Geodesic Shooting and Diffeomorphic Matching Via Textured Meshes

We propose a new approach in the context of diffeomorphic image matching with free boundaries. A region of interest is triangulated over a template, which is considered as a grey level textured mesh. A diffeomorphic transformation is then approximated by the piecewise affine deformation driven by the displacements of the vertices of the triangles. This provides a finite dimensional, landmark-type, reduction for this dense image comparison problem. Based on an optimal control model, we analyze and compare two optimization methods formulated in terms of the initial momentum: direct optimization by gradient descent, or root-finding for the transversality equation, enhanced by a preconditioning of the Jacobian. We finally provide a series of numerical experiments on digit and face matching.