Low rank priors for color image regularization

Abstract

In this work we consider the regularization of vectorial data such as color images. Based on the observation that edge alignment across image channels is a desirable prior for multichannel image restoration, we propose a novel scheme of minimizing the rank of the image Jacobian and extend this idea to second derivatives in the framework of total generalized variation. We compare the proposed convex and nonconvex relaxations of the rank function based on the Schatten-q norm to previous color image regularizers and show in our numerical experiments that they have several desirable properties. In particular, the nonconvex relaxations lead to better preservation of discontinuities. The efficient minimization of energies involving nonconvex and nonsmooth regularizers is still an important open question. We extend a recently proposed primal-dual splitting approach for nonconvex optimization and show that it can be effectively used to minimize such energies. Furthermore, we propose a novel algorithm for efficiently evaluating the proximal mapping of the ℓq norm appearing during optimization. We experimentally verify convergence of the proposed optimization method and show that it performs comparably to sequential convex programming.