Inclusion and Estimates for the Jumps of Minimizers in Variational Denoising

IF 2.1 3区数学 Q3 COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE

SIAM Journal on Imaging Sciences Pub Date : 2024-09-06 DOI:10.1137/23m1627948

Antonin Chambolle, Michał Łasica

引用次数: 0

Abstract

SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1844-1878, September 2024.
Abstract.We study stability and inclusion of the jump set of minimizers of convex denoising functionals, such as the celebrated “Rudin–Osher–Fatemi” functional, for scalar or vectorial signals. We show that under mild regularity assumptions on the data fidelity term and the regularizer, the jump set of the minimizer is essentially a subset of the original jump set. Moreover, we give an estimate on the magnitude of the jumps in terms of the data. This extends old results, in particular of the first author (with Caselles and Novaga) and of Valkonen, to much more general cases. We also consider the case where the original datum has unbounded variation, and we define a notion of its jump set which, again, must contain the jump set of the solution.

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变异去噪中最小值跃迁的包含与估计

SIAM 影像科学期刊》第 17 卷第 3 期第 1844-1878 页，2024 年 9 月。摘要：我们研究了标量或矢量信号的凸去噪函数（如著名的 "Rudin-Osher-Fatemi "函数）最小值的跳跃集的稳定性和包含性。我们证明，在数据保真度项和正则因子的温和正则性假设下，最小化的跳跃集本质上是原始跳跃集的子集。此外，我们还给出了数据跳变幅度的估计值。这就把以前的结果，特别是第一作者（与卡塞勒斯和诺瓦加）以及瓦尔科宁的结果，扩展到了更普遍的情况。我们还考虑了原始数据变化无界的情况，并定义了其跳跃集的概念，同样，跳跃集必须包含解的跳跃集。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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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.