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Theses

Analysis of geometric and functional shapes with extensions of currents : applications to registration and atlas estimation

Abstract : This thesis addresses several questions related to the recent field of computational anatomy. Broadly speaking, computational anatomy intends to analyse shape variability among populations of anatomical structures. In this work, we are focused, in the first place, on the case of datasets of curves, surfaces and more generally submanifolds. Our goal is to provide a mathematical and numerical setting to build relevant data attachment terms between those objects in the purpose of embedding it into the large diffeomorphic metric mapping (LDDMM) model for shape registration. Previous approaches have been relying on the concept of currents that represents oriented submanifolds. We first propose an extension of these methods to the situation of non-oriented shapes by adapting the concept of varifolds from geometric measure theory. In the second place, we focus on the study of geometrico-functional structures we call 'functional shapes' (or fshapes), which combine varying geometries across individuals with signal functions defined on these shapes. We introduce the new notion of fshape metamorphosis to generalize the idea of deformation groups in the pure geometrical case. In addition, we define the extended setting of 'functional currents' to quantify dissimilarity between fshapes and thus perform geometrico-functional registration between such objects. Finally, in the last part of the thesis, we move on to the issue of analyzing entire groups of individuals (shapes or fshapes) together. In that perspective, we introduce an atlas estimation variational formulation that we prove to be mathematically well-posed and build algorithms to estimate templates and atlases from populations, as well as tools to perform statistical analysis and classification. All these methods are evaluated on several applications to synthetic datasets on the one hand and real datasets from biomedical imaging on the other.
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https://tel.archives-ouvertes.fr/tel-00942078
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Submitted on : Tuesday, February 4, 2014 - 4:17:09 PM
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Nicolas Charon. Analysis of geometric and functional shapes with extensions of currents : applications to registration and atlas estimation. General Mathematics [math.GM]. École normale supérieure de Cachan - ENS Cachan, 2013. English. ⟨NNT : 2013DENS0045⟩. ⟨tel-00942078⟩

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