多频稀疏相位和无相位远场数据的电磁源散射反演问题
Inverse electromagnetic source scattering problems with multi-frequency sparse phased and phaseless far field data
摘要：
本文研究了多频稀疏相控或无相远场电磁源散射反演问题的唯一性、相位恢复和形状重建方法。利用相控数据，我们证明了以观测方向为法线的包含源支撑的最小条带可以由单一观测方向上的多频远场模式唯一确定。提出了一种分相直接采样的方法来重建条带。无相位远场数据与向外的能流密切相关，在实际中更容易测量。我们证明了无相位远场数据在源的平移下是不变的，这意味着源的位置不能由数据唯一地恢复。为了解决这个问题，我们同时考虑了具有一个固定源点和至多三个散射强度的磁偶极子的散射。利用这种技术，一个快速和稳定的相位恢复方法提出了一个简单的几何结果，提供了一个稳定的重建的平面上的一个点从三个距离给定点的基础上。提出了一种新的无相位直接采样方法来重构条带。相位恢复方法也可以与相位直接采样方法相结合来重建条带。最后，为了获得源支持，我们只需要将指示符相对于稀疏观测方向进行叠加。通过对含噪数据的三维扩展数值算例，进一步验证了所提出的相位恢复方法和直接采样方法的有效性和鲁棒性。
英文摘要：
This paper is concerned with uniqueness, phase retrieval and shape reconstruction methods for solving inverse electromagnetic source scattering problems with multi-frequency sparse phased or phaseless far field data. With the phased data, we show that the smallest strip containing the source support with the observation direction as the normal can be uniquely determined by the multi-frequency far field pattern at a single observation direction. A phased direct sampling method is also proposed to reconstruct the strip. The phaseless far field data is closely related to the outward energy flux, which can be measured more easily in practice. We show that the phaseless far field data is invariant under the translation of the sources, which implies that the location of the sources can not be uniquely recovered by the data. To solve this problem, we consider simultaneously the scattering of magnetic dipoles with one fixed source point and at most three scattering strengths. With this technique, a fast and stable phase retrieval approach is proposed based on a simple geometrical result which provides a stable reconstruction of a point in the plane from three distances to the given points. A novel phaseless direct sampling method is also proposed to reconstruct the strip. The phase retrieval approach can also be combined with the phased direct sampling method to reconstruct the strip. Finally, to obtain a source support, we just need to superpose the indicators with respect to the sparse observation directions. Extended numerical examples in three dimensions are conducted with noisy data, and the results further verify the effectiveness and robustness of the proposed phase retrieval approach and direct sampling methods.
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