Phased and Phaseless Domain Reconstructions in the Inverse Scattering Problem via Scattering Coefficients

In this work we first 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 the phaseless reconstructions in the linearized case. Stability is studied by estimating the condition number of the inversion process for both the phased and the 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 is new for the phased and phaseless inverse scattering problems and is 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.