Evolution equations on the group of diffeomorphisms, with applications in computational anatomy

Computational Anatomy (CA) introduces the idea of comparing anatomical structures via geodesic deformations generated by diffeomorphisms, to measure and analyze variations between healthy and diseased organs. Among these structures, landmarks and image outlines in CA have been shown to be singular solutions of the geodesic Euler-Poincare equation on diffeomorphisms, or EPDiff. This setting shares several similarities with the mechanics of perfect fluids, both involving a right-invariant stationary principle with an action in the fluid dynamics case, and a cost function in the CA case. Moreover, the momentum map for singular solutions of EPDiff yields a canonical Hamiltonian formulation, which provides a complete parametrization of the landmarks by their canonical positions and momenta.;In this thesis a model is proposed to speed up the calculation of EPDiff defined on sets of points representing smooth surfaces, involving a trilinear and tricubic interpolation of the time-dependent vector field. Special attention has been devoted to the conservation of its geometric properties. In particular, the linear and angular momentum preservation property of the trilinear model when using a scalar kernel is discovered, and proved.;The evolution of smooth closed curves in two dimensions under the action of a subgroup of the diffeomorphism group is also analyzed, and a model to approximate EPDiff on curves is presented. Additionally, bounds for the error of the approximation are estimated and an analysis of the difficulties of simulating this type of evolution is presented.;Finally, a study of the minimization of a segmentation energy defined on deformable sets is proposed. The general framework is reminiscent of variational shape optimization methods but remains closer to general developments on the deformable template theory of geometric flows. The metric that provides the gradient descent method is the same used to model EPDiff , defined on the tangent space at the identity of the group of deformations. To avoid local solutions and to mitigate the influence of several sources of noise, a finite set of control points is defined on the boundary of the template binary images, yielding a projected gradient descent on Diff.