Constructing probing functions for direct sampling methods for inverse scattering problems with limited-aperture data: finite space framework and deep probing network

Abstract

This work studies an inverse scattering problem when limited-aperture data are available that are from just one or a few incident fields. This inverse problem is highly ill-posed due to the limited receivers and a few incident fields employed. Solving inverse scattering problems with limited-aperture data is important in applications as collecting full data is often either unrealistic or too expensive. The direct sampling methods (DSMs) with full-aperture data can effectively and stably estimate the locations and geometric shapes of the unknown scatterers with a very limited number of incident waves. However, a direct application of DSMs to the case of limited receivers would face the resolution limit. To break this limitation, we propose a finite space framework with two specific schemes, and an unsupervised deep learning strategy to construct effective probing functions for the DSMs in the case with limited-aperture data. Several representative numerical experiments are carried out to illustrate and compare the performance of different proposed schemes.

Figures (8)

• Figure 1: The decay behaviour of $|K_\Gamma(z,y)|$ with $k=8, \alpha=\pi/3$ and different $\beta$.
• Figure 2: The architecture of the neural network used in the numerical experiments.
• Figure 3: Two different configurations considered in the numerical experiments, where the highlighted red parts denote the measurement area $\Gamma$.
• Figure 4: Reconstructions for Example 1.1, with different noise levels.
• Figure 5: The relative norm function $RN(z)=\frac{\Vert G_\Gamma(z,\cdot)\Vert_{L^2(\Gamma)}}{\Vert G^\infty(z,\cdot)\Vert_{L^2(\Gamma)}}$ of the probing functions $G_\Gamma(z,\hat{x})$ constructed by different methods.

Theorems & Definitions (8)

• Lemma 3.1
• proof
• Lemma 3.2
• Lemma 3.3
• proof
• Lemma 4.1
• proof
• Remark 1