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 and ARC projects named PlantScan3D.

Virtual 3D model of plants are required in many areas of plant modeling. 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 architectures in 3D [42] , [33] . 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 aim, we explore the use of pictures, video, laser scanner and direct sketching.

• Reconstruction of plant architecture from 3D laser scanner data. (Chakkrit Preuksakarn, Mathilde Balduzzi, Frédéric Boudon, Jean-Baptiste Durand, Christophe Godin, Xinghua Song [INRIA, Galaad], Bernard Mourrain [INRIA, Galaad], Dobrina Boltcheva [INRIA, Imagine], Franck Hetroy [INRIA, Morpheus], Marie-Paule Cani [Inria, Imagine], Pascal Ferraro [Labri, Bordeaux] )

  We investigate the possibility to use 3D laser scanners to automate plant digitizing. We are developping algorithms to reconstruct branching systems without leaves or foliage 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 Imagine from Grenoble, different INRA teams, UMR PIAF in Clermont Ferrand, UMR LEPSE and AFEF team in Montpellier and Lusignan, the University of Helsinki, Finland and the CFCC in England.

  We developed a reconstruction pipeline composed of several procedures. A contraction procedure, first aggregates points at the center of the point cloud. The team proposed a simple adaptive scheme to contract points. In parallel, the Galaad team explored uses of detection of circular patterns to be contracted toward their center. Comparison of these approaches has been carried out. In a second step, a skeleton procedure uses a Space Colonization Algorithm [41] to build the skeleton of the shape from the contracted point set. This method is adaptive to the local density of the point set. Then a pipe-model based procedure makes it possible to estimate locally diameters of the branches. Finally, an evaluation procedure has been designed to assess the accuracy of the reconstruction. Results Publication of this work is in progress.

  Figure 3. Reconstruction of a cherry tree. Left: photograph of the original tree. Right: 3D reconstruction from a laser scan rendered and integrated on the same background.

• Sketching of plants. (Frédéric Boudon, Christophe Godin, Steven Longuay [University of Calgary, Canada], Przemyslaw Prusinkiewicz [University of Calgary, Canada])

  Modeling natural elements such as trees in a plausible way, while offering simple and rapid user control, is a challenge. In a first collaboration with the EPI Evasion we developed a method based on the design of plants from silhouettes [43] . This sketching paradigm allows quick and intuitive specification of foliage at multiple scales. On this topic, we started a collaboration with S. Longuay and P. Prusinkiewicz who develop iPad tools to design plants based on SCA. Combination of multitouch interface, sketching paradigm and powerfull adaptive procedural model that generate realistic trees offer intuitive and flexible design tools. This work is part of the INRIA associated team with the University of Calgary.

• Reconstruction of vineyards from video. (Frédéric Boudon, Jerome Guenard [IRIT, Toulouse], Geraldine Morin [IRIT, Toulouse], Pierre Gurdjos [IRIT, Toulouse], Vincent Charvillat [IRIT, Toulouse])

  In this work, we investigate the reconstruction of constrained plant geometry of a vineyard from a set of pictures coming from video. Pictures are segmented to identify the different trees of a same row in the yard. From this segmentation, a number of parameters are estimated, which makes it possible to instantiate a virtual model of a vine tree. In particular, paths of main branches and leaf volumes and densities are estimated. A preliminary version of this work has been presented to the AFIG conference [21] .

• Reconstruction of virtual fruits from pictures. (Mik Cieslak, Nadia Bertin [Inra, Avignon], Frédéric Boudon, Christophe Godin, Christophe Pradal, Michel Genard [Inra, Avignon], Christophe Goz-Bac [Université Montpellier 2])

  This research theme is supported by the Agropolis project Fruit3D.

  Understanding the controlling factors of fruit quality development is challenging, because fruit quality is the result of the interplay between physical and physiological processes that are under the control of genes and the environment. Although process-based models have been used to make significant progress in understanding these factors, they to a large extent ignore the shape and internal structure of the fruit. To help characterizing effects of fruit shape and internal structure on quality, the creation of a 3D virtual fruit model that integrates fruit structure and function with growth governed by environmental inputs has been investigated. For this purpose, a modelling pipeline has been created that includes the following steps: creation of a 3D volumetric mesh of the internal and external fruit structure, calculation of the fruit’s physical properties from the resulting mesh, and integration of aspects of fruit physiology into the 3D structure. This pipeline has been applied to study tomato fruit (Solanum lycopersicum) by constructing 3D volumetric meshes from two images of perpendicular fruit slices and from MRI data, and integrating water and carbon transport processes into one of these meshes. To illustrate the tomato model, a simulation of one season’s of the fruit’s growth has been performed and its results compared with an already published process-based tomato fruit model. The results of the two models were in general agreement, but our model provided additional information on the internal properties of the fruit, such as a gradient in sugar concentration. Once the model is calibrated and evaluated, our approach will be suitable for studying the effects of internal fruit heterogeneity and overall shape on fruit quality development [18] .

Modeling the plant ontogenic programme

Participants : Christophe Godin, Yann Guédon, Evelyne Costes, Jean-Baptiste Durand, Pascal Ferraro, Anaëlle ambreville, Christophe Pradal, Catherine Trottier, Jean Peyhardi, Yassin Refahi, Etienne Farcot.

This research theme is supported by two PhD programmes.

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. [23] , [35] , [36] , [32] , 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 organizations such as "morphogenetic programme" [38] , "age state" [31] or "physiological age" [24] . 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 [31] , [24] . Here we develop computational methods to decipher these rules.

• Branching and axillary flowering structures of fruit tree shoots. (Yann Guédon, Evelyne Costes, David Da Silva [UC Davis], Anna Davidson [UC Davis], Ted DeJong [UC Davis], Claudia Negron [UC Davis]).

  In the context of a collaboration with Claudia Negron, Anna Davidson, 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 corresponding to different genetic backgrounds, environment conditions and horticultural practices 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 [40] , [3] ; see 6.1.3 .

• Genetic determinisms of the alternation of flowering in apple tree progenies. (Jean-Baptiste Durand, Jean Peyhardi, Baptiste Guitton [DAP, AFEF team], Catherine Trottier, Evelyne Costes, Yann Guédon)

  Previous approaches for a statistical quantification of the effect of factors on tree architecture were mainly oriented toward the structure of main axes, and environmental explanatory variables (see [34] , [27] ). To characterize genetic determinisms of the alternation of flowering in apple tree progenies at annual shoot (AS) scale, a model of the transitions between ASs was built. The ASs were of two types: flowering or vegetative. Two replications of each genotype were available. Our model operated on tree-structured data and relied on a second-order Markov tree. Generalized Linear Mixed Models (GLMMs) were used to model the effet of year, replications and genotypes (with their interactions with year or memories of the Markov model) on the transition probabilities. This work was the continuation of the Master 2 internship of Jean Peyhardi (Bordeaux 2 University) and was carried out in the contex of the PhD thesis of Baptiste Guitton.

  This PhD thesis also comprised the study of alternation in flowering at individual scale, with annual time step. To relate alternation of flowering at AS and individual scales, indices were proposed to characterize alternation at individual scale. The difficulty is related to early detection of alternating genotypes, in a context where alternation is often concealed by a substantial increase of the number of flowers over consecutive years. To separate correctly base effect from alternation in flowering, our model relied on a parametric hypothesis on the base effect (random slopes specific to genotype and replications), which translated into mixed effect modeling. Different indices of alternation were then computed on the residuals. Clusters of individuals with contrasted patterns of bearing habits were identified. Our models highlighted significant correlations between indices of alternation at AS and individual scales. The roles of local alternation and asynchronism in regularity of flowering were assessed using an entropy-based criterion, which characterized asynchronism.

  As a perspective of this work, patterns in the production of children ASs (numbers of flowering and vegetative children) depending on the type of the parent AS must be analyzed using branching processes and different types of Markov trees, in the context of Pierre Fernique's PhD Thesis.

• Modeling branching patterns in fruit tree shoots through the characterization of their demographic properties (Pierre Fernique, Jean-Baptiste Durand, Yann Guédon).

  To test the effect of some properties of a given parent shoot on the properties of its children shoots, statistical models based on multitype branching processes were developed. This kind of dependence between parent and children shoots is frequently at stake in fruit trees, for which the number of flowering or vegetative children of a parent shoot depends on its nature, with potential interactions with other factors. Thus, controlling demographic patterns of the shoots (through varietal selection or crop management strategies) is expected to bring substantial improvements in the quantity and quality of yields.

  Formally, the shoot properties are summed up using the notion of shoot state. The number of children shoots in each state is modeled through discrete multivariate distributions. Model selection procedures are necessary to specify parsimonious distributions. We developed an approach based on probabilistic graphical models to identify and exploit properties of conditional independence between numbers of children in different states, so as to simplify the specification of their joint distribution. The graph building stage was based on a Poissonian Generalized Linear Model for the contigency tables of the counts of joint children state configurations. Then, parametric families of distributions were implemented and compared statistically to provide probabilistic models compatible with the estimated independence graph.

  This work was carried out in the context of Pierre Fernique's Master 2 internship (Montpellier 2 University and AgroParisTech). It was applied to model dependencies between short or long, vegetative or flowering shoots in apple trees. The results highlighted contrasted patterns related to the parent shoot state, with interpretation in terms of alternation of flowering. This work will be continued during Pierre Fernique's PhD thesis, with extensions to other fruit tree species (mango trees) and other strategies to build probabilistic graphical models and parametric discrete multivariate distributions including covariates and mixed effects.

• Analyzing fruit tree phenology (Anaëlle Dambreville, Jean-Baptiste Durand, Yann Guédon, Christophe Pradal, Pierre-Eric Lauri [UMR AGAP], Frédéric Normand [UPR HortSys], Catherine Trottier)

  Mango is a tropical tree characterized by strong asynchronisms within and between trees. Causation networks explaining the vegetative and reproductive growths within and between growing cycles were studied on the basis of generalized linear models. We highlighted in this way marked interplays between structural and temporal components of tree structure development at three scales. At growth unit scale, a growth unit appeared early in the growing cycle had higher rate of burst compared to late appeared growth units. At growing cycle scale, a growth unit which flowered delayed its future vegetative growth compared to a vegetative growth unit. At tree scale, a fruiting tree delayed further vegetative growth and flowering compared to a non-fruiting tree. These results evidenced that tree phenology is strongly affected by structural components and not only by the environment.

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

  To study the redundancy of structures 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 want to find a self-nested tree that minimizes the distance of the original tree to the set of self-nested trees that embed the initial tree:

  ${\displaystyle NEST\left(T\right)=\underset{S\in {𝒮}^{+}\left(T\right)}{argmin}D(T,S),}$

  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 ${𝒮}^{+}\left(T\right)$ is the set of self-nested trees that contain $T$, i.e. that can be obtained from $T$ by inserting nodes only. In a previous work we showed that this problem can be solved in polynomial time and gave the corresponding algorithm [6] .

  Now, we continue this work along different directions:

  • Approximate compression including geometry (Anne-Laure Gaillard, Pascal Ferraro, Frédéric Boudon, Christophe Godin)

    we now investigate how to include branch geometry in the compression schemes. This problem constitutes a part of the PhD thesis of Anne-Laure Gaillard co-supervised with P. Ferraro. First results show that any tree architecture can be compressed at different degrees with a varying loss in the geometric information. A publication on this topic is currently in progress.

  • Search for the NST (Farah Ben-Naoum, Christophe Godin, Pascal Ferraro) The NEST algorithm constructs a closest self-nested tree that embeds a given tree $T$. This means in particular that $T\in NEST\left(T\right)$. For some trees this definition might be too restrictive and one may want to get rid of the latter constraint. In this case, the closest self-nested tree (NST) to a tree $T$ would be defined by:

    ${\displaystyle NST\left(T\right)=\underset{S\in 𝒮\left(T\right)}{argmin}D(T,S),}$

    where $𝒮\left(T\right)$ is now the set of all self-nested trees. Although it has not yet been demonstrated, the computation of $NST\left(T\right)$ is likely to be a NP-complete problem. Therefore we decided to develop a heuristic algorithm to carry out this optimization. This algorithm is based on evolutionary optimization algorithms. First results obtained at the University of Sidi Bel Abbes by Farah Ben Naoum are very encouraging. They should contribute to define a general method to compress tree-like structures based on self-similarity.

  • Stochastic NESTs (Jean-Baptiste Durand, Christophe Godin, Pascal Ferraro, Yann Guédon)

    One of the limits of the original NEST algorithm in defining a compressed structure is that it does not take into account possible small variations in the identification of similar tree patterns in the global tree structure. In real trees, it is however seldom the case that similar tree parts are exactly isomrophic. They may differ by small details while keeping globally similar. A general compression technique should account for such variations from a theoretical point of view. This is what we intend to do here by developing a stochastic NEST approach. The idea is to extend the definition of tree compression to include the notion of tree distribution in compressed representations of trees. A first formalism has been designed for stochastic tree compression.

• Statistical characterization of the branching properties of trees at individual scale – application to the quantification of approximate self-nestedness. (Jean-Baptiste Durand, Yann Guédon, Christophe Godin)

  To test different hypotheses related to the role of apical control on local branching properties within tree architecture, statistical models of the fates of the apical meristem and its production at a given scale were specified. They extend the hidden Markov tree models [29] to variable number of children of a given entity, and provide a model for their dependencies. Moreover, they allow for the comparison of different hypotheses regarding the relevance of the children ordering (absence of ordering, partial or total ordering). In the next years, these models will be used on mango and apple tree cultivars. Particularly, they will found new approaches to compare the effect of various strategies of culture.

  These models are based on a notion of state that extends the concept of equivalence classes for tree isomorphism used in the above paragraph, to the notion of classes of approximate isomorphism (i.e. isomorphism as a stochastic process). As a consequence, the hidden Markov tree models offer new insight for lossy compression of trees, which will be investigated in future work.

• Analyzing perturbations in Arabidopsis thaliana phyllotaxis. (Yann Guédon, Yassin Refahi, Etienne Farcot, 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 in the cytokinin signaling pathway. At the macroscopic scale, this mutation induces perturbations of the phyllotaxis, barely sensible on single plants. In order to characterize these perturbations, we designed a pipeline of models and methods [20] which decompose into three steps: (i) identification of perturbation patterns, (ii) characterization of perturbation patterns using hidden variable-order Markov chains and combinatorial mixture models both with von Mises observation distributions (Gaussian-like periodic distribution for circular variables), (iii) classification of plant phyllotaxis among wild-type and ahp6 mutant phyllotaxies. Using this pipeline of methods, we have shown that the perturbation patterns in both wild-type and mutant plants can be explained by permutations in the order of insertion along the stem of 2 or 3 consecutive organs. The number of successive synchronized organs between two permutations reveals unexpected patterns that depend on the nature of the preceding permutation (2- or 3-permutation). We identified significant individual deviations of the level of baseline segments with reference to 137.5°, which confirms theoretical model predictions. Finally, we highlighted a marked relationship between permutation of organs and defects in the elongation of the internodes in between these organs. All these results can be explained by the absence of a strict coupling between the timing of organ development and their angular and longitudinal position on the stem. A paper about these results is in revision for the journal Science. Another more methodological paper is in progress.

Analyzing the influence of the environment on the plant ontogenic programme

Participants : Jean-Baptiste Durand, Damien Fumey, Frédéric Boudon, Christophe Godin, Yann Guédon, Jean Peyhardi, Pierre Fernique, Christian Cilas, Evelyne Costes, Pascal Ferraro, 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 modulation 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 variables 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. (Yann Guédon, Jean-Baptiste Durand, Jean Peyhardi, Yves Caraglio [AMAP], 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 [34] , (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 Markov and semi-Markov switching linear mixed models [27] [2] . The underlying Markov or semi-Markov chain represents the succession of growth phases (endogenous component) while the linear mixed model attached to each state of the underlying Markov or 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. These integrative statistical models were in particular applied in a forest ecology context to characterize the opportunistic development of understory samplings in relation to light environment [15] . 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 Markov and semi-Markov switching linear mixed models, the response variable is constrained to be approximately normally distributed. We are now studying the statistical methodology for Markov and 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 AGAP])

  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. In the PhD of Damien Fumey we explored 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'. Results of this experiment [30] 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 [28] . A paper describing this model is currently in progress.