Projet de recherche doctoral numero :4627

Date depot: 1 janvier 1900
Titre: Learning methods for the spatiotemporal analysis of longitudinal data
Directeur de thèse: Stanley DURRLEMAN (ICM)
Domaine scientifique: Sciences et technologies de l'information et de la communication
Thématique CNRS : Non defini

Resumé: Keywords: statistical learning, spatiotemporal data, longitudinal data sets, personalization, classification, prediction, model of disease progression, brain imaging, neurological diseases Context: Longitudinal data sets are often acquired in biological and medical sciences to capture variable temporal phenomena, which are due for instance to growth, ageing or disease progression. They consist in the observation of several individuals, each of them being observed at multiple points in time. The statistical exploitation of such data sets is notably difficult since data of each individual follow a different trajectory of changes and at its own pace. This difficulty is further increased if observations take the form of structured data like images or measurements distributed at the nodes of a mesh, and if the measurements themselves are normalized data or positive definite matrices for which usual linear operations are not defined. Our team has contributed to the definition of a generic theoretical and algorithmic framework for learning typical trajectories from longitudinal data sets [1, 2, 3]. This framework is built on tools from the Riemannian geometry to describe trajectories of changes for any kind of data and their variability within a group. The inference is based on a stochastic EM algorithm coupled with Markov Chain simulation methods. The framework has been used so far to describe the dynamics of a set of biomarkers, namely unstructured data sets, which evolve in a sequential manner. Topic: The goal of the thesis is to extend this theoretical framework to deal with much more challenging data sets, called iconic-geometric data sets. These data take the form of measurements that are spatially distributed on a sub-manifold of any dimension (curve, surface or volume). In a discrete form, such a data takes the form of a measurement value, such as an intensity, a symmetric definite positive matrix or a ratio for instance, which is attached at each node of a geometric mesh. In brain imaging, the mesh may represent the cortical surface, where nodes represent precise anatomical locations, and signals may be the thickness of the cortical ribbon or a functional signal measuring glucose consumption at each anatomical location. The first objective of the thesis is to define a computational framework to track changes in such data sets. The main difficulty is that both the shape of the geometric support and the values of the measurements mapped onto the mesh vary at the same time. To this end, the framework of functional currents seems particularly interesting [4,5], as it defines a metric between iconic- geometric data sets built on a diffeomorphic deformation of the mesh to match the two geometric supports and a square integrable variation of the signal to match the measurements. The deformation and variation together follow a geodesic path on a certain Riemannian manifold of high- dimension. This framework needs to be adapted though, since, in our case, the variation of the measurement may not be defined by the addition of a square integrable function. Measurements like positive definite matrices live also on a Riemannian manifold and the variation should then take the form of a geodesic on this manifold. Furthermore, we will introduce structured variations, so that signal variations propagate along neighboring nodes, and not independently from each other. The second objective of the thesis is to include this computational model for iconic-geometric data set into the generic statistical model for longitudinal data that we have defined in [3]. In principle, this step is straightforward given that we provide the space of iconic-geometric data with a Riemannian structure. Nevertheless, several approximations will be investigated to make the inference tractable. In particular, we would need to derive an efficient numerical scheme to compute parallel transport on a manifold by borrowing ideas from [6], and an efficient computation of geodesic distances in a mesh that is continuously deformed. All in one, this computational and statistical framework will allow us to estimate typical trajectories of data changes from iconic-geometric data sets acquired in several individuals at multiple points in time. Extensions of this framework will include mixture models, which will allow us to estimate a family of typical trajectories indicative of clusters of individuals with similar spatiotemporal patterns. The thesis also aims to personalize the typical scenarios to data of new individuals in order to position them along a typical scenario and to predict their future evolution. References: [1] S. Durrleman, X. Pennec, A. Trouvé, J. Braga, G. Gerig, N. Ayache, Toward a comprehensive framework for the spatiotemporal statistical analysis of longitudinal shape data, International Journal of Computer Vision 103 (1), 22-59, 2013 [2] J.-B. Schiratti, S. Allassonnière, A. Routier, O. Colliot, S. Durrlem

Doctorant.e: Louis Maxime