电磁逆散射问题的定位成像理论与算法研究

Inverse scattering are key ingredients of many cutting-edge technologies such as non-destructive testing, medical imaging and geophysical exploration. The project concerns the analytical and numerical studies on target detection and imaging in inverse electromagnetic (EM) scattering problems with emphasis on uniqueness and reconstruction algorithms in complex scenarios. Though partial results have been achieved on the uniqueness of inverse scattering problems in terms of small, spherical or polygonal scatterers, there are quite many gaps in this active field due to the inherent complexity of wave motion. The project would be carried out step by step to bridge the following theoretical gaps. The first one is on uniquely determining isotropic EM medium by the fixed-incident-direction far-field data. The study is based on showing the discreteness of certain electromagnetic interior transmission eigenvalues with generic media. The second type of uniqueness is on the determination of an anisotropic EM medium in a gauge equivalence class. It is done by adapting the microlocal analysis technique developed for Calderon's problem and the shape analysis technique in optimization theory to the more challenging Maxwell case. The third one is on qualitatively determining the support of an EM scatterer disregarding its contents. Meanwhile, the project is numerically focused on problem-driven algorithmic research. They include qualitative schemes for fast locating and imaging multiple multiscale scatterers in the inhomogeneous space or on the non-flat ground. Thanks to the new uniqueness results, emphasis would be laid on the development of model-compatible, composite indicator function-based reconstruction algorithms. This project has profound significance both in theoretical development and in military or civil applications.

逆散射是众多前沿科技的基础，如无损探测，医学成像，地物勘探等。本项目针对电磁波逆散射中的定位成像问题，重点研究复杂场景中的唯一性理论和快速示性重构算法。尽管唯一性理论已有特殊情形下的部分结果，但波动的复杂性使其尚待完善。本项目拟填补以下空白：1）由固定方向入射场产生的远场数据唯一地确定各向同性的介质，此研究基于内传输特征值的离散特性；2）从散射体规范等价类中唯一识别出各向异性的介质，此基于微局部分析和形状分析技巧；3）在散射体内部结构未知的情况下定性地确定散射体的支集。同时，本项目拟开展应用驱动的算法研究，即对随机分布在非匀质天空中或非平坦地面上的若干多尺度散射体进行快速定位成像。其核心是以前述唯一性理论为指导，构造与模型问题匹配的、基于复合型示性函数的重构算法。此方法无需正则化且不用求解正问题，测量数据少，对误差鲁棒。该研究课题具有重要的理论意义及可行的军民用途。

本项目的研究背景是众多前沿科技如无损探测，医学成像，地物勘探中的逆散射现象及其应用的数学基础。针对电磁波逆散射中的定位成像关键问题，主要研究复杂场景中的唯一性理论和快速示性重构算法。尽管唯一性理论已有特殊情形下的部分结果，但波动的复杂性使其尚待完善。本项目的重要研究结果包括：1）由固定方向入射场产生的远场数据唯一地确定各向同性的介质，此研究基于内传输特征值的离散特性；2）从散射体规范等价类中唯一识别出各向异性的介质，此基于微局部分析和形状分析技巧；3）在散射体内部结构未知的情况下定性地确定散射体的支集。同时，本项目开展了应用驱动的算法研究，即对随机分布在非匀质天空中或非平坦地面上的若干多尺度散射体进行快速定位成像。其核心是以前述唯一性理论为指导，构造与模型问题匹配的、基于复合型示性函数的重构算法。此方法无需正则化且不用求解正问题，测量数据少，对误差鲁棒。该研究项目成果具有重要的理论意义及可行的民用用途。