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.

翻译：近年来，基于图像块的修复方法相较于传统变分方法展现出更优越的性能。本文深入探讨了基于图像块修复方法的数学基础，重点在利用图像块间自相似性假设的基础上，为基于图像块的图像修复建立恢复保证。为实现这一目标，我们将图像修复问题重构为结构化低秩矩阵补全问题，通过将可能重叠的图像块进行分组实现。在做出某些非相干性假设后，我们建立了恢复保证，条件为样本数量超过$r\log^2(N)$阶，其中$N\times N$表示图像尺寸，$r>0$表示每组图像块秩的总和。通过严密的数学分析，我们为基于图像块修复方法的理论基础提供了宝贵见解，揭示了其有效性原理，并为实际实现提供了指导准则。