Consistency of statistics in infinite dimensional quotient spaces

Loïc Devilliers 1, 2
2 ASCLEPIOS - Analysis and Simulation of Biomedical Images
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : In computational anatomy, organ shapes are assumed to be deformation of a common template. The data can be organ images but also organ surfaces, and the deformations are often assumed to be diffeomorphisms. In order to estimate the template, one often uses the max-max algorithm which minimizes, among all the prospective templates, the sum of the squared distance after registration between the data and a prospective template. Registration is here the step of the algorithm which finds the best deformation between two shapes. The goal of this thesis is to study this template estimation method from a mathematically point of view. We prove in particular that this algorithm is inconsistent due to the noise. This means that even with an infinite number of data, and with a perfect minimization algorithm, one estimates the original template with an error. In order to prove inconsistency, we formalize the template estimation: deformations are assumed to be random elements of a group which acts on the space of observations. Besides, the studied algorithm is interpreted as the computation of the Fréchet mean in the space of observations quotiented by the group of deformations. In this thesis, we prove that the inconsistency comes from the contraction of the distance in the quotient space with respect to the distance in the space of observations. Besides, we obtained a Taylor expansion of the consistency bias with respect to the noise level. As a consequence, the inconsistency is unavoidable when the noise level is high.
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Loïc Devilliers. Consistency of statistics in infinite dimensional quotient spaces. Other. Université Côte d'Azur, 2017. English. ⟨NNT : 2017AZUR4103⟩. ⟨tel-01683607v2⟩

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