Statistical models of currents for measuring the variability of anatomical curves, surfaces and their evolution. (Modèles statistiques de courants pour mesurer la variabilité anatomique de courbes, de surfaces et de leur évolution)

This thesis is about the definition, the implementation and the evaluation of statistical models of variability of curves and surfaces based on currents in the context of Computational Anatomy. Currents were recently introduced in medical imaging in order to define a metric between curves and surfaces which does not assume point correspondence between structures. This metric was used to drive the registration of anatomical data. In this thesis, we propose to extend this tool to analyze the variability of anatomical structures via the inference of generative statistical models. Besides the definition and discussion of these models, we provide a numerical framework to deal efficiently with their estimation. Several applications on real anatomical database in brain and cardiac imaging tend to show the generality and relevance of the approach. In the first part of the manuscript, we extend the computational framework of currents by introducing new numerical tools for approximation and compression purposes. First, a rigorous discretization framework based on linearly spaced grids is provided: it enables to give finite-dimensional projection of currents which converges to the initial continuous representation as the grids become finer. This leads to a generic way to derive robust and efficient algorithms on currents, while controlling the numerical precision. This gives for instance a more stable numerical implementation of the registration algorithm of currents. Then, we define an approximation algorithm which gives a sparse representation of any currents at any desired accuracy via the search of an adapted basis for currents decomposition. This sparse representation is of great interest to compress large sets of anatomical data and to give interpretable representation of statistics on such data sets. In the second part, we define an original statistical model which considers a set of curves or surfaces as the result of random deformations of an unknown template plus random residual perturbations in the space of currents. The inference of such models on anatomical data enables to decompose the variability into a geometrical part (captured by diffeomorphisms) and a "texture" part (captured by the residual currents). This approach allows us to address three anatomical problems: first, the analysis of variability of a set of sulcal lines is used to describe the variability of the cortex surface, second, the inference of the model on set of white matter fiber bundles shows that both the geometrical part and the texture part may contain relevant anatomical information and, third, the variability analysis is used in a clinical context for the prediction of the remodeling of the right ventricle of the heart in patients suffering from Tetralogy of Fallot. In the third part, we define statistical models for shape evolution. First, we define a spatiotemporal registration scheme which maps the sets of longitudinal data of two subjects. This registration does not only account for the morphological differences between subjects but also for the difference in terms of speed of evolution. Then, we propose a statistical model which jointly estimates a mean scenario of evolution from a set of longitudinal data along with its spatiotemporal variability in the population. This four-dimensional analysis opens up new possibilities for characterizing pathologies in terms of variations of the growth process of anatomical structures.