Asymptotic Analysis of Inpainting via Universal Shearlet Systems

TL;DR

This paper introduces universal shearlet systems with flexible scaling, analyzes their mathematical properties, and demonstrates their effectiveness in inpainting tasks, outperforming wavelets especially at fine scales.

Contribution

It develops a new class of shearlet systems adaptable to various scales and provides a theoretical framework for their use in image inpainting.

Findings

Universal shearlet systems form band-limited Parseval frames.

Nearly-perfect inpainting is achievable with these systems at fine scales.

The systems interpolate between wavelets and shearlets, offering flexible modeling.

Abstract

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 benefitallowing faithful implementations. Numerical results show that shearlets significantly outperform wavelets in inpainting tasks. One of those software packages, www.shearlab.org, even offers the flexibility of usingdifferent 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 $L^2(\mathbb{R}^2)$ consisting of Schwartz functions. Secondly, we analyze the performance for inpainting of this class of universal shearlet systems within a distributional model situation using an $\ell^1$-analysis minimization algorithm for reconstruction. Our main result in this part states that, provided the scaling sequence is comparable to the size of the (scale-dependent) gap, nearly-perfect inpainting is achieved at sufficiently fine scales.