精确重建和各向异性总变异噪声数据重建

IF 2.2 2区 数学 Q1 MATHEMATICS, APPLIED
Martin Holler, Benedikt Wirth
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引用次数: 0

摘要

SIAM 数学分析期刊》,第 56 卷,第 3 期,第 2938-2967 页,2024 年 6 月。摘要。众所周知,通过求解具有 Tikhonov 正则化的凸优化问题,可以从有限多个傅立叶测量精确重建具有足够相互距离的点源(这一特性有时被称为超分辨率)。在有噪声测量的情况下,我们可以用不平衡的瓦瑟斯坦距离或弱索波列夫类型规范来约束重建误差。一个很自然的问题是,超分辨率现象还可以延伸到哪些其他环境。在这里,我们保留了相同的测量算子,但将正则因子替换为各向异性总变异,这尤其适用于正则化具有水平和垂直边缘的片状常数图像。在水平边缘和垂直边缘之间有足够的相互距离的情况下,我们证明了精确的重建结果以及测量噪声的 [math] 误差边界。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Exact Reconstruction and Reconstruction from Noisy Data with Anisotropic Total Variation
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 2938-2967, June 2024.
Abstract. It is well-known that point sources with sufficient mutual distance can be reconstructed exactly from finitely many Fourier measurements by solving a convex optimization problem with Tikhonov-regularization (this property is sometimes termed superresolution). In the case of noisy measurements one can bound the reconstruction error in unbalanced Wasserstein distances or weak Sobolev-type norms. A natural question is to what other settings the phenomenon of superresolution extends. We here keep the same measurement operator but replace the regularizer to anisotropic total variation, which is particularly suitable for regularizing piecewise constant images with horizontal and vertical edges. Under sufficient mutual distance between the horizontal and vertical edges we prove exact reconstruction results and [math] error bounds in terms of the measurement noise.
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来源期刊
CiteScore
3.30
自引率
5.00%
发文量
175
审稿时长
12 months
期刊介绍: SIAM Journal on Mathematical Analysis (SIMA) features research articles of the highest quality employing innovative analytical techniques to treat problems in the natural sciences. Every paper has content that is primarily analytical and that employs mathematical methods in such areas as partial differential equations, the calculus of variations, functional analysis, approximation theory, harmonic or wavelet analysis, or dynamical systems. Additionally, every paper relates to a model for natural phenomena in such areas as fluid mechanics, materials science, quantum mechanics, biology, mathematical physics, or to the computational analysis of such phenomena. Submission of a manuscript to a SIAM journal is representation by the author that the manuscript has not been published or submitted simultaneously for publication elsewhere. Typical papers for SIMA do not exceed 35 journal pages. Substantial deviations from this page limit require that the referees, editor, and editor-in-chief be convinced that the increased length is both required by the subject matter and justified by the quality of the paper.
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