| Phase retrieval refers to the recovery of the missing phase information from the observed intensity information and reconstructing the original signal.This problem has a wide range of applications in the fields of X-ray crystallography,optical imaging and so on.The signal uses the electromagnetic wave as the carrier during the propagation process,which includes the intensity information and the phase information.Due to the limitations of existing measuring devices,only the signal intensity information can be measured.However,the phase information carries most of the structural information of the signal,which is of great significance for signal recovery.In this paper,we consider adding specific prior information to the phase retrieval problem to obtain high accurate solutions.The signal itself is sparsity,so we improve the accuracy of phase retrieval results by limiting sparsity to the recovery of signals within a sparsity set.This paper summarizes sparse phase retrieval as the parameter estimation problem of quadratic metric regression,and regression parameters are estimated by the regularized least squares estimation method and the regularized least squares estimation method respectively.The optimization theory and algorithm of the regularization problem are studied.The details are as follows:(1)For the regularized least absolute deviation estimation problem,we characterize its first-order optimality condition by defining the directional stationary point.Further,the specific form of the directional stationary point and the lower bound theory are analyzed.Based on these theories,we establish the equivalence conclusion of the solution to this problem and the corresponding L0 regularized problem.Finally,the smooth consistency theory is presented,and we propose the smoothing proximal gradient algorithm to solve this problem.Further,the convergence of the algorithm is established.(2)For the regularized least squares parameter estimation problem and the firstorder optimality condition of the problem is described by using the limit sub-differentiation,and the multi-stage convex relaxation algorithm is proposed to solve the problem,and the convergence of the algorithm is analyzed.(3)Through numerical simulation,the validity of our proposed theory and algorithm is verified.The numerical simulation results show that when the regression noise is non-heavy-tailed Gaussian noise,the multi-stage convex relaxation algorithm has good accuracy and robustness.When abnormal noise exists,the smoothing proximal gradient algorithm has good performance and noise immunity. |
