Localization of Point Scatterers via Sparse Optimization on Measures

IF 2.1 3区数学 Q3 COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE SIAM Journal on Imaging Sciences Pub Date : 2024-07-23 DOI:10.1137/24m1636265

Giovanni S. Alberti, Romain Petit, Matteo Santacesaria

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Abstract

SIAM Journal on Imaging Sciences, Volume 17, Issue 3, Page 1619-1649, September 2024.
Abstract.We consider the inverse scattering problem for time-harmonic acoustic waves in a medium with pointwise inhomogeneities. In the Foldy–Lax model, the estimation of the scatterers’ locations and intensities from far field measurements can be recast as the recovery of a discrete measure from nonlinear observations. We propose a “linearize and locally optimize” approach to perform this reconstruction. We first solve a convex program in the space of measures (known as the Beurling LASSO), which involves a linearization of the forward operator (the far field pattern in the Born approximation). Then, we locally minimize a second functional involving the nonlinear forward map, using the output of the first step as initialization. We provide guarantees that the output of the first step is close to the sought-after measure when the scatterers have small intensities and are sufficiently separated. We also provide numerical evidence that the second step still allows for accurate recovery in settings that are more involved.

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通过测量稀疏优化实现点散射体定位

SIAM 影像科学期刊》，第 17 卷第 3 期，第 1619-1649 页，2024 年 9 月。摘要：我们考虑了时谐声波在具有点状不均匀性介质中的反向散射问题。在 Foldy-Lax 模型中，根据远场测量结果对散射体位置和强度的估计可被视为从非线性观测结果中恢复离散度量。我们提出了一种 "线性化和局部优化 "的方法来进行这种重建。我们首先求解度量空间中的凸程序（称为 Beurling LASSO），其中涉及前向算子（玻恩近似中的远场模式）的线性化。然后，我们利用第一步的输出作为初始化，局部最小化涉及非线性前向图的第二个函数。我们保证，当散射体的强度较小且充分分离时，第一步的输出接近于所寻求的测量值。我们还提供了数值证据，证明第二步仍能在更复杂的情况下实现精确恢复。

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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.