A Second Order Non-Smooth Variational Model for Restoring Manifold-Valued Images

Miroslav Bačák, Ronny Bergmann, Gabriele Steidl, Andreas Weinmann

Published 2015-06-08Version 1

We introduce a new non-smooth variational model for the restoration of manifold-valued data which includes second order differences in the regularization term. First, we establish a suitable definition of absolute second order differences for signals and images with values in a manifold. Employing this definition, we introduce a variational denoising model based on first and second order differences in the manifold setup. In order to minimize the corresponding functional, we develop an algorithm using an inexact cyclic proximal point algorithm. We propose an efficient strategy for the computation of the corresponding proximal mappings in symmetric spaces utilizing the machinery of Jacobi fields. For the $n$-sphere and the manifold of symmetric positive definite matrices, we demonstrate the performance of our algorithm in practice. We prove the convergence of the proposed exact and inexact variant of the cyclic proximal point algorithm in Hadamard spaces. These results which are of interest on its own include, e.g., the manifold of symmetric positive definite matrices.