Restoration Guarantee of Image Inpainting via Low Rank Patch Matrix Completion

IF 2.1 3区数学 Q3 COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE

SIAM Journal on Imaging Sciences Pub Date : 2024-09-12 DOI:10.1137/23m1614456

Jian-Feng Cai, Jae Kyu Choi, Jingyang Li, Guojian Yin

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引用次数: 0

Abstract

SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1879-1908, September 2024.
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 [math], where [math] denotes the size of the image and [math] 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.

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通过低等级补丁矩阵完成图像绘制的恢复保证

SIAM 影像科学期刊》，第 17 卷第 3 期，第 1879-1908 页，2024 年 9 月。摘要.近年来，与传统的变分方法相比，基于补丁的图像复原方法表现出更优越的性能。本文深入探讨了基于补丁的图像复原方法的数学基础，重点是利用补丁间的自相似性假设，建立基于补丁的图像内绘的恢复保证。为了实现这一目标，我们将图像内绘问题重新表述为结构化低秩矩阵补全，通过对具有潜在重叠的图像补丁进行分组来完成。通过某些不一致性假设，我们建立了一种恢复保证，前提是样本数量超过 [math] 的数量级，其中 [math] 表示图像大小，[math] 表示每组图像补丁的等级总和。通过严谨的数学分析，我们对基于补丁的图像复原方法的理论基础提出了宝贵的见解，揭示了这些方法的功效，并为实际应用提供了指导。

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来源期刊

SIAM Journal on Imaging Sciences COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE-COMPUTER SCIENCE, SOFTWARE ENGINEERING

CiteScore

3.80

自引率

4.80%

发文量

审稿时长

>12 weeks

期刊介绍： SIAM Journal on Imaging Sciences (SIIMS) covers all areas of imaging sciences, broadly interpreted. It includes image formation, image processing, image analysis, image interpretation and understanding, imaging-related machine learning, and inverse problems in imaging; leading to applications to diverse areas in science, medicine, engineering, and other fields. The journal’s scope is meant to be broad enough to include areas now organized under the terms image processing, image analysis, computer graphics, computer vision, visual machine learning, and visualization. Formal approaches, at the level of mathematics and/or computations, as well as state-of-the-art practical results, are expected from manuscripts published in SIIMS. SIIMS is mathematically and computationally based, and offers a unique forum to highlight the commonality of methodology, models, and algorithms among diverse application areas of imaging sciences. SIIMS provides a broad authoritative source for fundamental results in imaging sciences, with a unique combination of mathematics and applications. SIIMS covers a broad range of areas, including but not limited to image formation, image processing, image analysis, computer graphics, computer vision, visualization, image understanding, pattern analysis, machine intelligence, remote sensing, geoscience, signal processing, medical and biomedical imaging, and seismic imaging. The fundamental mathematical theories addressing imaging problems covered by SIIMS include, but are not limited to, harmonic analysis, partial differential equations, differential geometry, numerical analysis, information theory, learning, optimization, statistics, and probability. Research papers that innovate both in the fundamentals and in the applications are especially welcome. SIIMS focuses on conceptually new ideas, methods, and fundamentals as applied to all aspects of imaging sciences.