Research On Efficient Algorithms Of Fourier Phase Retrieval Problem And Its Applications

Phase Retrieval(PR)problem,which recovers signal x with length n from m phaseless measurements.It is a highly ill-conditioned nonlinear inverse problem,which is arisen from many applications such as optical imaging,microscopic imaging,military reconnaissance,and remote sensing.In recent years,by utilizing the theoretical results of non-convex optimization and deep learning,a series of important breakthroughs have been made.However,suffering from the non-uniqueness of the solution space and the non-convexity of the model,there are several drawbacks in the Fourier PR problem which limit the applications of algorithms in engineering.In this paper,we relieve the ill condition of the FPR problem by introducing the background information and the prior information learned from the data.Then,we devise the ADMM algorithm,Douglas Rachford algorithm,DFPR-ADMM algorithm,and the Wirtinger flow gradient descent algorithm by utilizing the nonconvex optimization and deep learning techniques.By numerous simulation tests and real experiments,we demonstrate the effectiveness of the algorithms proposed.Specifically,the paper can be divided into three aspects:First,we fully analyze the solution space of the FPR problem and utilize the prior information of the signal.When m < 2n-1,the solution space of Fourier PR is proved to have a flower-like structure.We can also relieve the illness of the Fourier solution space and prove the uniqueness of the solution by utilizing the background information and the low dimension structure.Second,we also come up with a multiple under-sampling model and devise the ADMM algorithm to solve it with convergence guaranteed.At the same time,we also devise the Douglas-Rachford algorithm to solve the FPR problem with background information.We prove the algorithm can have a local R convergence rate.We also combine the ADMM algorithm with the denoising neural network to devise the DFPR-ADMM algorithm.Theoretically,we analyze its convergence and generalize it by different ways of operator splitting and different kinds of regularizers.Last,we also come up with a kind of ADMM algorithm to solve the FPR problem under the generative model and prove it can convergence.Large amounts of numerical tests and Coherent Diffraction Imaging experiments demonstrate that the algorithms proposed in this paper can have an obvious advantage over state-of-the-art methods.Last,in applications,we focus on the M(?)ssbauer spectroscopy in Synchrotron radiation light source.We establish the FPR model by the physical procedure and devise the Wirtinger flow algorithm to derive the M(?)ssbauer spectrum from the phaseless spectrum in time.By numerous simulations and M(?)ssbauer spectroscopy from the Spring-8synchrotron radiation light source,we can verify the algorithms proposed in this paper.