Processing Data in Lie Groups : An Algebraic Approach. Application to Non-Linear Registration and Diffusion Tensor MRI

Vincent Arsigny 1
1 ASCLEPIOS - Analysis and Simulation of Biomedical Images
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : Recently, the need for rigorous frameworks for the processing of non-linear data has grown considerably in medical imaging. In this thesis, we propose several general frameworks to process various types of non-linear data, which all belong to Lie groups. To this end, we rely on the algebraic properties of these spaces. Thus, we propose a general processing framework for symmetric and positive-definite matrices, named Log-Euclidean, very simple to use and which has excellent theoretical properties. It is particularly well-adapted to the processing of diffusion tensor MRI. We also propose several frameworks, called polyaffine, to parameterize locally rigid or affine transformations, in a way that guarantees their invertibility. Their use is illustrated in the case of the locally rigid registration of histological slices and of the locally affine 3D registration of MRIs of the human brain. This led us to propose two general frameworks for computing statistics in finite-dimensional Lie groups: first the Log-Euclidean one, which generalizes our work on tensors, and second a framework based on the novel notion of bi-invariant mean, whose properties generalize to Lie groups those of the arithmetic mean. Finally, we generalize our Log-Euclidean framework to diffeomorphic geometrical transformations, which opens the way to a general and consistent framework for statistics in computational anatomy.
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Vincent Arsigny. Processing Data in Lie Groups : An Algebraic Approach. Application to Non-Linear Registration and Diffusion Tensor MRI. Human-Computer Interaction [cs.HC]. Ecole Polytechnique X, 2006. English. ⟨tel-00121162⟩

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