GauGal: Gaussian-Galerkin Electromagnetic Inverse Scattering Imaging

Electromagnetic inverse scattering (EIS), which aims to recover object permittivity from measured scattered fields, is central to a wide range of applications, from medical diagnostics to security screening. However, it is a highly nonlinear and ill-posed problem. Despite substantial progress, existing approaches face a fundamental trade-off between physical fidelity, efficiency, and generalization: data-driven methods are fast but tied to training geometries and sensor setups, while physics-based optimization is accurate but computationally expensive. We introduce a Gaussian-Galerkin (GauGal) method that reformulates EIS from a dense point-wise field reconstruction to a compact, primitive-level physical solving. Rather than using Gaussian primitive merely as a material representation, GauGal take it as the computational units for wave scattering. By projecting the continuous EIS scattering equation into a Gaussian primitive space, the material modulation, Green propagation, source excitation, and receiver observation are all realized as primitive-level operators. This yields a compact differentiable forward solver for physics-consistent reconstruction. Our method achieves state-of-the-art accuracy on synthetic and real benchmarks while reducing runtime from over 30 minutes by leading physics-driven baselines to under 10 seconds. It also generalizes significantly better to unseen geometries and sensor configurations than leading data-driven baselines. Overall, this framework establishes an accurate, efficient, and generalizable paradigm for electromagnetic inverse imaging, enabling fast, physics-consistent imaging in practical settings.