Phased and phaseless domain reconstruction in inverse scattering problem via scattering coefficients

TL;DR

This paper reviews phased and phaseless inverse scattering problems, analyzing their stability, ill-posedness, and proposing algorithms for reconstruction, with a focus on the role of scattering coefficients and condition number estimates.

Contribution

It introduces a stability analysis based on condition numbers for both phased and phaseless inverse scattering problems, highlighting the increased ill-posedness of phaseless reconstruction and proposing strategies for optimal measurement.

Findings

Phaseless reconstruction is more nonlinear and severely ill-posed than phased.

Stability analysis using condition numbers provides new insights into problem ill-posedness.

Numerical experiments confirm the effectiveness of the proposed algorithms.

Abstract

In this work we shall review the (phased) inverse scattering problem and then pursue the phaseless reconstruction from far-field data with the help of the concept of scattering coefficients. We perform sensitivity, resolution and stability analysis of both phased and phaseless problems and compare the degree of ill-posedness of the phased and phaseless reconstructions. The phaseless reconstruction is highly nonlinear and much more severely ill-posed. Algorithms are provided to solve both the phased and phaseless reconstructions in the linearized case. Stability is studied by estimating the condition number of the inversion process for both the phased and phaseless cases. An optimal strategy is suggested to attain the infimum of the condition numbers of the phaseless reconstruction, which may provide an important guidance for efficient phaseless measurements in practical applications. To the best of our knowledge, the stability analysis in terms of condition numbers are new for the phased and phaseless inverse scattering problems, and are very important to help us understand the degree of ill-posedness of these inverse problems. Numerical experiments are provided to illustrate the theoretical asymptotic behavior, as well as the effectiveness and robustness of the phaseless reconstruction algorithm.