Asymptotic analysis of inpainting via universal shearlet systems

Recently introduced inpainting algorithms using a combination of applied harmonic analysis and compressed sensing have turned out to be very successful. One key ingredient is a carefully chosen representation system which provides (optimally) sparse approximations of the original image. Due to the common assumption that images are typically governed by anisotropic features, directional representation systems have often been utilized. One prominent example of this class are shearlets, which have the additional benefit of allowing faithful implementations. Numerical results show that shearlets significantly outperform wavelets in inpainting tasks. One of those software packages, ShearLab, even offers the flexibility of using a different parameter for each scale, which is not yet covered by shearlet theory. In this paper, we first introduce universal shearlet systems which are associated with an arbitrary scaling sequence, thereby modeling the previously mentioned flexibility. In addition, this novel construction allows for a smooth transition between wavelets and shearlets and therefore enables us to analyze them in a uniform fashion. For a large class of such scaling sequences, we first prove that the associated universal shearlet systems form band-limited Parseval frames for L2 (R2) consisting of Schwartz functions. Second, we analyze the inpainting performance of this class of universal shearlet systems within a distributional model situation using an ℓ1-analysis minimization algorithm for reconstruction. Our main result states that, provided that the scaling sequence is comparable to the size of the (scale-dependent) gap, asymptotically perfect inpainting is achieved.