Image inpainting using sparse multiscale representations
Demetrio Labate
Abstract: Image inpainting is concerned with the task of recovering missing blocks of data in images or videos. Among the several strategies proposed in the literature, sparse multiscale representations offer a powerful approach to analyze the image inpainting problem using a rigorous mathematical formulation. Here, we investigate inpainting in the continuous domain as a function interpolation problem in a Hilbert space, where only a masked version of an unknown image is known. Under the assumption that the unknown image is sparse with respect to an appropriate representation, we search for the sparsest admissible solution. Since images found in many applications are dominated by edges, we consider an image model consisting of distributions supported on curvilinear singularities and prove that the theoretical performance of the recovery depends on the sparsifying and microlocal properties of the representation system, namely, exact image recovery is achieved if the size of the missing singularity is smaller than the size of the structure elements of the representation system. As a consequence of this observation, we prove that a sparsity-based image inpainting algorithm based on the shearlet representation - a multiscale anisotropic system that provides nearly optimally sparse representation of piecewise smooth images - significantly outperforms wavelets and other conventional multiscale systems.