Multi-Scaling Differential Contraction Integral Method for Inverse Scattering Problems with Inhomogeneous Media

TL;DR

The paper tackles inverse scattering in inhomogeneous media by proposing a novel MS-DCIE method that merges a differential contraction integral equation (DCIE) framework with a multi-scaling (MS) regularization strategy. This integration localizes high-resolution reconstruction within Region of Interest (RoI) and reduces nonlinearity and ill-posedness, while avoiding the computation of the inhomogeneous-background Green's function by relying on the homogeneous Green's function $G_B$. A subspace-based optimization (SOM) with a singular-value decomposition (SVD)–driven decomposition separates deterministic and ambiguous currents, and a data/state mismatch cost is minimized via Polak-Ribiere conjugate gradients across successive RoIs. Numerical and experimental results show that MS-DCIE outperforms single-resolution DCIE and DLSIE-based methods across various contrasts, noise levels, and host-uncertainty scenarios, offering improved edge fidelity and reduced artifacts. The approach provides a computationally efficient path for high-resolution microwave imaging in complex backgrounds and lays groundwork for future 3D and biomedical extensions.

Abstract

Practical applications of microwave imaging often require the solution of inverse scattering problems with inhomogeneous backgrounds. Towards this end, a novel inversion strategy, which combines the multi-scaling (MS) regularization scheme and the Difference Contraction Integral Equation (DCIE) formulation, is proposed. Such an integrated approach mitigates the non-linearity and the ill-posedness of the problem to obtain reliable high-resolution reconstructions of the unknown scattering profiles. The arising algorithmic implementation, denoted as MS-DCIE, does not require the computation of the Green's function of the inhomogeneous background, thus it provides an efficient and effective way to deal with complex scenarios. The performance of the MS-DCIE are assessed by means of numerical and experimental tests, in comparison with competitive state-of-the-art inversion strategies, as well.