application of a set of fractal descriptors to 3D geometry, in particular for plant foliage. Based on our experience on plant multiscale organization, we chose to investigate the relation between ecophysiological models and plant structure from this point of view. We specifically studied the light interception for it is one of the major ecophys129
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Conclusion and prospects
iological process that drives plant development. This approach led us to the development, in chapter 6, of a general multiscale model of light interception: MµSLI M. This model allowed us to study the scale by scale behavior of plant organization toward the interception of light. Even though it does not yield a direct quantitative relation between structure and light interception capacity, it does, however, give informations on how to improve plant representation by refining measurement protocols. From this type of analysis, we can determine which scales are important, which ones can be approximated by global or statistical descriptions, and which ones can be neglected, thus allowing us to determine the simplest canopy description, i.e. field measurements, that gives an accurate estimation of light interception. The scalability of MµSLI M, as illustrated in chapter 7, allows a wide range of usage of this model in research topics that involves direct light interception. The main difficulty of characterizing the relation between the structure of a plant and its light interception abilities lies in the fact that we try to establish a relation between a threedimensional structure and a process, light-interception, that is inherently directional. MµSLI M combined with the fractal descriptors we presented constitute an new approach that brings us one step closer to the understanding of the intricate relation between plant architecture and light interception as I will explain below. The work realized during this thesis opens up on several prospects. As it has been illustrated in chapter 4, objects that have a null centered lacunarity at scale δ are manifold. A full characterization of the set of objects having this property with construction rules is an interesting challenge that links with discrete geometry. For assessing both the lacunarity and the box-dimension, the studied object is inserted in a grid. The voxels of this grid that contains the object is a discrete representation of the object. Clarifying this link is an interesting prospect, even more challenging if we consider it from a multiscale point of view. The centered lacunarity, at one scale, is a descriptor of a neighborhood relation, quite similar to Ripley’s K function. It could be considered as an alternative descriptor of spatial point process and could also be used to characterize the typology of stands. As it has been illustrated in chapter 7, MµSLI M is versatile and can be used with other tree species or plants, for isolated trees or large stand. It can be used as a simple light interception model to drive plant development in FSPMs or regeneration models. To date, the model does not consider the scattering of light (reflection and transmission), but extending it in that direction is a natural evolution of the model. It could include a local radiosity approach inspired from the one proposed by Soler et al. [2003], and/or, a Chelle et al. [1998]-like nested radiosity. In fact the underlying architecture of the model using a MTG-like representation allows to readily extend it with other light interception approaches. It can also be easily modified to take advantage of instantiation when computing the light interception of plants showing high-level of self-similarity or plants artificially compressed into a self-similar approximate structure Godin and Ferraro [2008]. Instantiation could also be used on plants generated with models using the fractal descriptors as input parameters. We could simplify the representation of plants by using the fractal dimension as the multiscale quantitative parameter to define how much plant should be created as a
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function of scale. To complete the description, a descriptor like the centered lacunarity could be used to address the spatial distribution of plant elements. The alternate version of centered lacunarity that uses continuous values instead of the current binary values for presence or absence may be more interesting when addressing ecophysiological processes. We could compute a leaf area centered lacunarity by using the leaf area in a voxel as it mass function. Another way to make simpler representation of a plant, would be to use a self-similar approximation based on its topology. The study of multiscale effect on light interception for such plants with compact description may be a lead to find out the parameters that summarize the plant characteristics related to light-interception. Finally, the methods, algorithms and models developed during this thesis are integrated to the OpenAlea platform as two open-software modules: Fractalysis and MµSLIM that will be part of the next OpenAlea public release.
Bibliography
Adam, B. (1999). Pol95 - software to drive a polhemus fastrak 3 space 3d digitiser. Technical report, UMR PIAF INRA-UBP, Clermont-Ferrand. (Cited page 27.) Adam, B., Dones, N., and Sinoquet, H. (2004). Vegestar v.3.1. a software to compute light interception and photosynthesis by 3d plant mock-ups. In 4th International Workshop on Functional-Structural Plant Models, pages 414+. CIRAD. (Cited page 73.) Adam, B., Sinoquet, H., Godin, C., and Donès, N. (1999). 3A-Software for the acquisition of plant architecture. INRA, Clermotn-Ferrand. (Cited page 25.) Adler, I., Barabe, D., and Jean, R. (1997). A history of the study of phyllotaxis. Annals of Botany, 80(3):231–244. (Cited page 5.) Allain, C. and Cloitre, M. (1991). Characterizing the lacunarity of random and deterministic fractal sets. Physical Review A, 44:3552–3558. (Cited pages 51, 53, 54, 56, 57, and 64.) Allen, M., Dejong, T., and Prusinkiewicz, P. (2006). L-peach, an l-systems based model for simulating the architecture and carbon partitioning of growing fruit trees. Acta Horticulturae, 707:71–76. (Cited page 9.) Andres, E., Nehlig, P., and Françon, J. (1997). Supercover of straight lines, planes and triangles. In Lecture Notes In Computer Science, volume 1347, pages 243–254. (Cited page 41.) Aphalo, P., Ballare, C., and Scopel, A. (1999). Plant-plant signalling, the shadeavoidance response and competition. J. Exp. Bot., 50(340):1629–1634. (Cited page 69.) Arvo, J. and Kirk, D. (1989). A survey of ray tracing acceleration techniques. In An introduction to ray tracing, pages 201–262. Academic Press Ltd., London, UK, UK. (Cited page 75.) Barnsley, M. (1988). Fractals Everywhere. Academic Press, Boston. (Cited pages 13, 15, 17, and 18.) Barthélémy, D. (1991). Levels of organization and repetition phenomena in seed plants. Acta Biotheoretica, 39(3):309–323. (Cited page 6.) Barthélémy, D. and Caraglio, Y. (2007). Plant architecture: A dynamic, multilevel and comprehensive approach to plant form, structure and ontogeny. Ann Bot, 99(3):375– 407. (Cited pages 3 and 5.) Barthélémy, D., Edelin, C., and Hallé, F. (1989). Architectural concepts for tropical trees. In Holm-Nielsen, L. B., Nielsen, I. C., and Balslev, E., editors, Symposium on Tropical Forest, pages 89–100. Academic Press, London. (Cited page 3.)
133
134
Bibliography
Boissard, P. (1985). Description de la géométrie d’une culture avec restriction aux parties vues. In 3rd International colloquium In Spectral signatures of object in remote sensing. (Cited page 25.) Borel, C. C., Gerstl, S. A. W., and Powers, B. J. (1991). The radiosity method in optical remote sensing of structured 3-d surfaces. Remote sensing of environment, 36:13–44. (Cited pages 73 and 78.) Boudon, F. (2004). Représentation géométrique multi-échelles de l’architecture des plantes. PhD thesis, Université de Montpellier 2. (Cited page 106.) Boudon, F., Godin, C., Pradal, C., Puech, O., and Sinoquet, H. (2006). Estimating the fractal dimension of plants using the two-surface method. an analysis based on 3d-digitized tree foliage. Fractals, 14(3). (Cited pages 8, 39, 46, 47, and 94.) Boudon, F. and Le Moguedec, G. (2007). Déformation asymétrique de houppiers pour la génération de représentations paysagères réalistes. Revue Electronique Francophone d’Informatique Graphique, 1(1). (Cited page 123.) Bradford-Barber, C., Dobkin, D. P., and Huhdanpaa, H. (1996). The quickhull algorithm for convex hulls. ACM Trans. Math. Softw., 22(4):469–483. (Cited page 109.) Brunner, A. (1998). A light model for spatially explicit forest stand models. Forest Ecology and Management, 107(1-3):19–46. (Cited page 86.) Carter, G. A. and Smith, W. K. (1985). Influence of shoot structure on light interception and photosynthesis in conifers. Plant Physiol., 79(4):1038–1043. (Cited page 93.) Casella, E. and Sinoquet, H. (2003). A method for describing the canopy architecture of coppice poplar with allometric relationships. Tree Physiology, 23:1153–1170. (Cited page 85.) Casella, E. and Sinoquet, H. (2007). Botanical determinants of foliage clumping and light interception in two-year-old coppice poplar canopies: assessment from 3-d plant mock-ups. Annals of Forest Science, 64:395–404. (Cited page 92.) Castrejon-Pita, J. R., Galan, S. A., and Castrejon-Garcia, R. (2002). Fractal dimension and self-similarity in asparagus plumosus. (Cited page 40.) Cescatti, A. (1997a). Modelling the radiative transfer in discontinuous canopies of asymmetric crowns. i. model structure and algorithms. Ecological Modelling, 101(2):263–274. (Cited pages 86 and 117.) Cescatti, A. (1997b). Modelling the radiative transfer in discontinuous canopies of asymmetric crowns. ii. model testing and application in a norway spruce stand. Ecological Modelling, 101(2):275–284. (Cited pages 85, 105, and 124.) Charles-Edwards, D. A. and Thornley, J. H. M. (1973). Light interception by an isolated plant a simple model. Ann Bot, 37(4):919–928. (Cited pages 92 and 98.) Chelle, M. (2006). Could plant leaves be treated as lambertian surfaces in dense crop canopies to estimate light absorption? Ecological Modelling, 198(1-2):219–228. (Cited page 75.)
Bibliography
135
Chelle, M. and Andrieu, B. (1998). The nested radiosity model for the distribution of light within plant canopies. Ecological Modelling, 111(1):75–91. (Cited pages 8, 71, 73, 75, 76, 78, and 112.) Chelle, M., Andrieu, B., and Bouatouch, K. (1998). Nested radiosity for plant canopies. The Visual Computer, 14(3):109–125. (Cited pages 78, 79, and 130.) Chen, J. M. and Black, T. A. (1992). Defining leaf area index for non-flat leaves. Plant, Cell and Environment, 15(4):421–429. (Cited page 81.) Chen, S. G., Ceulemans, R., and Impens, I. (1994). A fractal-based populus canopy structure model for the calculation of light interception. Forest Ecology and Management, 69(1-3):97–110. (Cited pages 13 and 101.) Chen, S. G., Impens, I., Ceulemans, R., and Kockelbergh, F. (1993). Measurement of gap fraction of fractal generated canopies using digitalized image analysis. Agricultural and Forest Meteorology, 65(3-4):245–259. (Cited page 73.) Cohen, M. F., Wallace, J., and Hanrahan, P. (1993). Radiosity and realistic image synthesis. Academic Press Professional, Inc., San Diego, CA, USA. (Cited page 77.) Cohen, S., Fuchs, M., Moreshet, S., and Cohen, Y. (1987). The distribution of leaf area, radiation, photosynthesis and transpiration in a shamouti orange hedgerow orchard. part ii. photosynthesis, transpiration, and the effect of row shape and direction. Agricultural and Forest Meteorology, 40(2):145–162. (Cited page 92.) Cohen, S., Mosoni, P., and Meron, M. (1995). Canopy clumpiness and radiation penetration in a young hedgerow apple orchard. Agricultural and Forest Meteorology, 76(3-4):185–200. (Cited pages 92, 101, and 104.) Combes, D. (2002). Comparaison de modèles de transferts radiatifs pour simuler la distribution du rayonnement actif sur la morphogénèse (MAR) au sein d’un peuplement vegetal à une échelle locale. PhD thesis, Université Blaise Pascal. (Cited page 79.) Cook, R. L. (1986). Stochastic sampling in computer graphics. ACM Trans. Graph., 5(1):51–72. (Cited page 74.) Cormen, T. H., Leiserson, C. E., Rivest, R., and Stein, C. (2001). Introduction to Algorithm. The MIT Press, Cambridge, Massasuchetts, USA, second edition. (Cited page 78.) Costes, E., Sinoquet, H., Kelner, J. J., and Godin, C. (2003). Exploring within-tree architectural development of two apple tree cultivars over 6 years. Ann Bot, 91(1):91– 104. (Cited pages 7 and 25.) Courbaud, B. (2003). Simulating radiation distribution in a heterogeneous norway spruce forest on a slope. Agricultural and Forest Meteorology, 116(1-2):1–18. (Cited page 86.) Cressie, N. A. C. (1993). Statistics for Spatial Data. Wiley Series in Probability and Statistics. Wiley-Interscience, New York, revised edition edition. (Cited page 125.)
136
Bibliography
Critten, D. L. (1997). Fractal dimension relationships and values associated with certain plant canopies. Journal of Agricultural Engineering Research, 67(1):61–72. (Cited page 40.) Da Silva, D., Balandier, P., Boudon, F., Marquier, A., Pradal, C., Godin, C., and Sinoquet, H. (2007). Modeling of light transmission under heterogeneous forest canopy: model description and validation. In 5th International Workshop on Functional-Structural Plant Models, pages 58–61. (Cited page 114.) Da Silva, D., Boudon, F., Godin, C., Puech, O., Smith, C., and Sinoquet, H. (2006). A critical appraisal of the box counting method to assess the fractal dimension of tree crowns. In Brebis, G., editor, Lecture Notes In Computer Science, pages 751–760. (Cited pages 38 and 40.) Dauzat, J., Rapidel, B., and Berger, A. (2001). Simulation of leaf transpiration and sap flow in virtual plants: model description and application to a coffee plantation in costa rica. Agricultural and Forest Meteorology, 109(2):143–160. (Cited page 9.) de Castro, F. and Fetcher, N. (1998). Three dimensional model of the interception of light by a canopy. Agricultural and Forest Meteorology, 90(3):215–233. (Cited page 82.) de Reffye, P., Edelin, C., Francon, J., Jaeger, M., and Puech, C. (1988). Plant models faithful to botanical structure and development. SIGGRAPH Comput. Graph., 22(4):151–158. (Cited page 9.) de Wit, C. T. (1965). Photosynthesis of leaf canopies. Technical report, Center for Agricultural Publication and Documentation, Wageningen, Netherlands. (Cited page 80.) Deering, W. and West, B. J. (1992). Fractal physiology. Engineering in Medicine and Biology Magazine, IEEE, 11(2):40–46. (Cited page 68.) Den Dulk, J. A. (1989). The interpretation of Remote Sensing, a feasibility study. PhD thesis, Wageningen university. (Cited pages 66, 73, and 121.) Disney, M. I., Lewis, P., and North, P. R. J. (2000). Monte carlo ray tracing in optical canopy reflectance modelling. Remote Sensing Reviews. (Cited page 74.) Duursma, R. A. and Mäkelä, A. (2007). Summary models for light interception and light-use efficiency of non-homogeneous canopies. Tree Physiology, 27(6):859–870. (Cited page 86.) Eshel, A. (1998). On the fractal dimensions of a root system. Plant, Cell & Environment, 21(2):247+. (Cited page 40.) Falconer, K. (1990). Fractal geometry : mathematical foundation and applications. John Wiley and Sons. (Cited pages 15, 16, 17, 33, 36, 37, 38, and 40.) Falconer, K. (1997). Techniques in fractal geometry. John Wiley and Sons. (Cited pages 16, 17, 33, and 34.) Ferraro, P. (2000). Méthodes algorithmiques de comparaison d’arborescences. Applications à la comparaison de l’architecture des plantes. PhD thesis, Institut National Polytechnique de Toulouse. (Cited page 12.)
Bibliography
137
Ferraro, P., Godin, C., and Prusinkiewicz, P. (2005). Toward a quantification of selfsimilarity in plants. Fractals, 2(12):91–109. (Cited page 94.) Fitter, A. H. (1986). The topology and geometry of plant root systems: Influence of watering rate on root system topology in trifolium pratense. Ann Bot, 58(1):91–101. (Cited page 9.) Fitter, A. H. (1987). An architectural approach to the comparative ecology of plant root systems. New Phytologist, 106(1):61–77. (Cited page 40.) Foley, J. D., van Dam, A., Feiner, S. K., and Hughes, J. F. (1995). Computer Graphics: Principles and Practice in C. Addison-Wesley Professional, second edition. (Cited page 91.) Ford, D. E., Avery, A., and Ford, R. (1990). Simulation of branch growth in the pinaceae: Interactions of morphology, phenology, foliage productivity, and the requirement for structural support, on the export of carbon. Journal of Theoretical Biology, 146(1):15–36. (Cited page 9.) Foroutan-Pour, K., Dutilleul, P., and Smith (1999). Advances in the implementation of the box-counting method of fractal dimension estimation. Applied Mathematics and Computation, 105(2):195–210. (Cited page 38.) Fournier, C. and Andrieu, B. (1999). Adel-maize: an l-system based model for the integration of growth processes from the organ to the canopy. application to regulation of morphogenesis by light availability. Agronomie, 19:313–327. (Cited page 73.) Früh, T. (1997). Simulation of water flow in the branched tree architecture. Silva Fennica, 31(3):275–284. (Cited page 9.) Gastellu-Etchegorry, J. P., Demarez, V., Pinel, V., and Zagolski, F. (1996). Modeling radiative transfer in heterogeneous 3-d vegetation canopies. Remote Sensing of Environment, 58(2):131–156. (Cited page 82.) Gautestad, A. O. and Mysterud, I. (1994). Fractal analysis of population ranges: Methodological problems and challenges. Oikos, 69(1):154–157. (Cited page 37.) Gefen, Y., Meir, Y., Mandelbrot, B. B., and Aharony, A. (1983). Geometric implementation of hypercubic lattices with noninteger dimensionality by use of low lacunarity fractal lattices. Physical Review Letters, 50(3):145+. (Cited page 53.) Giauque, P. (2003). Conduite du verger de pêcher. Recherche de la performance. Ctifl (Centre technique interprofessionnel des fruits et légumes). (Cited page 27.) Giavitto, J.-L. and Michel, O. (2003). Modeling the topological organization of cellular processes. Biosystems, 70(2):149–163. (Cited page 13.) Glassner, A. S. (1988). Space subdivision for fast ray tracing. In Tutorial: computer graphics; image synthesis, pages 160–167. Computer Science Press, Inc., New York, NY, USA. (Cited page 75.) Glassner, A. S. (1995). Principles of digital image synthesis. The Morgan Kaufmann Series in Computer Graphics and geometric modeling. Morgan Koaufmann Publishers, Inc., San Francisco, California, USA. (Cited pages 70, 71, 73, 74, 75, 77, and 78.)
138
Bibliography
Godin, C. (2000). Representing and encoding plant architecture: A review. Annals of Forest Science, 57:413–438. (Cited pages 3, 7, 8, and 9.) Godin, C. (2003). Introduction aux structures multi-échelles. Applications à la représentation des plantes. HDR,˜Université de Montpellier II. (Cited pages 15, 17, 18, 24, 29, 31, 32, and 33.) Godin, C. and Caraglio, Y. (1998). A multiscale model of plant topological structures. Journal of Theoretical Biology, 191(1):1–46. (Cited pages 5, 6, 9, 10, 11, 12, 39, and 94.) Godin, C., Costes, E., and Caraglio, Y. (1997). Exploring plant topology structure with the amapmod software : an outline. Silva Fennica, 31(3):355–366. (Cited pages 25 and 109.) Godin, C., Costes, E., and Sinoquet, H. (1999). A method for describing plant architecture which integrates topology and geometry. Ann Bot, 84(3):343–357. (Cited pages 25, 28, and 94.) Godin, C., Costes, E., and Sinoquet, H. (2005). Plant architecture modelling. Virtual plants and complex systems., volume 17 of Annual Plant Reviews, chapter 9, pages 238–287. CRC Press. (Cited pages 1, 11, 13, and 25.) Godin, C. and Ferraro, P. (2008). Quantifying the degree of self-nestedness of trees. application to the structural analysis of plants. IEEE Transactions on Computational Biology and Bioinformatics, Submitted. (Cited page 130.) Godin, C. and Sinoquet, H. (2005). Functional-structural plant modelling. New Phytologist, 166(3):705–708. (Cited page 13.) Goel, N. S., Rozehnal, I., and Thompson, R. L. (1991). A computer graphics based model for scattering from objects of arbitrary shapes in the optical region. Remote Sensing of Environment, 36(2):73–104. (Cited page 78.) Goreaud, F., Loussier, B., Ngo, and Allain, R. (2004). Simulating realistic spatial structure for forest stands : a mimetic point process. In Journées Interdisciplinaires de Statistiques Spatiales. (Cited page 124.) Greene, N. (1989). Voxel space automata: modeling with stochastic growth processes in voxel space. SIGGRAPH Comput. Graph., 23(3):175–184. (Cited page 7.) Hallé, F., Oldeman, R. A. A., and Tomlinson, P. B. (1978). Tropical trees and forests - an architectural analysis. Springer Verlag. (Cited pages 3 and 12.) Halley, J. M., Hartley, S., Kallimanis, A. S., Kunin, W. E., Lennon, J. J., and Sgardelis, S. P. (2004). Uses and abuses of fractal methodology in ecology. Ecology Letters, 7:254–271. (Cited pages 38, 42, 54, and 66.) Hanan, J. S. (1992). Parametric L-systems and their application to the modelling and visualization of plant. PhD thesis, University of Regina. (Cited page 21.) Harper, J. L. and White, L. (1986). The Growth and Form of Modular Organisms. Royal Society. (Cited page 5.)
Bibliography
139
Heeraman, D. A., Hopmans, J. W., and Clausnitzer, V. (1997). Three dimensional imaging of plant roots in situ with x-ray computed tomography. Plant and Soil, 189(2):167–179. (Cited page 25.) Honda, H. (1971). Description of the form of trees by the parameters of the tree-like body: Effects of the branching angle and the branch length on the shape of the tree-like body. Journal of Theoretical Biology, 31(2):331–338. (Cited page 9.) Hutchinson, J. (1981). Fractals and self-similarity. Indiana University Mathematics Journal, 30(5):713–747. (Cited page 16.) Ivanov, N., Boissard, P., Chapron, M., and Andrieu, B. (1995). Computer stereo plotting for 3-d reconstruction of a maize canopy. Agricultural and Forest Meteorology, 75(1-3):85–102. (Cited page 25.) Jackson, J. E. and Palmer, J. W. (1979). A simple model of light transmission and interception by discontinuous canopies. Annals of Botany, 44:381–383. (Cited page 93.) Janssen, J. M. and Lindenmayer, A. (1987). Models for the control of branch positions and flowering sequence of capitula in mycelis muralis l. dumont (compositae). New Phytologist, 105(2):191–220. (Cited page 9.) Johnson, N. L. and Leone, F. C. (1977). Statistics and experimental design in engineering and the physical sciences. John Wiley & Sons, New York, New York, USA, second edition. (Cited page 54.) Jones, H. G. (1992). Plants and Microclimate: A Quantitative Approach to Plant Physiology. Cambridge University Press. (Cited page 91.) Kajiya, J. T. (1985). Anisotropic reflection models. In SIGGRAPH ’85: Proceedings of the 12th annual conference on Computer graphics and interactive techniques, pages 15–21, New York, NY, USA. ACM. (Cited page 75.) Kajiya, J. T. (1986). The rendering equation. In SIGGRAPH ’86: Proceedings of the 13th annual conference on Computer graphics and interactive techniques, volume 20, pages 143–150, New York, NY, USA. ACM Press. (Cited pages 71 and 74.) Karwowski, R. (2002). Improving the process of plant modeling :The L+C modeling language. PhD thesis, University of Calgary. (Cited page 21.) Kenkel, N. C. and Walker, D. J. (1996). Fractals in the biological sciences. Website. (Cited page 40.) Kent, C. and Wong, J. (1982). An index of littoral zone complexity and its measurement. Canadian Journal of Fisheries and Aquatic Sciences, 39(6):847–853. (Cited page 37.) Kimes, D. S. and Kirchner, J. A. (1982). Radiative transfer model for heterogeneous 3-d scenes. Applied Optics, 21(22):4119+. (Cited pages 82 and 92.) Knyazikhin, Y., Mie[beta]en, G., Panfyorov, O., and Gravenhorst, G. (1997). Smallscale study of three-dimensional distribution of photosynthetically active radiation in a forest. Agricultural and Forest Meteorology, 88(1-4):215–239. (Cited pages 82 and 92.)
140
Bibliography
Lang, A. R. G. (1973). Leaf orientation of a cotton crop. Agricultural and Forest Meteorology, 11:37–51. (Cited page 24.) Lin, B. and Yang, Z. R. (1986). A suggested lacunarity expression for sierpinski carpets. Journal of Physics A: Mathematical and General, 19(2):L49–L52. (Cited page 53.) Lindenmayer, A. (1968). Mathematical models for cellular interaction in development, parts i and ii. Journal of Theoretical Biology, 18(3):280–315. (Cited pages 13 and 20.) Lyndon, R. F. (1998). The shoot apical meristem : its growth and development. Development and Cell Biology. Cambridge University Press, Cambridge, UK. (Cited page 5.) Mandelbrot, B. (1967). How long is the coast of britain? statistical self-similarity and fractional dimension. Science, 156(3775):636–638. (Cited page 15.) Mandelbrot, B. B. (1983). The fractal geometry of nature. Freeman˜. (Cited pages 1, 13, 25, 26, 29, 30, 33, 34, 40, 51, and 53.) Mann, J. E., Curry, G. L., and Sharpe, P. J. H. (1979). Light interception by isolated plants. Agricultural Meteorology, 20(3):205–214. (Cited pages 92 and 98.) Mech, R. (1997). Modeling and simulation of the interaction of plants with the environment using-L-systems and their extensions. PhD thesis, University of Calgary. (Cited page 21.) Meyer, A. and Neyret, F. (2000). Multiscale shaders for the efficient realistic rendering of pine-trees. In Graphics Interface, pages 137–144. Canadian Information Processing Society, Canadian Human-Computer Communications Society. (Cited page 75.) Monsi, M. and Saeki, T. (1953). Uber den lichtfaktor in den pflanzengesellschaften und seine bedeutung fur die stoffproduktion. Japanese Journal of Botany, 14(1):22–52. (Cited pages 70, 79, 81, and 91.) Moon, P. and Spencer, D. E. (1942). Illumination from a non-uniform sky. Transactions of the Illumination Engineering Society, 37. (Cited page 121.) Morse, D. R., Lawton, J. H., Dodson, M. M., and Williamson, M. H. (1985). Fractal dimension of vegetation and the distribution of arthropod body lengths. Nature, 314(6013):731–733. (Cited page 40.) Mottus, M., Sulev, M., and Lang, M. (2006). Estimation of crown volume for a geometric radiation model from detailed measurements of tree structure. Ecological Modelling, 198(3-4):506–514. (Cited page 85.) Moulia, B. and Sinoquet, H. (1993). Three-dimensional digitizing systems for plants canopy geometrical structure: a review. In Varlet-Grancher, C., Bonhomme, R., and Sinoquet, H., editors, Crop structure and light microclimate: Characterization and applications, Science Update. INRA, 147 rue de l’Université, 75338 Paris cedex 07, France. (Cited pages 24, 25, and 106.) Myneni, R. B. (1991). Modeling radiative transfer and photosynthesis in threedimensional vegetation canopies. Agricultural and Forest Meteorology, 55(3-4):323– 344. (Cited pages 82 and 92.)
Bibliography
141
Myneni, R. B., Ross, J., and Asrar, G. (1989). A review on the theory of photon transport in leaf canopies. Agricultural and Forest Meteorology, 45(1-2):1–153. (Cited pages 70 and 82.) Ngo, Ginisty, C., Goreaud, F., and Perot, T. (2006). First typology of oak and scots pine mixed stands in orléans forest (france), based on the canopy spatial structure. New Zealand Journal of forestry Science, 36(2):325–346. (Cited page 124.) Nicodemus, F. E., Richmond, J. C., Hsia, J. J., Ginsberg, I. W., and Limperis, T. (1977). Geometric considerations and nomenclature for reflectance. National Bureau of Standards. (Cited page 72.) Niinemets, U., Cescatti, A., and Rochelle, C. (2004). Constraints on light interception efficiency due to shoot architecture in broad-leaved nothofagus species. Tree Physiology, 24:617–630. (Cited pages 85 and 92.) Nikinmaa, E. (1992). Analyses of the growth of scots pine: matching structure with function. Acta Forestalia Fennica, 235:3–68. (Cited page 9.) Nilson, T. (1971). A theoretical analysis of the frequency of gaps in plant stands. Agricultural Meteorology, 8:25–38. (Cited pages 82, 85, 92, and 100.) Nilson, T. (1999). Inversion of gap frequency data in forest stands. Agricultural and Forest Meteorology, 98-99:437–448. (Cited page 86.) Norman, J. M. and Jarvis, P. G. (1975). Photosynthesis in sitka spruce (picea sitchensis (bong.) carr.): V. radiation penetration theory and a test case. The Journal of Applied Ecology, 12(3):839–878. (Cited pages 82, 86, 89, and 92.) Norman, J. M. and Welles, J. M. (1983). Radiative transfer in an array of canopies. Agronomy Journal, 75:481–488. (Cited pages 82, 85, and 105.) Obert, M. (1993). Numerical estimates of the fractal dimension d and the lacunarity l by the mass radius relation. Fractals, 1:711–721. (Cited page 53.) Oker-Blom, P., Kaufmann, M. R., and Ryan, M. G. (1991). Performance of a canopy light interception model for conifer shoots, trees and stands. Tree Physiology, 9(1):227–243. (Cited page 87.) Oker-Blom, P. and Kellomaki, S. (1983). Effect of grouping of foliage on within-stand and within-crown light regimes: comparison of radiation and grouping canopy models. Agricultural Meteorology, 28:143–155. (Cited pages 86, 89, 92, 99, and 105.) Oker-Blom, P. and Smolander, H. (1988). The ratio of shoot silhouette area to total needle area in scots pine. Forest Science, 34(4):894–906. (Cited page 93.) Oppelt, A. L., Kurth, W., Dzierzon, H., Jentschke, G., and Godbold, D. L. (2000). Structure and fractal dimensions of root systems of four co-occurring fruit tree species from botswana. Annals of Forest Science, 57:463–475. (Cited page 40.) Oppenheimer, P. E. (1986). Real time design and animation of fractal plants and trees. SIGGRAPH Computer Graphics, 20(4):55–64. (Cited page 13.)
142
Bibliography
Pearcy, R. W. and Yang, W. (1996). A three-dimensional crown architecture model for assessment of light capture and carbon gain by understory plants. Oecologia, 108(1):1–12. (Cited page 73.) Pfister, H., Zwicker, W., Baar, J. V., and Gross, M. (2000). Surfels: surface elements as rendering primitives. In Siggraph’00, Computer Graphics Proceedings, pages 335–342, Los Angeles, CA, USA. ACM Press. (Cited page 41.) Phattaralerphong, J., Sathornkich, J., and Sinoquet, H. (2006). A photographic gap fraction method for estimating leaf area of isolated trees: Assessment with 3d digitized plants. Tree Physiol, 26(9):1123–1136. (Cited page 98.) Phattaralerphong, J. and Sinoquet, H. (2005). A method for 3d reconstruction of tree crown volume from photographs: assessment with 3d-digitized plants. Tree Physiol, 25(10):1229–1242. (Cited page 8.) Piboule, A., Collet, C., Frochot, H., and Dhôte, J. F. (2005). Reconstructing crown shape from stem diameter and tree position to supply light models. i. algorithms and comparison of light simulations. Annals of Forest Science, 62:645–657. (Cited page 123.) Pinty, B., Widlowski, J. L., Taberner, M., Gobron, N., Verstraete, M. M., Disney, M., Gascon, F., Gastellu, J. P., Jiang, L., Kuusk, A., Lewis, P., Li, X., Ni-Meister, W., Nilson, T., North, P., Qin, W., Su, L., Tang, S., Thompson, R., Verhoef, W., Wang, H., Wang, J., Yan, G., and Zang, H. (2004). Radiation transfer model intercomparison (rami) exercise: Results from the second phase. Journal of Geophysical Research, 109:D06210+. (Cited page 85.) Planchais, I. and Sinoquet, H. (1998). Foliage determinants of light interception in sunny and shaded branches of fagus sylvatica (l.). Agricultural and Forest Meteorology, pages 241–253. (Cited page 73.) Plotnick, R. E., Gardner, R. H., Hargrove, W. W., Prestegaard, K., and Perlmutter, M. (1995). Lacunarity analysis: A general technique for analysis of spatial patterns. Physical Review E, 53(5):5461–5468. (Cited pages 55 and 56.) Plotnick, R. E., Gardner, R. H., and O’Neill, R. V. (1993). Lacunarity indices as measures of landscape texture. Landscape Ecology, 8:201–211. (Cited pages 26, 51, 53, 55, and 56.) Polhemus, A. (1993). 3SPACE FASTRAK User’s Manual, Revision F. Kaiser Aerospace and Electronics Company, PO Box 560, Colchester, Vermont, 05446, USA. (Cited page 25.) Pradal, C., Boudon, F., Nouguier, C., Chopard, J., and Godin, C. (2008a). Plantgl : a python-based geometric library for 3d plant modelling at different scales. Graphical Models. (Cited pages 41, 109, 115, 117, and 126.) Pradal, C., Dufour-Kowalski, S., Boudon, F., Fournier, C., and Godin, C. (2008b). Openalea: avisual programming and component-based software platform for plant modelling. Functional Plant Biology, 35. (Cited page 126.)
Bibliography
143
Prusinkiewicz, P. (1998). Modeling of spatial structure and development of plants: a review. Scientia Horticulturae, 74(1-2):113–149. (Cited page 9.) Prusinkiewicz, P. (2004a). Modeling plant growth and development. Current Opinion in Plant Biology, 7(1):79–83. (Cited pages 13 and 94.) Prusinkiewicz, P. (2004b). Self-similarity in plants: Integrating mathematical and biological perspectives. In Thinking inPatterns. Fractals and Related Phenomena in Nature., pages 103–118. Novak, M. (Cited pages 22 and 23.) Prusinkiewicz, P. and Hanan, J. (1989). Lindenmayer systems, fractals, and plants. Lecture Notes in Biomathematics, 75. (Cited page 13.) Prusinkiewicz, P., James, M., and Mech, R. (1994). Synthetic topiary. In SIGGRAPH ’94: Proceedings of the 21st annual conference on Computer graphics and interactive techniques, pages 351–358, New York, NY, USA. ACM Press. (Cited pages 9 and 21.) Prusinkiewicz, P. and Kari, L. (1996). Subapical bracketed l-systems. In Graph Grammars and Their Application to Computer Science, volume 1073 of Lecture Notes in Computer Science, pages 550–564. Springer Berlin / Heidelberg. (Cited page 20.) Prusinkiewicz, P. and Lindenmayer, A. (1990). The algorithmic beauty of plants. Springer-Verlag, New York. (Cited pages 9, 10, 13, 20, 21, and 23.) Reeve, R. (1992). A warning about standard errors when estimating the fractal dimension. Comput. Geosci., 18(1):89–91. (Cited page 42.) Richardson, L. F. (1961). The problem of contiguity : an appendix of statistics of deadly quarrels. In General systems yearbook, volume 6, pages 139+. Statistic of Deadly Quarrels. (Cited pages 29 and 36.) Robinson, D. F. (1996). A symbolic framework for the description of tree architecture models. Botanical Journal of the Linnean Society, pages 243–261. (Cited page 3.) Room, P., Hanan, J., and Prusinkiewicz, P. (1996). Virtual plants: new perspectives for ecologists, pathologists and agricultural scientists. Trends in Plant Science, 1(1):33–38. (Cited page 13.) Room, P. M., Maillette, L., and Hanan, J. S. (1994). Module and metamer dynamics and virtual plants. In Begon, M. and Fitter, A. H., editors, Advances in Ecological Research, volume 25 of Advances in Ecological Research, pages 105–157. Elsevier Ltd. (Cited pages 5 and 12.) Ross, J. (1981). The radiation regim and the architecture of plant stands. The Hague, The Netherlands. (Cited pages 3, 70, 81, 82, 91, and 95.) Ross, J. K. and Marshak, A. L. (1988). Calculation of canopy bidirectional reflectance using the monte carlo method. Remote Sensing of Environment, 24(2):213–225. (Cited pages 73 and 82.) Sillion, F. X. (1995). Hierarchical solution techniques for realistic rendering. In State of the Art Report - Graphicon’95. (Cited page 8.)
144
Bibliography
Sillion, F. X. and Puech, C. (1994). Radiosity and Global Illumination. The Morgan Kaufmann Series in Computer Graphics. Morgan Kaufmann Inc., San Francisco, California, USA. (Cited pages 70, 71, 73, 74, 75, 76, and 78.) Sinoquet, H. and Andrieu, B. (1993). The geometrical structure of plant canopies: characterization and direct measurement methods. In Varlet-Grancher, C., Bonhomme, R., and Sinoquet, H., editors, Crop structure and light microclimate: Characterization and applications, Science Update. INRA, 147 rue de l’Université, 75338 Paris cedex 07, France. (Cited page 80.) Sinoquet, H. and Bonhomme, R. (1992). Modeling radiative transfer in mixed and row intercropping systems. Agricultural and Forest Meteorology, 62(3-4):219–240. (Cited pages 7 and 82.) Sinoquet, H., Le Roux, X., Adam, B., Ameglio, T., and Daudet, F. A. (2001). Ratp: a model for simulating the spatial distribution of radiation absorption, transpiration and photosynthesis within canopies: application to an isolated tree crown. Plant, Cell & Environment, 24(4):395–406. (Cited page 82.) Sinoquet, H., Phattaralerphong, J., Kasemsap, P., and Thanisawanyangkura, S. (2004). Light and carbon acquisition partitioning between flushes of two-year-old mango trees . In 4th Workshop on Functional-Structural Plant Models (FSPM04), pages 146– 149. CIRAD. (Cited page 100.) Sinoquet, H. and Rivet, P. (1997). Measurement and visualization of the architecture of an adult tree based on a three-dimensional digitising device. Trees - Structure and Function, 11(5):265–270. (Cited pages 25, 28, 91, 92, and 94.) Sinoquet, H., Rivet, P., and Godin, C. (1997). Assessment of the three-dimensional architecture of walnut trees using digitising. Silva Fennica, 31:265–273. (Cited page 25.) Sinoquet, H., Sonohat, G., Phattaralerphong, J., and Godin, C. (2005). Foliage randomness and light interception in 3-d digitized trees: an analysis from multiscale discretization of the canopy. Plant, Cell & Environment, 28(9):1158–1170. (Cited pages 85, 92, and 97.) Sinoquet, H., Stephan, J., Sonohat, G., Lauri, P. E., and Monney, P. (2007). Simple equations to estimate light interception by isolated trees from canopy structure features: assessment with three-dimensional digitized apple trees. New Phytologist, 175(1):94–106. (Cited pages 86, 92, 93, and 100.) Sinoquet, H., Thanisawanyangkura, S., Mabrouk, H., and Kasemsap, P. (1998). Characterization of the light environment in canopies using 3d digitising and image processing. Ann Bot, 82(2):203–212. (Cited pages 8, 27, and 91.) Sinoquet, H., Varlet-Grancher, C., and Bonhomme, R. (1993). Modelling radiative transfer within homogeneous canopies: basic concepts. In Varlet-Grancher, C., Bonhomme, R., and Sinoquet, H., editors, Crop structure and light microclimate: Characterization and applications, Science Update. INRA, 147 rue de l’Université, 75338 Paris cedex 07, France. (Cited pages 79, 82, and 83.)
Bibliography
145
Smith, A. R. (1984). Plants, fractals, and formal languages. In SIGGRAPH ’84: Proceedings of the 11th annual conference on Computer graphics and interactive techniques, volume 18, pages 1–10. ACM Press. (Cited page 13.) Smith, H. and Whitelam, G. C. (1997). The shade avoidance syndrome: multiple responses mediated by multiple phytochromes. Plant, Cell and Environment, 20(6):840– 844. (Cited page 69.) Soler, C. (1998). Représentations hiérarchiques de la visibilité pour le contrôle de l’erreur en simulation de l’éclairage. PhD thesis, Université Joseph Fourier. (Cited page 78.) Soler, C., Sillion, F., Blaise, F., and de Reffye, P. (2003). An efficient instantiation algorithm for simulating radiant energy transfer in plant models. ACM Transactions On Graphics, 22(2). (Cited pages 78, 79, and 130.) Sonohat, G., Balandier, P., and Ruchaud, F. (2004). Predicting solar radiation transmittance in the understory of even-aged coniferous stands in temperate forests. Annals of Forest Science, 61:629–641. (Cited page 113.) Sonohat, G., Sinoquet, H., Kulandaivelu, V., Combes, D., and Lescourret, F. (2006). Three-dimensional reconstruction of partially 3d-digitized peach tree canopies. Tree Physiology, 26:337–351. (Cited pages 25, 27, 28, 105, and 106.) Southon, T. E. and Jones, R. A. (1992). Nmr imaging of roots: methods for reducing the soil signal and for obtaining a 3‐dimensional description of the roots. Physiologia Plantarum, 86(2):322–328. (Cited page 25.) Stenberg, P. (1996). Simulations of the effects of shoot structure and orientation on vertical gradients in intercepted light by conifer canopies. Tree Physiology, 16(1):99– 108. (Cited pages 85 and 86.) Stenberg, P. (2006). A note on the g-function for needle leaf canopies. Agricultural and Forest Meteorology, 136(1-2):76–79. (Cited page 81.) Sugihara, G. and May (1990). Applications of fractals in ecology. Trends in Ecology & Evolution, 5(3):79–86. (Cited page 37.) Takenaka, A. (1994). A simulation model of tree architecture development based on growth response to local light environment. Journal of Plant Research, 107(3):321–330. (Cited page 3.) Takenaka, A., Inui, Y., and Osawa, A. (1998). Measurement of three-dimensional structure of plants with a simple device and estimation of light capture of individual leaves. Functional Ecology, 12(1):159–165. (Cited page 24.) Thanisawanyangkura, S., Sinoquet, H., Rivet, P., Cretenet, M., and Jallas, E. (1997). Leaf orientation and sunlit leaf area distribution in cotton. Agricultural and Forest Meteorology, 86(1-2):1–15. (Cited page 106.) Tolle, C. R., Mcjunkin, T. R., Rohrbaugh, D. T., and Laviolette, R. A. (2003). Lacunarity definition for ramified data sets based on optimal cover. PhysicaD, 179:129–152. (Cited page 53.)
146
Bibliography
Tricot, C. (1993). Courbes et dimension fractale. Springer Verlag and Editions Sciences et Culture. (Cited pages 31 and 37.) Urban, L., Le Roux, X., Sinoquet, H., Jaffuel, S., and Jannoyer, M. (2003). A biochemical model of photosynthesis for mango leaves: evidence for the effect of fruit on photosynthetic capacity of nearby leaves. Tree Physiology, 23:289–300. (Cited page 27.) Valentine, H. T. (1985). Tree-growth models: Derivations employing the pipe-model theory. Journal of Theoretical Biology, 117(4):579–585. (Cited page 9.) Varlet-Grancher, C., Bonhomme, R., and Sinoquet, H., editors (1993). Crop structure and light microclimate: Characterization and applications. Science Update. INRA, 147 rue de l’Université, 75338 Paris cedex 07, France. (Cited page 69.) Verhoef, W. (1984). Light scattering by leaf layers with application to canopy reflectance modeling: The sail model. Remote Sensing of Environment, 16(2):125–141. (Cited page 82.) Villalobos, F. J., Orgaz, F., and Mateos, L. (1995). Non-destructive measurement of leaf area in olive (olea europaea l.) trees using a gap inversion method. Agricultural and Forest Meteorology, 73(1-2):29–42. (Cited page 98.) Walker, D. J. and Kenkel, N. C. (1993). Landscape complexity in space and time. Community Ecology, 2(1):109–119. (Cited page 37.) Walklate, P. J. (1989). A laser scanning instrument for measuring crop geometry. Agricultural and Forest Meteorology, 46(4):275–284. (Cited page 25.) Wang, Y. and Jarvis, P. (1990). Description and validation of an array model — maestro. Agricultural and Forest Meteorology, 51(3-4):257–280. (Cited page 86.) Weber, J. and Penn, J. (1995). Creation and rendering of realistic trees. In SIGGRAPH ’95: Proceedings of the 22nd annual conference on Computer graphics and interactive techniques, Annual Conference Series, pages 119–128, New York, NY, USA. SIGGRAPH, ACM. (Cited page 9.) White, J. (1979). The plant as a metapopulation. Annual Review of Ecology and Systematics, 10(1):109–145. (Cited page 5.) Whitehead, D., Grace, J. C., and Godfrey, M. J. (1990). Architectural distribution of foliage in individual pinus radiata d. don crowns and the effects of clumping on radiation interception. Tree Physiol, 7(1 2 3 4):135–155. (Cited pages 92 and 101.) Whitted, T. (1980). An improved illumination model for shaded display. Commun. ACM, 23(6):343–349. (Cited page 74.) Wiens, J. A. (1989). Spatial scaling in ecology. Functional Ecology, 3(4):385–398. (Cited page 37.) Willaume, M., Lauri, P. E., and Sinoquet, H. (2004). Light interception in apple trees influenced by canopy architecture manipulation. Trees, 18(6):705–713. (Cited page 101.)
Bibliography
147
Zeide, B. (1998). Fractal analysis of foliage distribution in loblolly pine crowns. Canadian Journal of Forest Research, 28(1):106–114. (Cited page 29.) Zeide, B. and Pfeifer, P. (1991). A method for estimation of fractal dimension of tree crowns. Forest Science, 37(5):1253–1265. (Cited pages 38 and 39.)
A
Papers and communication
A.1
International Symposium on Visual Computing’06 2nd International Symposium on Visual Computing Lake Tahoe, Nevada, USA November 6-8, 2006 Published in Advances in Visual Computing . Lecture Notes in Computer Science, 4291-4292 (2006), pp. 751-760.
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A Critical Appraisal of the Box Counting Method to Assess the Fractal Dimension of Tree Crowns D. Da Silva1,5 , F. Boudon2,5 , C. Godin2,3,5 , O. Puech4,5 , C. Smith4,5 , and H. Sinoquet6 1
5
Universit´e de Montpellier II 2 CIRAD 3 INRIA 4 INRA Virtual Plants Team, UMR AMAP TA/40E 34398 Montpellier Cedex 5, France 6 INRA-UBP, UMR PIAF, Domaine de Crouelle, 234 avenue du Br´ezet 63100 Clermont-Ferrand, France
Abstract. In this paper, we study the application of the box counting method (BCM) to estimate the fractal dimension of 3D plant foliage. We use artificial crowns with known theoretical fractal dimension to characterize the accuracy of the BCM and we extend the approach to 3D digitized plants. In particular, errors are experimentally characterized for the estimated values of the fractal dimension. Results show that, with careful protocols, the estimated values are quite accurate. Several limits of the BCM are also analyzed in this context. This analysis is used to introduce a new estimator, derived from the BCM estimator, whose behavior is characterized.
1
Introduction
Plant geometry is a key factor for modeling eco-physiological interaction of plant and the environment. These interactions may concern either the abiotic (resource capture, heat dissipation) or the biotic (disease propagation, insect movement) environment. Depending on applications, plant geometry has been abstracted in various ways [1] : simple volumic shapes (like ellipsoids, cones, or big leaves used in turbid medium approaches) or detailed models to render realistic trees. Global descriptions are simple and contain few parameters; however, they do not capture the irregular nature of plant shapes which severely limits the generalization capacity of the model. On the other hand, detailed descriptions tentatively address this problem but require over-parameterization of geometry, leading to non-parsimonious models. Characterizing the irregularity of plant shapes with a few parameters is thus a challenging problem. ractal geometry was introduced as a new conceptual framework to analyze and model the irregular nature of irregular shapes [2]. This framework has been applied in different occasions to the modeling of plant structure. Generative approaches use fractal concepts to illustrate how intricate vegetal-like structures can be generated using parsimonious models [3,4,5]. Such models were used to generate artificial plants in modeling applications [6,7]. Fractal geometry was G. Bebis et al. (Eds.): ISVC 2006, LNCS 4291, pp. 751–760, 2006. c Springer-Verlag Berlin Heidelberg 2006
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also used to analyze the irregularity of plants by determining their supposed fractal dimension. This parameter is of major importance in the study of irregularity: it characterizes the way plants physically occupy space. Most of these studies were carried out using the classical box counting method (BCM) [2] on woody structures, and especially on root systems [8,9,10]. This method consists of immersing the studied object in a grid with uniform cell size and studying the variation of the number of grid cells intercepted by the plant as the size of the cells decreases. For practical reasons, in most works, fractal dimension is estimated from 2D photographs [11,12]. Unfortunately, such a technique always under-estimates the actual fractal dimension [13], and so is not accurate. Recently BCM was used on 3D digitized root systems [10]; however, the accuracy of the estimated values could not be evaluated. In this paper, we study the application of the BCM to both artificial and real 3D plant foliage. We use artificial crowns with known theoretical fractal dimensions to characterize the accuracy of BCM and we extend the approach to 3D digitized plants. The limits of BCM is then analyzed and discussed in this context.
2
Plant Databases
Nine 3D plants were included in the study. Four real trees were digitized in the field and five additional plants were generated from theoretical models. The geometric scenes representing the plant crowns were designed using the PlantGL library [14]. Digitized Plants. Four four-year old Prunus Persica (peach) trees were digitized [15], but due to the high number of leaves (∼14,000), digitizing at leaf scale was impossible. A magnetic digitizing device was therefore used to record the spatial co-ordinates of the bottom and top of each leafy shoot. In addition, thirty shoots were digitized at leaf scale in order to derive the leaf angle distribution, and allometric relationships between number of leaves, shoot leaf area and shoot length. Leaves of each shoot were then generated from those data and additional assumptions for the internode length and the distribution of leaf size within a shoot. Theoretical Plants. Three fractal plants were generated from 3D iterated function systems (IFS) [4]. The generation process is illustrated in Fig. 2, and the finals artificial canopies are represented in Fig. 3. If the IFS satisfies the open set condition [16], the theoretical fractal dimension of the IFS attractor is the autosimilarity dimension, log n . (1) Da = log c A classical 3D cantor dust [2] was also generated using an IFS (n = 8, c = 3). Each IFS was developed over 5 iterations. In addition to these self-similar plants
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Fig. 1. Four four-year old peach trees (cv. August Red) were digitized in May 2001 in CTIFL Center, Nˆımes, South of France, at current-year shoot scale, one month after bud break
Fig. 2. Construction of an artificial crown. The initial object was a tapered ellipsoid and the IFS transformation was made of n = 5 duplications of a contracted object by a factor c = 3.
a stochastic 3D cantor dust was generated using a recursive algorithm derived from the method known as curdling and random trema generation [2,17]. Each iteration of the algorithm divides a given voxel into a set of subvoxels according to a specified subdivision factor. A fixed proportion of voxels eligible for the next iteration is chosen randomly from the subvoxels. At the end of the process, final voxels are considered to be leaves. The stochastic cantor dust is created by 8 as the proportion of chosen voxels specifying a subdivision factor of 3 and 27 for all 5 iteration levels. This object has the same theoretical dimension as the classical cantor dust.
3 3.1
Estimation of the Fractal Dimension Using the BCM The Box Counting Method
The BCM has been extensively used to estimate fractal dimension of objects embedded in the plane. Its adaptation to 3D consists of building a sequence of 3D grids dividing space in homogeneous voxels of decreasing size δ and counting
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Fig. 3. From left to right, the three artificial canopies : AC1 (n = 5, c = 3), AC2 (n = 7, c = 3), AC3 (n = 9, c = 3), on the top, the cantor dust and on the bottom a stochastic cantor dust
the number Nδ of grid voxels intercepted by the studied object. The estimator of the fractal dimension of the object is defined as log Nδ . δ→0 log 1 δ
Db = lim
(2)
To implement this estimator, we approximated all the geometric objects by triangular meshes. The intersection of each triangle with the grid voxels can then be computed in time proportional to the number of triangles in the mesh [18]. However, to decrease the overall complexity, we represent each triangle by a set of points [19]. The number of points used is chosen such as the distance between two points is small compared to the minimal voxel size. The intersection algorithm is thus reduced to checking whether a voxel contains at least one point. The grid sequence is obtained by dividing the original bounding-box size, δ0 , by a range of consecutive integers acting as subdivision factors. Thus the series of δn is a decreasing series formed by { Sδ0i }0≤i
Box Counting Method: Local Scale Variation Estimator
As pointed out in [22], a major problem of the BCM estimator is that the numbers of intercepted voxels at each scale are correlated positively, and the correlation structure is completely ignored in the estimation procedure. This violates the assumption of data independency used in regression analysis. The consequence is an underestimation of confidence interval associated with the estimated fractal dimension. To eliminate the correlation, we introduce a new estimator, namely local
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scale variation estimator (LSV), based on the relative increase of intercepted voxels against the relative decrease in scale. This estimator can be derived from the BCM estimator as follows. Assuming the power law is verified for each scale δ 1 Nδ ∝ ( )Db , δ
(3)
the differential form of this equation leads to 1 d log Nδ ∝ d(Db log( )), δ dδ dNδ ∝ −Db Nδ δ
(4)
which gives a variational interpretation of the fractal dimension. Db thus expresses the linear coefficient that corresponds to the ratio of new details due to a certain ratio of zoom in the structure. However, in this equation it is assumed that both dN and dδ 0, which is not usually the case for the scales used in BCM, except at very small scales. It is possible to generalize this variational principle to non-infinitely small quantities. Let Nδ be the number of intercepted voxels at scale δ. We define ΔNδ,Δδ as ΔNδ,Δδ = Nδ+Δδ − Nδ .
(5)
= ΔNδ,Δδ . Similarly, The relative increase in the number of boxes is denoted N Nδ Δδ we denote δ = δ the relative increase of zoom when passing from cell size δ to δ + Δδ. Thus, assuming Equation 3 is still satisfied, we have −Db −Db − δ −Db ∝ (δ + Δδ) N = 1 + δ − 1, δ −Db
(6)
which leads to a generalized form of Equation 4, where variations of Nδ and δ need not be infinitely small, ) ∝ −Db log(1 + δ). log(1 + N
(7)
) Db can thus be estimated by performing a linear regression between log(1 + N and log(1 + δ).
4 4.1
Results Number of Voxels as a Function of Scale
In general, we may expect that the number of intercepted voxels is a monotonously increasing function of scale. However this is not always the case due to a quantization effect which results from discrepancy between discretization with the 3D grid and space occupation of the plant at some scales. Fig. 4 contains plots of the
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number of voxels intercepted at the different scales for each object. The local variation of the curves comes from the fact that the number of intercepted voxels at one scale depends of the adjustment of the grid. Some shiftings, up to a factor δ in each direction, and reorientations of the grid may lead to overestimating the number of voxels at one scale, causing local variation of the curve. Thus, the discrete quantization of the 3D shape of the object into voxels introduces some fuzziness in its representation, depending on scale. It can be seen in Fig. 4 that the quantization effect is far more pronounced with the artificial crowns and Cantor dusts than the digitized peach trees. This difference is attributed to the less deterministically distributed foliage of the digitized trees.
Fig. 4. The number of intercepted voxels as a function of the scale
4.2
Estimating Fractal Dimension from the BCM
Scale Range. When the grid voxel size is smaller than the leaf size, the evaluation of the dimension is modified by the dimension of the leaves surfaces. √ To avoid this effect, a minimum voxel size, δmin , is determined such as δmin ≥ Al , where Al is the mean leaf area. Since every voxel size δi is obtained from the bounding box size δ0 δ0 as δi = Sδ0i , the minimum size must be δmin = Smax . Let Vbb be the bounding box volume. An uni-dimensional proportionality factor is defined by √ 3 Vbb (8) Smax = √ . Al Setting Smax as the upper bound for the subdivision factors {Si }0≤i
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the possible shifting configurations of the grid may cause the leaf also be included in any of the twenty-six neighboring voxels. Considering voxels of bigger sizes with a factor 3 can be seen as including the twenty-seven possible small voxels into the same large one and so limits the errors found in finer grids. Of course, the optimal grid for one leaf will not be the optimal grid for all leaves; therefore, artifact effects of grid adjustment may persist. We experimentally observed that this persistence is limited (see Fig. 5).
Fig. 5. Evolution of AC2 slope during BCM evaluation. The number of voxels inter , the cepted at various scales for AC2 with the slopes highlighted. In the range 0, Smax 3 slope is primarily influenced by the structure of AC2 and the fractal dimension Db = , S , the slope is also partially influenced by the fractal 1.765. In the range Smax max 3 dimension of individual leaves and is sensible to local variation due to grid adjustment. When this range is taken into account for the fractal dimension evaluation, Db drops from 1.765 to 1.584. Finally for grids with voxel sizes smaller than Smax , the slope is directly related to individual leaf fractal dimension (0 in our representation since we use points). With a naive range of evaluation including all points, the fractal dimension drops to 1.172.
Orientation of the Grid. Optimal voxel coverage of the plant depends on the orientation of the grid relative to the plant. For this, we made a sensitivity analysis to evaluate how the estimated fractal dimension is affected by changes in the grid’s orientation. A set of random grid orientations were selected and fractal dimension was estimated for each orientation. Table 1 gives the mean and variance of the estimated fractal dimension across orientations for all the considered plants. We can observe a low variability in the absolute values of the results: the standard deviations are inferior to one per cent of the mean values. From this, we conclude that the orientation of the grid has only limited effect on the BCM evaluation method. Error Characterization. To characterize the error made during the estimation, a comparison with theoretical fractal dimension can be used. In the case of plants corresponding to IFS attractors, the theoretical fractal dimension, D, is known. But there is no such dimension for real plants; however, it has been shown that,
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when plant’s topology is known, a faithful estimate of the plant fractal dimension can be obtained using the two-surface method [23]. This value will be used as reference value for the peach trees. A classical Student’s t-test on the computed Db distributions shows that a significant bias in the BCM estimation exists. However, results reported in Table 1 (cols 3-6) show that this bias is less than 3.1% of the theoretical value for the studied canopies. Table 1. Fractal dimension results for studied canopies and their properties. Da is the reference (theoretical) value of the fractal dimension. For Db estimation, Db gives the mean estimated value and σ the standard deviation over all considered rotations. The minimum standard error r 2 over all rotations is shown. All results are obtained with Smax as the upper limit. 3 Canopy AC1 AC2 AC3 Cantor Stoc. Cantor Peach 1 Peach 2 Peach 3 Peach 4
4.3
Da 1.47 1.77 2 1.89 1.89 2.33 2.36 2.38 2.33
BCM Db Db σ 1.4889 0.0056 1.7305 0.0053 1.97 0.0074 1.8835 0.0174 1.8896 0.0105 2.3221 0.0043 2.3516 0.0056 2.307 0.0064 2.3218 0.0076
r2 0.97 0.99 0.99 0.94 0.97 0.99 0.99 0.99 0.99
Relative Bias 0.0128 0.0223 0.015 0.0034 0.0002 0.0033 0.0035 0.0306 0.0035
LSV Db Db σ 1.8761 0.0457 1.9409 0.06 2.0705 0.0534 2.2286 0.0852 2.1218 0.0933 2.2832 0.0115 2.3416 0.0117 2.3022 0.0195 2.3147 0.0175
r2 0.33 0.58 0.74 0.09 0.17 0.97 0.97 0.97 0.98
√ 3
Vbb 1.83 2.29 1.85 0.99 2.43 2.97 2.97 3.04 2.61
√
Al 0.0143 0.0143 0.0143 0.0041 0.01 0.439 0.459 0.0463 0.0449
Smax 128 160 129 243 243 67 64 65 72
Estimating Fractal Dimension from the LSV Method
We use the LSV estimator of the box counting method, presented on section 3.2, on the theoretical and digitized plants. The δ values were defined using couple of successive scales δi+1 − δi 1 δ = = (9) δi δi values from the corresponding N values. Since it is based on a local estimaand N tion, it is sensible to the local variation of the number of box as a function of scales introduced by the quantization effect. The local variations in this estimation are reflected in the variance and standard error of the computed fractal dimensions, giving a better estimation of the reliability of the results compared to the classical box counting method. Experimentally, we observe that results on theoretical plants are very sensitive to quantization effect as shown by dispersion of the data in the Fig. 6 and the minimum standard error in Table 1 (cols 7-9). The minimum r2 for the estimated dimensions on these objects are between 0.09 to 0.74. This effect is much less important on real plants; the minimum r2 values are between 0.97 and 0.98. In this case, the results seems more relevant. The difference with theoretical values is small (less than 3.2%).
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Fig. 6. Estimated fractal dimension with the LSV method for AC2, Cantor Dust and Peach 2. This new estimator is very sensitive to quantization effect leading to a dispersion of the measurements in AC2 and Cantor Dust. On the contrary the method gives an estimation of D close to that obtained with the two-surface method (i.e D = 2.36) for Peach 2 tree.
5
Conclusion
In this paper the accuracy of the BCM for evaluating the fractal dimension of 3D crowns was studied. Several factors that may influence this accuracy were examined and practical solutions proposed. In particular a proper voxel size limit is determined dependent on leaf sizes and the BCM bias was quantified. The problem of data dependency used during the regression analysis was discussed and a new estimator, LSV, that does not violate the independence assumption is described. The LSV estimator appears to be an interesting indicator to determine whether the quantization effect disturb the fractal dimension estimation. Eventually it has to be improved to support more robust evaluations.
References 1. Godin, C.: Representing and encoding plant architecture: A review. Annals of Forest Science 57 (2000) 413–438 2. Mandelbrot, B.B.: The fractal geometry of nature. Freeman (1983) 3. Smith, A.R.: Plants, fractals, and formal languages. In: Siggraph’84, Computer Graphics Proceedings. Volume 18., ACM Press (1984) 1–10 4. Barnsley, M.: Fractals Everywhere. Academic Press, Boston (1988) 5. Prusinkiewicz, P., Hanan, J.: Lindenmayer systems, fractals, and plants. Lecture Notes in Biomathematics 75 (1989) 6. Chen, S., Ceulemans, R., Impens, I.: A fractal-based Populus canopy structure model for the calculation of light interception. Forest Ecology and Management 69(1-3) (1994) 97–110 7. Prusinkiewicz, P., Mundermann, L., Karwowski, R., Lane, B.: The use of positional information in the modeling of plants. In: Siggraph’01, Computer Graphics Proceedings, New York, NY, USA, ACM Press (2001) 289–300 8. Fitter, A.H.: An architectural approach to the comparative ecology of plant root systems. New Phytologist 106(1) (1987) 61–77
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9. Eshel, A.: On the fractal dimensions of a root system. Plant, Cell & Environment 21(2) (1998) 247+ 10. Oppelt, A.L., Kurth, W., Dzierzon, H., Jentschke, G., Godbold, D.L.: Structure and fractal dimensions of root systems of four co-occurring fruit tree species from Botswana. Annals of Forest Science 57 (2000) 463–475 11. Morse, D.R., Lawton, J.H., Dodson, M.M., Williamson, M.H.: Fractal dimension of vegetation and the distribution of arthropod body lengths. Nature 314(6013) (1985) 731–733 12. Critten, D.L.: Fractal dimension relationships and values associated with certain plant canopies. Journal of Agricultural Engineering Research 67(1) (1997) 61–72 13. Falconer, K.: Fractal geometry : mathematical foundation and applications. John Wiley and Sons (1990) 14. Boudon, F., Pradal, C., Nouguier, C., Godin, C.: Geom module manual: I user guide. Technical Report 3, CIRAD (2001) 15. Sonohat, G., Sinoquet, H., Kulandaivelu, V., Combes, D., Lescourret, F.: Threedimensional reconstruction of partially 3d-digitized peach tree canopies. Tree Physiol 26(3) (2006) 337–351 16. Falconer, K.: Techniques in fractal geometry. John Wiley and Sons (1997) 17. Plotnick, R.E., Gardner, R.H., O’Neill, R.V.: Lacunarity indices as measures of landscape texture. Landscape Ecology 8 (1993) 201–211 18. Andres, E., Nehlig, P., Franon, J.: Supercover of straight lines, planes and triangles. In: Proceedings of DGCI ’97, London, UK, Springer-Verlag (1997) 243–254 19. Pfister, H., Zwicker, W., Baar, J.v., Gross, M.: Surfels: surface elements as rendering primitives. In: Siggraph’00, Computer Graphics Proceedings, Los angeles, ACM Press (2000) 335–342 20. Foroutan-Pour, K., Dutilleul, P., Smith, D.L.: Advances in the implementation of the box-counting method of fractal dimension estimation. Applied Mathematics and Computation 105(2) (1999) 195–210 21. Halley, J.M., Hartley, S., Kallimanis, A.S., Kunin, W.E., Lennon, J.J., Sgardelis, S.P.: Uses and abuses of fractal methodology in ecology. Ecology Letters 7 (2004) 254–271 22. Reeve, R.: A warning about standard errors when estimating the fractal dimension. Comput. Geosci. 18(1) (1992) 89–91 23. Boudon, F., Godin, C., Pradal, P., Puech, O., Sinoquet, H.: Estimating the fractal dimension of plants using the two-surface method. an analysis based on 3d-digitized tree foliage. Fractals 14(3) (2006)
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Appendix A. Papers and communication
Functional-Structural Plant Models 5th International Workshop on Functional-Structural Plant Models Napier, New Zealand November 4-9, 2007
Modeling of light transmission under heterogeneous forest canopy: model description and validation
David Da SILVA‡ , Philippe BALANDIER1,† , Frédéric BOUDON2,‡ , André MARQUIER3,† , Christophe PRADAL2,‡ , Christophe GODIN4,‡ , Hervé SINOQUET3,† 1
CEMAGREF, Domaine des Barres, 45290 Nogent-sur-Vernisson, France
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3
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‡ †
CIRAD, Avenue Agropolis, 34398 Montpellier Cedex 5, France
INRA, Site de Crouël, 234 avenue du Brézet, 63100 Clermont-Ferrand, France INRIA, 2004 route des lucioles BP 93, 06902 Sophia Antipolis, France
Virtual Plants, UMR DAP, TA A-96/02, 34398 Montpellier Cedex 5, France
UMR PIAF, Site de Crouël, 234 avenue du Brézet, 63100 Clermont-Ferrand, France
Keywords : light transmission, heterogeneous canopy, foliage distribution, 3D reconstruction
Introduction Growth and survival of regeneration saplings and understorey vegetation development is closely related to light available below the forest trees. Manipulating the forest structure by thinning adult trees is a major tool to control light transmission to the understorey. The transmission is related to the attenuation of light which is usually estimated with the Beer-Lambert law assuming homogeneous foliage within the canopy. However forest canopies are far from homogeneous, which requires models that can take into account the eect of clumping between and within trees. In this work we present a model that can be readily used with both coarse or detailed parameterization to generate any type of stand and compute the distribution of light transmitted below the canopy. To evaluate the accuracy of the model, we compared model results with eld measurement from several stands of Pinus sylvestris L. in the French Massif Central.
Material The pine stands are located in the Chaîne des Puys, a mid-elevation volcanic mountain range (45◦ 42' N, 2◦ 58'E) at a place named Fontfreyde. The elevation is 900 m a.s.l., mean annual rainfall is about 820 mm, and mean annual temperature is about 7◦ C. The soil is a volcanic brown soil at pH 6.0 with no mineral de ciency. The pines were 30-year-old at time of measurement, with a density ranging from 500 to 4000 stem ha−1 . All trees in an area of 30 by 30 m were located by their x,y coordinates, and measured for their total height (14.1 ± 2 m mean ± SD) and DBH (16.3 ± 5 cm mean ± SD). Crown height was also measured. Crown extents was assessed by visually projecting to the soil its characteristic points (i.e. the points that better describe the crown irregularities) in, at least, four directions. The azimuth and distance of those points from trunk were then measured (see Figure 1).
Methods
Envelope reconstruction To reconstruct the 3D envelopes of the trees from the eld measurements, we used the PlantGL library [Boudon et al., 2001]. This library contains several geometric models, including dierent types of envelopes and algorithm to reconstruct the geometry of plants at dierent scales. For this particular case, we used the skinned surface implemented in PlantGL which is a generalization of surface of revolution with varying pro les being interpolated. This surface is thus built from any number of pro les with associated direction. The pro les we de ned were inspired by Cescatti work [Cescatti, 1997]. A pro le is supposed to pass through top and bottom points and an intermediate point at maximum radius. Two shape factors, CT and CB , are used to describe the shape of the pro les above and below the maximum width. Mathematically, two quarters of super-ellipse of degree CT and CB are used to de ne the top and bottom part of the pro les. Isopoints of the pro les are interpolated with B-Spline curves of given degree. Note that our envelopes can be viewed as extension of Cescatti 's asymmetric hull with pro les in any direction instead of restricted directions (cardinal directions). Flexibility of our model enables us to measure the most adequate pro les in case of irregular crowns.
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Figure 1: From left to right : Field data sample. Each circle locates a tree and each connected arrow de nes a speci c azimuth and distance from the tree trunk. Field data along with their matching projected crown. 3D reconstruction using skinned surface hulls from PlantGL library with maximum radius at the quarter of crown height and CT = CB = 2.
Light Transmission Model The presented model derives from Oker-Blom's model [Oker-Blom et al., 1989] where crowns are considered as porous envelopes but extends it to the more complex ones described above (p.1). Light interception by a crown C is related to the crown projection area, SΩ (C), in the sun direction Ω [Nilson, 1999]. In the case of an isolated tree, let E(C) be a convex hull of C , we will use the projected area of E(C) as the area of interest. Hence, the fraction of light intercepted by C , pΩ (C), can be expressed as the ratio between SΩ (C) and the area of interest, SΩ (E(C))
SΩ (C) = SΩ (E(C))pΩ (C)
(1)
pΩ (C) acting as an opacity factor for the envelope of C . In a crown, leaves can either be uniformly distributed or assigned to speci c spatial positions. In the case of uniform distribution the light attenuation is a function of the distance the solar beam travels within the crown. In the other case, the attenuation depends on the relative position of the beam and the leaf. Hence we discretize the volume E(C) using a set B of β parallelepipedic voxels of direction Ω representing light beams. The set of beams is large enough so that the discretization does not change volume or projected area. Let pbΩ (C) be the probability for the beam b to be intercepted by C (i.e. the opacity of C for the beam b), therefore the probability that b is not intercepted by C , 1 − pbΩ (C), is a function of the number of leaves in C and can be expressed using Beer-Lambert law or a binomial law if leaf size is to be taken into account : pbΩ (C) = 1 −
Y 1 − pbΩ (`)
(2)
`∈L(C)
where ` is a leaf, L(C) the set of leaves of C . The probability, pbΩ (`), for b to be intercepted by the leaf ` is known when spatial positions of leaves are taken into account; in the uniform distribution case, it can be shown that pbΩ (C) can be expressed as a function of projected leaf area, SΩ (`), volume of crown, V (C), and the distance b travels within the crown, lb (C) [Sinoquet et al., 2005]: Y SΩ (`)lb (C) b pΩ (C) = 1 − (3) 1− V (C) `∈L(C)
Hence we can compute the light transmission for each beam in both cases, when position of leaves is known and when we assume an homogeneous distribution, nally global opacity for C is simply the mean of beam opacity. Since Eq.(2) and (3) do not depend on scale, leaf scale is not mandatory and there is no restriction concerning the number of scales being used. This forest stand application illustrates the multi-scale approach with two scales, the nest one being the tree crown scale, and also illustrates how to take into account the nite size of a stand. Therefore, in this speci c case and using above notations, L(C) is 2
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the set of tree crowns in C and pbΩ (`) is the opacity of the crown ` for the beam b that can be either computed using Eq.(2) or (3) recursively if informations on nner scale is available or set with empirical or measured value.
Results Using this model, we computed the light attenuation for each cast beam and therefore generated a shadow map for dierent directions Ω; in this study we use the 46 directions sky discretization proposed in [Den Dulk, 1989]. Computations were done for the stand reconstruction (Fig.1), uniform foliage distribution hypothesis, and random positioning of reconstructed crowns in forest space. The results (Fig.2)
a.
b.
c.
Figure 2: Top view shadow maps with associated grey level histogram of: a. stand reconstruction, b. uniform foliage hypothesis and c. one example of random distribution of reconstructed crowns, in this case the histogram shows the mean light transmitance over a set of 20 random distributions. Azimuth is 180◦ and elevation is 90◦ for all views. Each tree crown has an overall direction opacity of 0.25. clearly show that if we are interested in light distribution in transmission classes, the hypothesis of uniform foliage distribution does not hold and thus cannot be used to model stands (Fig.2 b.). The distribution of light transmitted by stands generated using a random spatial ditribution of the reconstructed crowns (Fig.2 c.) was much closer to the distribution found with the real stand ( g.2 a.). However, the higher gap frequency in the random stand (see transmission class of 100 %) suggests that the tree distribution in real stands is slightly more regular than random. Further investigations on spatial distribution characterisation are being done in order to obtain more simple way of recreating stands with better light interception properties.
Concluding remarks and perspectives The integration of all those directional maps onto a ground projection yields a global shadow map that will allow us to study light intensity distribution over dierent time period and quantify the impact of clumping between and within trees. A light mesurement campaign is being done in order to obtain data for validation purpose. This better characterization and understanding of light transmission will hopefully lead us toward simpli ed and e cient models.
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References
[Boudon et al., 2001] Boudon, F., Pradal, C., Nouguier, C., and Godin, C. (2001). Geom module manual: I user guide. Technical Report 3. [Cescatti, 1997] Cescatti, A. (1997). Modelling the radiative transfer in discontinuous canopies of asymmetric crowns. ii. model testing and application in a norway spruce stand. Ecological Modelling, 101(2):275284. [Den Dulk, 1989] Den Dulk, J. A. (1989). thesis, Wageningen university.
The interpretation of Remote Sensing, a feasibility study. PhD
[Nilson, 1999] Nilson, T. (1999). Inversion of gap frequency data in forest stands. Meteorology, 98-99:437448.
Agricultural and Forest
[Oker-Blom et al., 1989] Oker-Blom, P., Pukkala, T., and Kuuluvainen, T. (1989). Relationship between radiation interception and photosynthesis in forest canopies: Eects of stand structure and latitude. Ecological Modelling, 9(1-2):7387. [Sinoquet et al., 2005] Sinoquet, H., Sonohat, G., Phattaralerphong, J., and Godin, C. (2005). Foliage randomness and light interception in 3-d digitized trees: an analysis from multiscale discretization of the canopy. Plant, Cell & Environment, 28(9):11581170.
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Multiscale framework for modeling and analyzing light interception by trees Published in a SIAM journal: Multiscale Modeling and Simulations, 7 (2) (2008), pp 910-933.
MULTISCALE MODEL. SIMUL. Vol. 7, No. 2, pp. 910–933
c 2008 Society for Industrial and Applied Mathematics
MULTISCALE FRAMEWORK FOR MODELING AND ANALYZING LIGHT INTERCEPTION BY TREES∗ ´ ERIC ´ DAVID DA SILVA† , FRED BOUDON† , CHRISTOPHE GODIN‡ , AND ´ SINOQUET§ HERVE Abstract. This paper presents a new framework for modeling light interception by isolated trees which makes it possible to analyze the influence of structural tree organization on light capture. The framework is based on a multiscale representation of the plant organization. Tree architecture is decomposed into a collection of components representing clusters of leaves at different scales in the tree crown. The components are represented by porous envelopes automatically generated as convex hulls containing components at a finer scale. The component opacity is defined as the interception probability of a light beam going through its envelope. The role of tree organization on light capture was assessed by running different scenarii where the components at any scale were either randomly distributed or localized to their actual three-dimensional (3D) position. The modeling framework was used with 3D digitized fruit trees, namely peach and mango trees. A sensitivity analysis was carried out to assess the effect of the spatial organization in each scale on light interception. This modeling framework makes it possible to identify a level of tree description that achieves a good compromise between the amount of measurement required to describe the tree architecture and the quality of the resulting light interception model. Key words. multiscale modeling, light interception, three-dimensional modeling, functionalstructural modeling, porous envelope, clumping, fruit trees AMS subject classifications. 93A30, 92C80 DOI. 10.1137/08071394X
1. Introduction. Light capture by plants is an essential process for plant growth and survival. Indeed light provides plants with energy which can be used for carbon fixation through foliage photosynthesis and for transpiration which allows water and nutriment transport within the plant [19]. Light interception by plant foliage is governed by simple basic principles: Photons coming from the sun direction (direct radiation) and the whole sky hemisphere (diffuse radiation) may be intercepted by the plant elements or transmitted onto the soil surface if they pass in the foliage gaps. Intercepted photons may then be absorbed or scattered in any direction. Scattered photons may then either be intercepted again or leave the canopy. The interception process can be seen as the intersection between the photon trajectory—a line—and the plant organ. It thus depends only on the canopy structure, i.e., the spatial distribution of the plant organs, and organ geometry, namely shape, size, and orientation [36]. If the detailed three-dimensional (3D) geometry is known, light interception can be easily and accurately computed by using 3D computer plant mock-up. Ray-tracing methods [13], the Z-buffer approach [13], or plant image processing [41] can be used. Although accurate, this class of computation ∗ Received by the editors January 21, 2008; accepted for publication (in revised form) May 15, 2008; published electronically August 27, 2008. The digitizing process in this work was supported by the AIP and PFI programmes of INRA and by the ATP Manguier programme from CIRAD. http://www.siam.org/journals/mms/7-2/71394.html † Virtual Plants, UMR DAP, TA A-96/02, 34398 Montpellier Cedex 5, France (david.da silva@ cirad.fr,
[email protected]). ‡ INRIA, 2004 route des lucioles BP 93, 06902 Sophia Antipolis, France, and Virtual Plants, UMR DAP, TA A-96/02, 34398 Montpellier Cedex 5, France (
[email protected]). § INRA and UMR PIAF, Site de Crou¨ el, 234 avenue du Br´ezet, 63100 Clermont-Ferrand, France (
[email protected]).
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methods shows several shortcomings. First, it does not allow one an understanding of which structural features are the main determinants of light interception by plants. Second, although methods exist to exhaustively measure the detailed 3D geometry— in particular 3D digitizing [38]—these techniques are very tedious and do not allow describing large sets of large trees. Third, if scattering is computed, the algorithm complexity dramatically increases due to multiple interception-scattering events and the high number of traced photons needed for convergence. Fourth, even though the basic processes are computed without any assumption, the simulation results are sensitive to measurement errors in the detailed canopy structure and in the optical properties of plant organs. For more than 50 years, more simple approaches have been proposed to estimate light interception by plants [23]. The most common approach abstracts the plant canopy as a turbid medium [36], i.e., a medium made of infinitely small foliage particles randomly dispersed in the vegetation volume and thus having a uniform optical density, i.e., transparency. In such a medium, light penetration can be expressed by the Beer–Lambert law; i.e., the probability that a photon crosses the vegetation volume without any interception can be written as (1.1)
p0 = exp(−G.LAD.L).
Hence, p0 is the probability of zero interception and corresponds to the canopy porosity. LAD is the leaf area density, and G is the extinction coefficient, namely the projection coefficient of plant elements on a plane perpendicular to the direction Ω [36]. G depends on the angle between Ω and the normal of the plant’s elements. The product G.LAD can be regarded as the optical density of the vegetation. Note that leaves are usually the only elements taken into account because they represent the solar collector of the plant. Finally L is the distance travelled by the photons in the canopy. If scattering is disregarded, which is the case in this study where we focus on the effect of canopy structure on light interception, the intercepted light is proportional to p = 1 − p0 , i.e., the probability of light interception that defines the canopy opacity. For a photon direction Ω, the distance L is constant for horizontally homogeneous vegetation canopies, e.g., grasslands. However, for tree crowns, L depends on the point where the photons enter the canopy. Usually, tree crowns are abstracted by envelopes, and beams are sent from a grid of points above the tree. The contribution of each beam to light interception is then summed up to compute total light interception probability in this direction. Finally the contribution of each direction Ω is summed up by weighting each directional probability with the fraction of incident radiation coming from direction Ω. Several light interception models for tree crowns are based on these principles [8, 22]. However, the assumption of uniform random distribution of leaf elements is rarely verified in actual tree crowns [46, 11, 38]. Indeed leaves are grouped around shoots, with higher density at the crown periphery. This leads to an overall clumped dispersion of the foliage, nonuniform LAD distribution, and lesser interception by crowns showing foliage clumping. The simplest way to deal with the nonrandom location of leaf elements is to introduce a leaf dispersion parameter μ in the Beer–Lambert equation (1.2)
p0 = exp(−G.μ.LAD.L).
Parameter μ equals 1 for random distribution. It is less than 1 for clumped foliage, i.e., crown porosity, p0 , is higher, and μ could be greater than 1 if foliage would show
Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.
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regular dispersion. Indeed the product μ.LAD can be regarded as the LAD of an equivalent random canopy showing the same porosity. Thus a clumped canopy shows a higher porosity and therefore a lower equivalent LAD, i.e., μ < 1. The parameter μ possibly changes with direction Ω [27] and botanical parameters [26, 6]. However, there is not yet a clear knowledge about the structural parameters determining the degree of foliage clumping, although Sinoquet et al. [40] have shown that μ is related to the spatial variations of LAD. Two other approaches have been proposed to deal with nonrandom and nonuniform foliage. In the first one, the crown volume is divided into voxels, and a value of LAD is assigned to each voxel [20, 10, 25]. This approach shows two shortcomings. On the one hand, computed light interception depends on voxel size [21, 39]. On the other hand, assigning LAD values to each voxel needs a huge number of field measurements [10]. The second approach is based on the botanical multiscale structure of trees and was applied to conifer species. Norman and Jarvis [28] assumed that spruce crowns were made of whorls regularly distributed along the trunk, with shoots randomly distributed in whorls and needles randomly dispersed in shoots. Oker-Blom and Kellomaki [30] proposed a simplification of the Norman and Jarvis approach, where shoots were directly distributed at random in the tree crown of Scots pines. This kind of model, called grouping model, better takes into account the foliage distribution according to the plant organization at several scales. It allows better rendering of foliage clumping without introducing a calibration parameter μ. The objective of this study was to develop a general modeling framework for computing light interception by single tree crowns. This framework includes most of the previously proposed methods in a unifying formalism: 3D plant mock-ups vs. turbid medium, mono- vs. multiscale approaches. This modeling framework is aimed at better understanding the effects of the crown organization on light capture at the whole tree scale, i.e., giving meaning to μ. The expected outcome of this study is to define ways of describing canopy structure as simple as possible and allowing accurate estimation of light interception, without the need of introducing any empirical dispersion parameter μ. In this paper, this approach was applied to a collection of fruit trees, namely four peach and four mango trees. 2. Modeling framework. At a macroscopic level, the problem consists of estimating the amount of direct radiation intercepted by a vegetal component x (representing either the entire plant crown or a subbranching system) for each direction Ω of incident light. Light interception is computed in terms of ST AR, silhouette to total area ratio: P LA (2.1) ST AR = , T LA where P LA (m2 ) is the projected leaf area on a plane perpendicular to the incident direction Ω (i.e., the leaf area which intercepts light in direction Ω) and T LA (m2 ) is the total leaf area [5, 31]. The ST AR is thus the relative irradiance of the leaf area. To take into account the clumping of leaves in plant crowns, this definition can be extended to the case where a canopy is decomposed into groups of leaves rather than directly into leaves. Groups of leaves can be regarded as macroscopic plant components, corresponding, for instance, to particular branching systems in the plant. In this case, the notion of P LA must be redefined since it now refers to the projected area of these coarser components, which are not entirely opaque. For this, we assume that the shape of a component x can be globally characterized by its convex envelope. According to [18], the P LA of x, denoted P LAx , can then be defined
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from the projected surface of the component envelope by introducing its opacity px in direction Ω [40]: (2.2)
P LAx = P EAx .px ,
where P EAx is the projected envelope area of x in the direction Ω and px can be regarded as the probability of photon interception in the projected envelope. Reciprocally, 1 − px is the envelope porosity. In the case of such multiscale plant structures, our original question thus amounts to estimating the opacity of the coarse components that are identified at different scales. Intuitively, the opacity in a particular direction Ω of such components, themselves composed of subcomponents (such as leaves or smaller branching systems) with defined shapes, is controlled by two independent factors: 1. On the one hand, it depends on the opacity of the subcomponents themselves. 2. On the other hand, it depends on the spatial distribution of the subcomponents and, more precisely, on how much the subcomponent silhouettes overlap when observed from direction Ω. In the case of opaque subcomponents (e.g., opaque leaves in a tree crown) the opacity of the composed object in a given direction Ω is solely a function of the directional overlapping. In the case of porous subcomponents, the opacity is the result of these two factors applied to the smaller components. Possibly, the subcomponents themselves can be decomposed into smaller components with their own opacity. In this section, we first briefly recall how the structural organization of a plant can be formalized within a multiscale framework [15, 3]. We then show how to compute the porosity factors of these elements and use the resulting hierarchical structure to compute light interception. We then show how sensitivity analysis of the model can be carried out to determine the influence of each scale of the hierarchy in the light interception. 2.1. Multiscale representation of plants. Plant 3D mock-ups are represented by sets of geometric components for which the shape, size, spatial coordinates, and orientation of each component are well defined (Figure 2.1(a)). This information can be obtained either from direct measurements [38, 17] or from simulation models of plant architecture [35, 12]. In both cases, the multiscale structural information is described as a multiscale tree graph (MTG) [15]. At each scale i (i = 1 . . . n), the plant is regarded as a set of botanical components (e.g., branches, shoots, leaves) arranged as a rooted tree graph. Components at a scale i are made of components at scale i + 1 and together define a partition of the set of components at scale i + 1. Scale 1 corresponds to the whole tree and scale n to the set of leaves (an MTG includes at least these two scales) (Figure 2.1(d)). Each component is associated with a shape. At the leaf scale (i = n), components (leaves) are represented with a set of polygons. At the other scales (i = n), component shapes are convex hulls containing the shapes of their subcomponents (Figure 2.1). 2.2. Multiscale model of light interception. Using this framework, for each component j at any scale i, both ST ARi,j and P LAi,j values can be derived from (2.1) and (2.2): (2.3) (2.4)
ST ARi,j =
P LAi,j , T LAi,j
P LAi,j = P EAi,j .pi,j ,
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Fig. 2.1. Four-scale decomposition of an artificial tree. Leaves are represented by a set of geometric models (a). Leafy modules are defined here by botanical branching order 2 (b), 1 (c), and 0 (d). Illustration from [F. Boudon, C. Godin, C. Pradal, O. Puech, and H. Sinoquet, Estimating the fractal dimension of plants using the two-surface method. An analysis based on 3D-digitized tree foliage, Fractals, 14 (3) (2006), pp. 149–163]. Reprinted with permission from World Scientific Publishing Company.
where P EAi,j is the projected envelope area of the component j at scale i and pi,j is its envelope opacity. For leaves (i.e., components at scale n), P LAn,j and ST ARn,j are simply the projected area of the leaf and the extinction coefficient of the leaf, respectively [36]. At other scales i (i = n), components j are porous objects containing subcomponents at scale i + 1. In what follows, we shall show how such a multiscale organization of plants can be used to compute recursively the opacity of a plant crown and to interpret the light interception properties of the plant at different scales. At each scale i and for each component j at this scale, the envelope opacity pi,j can be estimated by casting a set of regularly spaced beams in the envelope. Let Bi,j be the set of beams intersecting with component j at scale i. The origin of beams b (b = 1 . . . Bi,j ) can be obtained from the cell centers of a regular grid perpendicular to the direction Ω and positioned above the tree. Each beam is affected with a crosssection area, Ab , corresponding to the surface area of a grid unit element; see Figure b 2.2. Consequently, we can associate a volume Vi,j with each beam b: b = Lbi,j .Ab , Vi,j
(2.5)
where Lbi,j is the travelling distance of beam b in component j. Based on the beam i,j and P EAi,j of the volume and projected area sampling we can define estimators V of the envelope of component j at scale i: (2.6)
i,j = V
Bi,j
b=1
b Vi,j
and P EAi,j =
Bi,j
P EAbi,j = Bi,j .Ab .
b=1
i,j and P The beam sampling must be dense enough to verify that V EAi,j provide good estimates of Vi,j and P EAi,j . With such a beam sampling, an estimator of the
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Fig. 2.2. Beam sampling illustration using a three-scale component representation. A component j at scale i is discretized using an Ω-oriented beam sampling where each beam b has the same cross-section area, Ab , but a specific length in its envelope, Lbi,b , and therefore a specific volume, b . P EA Vi,j i,j is the projected envelope area onto a plane orthogonal to the direction Ω.
envelope opacity, pi,j , can be defined as the mean of beam opacities pbi,j : (2.7)
pi,j =
Bi,j 1 b pi,j . Bi,j b=1
At any scale (i < n), the model includes two options for computing opacity of a component. The first one (called option A) takes into account the Actual geometry of subcomponents in the component envelope. The second one (option R) uses the turbid medium analogy, i.e., assumes subcomponents to be Randomly distributed in the envelope volume with uniform density. Opacity computation in option A. The actual geometry of each subcomponent k in any envelope j at scale i is taken into account. For each beam of the grid positioned above the tree crown, the opacity pbi+1,k of subcomponent k, defined as its light interception probability, can be computed from the intersection between the beam and the subcomponent. If the subcomponent is intersected by the beam, i.e., pbi+1,k 0, the value of pbi+1,k is 1 for a solid object, e.g., a leaf, and less than 1 for a porous object. If the subcomponent k is not intersected by the beam pbi+1,k = 0, therefore the contribution pbi,j of beam b to the opacity of envelope j is computed by taking into account all subcomponents k, of scale i + 1, included in j: (2.8)
pbi,j = 1 −
nj 1 − pbi+1,k . k=1
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Fig. 2.3. Recursive pattern for computing the P LA of an element through its envelope opacity. The combined options of interscale distribution determine the path to follow in this equation scheme. The recursion ends when the last scale is reached; this scale must have a known or fixed opacity. For instance in this study leaves are considered opaque, i.e., opacity = 1. Hence for the all-A scenario the value of every pbi,j is either 0 or 1 for all i and j.
Note that the product is due to the sequence of subcomponents intersected by the beam b, but the restriction to these particular subcomponents is made through the value of their beam opacities. Indeed, the above equation shows that pbi,j equals 0 if all pbi+1,k are equal to 0, and pbi,j equals 1 as soon as a subcomponent k intersected by beam b is opaque, pbi+1,k = 1. For instance, the latter happens when subcomponents are leaves, i.e., i + 1 = n. When i + 1 = n the value of pbi+1,k is in [0, 1], depending on the options of finer scales; see Figure 2.3. Opacity computation in option R. In option R, subcomponents are randomly and uniformly distributed. The probability that the beam b is not intercepted by the component j at scale i is computed from the product of gap fractions produced by each subcomponent k, of scale i + 1, included in j [39]. The gap fraction for a beam b due to the subcomponent k at scale i + 1 is defined as the fraction of Ab free of the subcomponent projection, i.e., Ab − P LAbi+1,k , Ab where P LAbi+1,k is the portion of component k area projected onto the beam crosssection area, i.e., the restriction of P LAi+1,k to the beam b. The probability of no interception, assuming independence between events, is nj P LAbi+1,k b 1− . (2.9) p0 i,j = Ab k=1
In (2.9), the product expresses the effect of the uniform random distribution of leaves on the beam opacity in envelope j. Moreover, one can see that the probability of beam interception by component k: (2.10)
pbi+1,k =
P LAbi+1,k Ab
is exactly
P LAbi+1,k ; Ab
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this leads to (2.11)
pbi,j = 1 − p0 bi,j = 1 −
nj 1 − pbi+1,k . k=1
Equation (2.11) is exactly the same as (2.8) found in the case of actual distributions of components k in the envelope j. However, they differ by the interpretation of the product: In option A, the product expresses the sequence of components k intersected by the beam b, while in option R, it expresses the random position of components. This analogy between both equations is further investigated in section 5.3. In option R, we thus need to compute P LAbi+1,k for each beam. To carry out this computation, we use the assumption of uniform density of foliage which makes it possible to write P LAbi+1,k
(2.12)
b Vi,j
=
P LAi+1,k Vi,j
b as Vi,j = Ab Lbi,j ; (2.10) and (2.12) can be combined to express opacity pbi+1,k
(2.13)
pbi+1,k =
b P LAi+1,k Vi,j P EAi+1,k pi+1,k Lbi,j = ; Ab Vi,j Vi,j
and (2.11) can then be written (2.14)
pbi,j = 1 −
nj k=1
P EAi+1,k pi+1,k Lbi,j 1− . Vi,j
To summarize, we showed that the opacity of component j at scale i can be expressed by a unique set of recursive equations that enables us to express P LAi,j for each (i, j) as (2.15)
P LAi,j
Bi,j 1 b = P EAi,j pi,j . Bi,j b=1
This makes it possible to evaluate ST ARi,j from (2.3) based on an estimate of the plant total leaf area [45, 32]. The scheme described in Figure 2.3 illustrates the recursive procedure that uses (2.8) or (2.14), depending on the scenario. It can be used with any number of scales, and either option A or R can be independently used at each scale. 2.3. Assessing light interception. Using this modeling framework, any particular scale can be added or removed or its spatial distribution switched from one option to another, e.g., changing the actual position of leaves (option A) to the hypothesis of random distribution of leaf area density (option R). This will be used to analyze the influence of one specific scale in light interception. In case of a crown filled with leaves, i.e., the number of scales is two, option A corresponds to the actual leaf distribution within the crown, e.g., as digitized in the field. Otherwise, option R leads to the basic random distribution of leaf area in the tree crown, as used in many turbid medium models [8, 22]. In case of more scales, the scenario corresponding to the selected combination of options A and R is encoded as a 6 string of characters,
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AA
AR
RA
RR
-R
Fig. 2.4. Example of distribution options for an object with three scales, namely crown, shoot, and leaves. Hence a 2-character string represents interscale distribution options. AA represents the actual distribution of leaves in the crown. With AR the actual position of shoots are used, but the opacity of their envelopes is computed with the turbid medium analogy, whereas RA uses the turbid medium analogy for shoots but with their real opacity taken into account. The RR distribution corresponds to the model where leaves and shoots are randomly distributed in shoots and in the crown, respectively, i.e., grouping model. Finally, -R uses the turbid medium analogy and supposes a uniform distribution of leaves in the crown.
containing the number of scales minus 1 character. The first and last characters of the string refer to scale 2 and scale n, respectively. The option for scale 1 can only be A, i.e., the actual tree crown. This is why scale 1 is not included in the string. For example, let us consider a tree organized in three scales, e.g., crown, shoots, and leaves. A scenario AA would correspond to actual arrangement of shoots within the crown and to actual arrangement of leaves within shoots. This corresponds to the real plant structure and is assumed to be the true value. Another example illustrated in Figure 2.4 is the scenario RR corresponding to the Oker-Blom and Kellomaki model [30], where leaves and shoots are randomly distributed in shoots and in the crown, respectively. In addition, any inner scale (i ∈ [2, n − 1]) can be discarded to distribute components at scale i directly in envelopes at scale i − 2. Discarded scales are denoted “-” in the scenario string. For instance, in the example above, the scenario -R means that leaves are directly randomly distributed in the tree crown. Finally, note that discarding a scale and then using the actual position is equivalent to using the actual distribution for the discarded scale, i.e., -A ≡ AA. As presented in section 2.2, our goal is to estimate each plant’s global light interception efficiency in terms of ST AR. This efficiency is evaluated at the plant scale, i.e., i = 1 and j = 1, by integrating directional ST AR values over the sky directions Ω. In the following i and j indexed notations will thus be omitted and the Ω indexed notation will indicate directional quantities, whereas Ω-free notation will stand for integrated values. Ω-integrated values for ST AR and P EA are obtained by summing up the directional values weighted by coefficients, ωΩ , derived from the standard overcast sky radiance distribution [37]: (2.16)
ST AR =
Ω
ST ARΩ .ωΩ
and P EA =
P EAΩ .ωΩ .
Ω
ST ARΩ values are computed according to (2.1) from P LAΩ whose evaluation is de-
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scribed in Figure 2.3. Originally the leaf dispersion parameter, μ, introduced in the Beer–Lambert equation as a LAD modifier, expresses the departure of the actual crown gap fraction from the gap fraction of a crown with random distribution of leaves and equivalent leaf area density. Indeed, μ is 1 for random distribution of leaves, whereas leaf clumping leading to higher crown porosity yields a μ strictly less than 1. It can be extended to compare two scenarii, expressing a relative dispersion coefficient, μ, of a test porosity, p0 , against a reference one, p0 ref . By analogy with the Beer–Lambert formalism, where p0 ref would be expressed using (1.1) and p0 using (1.2), the following relationship is defined [40]: (2.17)
p0 = p0 ref μ .
Using the complementary relationship between opacity and porosity and (2.1) and (2.2), an Ω-integrated μ value can be expressed as a function of ST AR and P EA:
T LA log 1 − ST AR. P EA . (2.18) μ= ST ARref . T LA log 1 − P EA It allows us to study the global or scale-by-scale spatial organization through the computation of μ from a specific scenario. For instance, in the example illustrated in Figure 2.4, the definition of μ as introduced by Nilson [27] corresponds to comparing the situation AA to the reference situation -R and is therefore noted μ(AA/-R). The ST AR of these scenarii are noted ST ARAA and ST AR-R , respectively. The scaleby-scale organization is given by scenarii which differ by only one letter corresponding to the studied scale. For instance the shoot distribution within the crown and the leaf distribution in the shoots are given by the ratio of ST ARAA to ST ARRA and ST ARAA to ST ARAR , respectively. 3. Plant database. 3.1. Plant material. Four 4-year old peach trees (variety August Red ) were digitized one month after bud break in May 2001 in CTIFL-Balandran in southeast France (43◦ 83 N, 4◦ 35 E). Two of them were trained as a tight goblet (TG) system and the two others as wide double-Y (WDY). TG is an open center (goblet-shaped) structure employing several primary scaffolds. Planting distances in the TG system were 6 × 3 m. WDY derives from the goblet, with larger planting distances, 7 × 4 m, and four primary scaffolds arranged by pairs [14]. Tree height was about 2.5 m. Four 3-year old mango trees grown in a commercial farm were digitized in March 2001 in Saint-Paul, La R´eunion Island (20◦ 53 S, 55◦ 32 E). Two of them belonged to variety Lirfa and the two others to variety Jos´e. A square planting system was used with a 6-meter distance between trees and a northwest–southeast row orientation. Tree height was about 1.5 m. Full information on peach and mango trees can be found in [42] and [44], respectively. 3.2. 3D plant digitizing and reconstruction. All trees were 3D digitized with a 3Space Fastrak electromagnetic device (Polhemus Inc., Colchester, VT, USA). The digitizer includes a magnetic field generator and a pointer to be positioned by an operator at each point to be measured. The pointer includes coils where electrical currents are induced when located in the magnetic fields. The spatial coordinates and orientation angles in three dimensions can be derived from the values of induced currents [33]. Plant digitizing was driven by software Pol95 [1].
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Fig. 3.1. Mango tree scale decomposition. Top: top view; bottom: side view and from left to right: leaves, CYS, scaffold, crown.
Fig. 3.2. Peach tree scale decomposition. Top: top view; bottom: side view and from left to right: leaves, CYS, OYOS, scaffold, crown.
Mango trees were digitized at leaf scale according to Sinoquet et al.’s method [41]. For each leaf, the pointer was located at the junction between the lamina and petiole, with pointer X-axis parallel to the midrib and pointer X-Y plane parallel to the lamina. With this configuration, measured angles were the midrib azimuth, the midrib elevation, and the rolling angle of the lamina around the midrib. In addition, leaf length was measured with a ruler. 50 leaves were harvested for leaf area (A cm2 ) measurement with a planimeter, and leaf length (L cm) was measured with a ruler. This sample was used to set a relationship between A and L2 : (3.1)
A = k.L2 .
Coefficient k was constant for both varieties, with a value of 0.1826. Peach trees were digitized at the leafy shoot scale according to Sinoquet and Rivet’s method [38]. For each leafy shoot, the spatial bottom and top coordinates of the shoot were recorded with the digitizer. For each shoot type in each cultivar, namely short and long (> 5 cm) shoots, 15 to 30 shoots were digitized at leaf scale, as described above. These data were used to establish foliage reconstruction rules for all digitized shoots.
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Nb elmt 1 4 86 1966 14405 1 5 88 1522 13709 1 5 93 1583 15118 1 6 101 1990 15589
Volume (dm3 ) 11811.50 2832.70 ± 1214.52 14.02 ± 29.27 0.97 ± 1.54 12200.89 1915.22 ± 507.09 13.98 ± 22.88 1.40 ± 2.06 12442.29 1934.32 ± 935.55 15.51 ± 24.89 1.52 ± 2.30 15911.77 2060.11 ± 773.34 12.48 ± 20.19 1.17 ± 1.91 -
Leaf area (dm2 ) 2773.85 693.45 ± 232.08 9.10 ± 14.85 1.41 ± 1.95 0.19 ± 0.05 2889.69 577.74 ± 190.48 9.99 ± 12.51 1.90 ± 2.32 0.21 ± 0.55 3242.96 648.58 ± 266.30 11.30 ± 15.24 2.05 ± 2.78 0.21 ± 0.55 3144.29 524.04 ± 186.99 8.30 ± 9.99 1.58 ± 2.01 0.20 ± 0.06
LAD (dm−1 ) 0.24 0.253 ± 0.04 1.31 ± 0.83 1.90 ± 0.58 0.24 0.30 ± 0.04 1.22 ± 0.72 1.77 ± 0.53 0.26 0.34 ± 0.05 1.35 ± 1.00 1.77 ± 0.52 0.20 0.26 ± 0.04 1.39 ± 1.10 1.84 ± 0.57 -
Mango a19
Crown Scaffold CYS Leaf
1 3 139 1475
1146.66 417.03 ± 68.42 6.07 ± 4.81 -
631.64 210.54 ± 11.95 4.54 ± 2.64 0.43 ± 0.17
0.55 0.51 ± 0.07 1.06 ± 0.72 -
Mango b7
Crown Scaffold CYS Leaf
1 3 252 2759
2073.24 793.35 ± 229.78 9.00 ± 6.47 -
1221.87 407.29 ± 126.59 4.85 ± 2.70 0.44 ± 0.19
0.59 0.51 ± 0.02 0.73 ± 0.52 -
Mango f21
Crown Scaffold CYS Leaf
1 2 147 1367
1188.56 614.09 ± 93.89 4.80 ± 4.93 -
509.19 254.60 ± 55.84 3.46 ± 2.40 0.37 ± 0.18
0.43 0.41 ± 0.03 0.77 ± 0.48 -
Crown Scaffold CYS Leaf
1 7 99 1135
869.39 146.10 ± 68.07 8.27 ± 6.92 -
492.87 70.41 ± 32.48 4.98 ± 3.19 0.43 ± 0.19
0.57 0.48 ± 0.05 0.83 ± 0.50 -
Peach 4
Peach 3
Peach 2
Peach 1
Scale Crown Scaffold OYOS CYS Leaf Crown Scaffold OYOS CYS Leaf Crown Scaffold OYOS CYS Leaf Crown Scaffold OYOS CYS Leaf
Mango g5
Table 3.1 Scale-by-scale component demographies and their envelope volume, contained leaf area, and LAD. Abbreviations: OYOS = one-year-old shoot; CYS = current-year shoot; LAD = leaf area density.
Reconstruction rules included allometric relationships at the shoot and leaf scale, random sampling in leaf angle distributions, and additional hypotheses. The reconstruction method has been fully presented and assessed in [42]. At the same time as digitizing, the position of the digitized organ in the multiscale tree organization was recorded using the method of Godin, Costes, and Sinoquet [17]. For mango trees four scales were used: plant, scaffold, current-year shoot (CYS), i.e., leafy shoot, and leaf. One additional scale was used for peach trees, i.e., one-year-old shoot (OYOS) between the scaffold and CYS scales. Of course reconstructed peach leaves were assigned to the corresponding digitized shoot. Finally a database was obtained for each tree as a collection of leaves explicitly distributed in 3D space and related to the multiscale organization of the tree.
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4. Results and clumping analysis. Ω-integrated values for ST AR and μ according to several hypotheses are shown in Tables 4.1 and 4.2 for peach and mango trees, respectively. Actual ST AR values (line 0) for peach trees did not show a large range, i.e., between 0.245 and 0.275. This means that foliage irradiance was about 25% of the available light above the tree. For mango trees the range of ST AR values was greater, i.e., between 0.243 and 0.364. These values are in the interval commonly reported for fruit trees [47]. For both species, the classical assumption of random distribution of leaves in crowns (row 1) led to large overestimation of the ST AR values. Therefore light interception models based on this assumption are biased, as already reported by [46, 9, 11]. This confirms that the actual foliage distribution shows significant departure from randomness. The values of the dispersion parameter μ were between 0.55 and 0.61 and between 0.70 and 0.81 for peach and mango trees, respectively, expressing a clumped organization of the foliage and higher clumping for peach trees than for mango trees. Table 4.1 Ω-integrated ST AR for peach trees.
0 1 2 3 4 5 6 7 8 9 10 11 12 13
AAAA - - -R μ (AAAA/- - -R) RRRR μ (AAAA/RRRR) AAAR μ (AAAA/AAAR) AARA μ (AAAA/AARA) ARAA μ (AAAA/ARAA) RAAA μ (AAAA/RAAA) RRRA μ (RRRA/RRRR) RRAR μ (RRAR/RRRR) RARR μ (RARR/RRRR) ARRR μ (ARRR/RRRR) -RRR μ (AAAA/-RRR) - -RR μ (AAAA/- -RR) AA-R μ (AAAA/AA-R)
Peach1 0.2477 0.3389 0.5482 0.2422 1.0368 0.2372 1.071 0.2480 0.9987 0.2594 0.9281 0.2349 1.0868 0.2518 1.0643 0.2421 0.9995 0.2261 0.8999 0.2487 1.0432 0.2803 0.8117 0.3061 0.6871 0.2522 0.9717
Peach2 0.2715 0.3483 0.6127 0.2347 1.2582 0.2497 1.1445 0.2697 1.01 0.2890 0.8971 0.2412 1.2071 0.2519 1.1145 0.2352 1.003 0.2235 0.9304 0.2650 1.2087 0.2789 0.9554 0.3035 0.8202 0.2578 1.0882
Peach3 0.2456 0.3239 0.5734 0.2177 1.2110 0.2312 1.1036 0.2442 1.0096 0.2586 0.9154 0.2226 1.1782 0.2300 1.0884 0.2184 1.0048 0.2100 0.9466 0.2416 1.1782 0.2639 0.8835 0.2898 0.7386 0.2443 1.0094
Peach4 0.2754 0.3591 0.6082 0.2504 1.1599 0.2542 1.1331 0.2719 1.021 0.2927 0.904 0.2585 1.1257 0.2676 1.1082 0.2514 1.0066 0.2379 0.9274 0.2703 1.1257 0.3078 0.8280 0.3221 0.7615 0.2651 1.0627
The full grouping model (all-R, row 2) underestimated light interception (for all trees but Peach1). Indeed μ values above 1 means that the leaves in the actual canopy were less overlapping than they would have been in the canopy generated from the full grouping model. Consequently, actual ST AR values were comprised between these two most usual modeling approaches. However, the two species showed different behavior: Actual ST AR of peach trees was overestimated by 30% with scenario - --R, whereas scenario all -R underestimated actual ST AR by only 10%. In the case of mango trees, the overestimation given by scenario --R was around 12% and equal to
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MULTISCALE LIGHT INTERCEPTION MODEL Table 4.2 Ω-integrated ST AR for mango trees.
0 1 2 3 4 5 6 7 8 9 10
μ μ μ μ μ μ μ μ μ μ
AAA - -R (AAA/- -R) RRR (AAA/RRR) AAR (AAA/AAR) ARA (AAA/ARA) RAA (AAA/RAA) RRA (RRA/RRR) RAR (RAR/RRR) ARR (ARR/RRR) -RR (AAA/-RR) A-R (AAA/A-R)
Mango a19 0.2824 0.3282 0.7039 0.2544 1.2157 0.2736 1.064 0.2915 0.9366 0.2507 1.2465 0.2608 1.0458 0.2444 0.9335 0.2828 1.2195 0.3018 0.8683 0.3068 0.8363
Mango b7 0.2434 0.2717 0.7315 0.2193 1.2552 0.2347 1.0886 0.2497 0.9391 0.2124 1.3346 0.2222 1.0267 0.2065 0.8922 0.2456 1.2830 0.2582 0.8577 0.2587 0.8537
Mango f21 0.3637 0.4025 0.8112 0.3159 1.2809 0.3478 1.0859 0.3643 0.9964 0.3191 1.2596 0.3252 1.0495 0.3076 0.9576 0.3519 1.2047 0.3670 0.9829 0.3846 0.8949
Mango g5 0.3235 0.3564 0.7982 0.2811 1.3077 0.3084 1.1019 0.3053 1.1240 0.2927 1.2163 0.2892 1.0519 0.2820 1.0058 0.2956 1.0949 0.3262 0.9825 0.3190 1.0298
the underestimation given by scenario all -R. A deeper analysis was made by comparing the actual ST AR values to those obtained by switching one scale option from A to R. This allows studying the dispersion pattern at the switched scale. For instance, the leaf dispersion within CYS in peach trees was assessed by comparing scenario AAAA to scenario AAAR. For peach trees (Table 4.1), only CYS in OYOS (row 4) showed a clear random dispersion (μ 1). ST ARAAAR and ST ARRAAA (rows 3 and 6, respectively) were slightly less than the actual value; thus the distribution of leaves in CYS and scaffolds in crowns was slightly regular (μ 1.11 and μ 1.15, respectively). Finally ST ARARAA (row 5) was a bit greater than the actual value, showing a small clumping trend of OYOS in scaffolds (μ 0.91). For mango trees (Table 4.2), scenarii AAR and ARA (rows 3 and 4) led to small under- and overestimation of actual ST AR, expressing slight regularity and clumpiness of leaves in CYS and CYS in scaffolds, respectively. By contrast, scenario RAA (row 5) markedly underestimated actual ST AR, meaning that the actual distribution of scaffolds in the crown was regular (μ 1.26). The dispersion pattern for each scale was also assessed by comparing the ST AR behavior values of the full grouping scenario (all -R) to those obtained by switching one scale option from R to A (rows 7 to 10 in Table 4.1 and rows 6 to 8 in Table 4.2 for peach and mango trees, respectively). This analysis confirmed the dispersion previously found at each scale. As the scaffold scale was shown to be regular, we tested a scenario where this scale was discarded (rows 11 and 9 for peach and mango trees, respectively). Theoretically this should lead to ST AR values closer to the actual ones because the scaffold subcomponents are no longer confined within the scaffold envelopes but distributed in the full crown volume. It is expected to be a way to simulate the regularity of scaffold distribution in the crown. Indeed ST AR values in scenarios -RRR and -RR were greater than values given by the full grouping model for peach and mango trees,
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D. DA SILVA, F. BOUDON, C. GODIN, AND H. SINOQUET
respectively. They were also greater than ST AR values given by scenarii ARRR and ARR, meaning that disregarding the scaffold scale led to overestimating the actual regular dispersion of scaffolds. For all trees the actual ST AR value was in between the values given by the full grouping scenario and that disregarding the scaffold scale. For peach trees scenario AA-R (row 13) gave ST AR values very close to the actual ones. The profit of this scenario will be discussed in the next section. 5. Discussion. 5.1. Requirements in canopy structure description for an accurate estimation of light interception. Exhaustive measurement of the canopy structure, e.g., to get data for scenario all -A, is extremely tedious, especially for the peach trees where the number of leaves was about 15,000 per tree. Conversely, the simplest scenario ---R uses only a few data to describe the tree structure, namely crown shape and volume and total leaf area. However, the computed light interception was shown to be largely overestimated. This confirms that the actual tree foliage distribution in crowns is not uniform and shows high clumping [11]. Consequently, the simulation models using this scenario [29, 7] should include a calibration parameter μ. By contrast, the full grouping scenario underestimated light interception for both species. Moreover, the Oker-Blom and Kellomaki [30] model, i.e., --RR and -RR for peach and mango trees, respectively, yielded ST AR values close to the actual ones but only for mango trees. As the Oker-Blom and Kellomaki model can be regarded as a partial grouping model (i.e., leaves randomly distributed in shoots and shoots directly distributed at random in the tree), this questions the number of botanical scales that must be used in grouping models. However, the Oker-Blom and Kellomaki model was reported to be successful for conifer species [30], and our results show that this approach can also be used for some, but not all, fruit tree species. In the case of peach trees, the grouping scenarii did not work satisfactorily; however, the scenario AA-R gave nice results. This means that good estimations of light interception can be obtained by measuring the actual position of OYOS in the crown. In practice, this kind of measurement is really tractable [42], as a peach tree usually includes only about 100 OYOS (see Table 3.1). Moreover, the OYOS scale corresponds to the management unit used to train the tree (i.e., the fruit grower selects a given number of OYOS in the sunlit zones of the tree and prunes the other ones). Therefore an interesting use of the proposed modeling framework is to find out scenarii of canopy structure description as simple as possible that enables an accurate estimation of light interception without artificially using leaf dispersion parameters μ (especially because there is almost no means to estimate μ). 5.2. Sensitivity analysis. Measurement errors effect. Magnetic digitizing is prone to measurement errors both on the location and orientation of organs. The error on spatial coordinates is typically less than 1 mm in a controlled environment [24] and is less than 1 cm for field measurements [43]. The impact of such errors on ST AR computation has already been assessed on peach trees by Sonohat et al. [42]. In this case, measurements were conducted at the shoot scale, and leaf position and shape were reconstructed using allometric rules. It was shown that the ST AR estimation at the shoot scale was inaccurate while it was satisfactory at the plant scale. In the case of mango trees, which were digitized at the leaf scale, we quantified the error in ST AR estimation by generating mock-ups where random errors were introduced in leaf spatial coordinates and orientation angles. We designed two procedures for this. We first used a uniform
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MULTISCALE LIGHT INTERCEPTION MODEL
(a)
(b)
Fig. 5.1. Influence of measurement errors on the ST AR estimation. (a) Effect of increasing measurement error. Random error generated by a uniform distribution (dark grey) or Student’s distribution (light grey) is introduced on leaf spatial coordinates so that the new position is in a sphere of radius 0.1, 0.2, 0.5, 1, 2, and 5 cm centered on its original position. (b) 1 cm positioning error of leaves on several scenarii using the R option. Errors are generated using a uniform distribution (dark grey) or Student’s distribution (light grey). The ST AR values from the original tree are taken as the reference.
distribution in the range [−ε, ε] to modify the coordinates and the orientation of each leaf. Alternatively, we used Student’s distribution, which yields a greater proportion of small errors. The impact of these two distributions on ST AR computation was evaluated on a set of 100 mock-ups with ε values defined so that the new position of each leaf is in a sphere of radius 0.1, 0.2, 0.5, 1, 2, and 5 cm centered on its original position. Orientation angles were modified similarly with ε ranging from 5◦ to 45◦ . Results showed that positioning errors less than 1 cm had little effect on light interception capacities at the plant scale (Figure 5.1(a)), and for 1 cm, corresponding to the field measurement error, the ST AR value error using scenario all-A was less than 5% regardless of the distribution used. For measurement errors greater than 1 cm the foliage rapidly tended to occupy a more important volume, leading to a lower leaf area density and thus a greater ST AR. This effect was much more marked when errors were generated using a uniform distribution (Figure 5.1(a)). Note, however, that errors greater than 1 cm should be avoided if measurements are carefully conducted. Orientation errors had only a neglectable effect on ST AR values integrated over the sky vault (less than 0.5% for all ε values; data are not shown). Positioning errors in plant organs may also affect the convex envelope of plant components. To assess the impact of this phenomenon on the ST AR computation using scenarii with option R, we created a set of five mango mock-ups by introducing a 1 cm error on leaf position with the previously described distributions. For each of these trees, integrated ST AR values were computed for scenarii AAA, AAR, ARR, RRR, and --R to assess the cumulative impact of option R. Results are shown in Figure 5.1(b), with the ST AR value of the all-A scenario of the original tree taken as the reference. An increase in the ST AR estimation error with the use of the R option at different scales is clearly visible (the effect is slightly reduced for Student’s distribution) and is due to the fact that the porosity of a component results from the product of its subcomponents’ porosities, (2.8). Using other types of bounding envelopes such as boxes, spheres, or ellipsoids induces a larger envelope volume [2] and thus leads to similar error effects. If the all-A scenario ST AR value of the mock-up
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Fig. 5.2. Influence of beam sampling lineic density on the ST AR estimation. The bounding box upper face surface S is considered as the unitary area, and the ST AR value obtained with a density of 300 is taken as the reference for each scenario.
with leaf positioning errors is taken as the reference for the multiscale organization analysis, then we still find the same result trends. Beam sampling effect. As mentioned in section 2.2, the density of the beam sampling may influence ST AR estimation. The plant is included in a bounding box such that its upper face is orthogonal to the light direction, Ω. This face is subdivided into cells of a regular grid whose centers define the locations of the beams. This defines a beam sampling density as B S , B being the total number of beams and S the area of the upper face. In the case of our regular grid we control the sampling density by the lineic density of the beams, d, defined as
B , (5.1) d= S representing the number of beams per unit length along the grid axes. We assessed the effect of the sampling density by computing ST AR values for several scenarii using different lineic densities: 20, 50, 100, 200, and 300, the latest being the one used to conduct the multiscale analysis of section 4. For each value of d, integrated ST AR values were computed for scenarii AAA, AAR, ARR, RRR, and --R. Results show that convergence was rapidly obtained with increasing density (the error was less than 0.5% for densities over 100); see Figure 5.2. Using a lineic density over 100 guarantees a good ST AR estimation regardless of the scenario used. Moreover, since the complexity of our algorithm is quadratic in the beam sampling lineic density (see section 5.4 for details), large gain in computation time is achieved by decreasing d from 300 to 100 with almost no loss of quality in the results. 5.3. A unifying approach. In section 2.2, we observed that, surprisingly, the recursive equations (2.8) and (2.11) relating component porosities at different scales were actually identical despite the fact that they were derived from two different hypotheses. This raises the question as to whether these equations can be derived from a single unifying framework. To answer this question, let us call {x1 . . . xn } the set of
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subcomponents of a component c, and Ib (c) indicates whether the beam b intersects with the envelope of c (1 = yes, 0 = no). Let us also denote Xb (c) the random variable such that Xb (c) = 1 if the beam b is intercepted by an elementary component of c, i.e., it interacts with c, and Xb (c) = 0 otherwise. With these definitions the porosity of a component c for the beam b is defined as (5.2)
pb0 (c) = P (Xb (c) = 0|Ib (c) = 1);
reciprocally, its opacity is thus (5.3)
pb (c) = 1 − p0 b (c) = P (Xb (c) = 1|Ib (c) = 1).
Let us now consider a beam b that intersects a component c, i.e., P (Ib (c) = 0) = 0; thus the decomposition P (Xb (c) = 0) = P (Xb (c) = 0|Ib (c) = 1)P (Ib (c) = 1) + P (Xb (c) = 0|Ib (c) = 0)P (Ib (c) = 0) yields pb0 (c) = P (Xb (c) = 0|Ib (c) = 1) = P (Xb (c) = 0). A beam b does not interact with a component c if and only if b does not interact with any subcomponent of c. Since going through a subcomponent without being intercepted does not modify the beam, all P (Xb (xi ) = 0) are independent, and thus we have
(5.4)
P (Xb (c) = 0) = P (Xb (x1 ) = 0, . . . , Xb (xn ) = 0) n n = P (Xb (xk ) = 0) = [1 − P (Xb (xk ) = 1)] . k=1
k=1
Using the fact that the probability for a beam b to interact with a subcomponent that it is not intersecting, P (Xb (xk ) = 1|Ib (xk ) = 0), is obviously 0, the decomposition of P (Xb (xk ) = 1) yields (5.5)
P (Xb (xk ) = 1) = P (Xb (xk ) = 1|Ib (xk ) = 1)P (Ib (xk ) = 1).
Using expression (5.5) to replace P (Xb (xk ) = 1) in the component porosity equation (5.4), and given the complementary relationship between opacity and porosity equation (5.3), the expression of opacity becomes (5.6) P (Xb (c) = 1|Ib (c) = 1) = 1−
n
[1−P (Xb (xk ) = 1|Ib (xk ) = 1)P (Ib (xk ) = 1)] ,
k=1
which illustrates our previous remark stating that opacity is controlled by two factors. The first is the opacity of the subcomponents: P (Xb (xk ) = 1|Ib (xk ) = 1). The second is their spatial distribution: P (Ib (xk ) = 1). Indeed, when using option A, P (Ib (xk ) = 1) is equal either to 0 (the beam does not intersect the component) or 1 (the beam intersects the component). Therefore this term acts as a filter to disregard all subcomponents not intersected by the beam b. This is equivalent to (2.8), where the opacity of subcomponents not intercepted is equal to 0.
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When using option R the spatial distribution of subcomponents can only be estimated. The probability of intersecting a subcomponent can be expressed as P (Ib (xk ) = 1) =
P EAb (xk ) , P EAb (c)
whereas the expression of the porosity of the subcomponent for the beam b is P (Xb (xk ) = 1|Ib (xk ) = 1) =
P LAb (xk ) . P EAb (xk )
Since P EAb (c) is the beam cross-section, Ab , the product of these two quantities in (5.6) yields the expression of the beam interception probability: P LAb (xk ) P EAb (xk ) P EAb (xk ) P EAb (c) P LAb (xk ) = , Ab
P (Xb (xk ) = 1|Ib (xk ) = 1)P (Ib (xk ) = 1) =
which leads us to the second equation, (2.10), that was derived in case R. Consequently, we showed that it was possible to derive equations for both cases A and R from a unique expression, (5.6), that unify both situations. 5.4. Implementation issues and complexity. This software has been written in Python and C++. It is a stand-alone module part of the VPlants software project (successor of AMAPmod [16]). In a first step a 3D shape is associated with each component of the MTG at each scale. The leaf geometry is defined using the PlantGL library [34], and convex hulls are computed with the QuickHull algorithm [4] available in this library. The multiscale computation of opacity is carried out for each direction Ω as follows. It starts with a double Z-buffer approach: Two opposite orthographic cameras oriented along the Ω direction are used to generate two images of each geometric component of the plant from the same distance. Each pair of facing pixels from the two orthographic views represents the same beam b, and its cross-section area, Ab , is the area represented by a pixel expressed in metric units. The Z-values of each pixel yield the beam inand out-points in the component. The travel distance of a beam within the envelope of component j at scale i, Lbi,j , is deduced from these two values. This length is null for components associated with planar shape, e.g., leaves. Moreover, this double Z-buffer approach allows us to identify the beams intercepted by each component which is required for the use of option A, i.e., when actual component positions are used. In this approach convex envelopes are used since multiple in- and out-points are not taken into account; hence nonconvex envelopes will be treated as convex ones. This step also provides the projected area of the component shape by multiplying Ab with the number of pixels within the orthographic image. All of these quantities are scenario independent and thus are computed only once for each direction Ω. The volume of a convex hull, Vi,j , is computed separately using routines implemented in the PlantGL library [34]. Next the recursive scheme described in Figure 2.3 is applied to compute the opacity according to the scenario used. The two recursive procedures used to compute the opacity of a component, pi,j , and the beam opacity of a component, pbi,j , are described by Algorithms 1 and 2, respectively. The recursion starts from macroscopic components towards microscopic ones and ends when leaves are reached. It is important to
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Algorithm 1. Opacity(c, Ω, s) Input: component c, direction Ω, scenario s /* s = string in {A,R} */ Output: mean opacity of c if c == leaf then return 1 else sum = 0 ; foreach intercepted beam b do sum+ = OpacityBeam(c, Ω, s, b) return
sum size(intercepted beams)
Algorithm 2. OpacityBeam(c, Ω, s, b) Input: component c, direction Ω, scenario s, beam b Output: beam opacity of c if c == leaf then return 1 else option = s[0] ; /* get the current scale option */ s = s[1..n] ; /* and remove it from scenario */ porosity = 1 ; if option == A then foreach subcomponent x do if b ∩ x = ∅ then porosity ∗ = OpacityBeam(x, Ω, s, b) ; if option == R then foreach subcomponent x do px = Opacity( x, Ω, s ) ; porosity
∗=1−
P EAx px Lbc Vc
return 1 − porosity
note that in practice each value is computed once—the first time needed—and then stored for future use to save computation time. Leaves are considered as opaque components; thus their opacity does not need to be computed. The test of whether or not a beam intersects a leaf is considered as the atomic operation. Therefore, the computation cost of one component opacity depends on the number of intersected beams, i.e., the number of pixels of its envelope projection in direction Ω, and the number of its subcomponents. In the worst-case scenario, every component is intersected by all beams. A very simple model allows us to evaluate the computation cost of our algorithm. Let us denote the number of scales by k and assume that the number of subcomponents, N , is identical for all components at every scale. The total number of components is consequently (5.7)
1 + N + N 2 + · · · + N k−1 = O(N k−1 ) = O(n),
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where n = N k−1 is the number of leaves, i.e., components of the last scale. Let us define a size ratio, δ, by comparing the size of a component to the (smaller) size of its subcomponents. We also assume that the size ratio between two scales is constant and less than 1, that is, (5.8)
∀i, j, l
Vi+1,j = δ 3 Vi,l
and P EAi+1,j = δ 2 P EAi,l .
Let Bi be the number of beams intercepted by a component at scale i with B0 = B. As a consequence of a constant size ratio, the number of intercepted beams at one scale is related to the number of intercepted beams at the previous scale: (5.9)
Bi+1 = δ 2 Bi = δ 2i B.
Consequently, the total cost γ is the sum of each component cost at every scale except the leaf scale:
(5.10)
γ = 1 × N B + N × N Bδ 2 + N 2 × N Bδ 4 + · · · + N k−2 × N Bδ 2(k−2) = O N k−1 Bδ 2(k−2) = O nBδ 2(k−2) .
The worst case is reached if k = 2; thus the total cost is in O (nB). However, when k > 2, since δ < 1 the gain in complexity due to the hierarchical structure is proportional to δ 2(k−2) . For example, using the plant illustrated in Figure 2.1, adding 1 scale to the 2 basic scales will reduce the complexity by 19 (δ = 13 ), and for 1 2 additional scales, the complexity will be reduced by 81 . 6. Conclusion. This paper presented a new framework for modeling efficiently light interception by isolated trees. The turbid medium approach, usually limited to large canopies because of its statistical description of plants, has been adapted to isolated trees. The modeling was based on a multiscale representation of plants and on a porous envelope hypothesis. We defined recursive expressions to compute the opacity of components with two types of spatial distribution hypotheses that can be chosen at each scale independently. The combination of these options defined scenarii that allowed us to analyze the influence of the plant architecture on light interception through the generalization of a dispersion parameter μ which expresses the departure of a plant foliage from randomness. This model was then assessed on 3D digitized peach and mango trees. Peach trees were markedly more clumped than mango trees. The two species showed different clumping behaviors but with the same trend for scaffold branches toward regular positioning. We showed that Oker-Blom and Kellomaki’s partial grouping model can be used for mango trees but not for peach trees which shows a different type of clumping that does not fit their original assumption. Our model alleviates this problem by making it possible to use a variable number of scales and can thus be applied to both situations. Moreover, the true ST AR values were always comprised between the values from the full grouping scenario and the partial grouping one disregarding the scaffold scale. This suggests that the multiscale organization is not the only factor involved in the light interception strategy of trees. The proposed multiscale framework may be used to optimize plant architecture measurement in the context of modeling light interception by plants. It also defines a versatile and incremental procedure to compute light interception up to a desired level of accuracy, ranging from coarse descriptions, i.e., using the turbid medium hypothesis at the plant scale, to detailed descriptions, e.g., obtained by 3D digitizing, with a
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series of intermediate options defined by the number of scales taken into account and their relative positioning. The multiscale organization of components represented by porous envelopes provides another advantage in terms of model flexibility; depending on the availability of field measurements the envelopes and their positions can either correspond to the actual field values or be inferred from statistical assumptions. Acknowledgments. We thank Yann Gu´edon for a thorough reading of the paper. We thank Gabriela Sonohat, Didier Combes, and Vengadessan Kulandaivelu for their help in digitizing the peach trees. REFERENCES [1] B. Adam, POL95—Software to Drive a Polhemus Fastrak 3 SPACE 3D Digitiser, Technical report, UMR PIAF INRA-UBP, Clermont-Ferrand, France, 1999. [2] F. Boudon, Repr´ esentation g´ eom´ etrique de l’architecture des plantes, Ph.D. thesis, Universit´e Montpellier 2, Montpellier, France, 2004. [3] F. Boudon, C. Godin, C. Pradal, O. Puech, and H. Sinoquet, Estimating the fractal dimension of plants using the two-surface method. An analysis based on 3D-digitized tree foliage, Fractals, 14 (2006), pp. 149–163. [4] C. Bradford-Barber, D. P. Dobkin, and H. Huhdanpaa, The quickhull algorithm for convex hulls, ACM Trans. Math. Software, 22 (1996), pp. 469–483. [5] G. A. Carter and W. K. Smith, Influence of shoot structure on light interception and photosynthesis in conifers, Plant Physiol., 79 (1985), pp. 1038–1043. [6] E. Casella and H. Sinoquet, Botanical determinants of foliage clumping and light interception in two-year-old coppice poplar canopies: Assessment from 3-D plant mock-ups, Ann. For. Sci., 64 (2007), pp. 395–404. [7] A. Cescatti, Modelling the radiative transfer in discontinuous canopies of asymmetric crowns. II. Model testing and application in a Norway spruce stand, Ecol. Modell., 101 (1997), pp. 275–284. [8] D. A. Charles-Edwards and J. H. M. Thornley, Light interception by an isolated plant a simple model, Ann. Bot., 37 (1973), pp. 919–928. [9] S. G. Chen, R. Ceulemans, and I. Impens, A fractal-based Populus canopy structure model for the calculation of light interception, For. Ecol. Manage., 69 (1994), pp. 97–110. [10] S. Cohen, M. Fuchs, S. Moreshet, and Y. Cohen, The distribution of leaf area, radiation, photosynthesis and transpiration in a Shamouti orange hedgerow orchard. Part II. Photosynthesis, transpiration, and the effect of row shape and direction, Agric. For. Meteorol., 40 (1987), pp. 145–162. [11] S. Cohen, P. Mosoni, and M. Meron, Canopy clumpiness and radiation penetration in a young hedgerow apple orchard, Agric. For. Meteorol., 76 (1995), pp. 185–200. [12] P. Ferraro, C. Godin, and P. Prusinkiewicz, Toward a quantification of self-similarity in plants, Fractals, 2 (2005), pp. 91–109. [13] J. D. Foley, A. van Dam, S. K. Feiner, and J. F. Hughes, Computer Graphics: Principles and Practice in C, 2nd ed., Addison–Wesley, Reading, MA, 1995. [14] P. Giauque, Conduite du verger de pˆ echer. Recherche de la performance, CTIFL (Centre Technique Interprofessionnel des Fruits et L´egumes) Editions, Paris, 2003. [15] C. Godin and Y. Caraglio, A multiscale model of plant topological structures, J. Theoret. Biol., 191 (1998), pp. 1–46. [16] C. Godin, E. Costes, and Y. Caraglio, Exploring plant topology structure with the AMAPmod software: An outline, Silva Fennica, 31 (1997), pp. 355–366. [17] C. Godin, E. Costes, and H. Sinoquet, A method for describing plant architecture which integrates topology and geometry, Ann. Bot., 84 (1999), pp. 343–357. [18] J. E. Jackson and J. W. Palmer, A simple model of light transmission and interception by discontinuous canopies, Ann. Bot., 44 (1979), pp. 381–383. [19] H. G. Jones, Plants and Microclimate: A Quantitative Approach to Plant Physiology, Cambridge University Press, Cambridge, UK, 1992. [20] D. S. Kimes and J. A. Kirchner, Radiative transfer model for heterogeneous 3-D scenes, Appl. Opt., 21 (1982), pp. 4119–4129. [21] Y. Knyazikhin, G. Mießen, O. Panfyorov, and G. Gravenhorst, Small-scale study of three-dimensional distribution of photosynthetically active radiation in a forest, Agric. For. Meteorol., 88 (1997), pp. 215–239.
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[22] J. E. Mann, G. L. Curry, and P. J. H. Sharpe, Light interception by isolated plants, Agric. Meteorol., 20 (1979), pp. 205–214. [23] M. Monsi and T. Saeki, Uber den Lichtfaktor in den Pflanzengesellschaften und seine Bedeutung fur die Stoffproduktion, Japan. J. Bot., 14 (1953), pp. 22–52. [24] B. Moulia and H. Sinoquet, Three-dimensional digitizing systems for plant canopy geometrical structure: A review, in Crop Structure and Light Microclimate: Characterization and Applications, INRA, Paris, 1993, pp. 183–193. [25] R. B. Myneni, Modeling radiative transfer and photosynthesis in three-dimensional vegetation canopies, Agric. For. Meteorol., 55 (1991), pp. 323–344. [26] U. Niinemets, A. Cescatti, and C. Rochelle, Constraints on light interception efficiency due to shoot architecture in broad-leaved Nothofagus species, Tree Physiol., 24 (2004), pp. 617–630. [27] T. Nilson, A theoretical analysis of the frequency of gaps in plant stands, Agric. Meteorol., 8 (1971), pp. 25–38. [28] J. M. Norman and P. G. Jarvis, Photosynthesis in Sitka spruce (Picea sitchensis (Bong.) Carr.): V. Radiation penetration theory and a test case, J. Appl. Ecol., 12 (1975), pp. 839–878. [29] J. M. Norman and J. M. Welles, Radiative transfer in an array of canopies, Agron. J., 75 (1983), pp. 481–488. [30] P. Oker-Blom and S. Kellomaki, Effect of grouping of foliage on within-stand and withincrown light regimes: Comparison of radiation and grouping canopy models, Agric. Meteorol., 28 (1983), pp. 143–155. [31] P. Oker-Blom and H. Smolander, The ratio of shoot silhouette area to total needle area in Scots pine, For. Sci., 34 (1988), pp. 894–906. [32] J. Phattaralerphong, J. Sathornkich, and H. Sinoquet, A photographic gap fraction method for estimating leaf area of isolated trees: Assessment with 3D digitized plants, Tree Physiol., 26 (2006), pp. 1123–1136. [33] Polhemus Inc., 3SPACE FASTRAK User’s Manual, Revision F, Colchester, VT, 1993. [34] C. Pradal, F. Boudon, C. Nouguier, J. Chopard, and C. Godin, PlantGL: A Python-Based Geometric Library for 3D Plant Modelling at Different Scales, Technical report, INRIA, Sophia Antipolis, France, 2007. [35] P. Prusinkiewicz, Self-similarity in plants: Integrating mathematical and biological perspectives, in Thinking in Patterns. Fractals and Related Phenomena in Nature, M. Novak, ed., World Scientific, Singapore, 2004, pp. 103–118. [36] J. Ross, The Radiation Regime and Architecture of Plant Stands, Dr. W. Junk, The Hague, Boston, London, 1981. [37] H. Sinoquet, J. Phattaralerphong, P. Kasemsap, and S. Thanisawanyangkura, Light and carbon acquisition partitioning between flushes of two-year-old mango trees, in Proceedings of the 4th Workshop on Functional-Structural Plant Models (FSPM04), CIRAD, Montpellier, France, 2004, pp. 146–149. [38] H. Sinoquet and P. Rivet, Measurement and visualization of the architecture of an adult tree based on a three-dimensional digitising device, Trees - Structure and Function, 11 (1997), pp. 265–270. [39] H. Sinoquet, G. Sonohat, J. Phattaralerphong, and C. Godin, Foliage randomness and light interception in 3-D digitized trees: An analysis from multiscale discretization of the canopy, Plant, Cell & Environment, 28 (2005), pp. 1158–1170. [40] H. Sinoquet, J. Stephan, G. Sonohat, P. E. Lauri, and P. Monney, Simple equations to estimate light interception by isolated trees from canopy structure features: Assessment with three-dimensional digitized apple trees, New Phytologist, 175 (2007), pp. 94–106. [41] H. Sinoquet, S. Thanisawanyangkura, H. Mabrouk, and P. Kasemsap, Characterization of the light environment in canopies using 3D digitising and image processing, Ann. Bot., 82 (1998), pp. 203–212. [42] G. Sonohat, H. Sinoquet, V. Kulandaivelu, D. Combes, and F. Lescourret, Threedimensional reconstruction of partially 3D-digitized peach tree canopies, Tree Physiol., 26 (2006), pp. 337–351. [43] S. Thanisawanyangkura, H. Sinoquet, P. Rivet, M. Cretenet, and E. Jallas, Leaf orientation and sunlit leaf area distribution in cotton, Agric. For. Meteorol., 86 (1997), pp. 1–15. [44] L. Urban, X. Le Roux, H. Sinoquet, S. Jaffuel, and M. Jannoyer, A biochemical model of photosynthesis for mango leaves: Evidence for the effect of fruit on photosynthetic capacity of nearby leaves, Tree Physiol., 23 (2003), pp. 289–300. [45] F. J. Villalobos, F. Orgaz, and L. Mateos, Non-destructive measurement of leaf area in
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olive (Olea europaea L.) trees using a gap inversion method, Agric. For. Meteorol., 73 (1995), pp. 29–42. [46] D. Whitehead, J. C. Grace, and M. J. Godfrey, Architectural distribution of foliage in individual Pinus radiata D. Don crowns and the effects of clumping on radiation interception, Tree Physiol., 7 (1990), pp. 135–155. [47] M. Willaume, P. E. Lauri, and H. Sinoquet, Light interception in apple trees influenced by canopy architecture manipulation, Trees, 18 (2004), pp. 705–713.
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B
Reference Manual for OpenAlea Fractalysis and MµSLIM modules
Several methods, models and algorithm were implemented during this thesis and are available in the OpenAlea platform. The next release of this work will be integrated as Fractalysis and MuSLIM modules, but are currently available as fractalysis.engine and fractalysis.light. This appendix presents the main classes and methods of these modules as they should appear after the refactoring process.
B.1
FR A C T A L Y S I S
B.1.1
BCM This module provides functions to estimate the box-counting dimension of a 3D scene described using PlantGL. computeGrid(scene, gridSize) Compute the number of intercepted voxels of the regular grid generated by subdividing the scene bounding-box size by the gridSize parameter. Parameters: scene : (PlantGL scene) - the scene to be embedded in the grid. gridSize : (int) - the subdividing factor of the bounding-box. Returns: (int, float) A tuple constituted of the number of intercepted voxels, and their size definied as the cubic root of the voxels volume. computeGrids(scene, maxGridsize) This method is defined for user convenience and simply provide a simple way for calling computeGrid with the gridSize parameter ranging from 2 to maxGridSize. Parameters:
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192 Appendix B. Reference Manual for OpenAlea Fractalysis and MµSLIM modules scene : (PlantGL scene) - the scene to be embedded in the set of grids. maxGridSize : (int) - the maximum value that will be used to defined the range of subdividing factors. Returns: [(int, float)] A liste of tuples yielded by the computeGrid calls.
B.1.2
MatrixLac This class allow to study various lacunarities of a n-dimensional square matrix.
B.1.2.1 Factory functions lactrix_fromScene(scene, fileName, gridSize, density, savePath) Generates a MatrixLac instance from a PlantGL scene. Parameters: scene : (PlantGL scene) - the scene to be discretized into a matrix. fileName : (string) - name of the object that will be used to identify the saved results. gridSize : (int) - the subdividing factor of the bounding-box that will define the size of the matrix. density : (boolean) - when True each non-empty voxel is associated with a proper value (e.g. leaf area density inside each voxel). savePath : (string) - the absolute path where the results will be saved. Returns: MatrixLac A MatrixLac instance generated from the scene. lactrix_fromPix(imagePath, pixWidth, savePath,th) Generates a MatrixLac instance from a square image. Parameters: imagePath : (string) - absolute path to the PNG image (temporary restriction). pixWidth : (float) - pixel representing size defining the grid step. savePath : (string) - the absolute path where the results will be saved. th : (int) - threshold value to decide object pixels from void pixels. Returns: MatrixLac A MatrixLac instance generated from the image.
B.2. MµSLIM
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B.1.2.2 Class methods lacunarity(radiusStart, radiusStop, radiusStep, lacType) MatrixLac method that computes lacunarities using the gliding-box algorithm for a set of gliding-box sizes. Parameters: radiusStart : (int) - the beginning size of the gliding-box set. radiusStop : (int) - the final size of the gliding-box set. radiusStep : (int) - the size step between two gliding-box sizes. lacType : (string) - the type of lacunarity to be computed. The value can be either ’ac’ for Alain & Cloitre lacunarity, or ’ctrd’ for centered lacunarity. Returns: [(float, float)] A liste of tuples consituted of the size of the gliding boxes and the lacunarity value, i.e. (scale, lacunarity).
B.2
MµSLIM
B.2.1
scaledStruct This class is the data structure used in MµSLIM model as described in chapter 6.
B.2.1.1 Factory functions ssFromDict(name, scene, decompositionList, hullType) Generates a scaledStruct from a from a PlantGL scene and a list of scale by scale decomposition. Parameters: name : (string) - the name of the studied object. Will be used for optional serialization. scene : (PlantGL scene) - the scene representing the object at its finest scale.. decompositionList : (list of maps) - the list of decomposition relation between two scales. The list length is one less than the number of scales. hullType : (string) - the type of hulls generated for macro-components. The default value is CvxHull for a convexe hull but can be either Box, Sphere or Ellipse. Returns: scaledStruct A scaledStruct instance generated from the PlantGL scene and the multi-scale decomposition.
194 Appendix B. Reference Manual for OpenAlea Fractalysis and MµSLIM modules B.2.1.2 Attributes depth The number of scales of the scaledStruct. name The name of the scaledStruct. B.2.1.3 Class methods genGlobalScene() Generates a PlantGL scene with all scales. Returns: PlantGL scene The scene representing the object with the representation of all scales. genScaleScene(scale) Generates a PlantGL scene of one specific scale. Parameters: scale : (int) - the scale to be represented. Returns: PlantGL scene The scene representing the object at the required scale. get1Scale(scale) Retrieve the id of all components of one specific scale. Parameters: scale : (int) - the scale for which components id are required. Returns: [int] A list of id. removeScale(scale) Remove the scale from the scaledStruct. Parameters: scale : (int) - the scale to be removed.
B.2. MµSLIM
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computeDir(az, el, wg, scenarii, skt, width, height, dFactor, pth, saveData) Compute the light intercepted by the object for one light direction, according to the given scenarii. Parameters: az : (float) - azimuth of light direction. el : (float) - elevation of light direction. wg : (float) - weight in SOC meaning of light direction. scenarii : ([[char]]) - list of scenario. Default value is all-A scenario. skt : (int) - if defined, it replace the light direction information by the ones from the associated sky-turtle direction (1-46). width : (int) - the width of the window defining the number of beams. Default value is 150. height : (int) - the height of the window defining the number of beams. Default value is 150. dFactor : (float) - ad-hoc parameter to set camera position so the entire scene is always seen. Value depends of beams window size. Default value is set to 8 for a beams window size of 150x150. pth : (string) - path used for optional serialization of results. saveData : (Bool) - wether or not the results should be serialized. If true, pth and scaledStruct’s name will be used. Returns: map A map of computed results with Star_turbid as one key and for each scenario s a STAR value and an image identified as Star_s and Pix_s.
196 Appendix B. Reference Manual for OpenAlea Fractalysis and MµSLIM modules
This document was realised using GNU text editor Kile and LATEX typeseting.
Titre Caractérisation de la nature multi-échelles des plantes par des outils de géométrie fractale, application à la modélisation de l’interception de la lumière. Résumé Dans le contexte du développement durable ou des changements climatiques, la compréhension et le contrôle de la croissance des plantes sont devenus des enjeux de société importants. La recherche dans ce domaine s’appuie de plus en plus sur des modèles informatiques, dits “structure-fonction”, qui prennent en compte l’architecture des plantes. Contrairement aux modèles agronomiques plus classiques fondés sur les relations entre un petit nombre de variables d’entrée-sortie, ces modèles permettent d’étudier la relation entre la structure tridimensionnelle des plantes et les processus physiques et écohysiologiques qui contrôlent leur développement. Cependant, l’architecture d’une plante, et en particulier sa géométrie, est souvent complexe et peut être décrite à différents niveaux de détails. Dans cette thèse nous avons cherché à caractériser la complexité de la géométrie multi-échelles des plantes avec peu de descripteurs afin de pouvoir prendre en compte la structure dans des modèles simples d’interception de la lumière. Nous avons donc dans un premier temps tenté de développer des outils mathématiques permettant de caractériser la géométrie multi-échelles d’une plante avec des descripteurs inspirés de la géométrie fractale. La dimension fractale permet de caractériser le taux de croissance des détails géométriques d’une plante en fonction de l’échelle à laquelle on l’observe. Pour caractériser plus précisément cette géométrie, la notion de dimension fractale doit être complétée par une notion permettant de décrire de manière spatiale la densité des détails géométriques à chaque échelle: la lacunarité. Dans chaque cas, nous rappelons les estimateurs classiques de ces grandeurs, étudions leurs limites et proposons des estimateurs alternatifs, mieux adaptés à la description de structures végétales. Ces différents estimateurs sont ensuite évalués sur des bases de données de plantes artificielles et réelles. Dans une deuxième partie, nous développons un modèle d’interception de la lumière basé sur l’organisation multi-échelles des plantes. Ce modèle permet d’estimer l’interception de la lumière à différentes échelles, mais également d’analyser, échelle par échelle, la relation entre l’organisation de la plante et sa capacité d’interception. Ce modèle est ensuite appliqué et évalué sur des arbres isolés et sur des forêts hétérogènes.
Mots-clés multi-échelles, fractal, transfert radiatif, plantes virtuelles, modèles Title Characterizing multiscale nature of plants using fractal geometry descriptors, application on light-interception modeling.
Abstract With concerns such as sustainable development or climat changes, controling and understanding plant growth has become important society matters. Computer models that uses plant architecture, called “functional-structural plant models”(FSPMs), have become more and more widespread. Contrariwise to agronomic models based on the relations between few parameters, FSPMs allow to assess the relation between the three-dimensional structure of plants and the physical and ecophysiological processes that drive their development. However, plant architecture, and particularly its geometry, is rather complex and can be described at different detail levels. In this thesis we wanted to characterize the complex multiscale geometry of plants with few descriptors in order to be able to acknowledge the structure in simple models of light interception. First, we developped mathematical methods to characterize the multiscale geometry of plants using descriptors from fractal geometry. The fractal dimension allows one to characterize the way plants physically penetrate space as a function of the observation scale. To characterize the plant geometry more thoroughly, the fractal dimension needs to be complemented with a description of spatial density of geometric details at each scale: the lacunarity. We recall the usual definitions for both the fractal dimension and the lacunarity, analyze their limits, and propose variants of these descriptors that better suit the characterization of plant organization. These different definitions are then appraised on data bases of virtual and real plants. Second, we create a light interception model based on the plant multiscale organization. This model computes light interception at each scale, and allows to analyze the scale by scale relation between plant structure and its light interception ability. This model is then used and evaluated on isolated trees and heterogeneous canopies.
Keywords multiscale, fractal, radiatif transfer, virtual plants, models
