Section: New Results

Analysis of structures resulting from meristem activity

Acquisition and design of plant geometry

Participants : Chakkrit Preuksakarn, Frédéric Boudon, Christophe Pradal, Christophe Godin.

This research theme is supported by RTRA project PlantScan3D.

Virtual 3D model of plants are required in many areas of plant modelling. They can be used for instance to simulate physical interaction of real plant structures with their environment (ligth, rain, wind, pests, ...), to set up initial conditions of growth models or to assess their output against real data. In the past decade, methods have been developed to digitize plant architecture in 3D [40] , [28] . These methods are based on direct measurements of position and shape of every plant organ in space. Although they provide accurate results, they are particularly time consuming. More rapid and automated methods are now required in order to collect plant architecture data of various types and sizes in a systematic way. In this topic, we explore the use of laser scanner and direct sketching. Resulting models should be parsimonious. We also consider the definition of methods to post-process these complex representations into compact ones and apply them on streaming.

• Reconstruction of plant architecture from 3D laser scanner data. (Chakkrit Preuksakarn, Frédéric Boudon, Christophe Godin, Jean-Christophe Chambellan [UMR PIAF, Clermont-Ferrand], Bernard Mourrain [INRIA, Galaad], Hervé Sinoquet [UMR PIAF, Clermont-Ferrand], Wenping Wang [University of Hong Kong], Dong-Ming Yan [University of Hong Kong])

  In this topic, we investigate the possibility to use 3D laser scanners to automate plant digitizing. We are currently designing algorithms to reconstruct branching systems without leaves from scanner data or from scan simulated on plant mock-up obtained using different digitizing method. For this we collaborate with the EPI GALAAD from Sophia-Antipolis, the EPI Evasion from Grenoble, different INRA team : PIAF-Clermont Ferrand, LEPSE- and AFEF-Montpellier and Lusignan, the University of Helsinki, Finland and the CFCC in England. The project PlantScan3D has been funded in 2009 by the Agropolis Fundation and started in septembre. A new proposal to the 2009 ARC INRIA call has been made to obtain grants for postdoctoral positions.

  With the GALAAD team, we already proposed a first reconstruction pipeline made up of three main steps: segmentation, reconstruction and modeling. Based on a variational k-means clustering algorithm, cylindrical components and branching regions of data points are identified and located. An adjacency graph is then built from neighborhood information of components. Simple heuristics allow us to extract a tree structure and identified branches from the graph. Finally, a B-spline model is computed to give a compact and accurate reconstruction of the branching system. This work has been accepted for publication as short paper at the IEEE International conference on Computer-Aided Design and Computer Graphics [18] .

  We continue this work by investigating new methods based on Space Colonization Algorithm [39] in the context of the PhD of Chakkrit Preuksakarn. An evaluation procedure to assess accuracy of the reconstruction is also currently being investigated. It is based on topological edit distance [25] to compare two tree graphs weighted with geometrical information.

• Sketching of plants. (Frédéric Boudon, Christophe Godin, Jamie Wither [INRIA, Evasion], Marie-Paule Cani [INRIA, Evasion])

  Modeling natural elements such as trees in a plausible way, while offering simple and rapid user control, is a challenge. In collaboration with the EPI Evasion we developed a method based on the design of plants from silhouettes. This sketching paradigm allows quick and intuitive specification of foliage at multiple scales. This choice allows us to incorporate botanical knowledge to infer branches that connect in a plausible way to their parent branch and have a correct 3D distribution. We illustrated these ideas by building a seamless sketch-based interface, used for sketching foliage silhouettes from the scale of an entire tree to the scale of a leaf. Each sketch serves for inferring both the branches at that level and construction lines to support sub-silhouette refinement. When the user finally zooms out, the style inferred for the branching systems he has refined (in terms of branch density, angle, length distribution and shape) is duplicated to the unspecified branching systems at the same level. Meanwhile, knowledge from botany is again used for extending the branch distribution to 3D, resulting in a full, plausible 3D tree that fits the user-sketched contours. As our results show, this system can be of interest to both experts and novice users: while experts can fully specify all parts of a tree and over-sketch specific branches if required, any user can design a basic 3D tree in one or two minutes, as easily as sketching it with paper and pen, as illustrated on Figure 3 . This work has been accepted at the Eurographics 2009 conference and has been published in Computer Graphics Forum [17] . The next step will consist of sketching growing plants. In this case, the user refinement will correspond to addition of elements through time.

  Figure 3. Output of an eucalytpus, as produced from the sketch pictured on the upper-right.

• Compression and streaming of plant geometry for distributed environments. (Frédéric Boudon, Christophe Pradal, Wei Cheng [National University of Singapour], Romulus Grigoras [IRIT, Toulouse], Jean-Christophe Hoelt [IRIT, Toulouse], Sebastien Mondet [IRIT, Toulouse], Geraldine Morin [IRIT, Toulouse], Wei Tsang Ooi [National University of Singapour])

  Much effort has been made in realistic modeling of plants for instance as support of physical simulation. As the trend moves towards networked and distributed virtual environment, however, the current models are inadequate as they are not designed for progressive transmissions. In this work, we fill in this gap by proposing a progressive representation for plants based on generalized cylinders. We proposed a differential coding of plants: an average branch is computed for any chosen group of branches and then, for each branch, we only need to code some transformations for instantiation and differences for geometry refinement. To be able to stream, we identify and take advantage of two types of dependencies: topological and dependencies due to differential coding. We obtain a progressive model that makes it possible to select a lightweight representation of a plant while preserving branch density. To facilitate the transmission of the plants in the network, we quantify the visual contribution of each branch and use this weight in packet scheduling. We show the efficiency of our representations and effectiveness of our packet scheduler through simulations. This work was presented at ACM Multimedia 2008 Conference [35] and received the Best Paper Award. An extended version of this work was published this year in ACM TOMCCAP [14] . Moreover, a technical demonstration of an implementation of this technique for Mobile, named MobiTree [16] , has been made at the ACM Multimedia 2009 Conference.

Modeling the plant ontogenic programme

Participants : Christophe Godin, Yann Guédon, Evelyne Costes, Jean-Baptiste Durand, Pascal Ferraro, Yassin Refahi, Etienne Farcot.

This research theme is supported by a PhD programme.

The remarkable organization of plants at macroscopic scales may be used to infer particular aspects of meristem functioning. The fact that plants are made up of the repetition of many similar components at different scales, and the presence of morphological gradients, e.g. [22] , [30] , [31] , [6] , provides macroscopic evidence for the existence of regularities and identities in processes that drive meristem activity at microscopic scales. Different concepts have been proposed to explain these specific organisations such as "morphogenetic programme" [36] , "age state" [27] or "physiological age" [23] . All these concepts state that meristem fate changes according to position within the plant structure and during its development. Even though these changes in meristem fate are specific to each species and lead to the differentiation of axes, general rules can be highlighted [27] , [23] . Here we develop computational methods to decipher these rules.

• Branching and axillary flowering structures of fruit tree shoots. (Yann Guédon, Evelyne Costes, Ted DeJong (UC Davis), Claudia Negron (UC Davis))

  In the context of a collaboration with Claudia Negron, David Da Silva and Ted DeJong, stochastic models (hidden semi-Markov chains) for the branching and axillary flowering structures of different categories of peach and almond shoots were built. These stochastic models have been integrated in simulation systems which combine stochastic models with different mechanistic models of biological function, in particular carbon partitioning models. This collaboration extends the work initiated on apple trees [38] , [24] ; see 6.1.3 .

• Axis structure of fruit trees. (Yann Guédon, Evelyne Costes)

  During the past years, a set of methods was proposed to characterize the growth components (mainly ontogenetic and environmental components) of forest trees [29] , [11] on the basis of main axes measured at the growth unit or annual shoot scale. We started to study axis structure of fruit trees. The agronomic context (grafted cultivars, irrigated orchard) renders very rapid the ontogenetic changes and reduces the environmental influence compared to forest trees. We studied in particular the structures of apple tree cultivars at the growth unit (GU) scale. Hidden variable-order Markov chains were applied to identify repeated patterns corresponding to the alternation between vegetative and flowering GUs along axes. In these models, GU categories are deduced from morphological characteristics (number of nodes and presence/absence of flowering) and their succession modeled by a non-observable variable-order Markov chain. This integrative statistical model enabled us to reveal a two-scale structuring of the successive differentiation stages during apple tree ontogeny, a coarser scale corresponding to the succession of two developmental phases and a finer scale corresponding to the alternation between flowering and vegetative GUs of a given type. This approach led us to propose a synthetic scheme of apple tree ontogeny that combines polycyclism, flowering and morphogenetic gradients.

• Hidden Markov tree models for investigating physiological states within plants. (Jean-Baptiste Durand, Patrick Heuret (UMR AMAP))

  During plant ontogeny, different kinds of patterns emerge, among which homogeneous stages of growth, or remarkable succession or alternation of given features. To unravel such patterns that characterize plant architecture, statistical analyses have been performed, based on hidden Markov tree (HMT) models. The global aim of these analyses is to infer several levels of differentiation of the meristems from the structures they produce [3] . Since these levels are assumed to be ordered, they are refered to as "physiological state of the meristems".

  Based on HMT models, a study was conducted on Symphonia globulifera (Clusiaceae) (a tree species of the rain forest in French Guyana) to characterize the effect of light on the architecture, and more specifically to quantitatively assess the connection between illuminance, the number of growth cycles necessary for the plant to reach the canopy, and the rate of synchronism among the plant entities.

• Self-nested structure of plants. (Christophe Godin, Pascal Ferraro)

  To study the redundancy of structure embedded at various levels in tree architectures, we investigated the problem of approximating trees by trees with particular self-nested structures. Self-nested trees are such that all their subtrees of a given height are isomorphic. We show that these trees present remarkable compression properties, with high compression rates. In order to measure how far a tree is from being a self-nested tree, we introduced a quantitative measure of the degree of self-nestedness for any tree. For this, we construct a self-nested tree that minimizes the distance of the original tree to the set of self-nested trees that embed the initial tree:

  

  where T is a tree, D(·, ·) is a distance on the set of trees (chosen so as to preserve certain structural properties between the compared trees) and is the set of self-nested trees that contain T , i.e. that can be obtained from T by inserting nodes only.

  To solve this optimization problem, we designed a polynomial-time algorithm that makes it possible to quantify the degree of self-nestedness of a tree in a precise manner. The distance to this nearest embedding self-nested tree (NEST) is then used to define compression coefficients that reflect the compressibility of a tree. In the context of the structural analysis of botanical plants, it is possible to give a strong biological interpretation of the NEST of a tree based on the hypothesis that isomorphic tree structures at macroscopic levels are actually produced by meristems in identical physiological states (scaling hypothesis). This makes it possible to show that the reduction graph of the NEST of a plant may be interpreted as the maximum sequence of differentiation states that any meristem of a plant may go through. We characterized this approach on both a database of artificial plants with degraded degree of self-nestedness and on a real plant (a rice panicle) whose structure was completely measured, see Figure 4 . We showed that the NEST of this plant may be interpreted in biological terms and reveals important aspects of the plant growth. This work has been accepted by IEEE Transactions on Computational Biology and Bioinformatics (IEEE/ACM TCBB).

  Figure 4. (a) Photo of a rice panicle (b) tree representing the topological structure of the panicle (c) Reduced graph of the panicle tree (d) Corresponding NEST that can be interpreted as the maximal sequence of differentiation states that a meristem can go through when building the panicle (e) color map of the meristem differentiation states obtained by projecting the NEST states back to the original topological tree

• Perturbed phyllotaxis of Arabidopsis thaliana AHP6 mutant. (Etienne Farcot, Yann Guédon, Yassin Refahi, Christophe Godin, Fabrice Besnard (RDP, Lyon), Teva Vernoux (RDP, Lyon))

  The cytokinin hormones are known to play a significant role in the regulation of phyllotaxis. To investigate this, Fabrice Besnard and Teva Vernoux are studying Arabidopsis thaliana ahp6 mutants, AHP6 being a protein known for its inhibitory effect on the cytokinin signaling pathway. At the macroscopic scale, this mutation induces perturbations of the phyllotaxis, barely sensible on single plants. Based on a sample of approximately 40 wild-type and 40 mutant plants, we have developed a method to characterize these perturbations. In both wild-type and mutant plants, the theoretical spiral of organs along the main axis is in fact affected by local permutations (i.e. the interchange of adjacent organs in the sequence). Using a statistical model (hidden variable-order Markov chain) and a combinatorial model, we have shown that a regular spiral with such permutations provides an appropriate description for more than 95% of the data, the mutant plants exhibiting significantly more local permutations than the wild-type.

Analyzing the influence of the environment on the plant ontogenic programme

Participants : Florence Chaubert-Pereira, David Da Silva, Damien Fumey, Frédéric Boudon, Christophe Godin, Yann Guédon, Christian Cilas, Evelyne Costes, Pascal Ferraro, Christian Lavergne, Hervé Sinoquet, Catherine Trottier.

This research theme is supported by a CIFRE contract and two PhD programmes.

The ontogenetic programme of a plant is actually sensitive to environmental changes. If, in particular cases, we can make the assumption that the environment is a fixed control variable (see section 6.1.2 ), in general the structure produced by meristem results from a tight interaction between the plant and its environment, throughout its lifetime. Based on observations, we thus aim to trace back to the different components of the growth (ontogenetic development and its control by the environment). This is made using two types of approaches. On the one hand, we develop a statistical approach in which stochastic models are augmented with additional time-varying explanatory variates that represent the environment variations. The design of estimation procedures for these models make it possible to separate the plant ontogenetic programme from its modulation by the environment. On the other hand, we build reactive models that make it possible to simulate in a mechanistic way the interaction between the plant development and its environment.

• Analyzing growth components in trees. (Florence Chaubert-Pereira, Yann Guédon, Yves Caraglio (AMAP), Christian Lavergne, Emilie Lebarbier (AgroParisTech), Catherine Trottier, Olivier Taugourdeau (AMAP))

  Observed growth, as given for instance by the length of successive annual shoots along a forest tree trunk, is assumed to be mainly the result of three components: (i) an endogenous component assumed to be structured as a succession of roughly stationary phases separated by marked change points that are asynchronous between individuals [29] , (ii) a time-varying environmental component assumed to take the form of fluctuations that are synchronous between individuals, (iii) an individual component corresponding to the local environment of each tree. This environmental component is thus assumed to be a "population" component as opposed to the individual component. In order to identify and characterize these three components, we proposed to use semi-Markov switching linear mixed models [11] , [12] . The underlying semi-Markov chain represents the succession of growth phases (endogenous component) while the linear mixed model attached to each state of the underlying semi-Markov chain represents -in the corresponding growth phase- both the influence of time-varying climatic explanatory variables (environmental component) as fixed effects, and inter-individual heterogeneity (individual component) as random effects. We investigated the estimation of Markov and semi-Markov switching linear mixed models in a general framework using MCEM-like algorithms. Concerning the application to forest trees, the proposed statistical modeling approach relies on the availability of climatic data. In the case where climatic data are not available, we are studying Markov and semi-Markov switching linear mixed models with year random effects common to all the trees to model the synchronous part of the growth fluctuations. With semi-Markov switching linear mixed models, the response variable is constrained to be approximately normally distributed. We are now studying the statistical methodology for semi-Markov switching generalized linear mixed models to take into account non-normally distributed response variables (e.g. number of growth units, apex death/life, non-flowering/flowering character). It should be noted that the estimation algorithms proposed for Markov switching linear mixed models can be directly transposed to other families of hidden Markov models such as, for instance, hidden Markov tree models; see Section 6.1.2 .

• Coupling stochastic models with mechanistic models for plant development simulation. (Damien Fumey, Yann Guédon, Christophe Godin, Thomas Cokelaer, Evelyne Costes, Pierre-Eric Lauri (UMR DAP))

  Arboricultural practices such as pruning, artificial bending or fruit thinning are crucial interventions in orchard management and are used for controlling tree size, penetration of light into the canopy and the equilibrium between vegetative and reproductive growth. The aim of the PhD of Damien Fumey is to explore the possibility of integrating such practices in a model of apple tree development. To this end, a field experiment was designed to study the effects of pruning (thinning or heading cuts) on two apple cultivars with contrasted architecture, 'Fuji' and 'Braeburn'. The first results of this experiment [26] showed that thinning cuts of laterals tended to be compensated by an increase in lateral branching. Based on these field experiments, a model is currently being developed to account for pruning practices on fruit trees. This model relies on a formalization of the competition of meristems by combining a carbon allocation strategy and a competition mechanism in a stochastic manner. The resulting model is reactive to human interventions and should enable us to capture plant reactions to pruning practices in a robust way.