Résumé en anglais

This article studies regularization schemes that are defined using a lifting of the image pixels in a high dimensional space. For some specific classes of geometric images, this discrete set of points is sampled along a low dimensional smooth manifold. The construction of differential operators on this lifted space allows one to compute PDE flows and perform variational optimizations. All these schemes lead to regularizations that exploit the manifold structure of the lifted image. Depending on the specific definition of the lifting, one recovers several well-known semi-local and non-local denoising algorithms that can be interpreted as local estimators over a semi-local or a non-local manifold. This framework also allows one to define thresholding operators in adapted orthogonal bases. These bases are eigenvectors of the discrete Laplacian on a manifold adapted to the geometry of the image. Numerical results compare the efficiency of PDE flows, energy minimizations and thresholdings in the semi-local and non-local settings. The superiority of the non-local computations is studied through the performance of non-linear approximation in orthogonal bases.