Computational inverse scattering with internal sources: a reproducing kernel Hilbert space approach

TL;DR

This work addresses reconstructing the dielectric susceptibility $\eta$ of an inhomogeneous medium from measurements produced by internal sources. It develops a reproducing kernel Hilbert space (RKHS) framework with regularization, enabling a finite-dimensional representation of the reconstruction via the representer theorem and Sobolev kernels. The method is applied to 2D and 3D inverse scattering, with detailed forward-model handling, averaging over detectors, and explicit differentiation of the scattering amplitude to recover $\eta$; the approach is validated through three numerical experiments, including a three-ball model and a neuron, and compared favorably to prior methods. The results demonstrate accurate reconstructions with reduced detector requirements and establish the method's potential for high-resolution imaging in contexts such as super-resolution microscopy and ultrasound localization, where internal sources are available.

Abstract

We present a method to reconstruct the dielectric susceptibility (scattering potential) of an inhomogeneous scattering medium, based on the solution to the inverse scattering problem with internal sources. We employ the theory of reproducing kernel Hilbert spaces, together with regularization to recover the susceptibility of two- and three-dimensional scattering media. Numerical examples illustrate the effectiveness of the proposed reconstruction method.

Figures (6)

• Figure 1: Susceptibility reconstruction of three spherical scatterers on each layer by the two-dimensional method.
• Figure 2: Neuron model and susceptibility reconstructions on each layer by the two-dimensional method.
• Figure 3: Neuron reconstructions by two-dimensional method compared to the groundtruth.
• Figure 4: Error behavior with different numbers of sources
• Figure 5: Error behavior using different numbers of detectors