Geometric statistics for computational anatomy

Nina Miolane 1
1 ASCLEPIOS - Analysis and Simulation of Biomedical Images
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : This thesis develops Geometric Statistics to analyze the normal andpathological variability of organ shapes in Computational Anatomy. Geometricstatistics consider data that belong to manifolds with additional geometricstructures. In Computational Anatomy, organ shapes may be modeled asdeformations of a template - i.e. as elements of a Lie group, a manifold with agroup structure - or as the equivalence classes of their 3D configurations underthe action of transformations - i.e. as elements of a quotient space, a manifoldwith a stratification. Medical images can be modeled as manifolds with ahorizontal distribution. The contribution of this thesis is to extend GeometricStatistics beyond the now classical Riemannian and metric geometries in orderto account for these additional structures. First, we tackle the definition ofGeometric Statistics on Lie groups. We provide an algorithm that constructs a(pseudo-)Riemannian metric compatible with the group structure when itexists. We find that some groups do not admit such a (pseudo-)metric andadvocate for non-metric statistics on Lie groups. Second, we use GeometricStatistics to analyze the algorithm of organ template computation. We show itsasymptotic bias by considering the geometry of quotient spaces. We illustratethe bias on brain templates and suggest an improved algorithm. We then showthat registering organ shapes induces a bias in their statistical analysis, whichwe offer to correct. Third, we apply Geometric Statistics to medical imageprocessing, providing the mathematics to extend sub-Riemannian structures,already used in 2D, to our 3D images
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Nina Miolane. Geometric statistics for computational anatomy. Human health and pathology. Université Côte d'Azur, 2016. English. ⟨NNT : 2016AZUR4146⟩. ⟨tel-01411886v2⟩

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