A Review of Inverse Scattering Imaging Methods Based on Transmission Eigenfunctions

Designing imaging methods is one of the important issues in inverse scattering problems. In recent years, some studies have shown that the transmission eigenfunctions contain important qualitative and quantitative information about the unknown scatterers. These spectral properties are closely related to the intrinsic nature of the scatterers and provide a connection between the scattering data and the geometry and the material characteristics of the target objects. This paper reviews recent developments in inverse scattering imaging methods that utilize local and global geometric structures of the transmission eigenfunctions. We first summarize the theoretical properties of these eigenfunctions, and then discuss the imaging algorithms based on them. Particular emphasis is placed on the theoretical justification of these imaging methods, including comparisons between them and traditional qualitative reconstruction methods. Finally, we discuss the current challenges and open problems, such as issues related to the theoretical explanation of super-resolution effect, limited-aperture data, and the extension to more complex physical models.