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Computational methods for statistical estimation on Riemannian manifolds and application to the study of the cardiac deformations

Nicolas Guigui 1 
Abstract : The study of anatomical shapes and deformations is of major interest in cardiology where diseases such as arrhythmia or pulmonary hypertension cause abnormalities in the way the heart contracts or of its shape. The characterization of these abnormalities allows to evaluate the gravity of the disease or the impact of a treatment. A mathematical framework that accounts for the nonlinearity and invariance properties of shape spaces and shape deformations is needed to obtain such characterizations, perform population statistics and disentangle the two factors of variability: shape and deformation.Riemannian manifolds are a natural setting that fill in these desiderata, and parallel transport is commonly used to normalize the deformations of the cerebral cortex, allowing to decorrelate its evolution from the initial shape. More generally, data lying on Riemannian manifolds are ubiquitous, yet, the development of computational tools from the basic theory of Riemannian geometry is tedious. One of the aims of this thesis is to provide the computational tools to perform statistics and machine learning on Riemannian manifolds to the community. The work presented here forms one of the main contributions to the open-source project geomstats, that consists in a Python package providing efficient implementations of the concepts of Riemannian geometry, both for mathematicians and for applied scientists for whom most of the difficulties are hidden under high-level functions.An exposition of Riemannian geometry is first given, with illustrations and examples. We adopt a computational point of view to then demonstrate how these concepts are implemented in geomstats, explaining the choices made during its development. The use of geomstats to perform statistics on manifolds is briefly exemplified.Particular attention is given to the implementation of invariant metrics on Lie groups for which a new formulation of the parallel transport is given, and on a general implementation of quotient metrics with applications to Kendall shape spaces. Our new implementation of parallel transport in these spaces outperforms the state of the art in speed and precision.We then focus on the study of ladder methods that consist in reproducing the construction of small geodesic parallelograms to approximate parallel transport on manifolds. These are now used in applications such as robotics, medical imaging or computer vision. However, the literature lacks a clear analysis of their convergence performance. We give Taylor approximations of the constructions with respect to the curvature of the space and prove that these methods can be iterated to converge with quadratic speed, even when geodesics are approximated by numerical schemes. This rate is precisely observed on common manifolds. The special Euclidean group with an anisotropic invariant metric is of particular interest to illustrate the effect of the covariant derivative of curvature.Finally, we apply these results to the problem of studying cardiac deformations. We use the Large Deformation Diffeomorphic Metric Mapping framework and parallel transport to reorient deformations of the right-ventricle. However, we find undesirable effects of the large volume differences that occur between patients and the control group. We propose a normalization procedure for the amplitude of the deformation by a scaling step that conserves the relative volume change. This approach proves effective, and we exhibit a significant relation between the volume changes and the scaling parameter. The normalized deformations are then studied by summarizing each deformation with a spline regression and performing statistics on the parameters of the splines that reveal significant differences between each disease and the control group, reflecting the dynamics of each disease.
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Submitted on : Thursday, February 10, 2022 - 10:03:09 AM
Last modification on : Tuesday, October 25, 2022 - 4:22:58 PM

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Nicolas Guigui. Computational methods for statistical estimation on Riemannian manifolds and application to the study of the cardiac deformations. Differential Geometry [math.DG]. Université Côte d'Azur, 2021. English. ⟨NNT : 2021COAZ4081⟩. ⟨tel-03563980v2⟩

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