IML FISTA：不精确和惯性前向-后向的多层次框架。应用于图像复原

IF 2.1 3区数学 Q3 COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE

SIAM Journal on Imaging Sciences Pub Date : 2024-07-01 DOI:10.1137/23m1582345

Guillaume Lauga, Elisa Riccietti, Nelly Pustelnik, Paulo Gonçalves

{"title":"IML FISTA：不精确和惯性前向-后向的多层次框架。应用于图像复原","authors":"Guillaume Lauga, Elisa Riccietti, Nelly Pustelnik, Paulo Gonçalves","doi":"10.1137/23m1582345","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1347-1376, September 2024. <br/> Abstract. This paper presents a multilevel framework for inertial and inexact proximal algorithms that encompasses multilevel versions of classical algorithms such as forward-backward and FISTA. The methods are supported by strong theoretical guarantees: we prove both the rate of convergence and the convergence of the iterates to a minimum in the convex case, an important result for ill-posed problems. We propose a particular instance of IML (Inexact MultiLevel) FISTA, based on the use of the Moreau envelope to build efficient and useful coarse corrections, fully adapted to solve problems in image restoration. Such a construction is derived for a broad class of composite optimization problems with proximable functions. We evaluate our approach on several image reconstruction problems, and we show that it considerably accelerates the convergence of the corresponding one-level (i.e., standard) version of the methods for large-scale images.","PeriodicalId":49528,"journal":{"name":"SIAM Journal on Imaging Sciences","volume":"112 1","pages":""},"PeriodicalIF":2.1000,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"IML FISTA: A Multilevel Framework for Inexact and Inertial Forward-Backward. Application to Image Restoration\",\"authors\":\"Guillaume Lauga, Elisa Riccietti, Nelly Pustelnik, Paulo Gonçalves\",\"doi\":\"10.1137/23m1582345\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1347-1376, September 2024. <br/> Abstract. This paper presents a multilevel framework for inertial and inexact proximal algorithms that encompasses multilevel versions of classical algorithms such as forward-backward and FISTA. The methods are supported by strong theoretical guarantees: we prove both the rate of convergence and the convergence of the iterates to a minimum in the convex case, an important result for ill-posed problems. We propose a particular instance of IML (Inexact MultiLevel) FISTA, based on the use of the Moreau envelope to build efficient and useful coarse corrections, fully adapted to solve problems in image restoration. Such a construction is derived for a broad class of composite optimization problems with proximable functions. We evaluate our approach on several image reconstruction problems, and we show that it considerably accelerates the convergence of the corresponding one-level (i.e., standard) version of the methods for large-scale images.\",\"PeriodicalId\":49528,\"journal\":{\"name\":\"SIAM Journal on Imaging Sciences\",\"volume\":\"112 1\",\"pages\":\"\"},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2024-07-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"SIAM Journal on Imaging Sciences\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1137/23m1582345\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Imaging Sciences","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/23m1582345","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE","Score":null,"Total":0}

引用次数: 0

摘要

SIAM 影像科学杂志》，第 17 卷第 3 期，第 1347-1376 页，2024 年 9 月。摘要本文提出了一种惯性和非精确近似算法的多层次框架，其中包含经典算法的多层次版本，如前向-后向和 FISTA。这些方法有强有力的理论保证：我们证明了收敛率和迭代在凸情况下收敛到最小值，这对于问题不明确的情况是一个重要结果。我们提出了 IML（非精确多级）FISTA 的一个特殊实例，它基于莫罗包络来建立高效有用的粗校正，完全适用于解决图像复原问题。这种构造适用于一大类具有近似函数的复合优化问题。我们在几个图像重建问题上对我们的方法进行了评估，结果表明它大大加快了相应的单级（即标准）版本方法对大规模图像的收敛速度。

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

IML FISTA: A Multilevel Framework for Inexact and Inertial Forward-Backward. Application to Image Restoration

SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1347-1376, September 2024.
Abstract. This paper presents a multilevel framework for inertial and inexact proximal algorithms that encompasses multilevel versions of classical algorithms such as forward-backward and FISTA. The methods are supported by strong theoretical guarantees: we prove both the rate of convergence and the convergence of the iterates to a minimum in the convex case, an important result for ill-posed problems. We propose a particular instance of IML (Inexact MultiLevel) FISTA, based on the use of the Moreau envelope to build efficient and useful coarse corrections, fully adapted to solve problems in image restoration. Such a construction is derived for a broad class of composite optimization problems with proximable functions. We evaluate our approach on several image reconstruction problems, and we show that it considerably accelerates the convergence of the corresponding one-level (i.e., standard) version of the methods for large-scale images.

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

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.