Reconciling landmarks and level sets: geometric shape warping and matching using generalized gradients and correspondence-augmented implicit representations

Shape warping is a key problem in statistical shape analysis. This paper proposes a framework for geometric shape warping based on both shape distances and landmarks. Taking advantage of the recently proposed spatially coherent flows, our method is mathematically well-posed and uses only intrinsic shape information, namely some similarity measure between shapes and the correspondence of landmarks provided on the shape surface. No extrinsic quantity is considered, neither a diffeomorphism of the embedding space nor point correspondences in this space. Thanks to a recent extension of the level set method allowing point tracking and tangential velocities, our method is compatible with implicit representations. Moreover, a matching between shape surfaces is provided at no additional cost. Although some recent work deals with implicit representations and landmarks, it is, to our knowledge, the first time that landmarks and shape distances are reconciled in a pure geometric level set framework. The feasibility of the method is demonstrated with two- and three-dimensional examples. Combining shape distance and landmarks, our approach reveals to need only a small number of landmarks to obtain improvements on both warping and matching.