Single- and Multi-Frequency Direct Sampling Methods in a Limited-Aperture Inverse Scattering Problem

Although the direct sampling method (DSM) has demonstrated its feasibility and robustness for imaging of small inhomogeneities, mathematical analyses of DSM have been conducted only on the full-aperture inverse scattering problem. Numerous studies have shown that DSM can also be applied in the limited-aperture inverse scattering problem, but most of its applications are still heuristic. This study considers an application, mathematical analysis, and improvement of DSM with a single-incident field only in the limited-aperture inverse scattering problem. First, we introduce a traditional indicator function of DSM at a single frequency, establish its mathematical structure, and examine its inherent limitation. To demonstrate the theoretical result, various results of numerical simulations with synthetic and experimental data are presented. Next, we consider the multi-frequency indicator function of DSM with a single-incident direction to improve imaging performance. For this, we design a multi-frequency indicator function of MDSM, analyze its mathematical structure, and theoretically explain the improvement of the imaging of single inhomogeneity and the limitation on the identification of multiple inhomogeneities. Various numerical simulations with synthetic and experimental data are presented to validate our results.

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