Phase Retrieval Based On Fractional Fourier Transform

Phase retrieval is to recover the signal from its Fourier magnitudes and it has attracted much attention because of its importance in optical imaging,image processing and astronomical observation.However,the classical phase retrieval in only based on the time/frequency-domain magnitudes,which is not enough to determine the signal uniquely.The fractional Fourier transform(FRFT)is a generalized transform of Fourier transform,which can be used to analyze signals from more aspects besides time domain and frequency domain.FRFT also has a wide application in Engineering.On the other hand,the problem about phase retrieval from FRFT-domain magnitudes has been investigated deeply in recent years.However,most algorithms only consider reconstruct the signal from its time-domain and frequency-domain magnitudes,which has been proved to be not enough to ensure a unique recovery.To solve this problem,this thesis considers using FRFT magnitudes of signals to implement the phase retrieval,since more FRFT magnitudes can be obtained through adjusting the orders of FRFT.Therefore phase retrieval from multiple FRFT magnitudes is investigated theoretically and the numerical algorithm,matrix completion method is also proposed.To be more specific,our main works are as following:1.We prove that in the continuous case,phase retrieval from magnitudes of two FRFTs,whose difference of orders is not an integer,is identical to the traditional phase retrieval from time-domain and frequency-domain magnitudes.Furthermore,it is also proved that the non-uniqueness of phase retrieval from three FRFTs magnitudes.2.The numerical investigations are demonstrated.The matrix completion method for FRFT phase retrieval is proposed and compared with Gerchberg-Saxton method,which indicates the superiority of matrix completion method in terms of convergency.