The Research Of The Low Complexity Phase Retrieval Problem Based On The Priori Information

Phase retrieval(PR)is to recover x?C~nfrom b_i=|a_i~*x|~2where a_i?C~n,i=1,···,m are the measurement vectors.PR is a nonlinear and non-convex inverse problem which arises in optics imaging,military remote sensing and astronomy.In the last five years,there are several theories and numerical algorithms sprung up.But most of the works are based on the random measurement besides the sample complexity of the state-of-the-art algorithms is far beyond the theoretical bound which can't satisfy the demand of high accuracy of the real-time observation.Based on the priori information of the signal,we relieve the ill condition of the PR problem by adding the background information,constructing a model of the binary signal and utilizing the sparsity constraint besides proving the unique solution of PR with a high probability.Further,we propose highly efficient algorithms such as ER,RWF and SWF in Fourier measurements and random measurements besides giving the convergence con-ditions and analyzing the complexity of algorithms.Numerical tests show that the sample complexity m of algorithms in this paper can decrease to O(n)for the general signal and O(k~2logn)for the k sparse signal which approximates to the theoretical lower bound.