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. $\mathbb {R}$ for a grayscale image, $S^2$ for the chromaticity component of an RGB-image or ${\rm SPD}(3)$ , the set of positive definite $3\times 3$ 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.