An Iterative Algorithm for Nonlinear Inverse Problems with Joint Sparsity Constraints in Vector Valued Regimes and an Application to Color Image Inpainting

This paper is concerned with nonlinear inverse problems where data and solution are vector valued and, moreover, where the solution is assumed to have a sparse expansion with respect to a preassigned frame. We especially focus on such problems where the different components of the solution exhibit a common or so–called joint sparsity pattern. Joint sparsity means here that the measure (typically constructed as weighted `1 norms of componentwise `q norms of the frame coefficients) promotes a coupling of non–vanishing components. Quite recently, an iterative strategy for linear inverse problems with such joint sparsity constraints was presented. Here we develop an iterative concept for nonlinear inverse problems with joint sparsity constraints for which we show convergence and regularization properties. Moreover, we demonstrate the capabilities of the proposed algorithm in the field of color image inpainting. MSC: 15A29, 47A52, 68U10, 94A08, 94A40