Inverse Obstacle Scattering from Multi-Frequency Near-Field Backscattering Data

This paper addresses the inverse obstacle scattering problem of simultaneously reconstructing the obstacle geometry and boundary conditions from multi-frequency near-field backscattering data. We first establish rigorous high-frequency asymptotic expansions for the scattered near-field, leveraging pseudo-differential operators (PDOs) to characterize the interaction between wavefront propagation and obstacle boundaries, where the principal symbol of the PDO governs the leading-order behavior of the scattering field. Based on these asymptotic results, we prove a global uniqueness theorem for the simultaneous recovery of the obstacle shape and impedance boundary condition under convexity assumptions. Furthermore, we develop a three-stage numerical reconstruction framework: (1) qualitative shape reconstruction via the direct sampling method; (2) quantitative boundary refinement via shape optimization; and (3) decoupled reconstruction of the boundary condition. A highlight of this algorithm is that all the three steps avoid computing the direct problem. Numerical experiments are presented to verify the robustness and efficiency of the proposed algorithm.