Infimal Convolution Coupling of First and Second Order Differences on Manifold-Valued Images

Abstract

Recently infimal convolution type functions were used in regularization terms of variational models for restoring and decomposing images. This is the first attempt to generalize the infimal convolution of first and second order differences to manifold-valued images. We propose both an extrinsic and an intrinsic approach. Our focus is on the second one since the summands arising in the infimal convolution lie on the manifold themselves and not in the higher dimensional embedding space. We demonstrate by numerical examples that the approach works well on the circle, the 2-sphere, the rotation group, and the manifold of positive definite matrices with the affine invariant metric.