通过总变异正则化精确恢复片断常数图像的支持度

IF 2 2区数学 Q1 MATHEMATICS, APPLIED

Inverse Problems Pub Date : 2024-09-11 DOI:10.1088/1361-6420/ad75b1

Yohann De Castro, Vincent Duval and Romain Petit

{"title":"通过总变异正则化精确恢复片断常数图像的支持度","authors":"Yohann De Castro, Vincent Duval and Romain Petit","doi":"10.1088/1361-6420/ad75b1","DOIUrl":null,"url":null,"abstract":"This work is concerned with the recovery of piecewise constant images from noisy linear measurements. We study the noise robustness of a variational reconstruction method, which is based on total (gradient) variation regularization. We show that, if the unknown image is the superposition of a few simple shapes, and if a non-degenerate source condition holds, then, in the low noise regime, the reconstructed images have the same structure: they are the superposition of the same number of shapes, each a smooth deformation of one of the unknown shapes. Moreover, the reconstructed shapes and the associated intensities converge to the unknown ones as the noise goes to zero.","PeriodicalId":50275,"journal":{"name":"Inverse Problems","volume":null,"pages":null},"PeriodicalIF":2.0000,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Exact recovery of the support of piecewise constant images via total variation regularization\",\"authors\":\"Yohann De Castro, Vincent Duval and Romain Petit\",\"doi\":\"10.1088/1361-6420/ad75b1\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"This work is concerned with the recovery of piecewise constant images from noisy linear measurements. We study the noise robustness of a variational reconstruction method, which is based on total (gradient) variation regularization. We show that, if the unknown image is the superposition of a few simple shapes, and if a non-degenerate source condition holds, then, in the low noise regime, the reconstructed images have the same structure: they are the superposition of the same number of shapes, each a smooth deformation of one of the unknown shapes. Moreover, the reconstructed shapes and the associated intensities converge to the unknown ones as the noise goes to zero.\",\"PeriodicalId\":50275,\"journal\":{\"name\":\"Inverse Problems\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.0000,\"publicationDate\":\"2024-09-11\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Inverse Problems\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1088/1361-6420/ad75b1\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Inverse Problems","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1088/1361-6420/ad75b1","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

摘要

这项研究涉及从噪声线性测量中恢复片状常数图像。我们研究了一种基于总（梯度）变化正则化的变分重建方法的噪声鲁棒性。我们的研究表明，如果未知图像是几个简单形状的叠加，并且如果非退化源条件成立，那么在低噪声条件下，重建图像具有相同的结构：它们是相同数量形状的叠加，每个形状都是其中一个未知形状的平滑变形。此外，当噪声为零时，重建的形状和相关的强度都会趋近于未知形状。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

Exact recovery of the support of piecewise constant images via total variation regularization

This work is concerned with the recovery of piecewise constant images from noisy linear measurements. We study the noise robustness of a variational reconstruction method, which is based on total (gradient) variation regularization. We show that, if the unknown image is the superposition of a few simple shapes, and if a non-degenerate source condition holds, then, in the low noise regime, the reconstructed images have the same structure: they are the superposition of the same number of shapes, each a smooth deformation of one of the unknown shapes. Moreover, the reconstructed shapes and the associated intensities converge to the unknown ones as the noise goes to zero.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Inverse Problems 数学-物理：数学物理

CiteScore

4.40

自引率

14.30%

发文量

115

审稿时长

2.3 months

期刊介绍： An interdisciplinary journal combining mathematical and experimental papers on inverse problems with theoretical, numerical and practical approaches to their solution. As well as applied mathematicians, physical scientists and engineers, the readership includes those working in geophysics, radar, optics, biology, acoustics, communication theory, signal processing and imaging, among others. The emphasis is on publishing original contributions to methods of solving mathematical, physical and applied problems. To be publishable in this journal, papers must meet the highest standards of scientific quality, contain significant and original new science and should present substantial advancement in the field. Due to the broad scope of the journal, we require that authors provide sufficient introductory material to appeal to the wide readership and that articles which are not explicitly applied include a discussion of possible applications.