Phase Retrieval Based On Wirtinger Flow

Having been applied in the study of microstructure of materials,diffraction imaging is a significant technology,in which there is an essential process to recover the signals from their diffraction patterns.However,in this part,there is a hard issue that we cannot detect the phases but the intensities of diffraction patterns because of the limitation of detectors,for which we do not have enough information.On the other hand,the phase information outweighs intensity information,therefore,the recoveries of signals become more difficult.In general,this kind of recovery problem is called Phase Retrieval.This problem not only include in the process of diffraction imaging but also in the recoveries of other signal.Many researchers proposed several methods or algorithms to solve this problem.In this paper,we have reviewed some of their methods,and summarized their characteristics.Recently,a new effective algorithm,Wirtinger flow is proposed by Candes(2015).We learned this algorithm,and presented its detailed demonstration of its reliability.Based on the algorithm,a new measuring pattern is proposed,called 1-1 pattern,which can be applied practically in diffraction imaging.Then through simulation,the performances of the new model is compared with other two models based on WF on the recoveries of three types of signals: real and nonnegative ones,complex ones and the ones of physical microstructure.The results show that all of three patterns have good performances with lower noise but have bad performances with higher noise,proving the reliability of 1-1 pattern which will be applicable well on diffraction imaging.Finally,it is applied to solve the problem of quantum tomography.Because of Wirtinger flow’s only application for vector recovery,we select the density matrix to stimulate whose rank is equal to 1,and convert it to the problem of vector recovery.Through the simulation,it seems to be able to get good results no matter on noiseless condition or low-noisy condition.