SIAM Journal on Imaging Sciences, Volume 13, Issue 3, Page 1467-1510, January 2020.
The purpose of this work is to provide a rigorous mathematical analysis of the expected superresolution phenomenon in the time-reversal imaging of electromagnetic (EM) radiating sources embedded in a high contrast medium. It is known that the resolution limit is essentially determined by the sharpness of the imaginary part of the EM Green's tensor for the associated background. We first establish the close connection between the resolution and the material parameters and the resolvent of the electric integral operator, via the Lippmann--Schwinger representation formula. We then present an insightful characterization of the spectral structure of the integral operator for a general bounded domain and derive the pole-pencil decomposition of its resolvent in the high contrast regime. For the special case of a spherical domain, we provide some quantitative asymptotic behavior of the eigenvalues and eigenfunctions. These mathematical findings shall enable us to provide a concise and rigorous illustration of the superresolution in the EM source reconstruction in high contrast media. Some numerical examples are also presented to verify our main theoretical results.

中文翻译：

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在高对比度介质中回收嵌入式电磁源的超分辨率

SIAM影像科学杂志，第13卷，第3期，第1467-1510页，2020年1月。
这项工作的目的是对嵌入高对比度介质中的电磁（EM）辐射源的时间反转成像中的预期超分辨率现象进行严格的数学分析。众所周知，分辨率极限基本上由相关背景的EM Green张量的虚部的清晰度决定。我们首先通过Lippmann-Schwinger表示公式在分辨率和材料参数以及电积分算子的解析度之间建立紧密联系。然后，我们为一般有界域提出了积分算子的光谱结构的深刻见解，并推导了其在高对比度状态下的分解物的极铅笔分解。对于球形域的特殊情况，我们提供了特征值和特征函数的一些定量渐近行为。这些数学发现将使我们能够提供高对比度介质中EM源重构中超分辨率的简明而严格的图示。还提供了一些数值示例来验证我们的主要理论结果。