Diffraction Tomography with Helmholtz Equation: Efficient and Robust Multigrid-Based Solver

TL;DR

This paper introduces a Helmholtz equation-based nonlinear model for diffraction tomography, along with a multigrid solver, offering a more efficient and robust alternative to the Lippmann-Schwinger model, especially for highly scattering objects.

Contribution

The paper presents a novel Helmholtz-based inverse scattering model and a multigrid solver that improves efficiency and robustness over existing Lippmann-Schwinger approaches.

Findings

The Helmholtz model performs well on simulated and real data.

The multigrid solver significantly accelerates computation.

The method is particularly effective for strongly scattering objects.

Abstract

Diffraction tomography is a noninvasive technique that estimates the refractive indices of unknown objects and involves an inverse-scattering problem governed by the wave equation. Recent works have shown the benefit of nonlinear models of wave propagation that account for multiple scattering and reflections. In particular, the Lippmann-Schwinger~(LiS) model defines an inverse problem to simulate the wave propagation. Although accurate, this model is hard to solve when the samples are highly contrasted or have a large physical size. In this work, we introduce instead a Helmholtz-based nonlinear model for inverse scattering. To solve the corresponding inverse problem, we propose a robust and efficient multigrid-based solver. Moreover, we show that our method is a suitable alternative to the LiS model, especially for strongly scattering objects. Numerical experiments on simulated and real data demonstrate the effectiveness of the Helmholtz model, as well as the efficiency of the proposed multigrid method.