The [math]-Laplace “Signature” for Quasilinear Inverse Problems

IF 2.1 3区数学 Q3 COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE

SIAM Journal on Imaging Sciences Pub Date : 2024-02-15 DOI:10.1137/22m1527192

Antonio Corbo Esposito, Luisa Faella, Gianpaolo Piscitelli, Vincenzo Mottola, Ravi Prakash, Antonello Tamburrino

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引用次数: 0

Abstract

SIAM Journal on Imaging Sciences, Volume 17, Issue 1, Page 351-388, March 2024.
Abstract. This paper refers to an imaging problem in the presence of nonlinear materials. Specifically, the problem we address falls within the framework of Electrical Resistance Tomography and involves two different materials, one or both of which are nonlinear. Tomography with nonlinear materials is in the early stages of development, although breakthroughs are expected in the not-too-distant future. The original contribution this work makes is that the nonlinear problem can be approximated by a weighted [math]-Laplace problem. From the perspective of tomography, this is a significant result because it highlights the central role played by the [math]-Laplacian in inverse problems with nonlinear materials. Moreover, when [math], this result allows all the imaging methods and algorithms developed for linear materials to be brought into the arena of problems with nonlinear materials. The main result of this work is that for “small” Dirichlet data, (i) one material can be replaced by a perfect electric conductor and (ii) the other material can be replaced by a material giving rise to a weighted [math]-Laplace problem.

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准线性反问题的[math]-拉普拉斯 "签名

SIAM 影像科学杂志》，第 17 卷第 1 期，第 351-388 页，2024 年 3 月。摘要本文涉及非线性材料存在时的成像问题。具体来说，我们要解决的问题属于电阻断层成像框架，涉及两种不同的材料，其中一种或两种都是非线性材料。使用非线性材料的层析成像技术正处于发展的早期阶段，但在不远的将来有望取得突破性进展。这项工作的原创性贡献在于，非线性问题可以用加权[数学]-拉普拉斯问题来近似。从层析成像的角度来看，这是一个重要的结果，因为它突出了[math]-拉普拉斯问题在非线性材料逆问题中的核心作用。此外，当[math]时，这一结果允许将针对线性材料开发的所有成像方法和算法带入非线性材料问题的领域。这项工作的主要成果是，对于 "小 "迪里夏特数据，(i) 一种材料可以由完美电导体代替，(ii) 另一种材料可以由引起加权[math]-拉普拉斯问题的材料代替。

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.