高对比度介质中埋藏电磁源的超分辨恢复
Super-resolution in recovering embedded electromagnetic sources in high contrast media
摘要：
这项工作的目的是提供一个严格的数学分析的预期超分辨率现象的时间反演成像的电磁（EM）辐射源嵌入在一个高对比度介质。众所周知，分辨率极限基本上由相关背景的EM格林张量的虚部的锐度确定。我们首先通过Lippmann-Schwinger表示公式建立了解析度与材料参数和电积分算子的预解式之间的紧密联系。然后，我们提出了一个深刻的表征的频谱结构的积分算子的一般有界域，并推导出极束分解的高对比度制度的预解式。对于球面区域的特殊情况，我们给出了特征值和特征函数的一些定量渐近性质。这些数学发现将使我们能够提供一个简洁和严格的说明在高对比度介质中的EM源重建的超分辨率。最后给出了一些数值例子来验证我们的主要理论结果。
英文摘要：
The purpose of this work is to provide a rigorous mathematical analysis of the expected super-resolution 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 super-resolution in the EM source reconstruction in high contrast media. Some numerical examples are also presented to verify our main theoretical results.
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