Restoration Guarantee of Image Inpainting via Low Rank Patch Matrix Completion

TL;DR

This paper provides a theoretical guarantee for patch-based image inpainting by reformulating it as a low-rank matrix completion problem, establishing conditions under which accurate restoration is possible.

Contribution

It introduces a mathematical framework linking patch-based image inpainting to low-rank matrix completion, with proven guarantees under certain incoherence and sampling conditions.

Findings

Restoration guarantee holds when samples exceed r log^2(N)

Reformulation as structured low-rank matrix completion

Provides theoretical insights into patch-based image restoration

Abstract

In recent years, patch-based image restoration approaches have demonstrated superior performance compared to conventional variational methods. This paper delves into the mathematical foundations underlying patch-based image restoration methods, with a specific focus on establishing restoration guarantees for patch-based image inpainting, leveraging the assumption of self-similarity among patches. To accomplish this, we present a reformulation of the image inpainting problem as structured low-rank matrix completion, accomplished by grouping image patches with potential overlaps. By making certain incoherence assumptions, we establish a restoration guarantee, given that the number of samples exceeds the order of $rlog^2(N)$, where $N\times N$ denotes the size of the image and $r > 0$ represents the sum of ranks for each group of image patches. Through our rigorous mathematical analysis, we provide valuable insights into the theoretical foundations of patch-based image restoration methods, shedding light on their efficacy and offering guidelines for practical implementation.