Priors with Coupled First and Second Order Differences for Manifold-Valued Image Processing

We generalize discrete variational models involving the infimal convolution (IC) of first and second order differences and the total generalized variation (TGV) to manifold-valued images. We propose both extrinsic and intrinsic approaches. The extrinsic models are based on embedding the manifold into an Euclidean space of higher dimension with manifold constraints. An alternating direction methods of multipliers can be employed for finding the minimizers. However, the components within the extrinsic IC or TGV decompositions live in the embedding space which makes their interpretation difficult. Therefore, we investigate two intrinsic approaches: for Lie groups, we employ the group action within the models; for more general manifolds, our IC model is based on recently developed absolute second order differences on manifolds, while our TGV approach uses an approximation of the parallel transport by the pole ladder. For computing the minimizers of the intrinsic models, we apply gradient descent algorithms. Numerical examples demonstrate that our approaches work well for certain manifolds.