Projection onto manifolds, manifold denoising & application to image segmentation with non linear shape priors

We introduce a non-linear shape prior for the deformable model framework that we learn from a set of shape samples using recent manifold learning techniques. We model a category of shapes as a finite dimensional manifold which we approximate using Diffusion maps, that we call the shape prior manifold. Our method computes a Delaunay triangulation of the reduced space, considered as Euclidean, and uses the resulting space partition to identify the closest neighbors of any given shape based on its Nystrom extension. Our contribution lies in three aspects. First, we propose a solution to the preimage problem and define the projection of a shape onto the manifold. We then introduce a shape prior term for the deformable framework through a non-linear energy term designed to attract a shape towards the manifold at given constant embedding. Finally, we describe a variational framework for manifold denoising, based on closest neighbors for the Diffusion distance. Results on shapes of cars and ventricule nuclei are presented and demonstrate the potentials of our method.