Convex Relaxation of Vectorial Problems with Coupled Regularization

We propose convex relaxations for nonconvex energies on vector-valued functions which are tractable yet as tight as possible. In contrast to existing relaxations, we can handle the combination of nonconvex data terms with coupled regularizers such as l-regularizers. The key idea is to consider a collection of hypersurfaces with a relaxation that takes into account the entire functional rather than separately treating the data term and the regularizers. We provide a theoretical analysis, detail the implementations for different functionals, present run time and memory requirements, and experimentally demonstrate that the coupled l-regularizers give systematic improvements regarding denoising, inpainting, and optical flow estimation.