Distortion Minimizing Geodesic Subspaces in Shape Spaces and Computational Anatomy

The estimation of finite dimensional nonlinear submanifold representing shape samples is of paramount importance in many applications. The Distortion Minimizing Geodesic Submanifold (DMGS) approach allows to select the most accurate submanifolds in term of distortion under a dimensionality constraint for shape spaces. We show that the computation of DMGS is widely compatible with the Large Deformation Diffeomorphic Metric Mapping (LDDMM) framework and the varifold distortion for application to computational anatomy. It allows the estimation of finite dimensional geodesic submanifolds in the difficult situation where we do not assume any one to one correspondance between shapes (parametrisation invariance). Unlike regular Tangent PCA, the computation of DMGS does not need to deal with the classical balance between the deformation cost from the template to target and the resulting distortion. On the contrary, the greedy minimization of the distortion under dimensionality constraints, hiding the deformation metric in the exponential map, suggests a new way to select between alternative metrics and shape spaces under the unifying point of view of the dimension/distortion curves in the spirit of the rate/distortion curves in information theory. Proof of concept on 2D and 3D experiments are discussed.