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 rlog2(N), where N×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.
@article{arxiv.2309.01328,
title = {Restoration Guarantee of Image Inpainting via Low Rank Patch Matrix Completion},
author = {Jian-Feng Cai and Jae Kyu Choi and Jingyang Li and Guojian Yin},
journal= {arXiv preprint arXiv:2309.01328},
year = {2023}
}