Total variation regularization on Riemannian manifolds by iteratively reweighted minimization

We consider the problem of reconstructing an image from noisy and/or incomplete data. The values of the pixels lie on a Riemannian manifold M, e.g. R for a grayscale image, S-2 for the chromaticity component of an RGB-image or SPD(3), the set of positive definite 3 x 3 matrices, for diffusion tensor magnetic resonance imaging. We use the common technique of minimizing a total variation functional J. To this end we propose an iteratively reweighted minimization (IRM) algorithm, which is an adaption of the well-known iteratively reweighted least squares algorithm, to minimize a regularized functional J epsilon, where epsilon > 0. For the case of M being a Hadamard manifold we prove that J epsilon has a unique minimizer, and that IRM converges to this unique minimizer. We further prove that these minimizers converge to a minimizer of J if epsilon tends to zero. We show that IRM can also be applied for M being a half-sphere. For a simple test image it is shown that the sequence generated by IRM converges linearly. We present numerical experiments where we denoise and/or inpaint manifold-valued images, and compare with the proximal point algorithm of Weinmann et al. (2014, SIAM J. Imaging Sci., 7, 2226-2257). We use the Riemannian Newton method to solve the optimization problem occurring in the IRM algorithm.