Study Of Convex Optimization Phase Retrieval Algorithm Based On Transport Of Intensity Equation

Phase retrieval for complex-valued signals is to recover the original signals,including phase information,only from their magnitudes of Fourier transform.Since it is hard to measure the phase of light wave,phase retrieval is often adopted to determine plenty of shape and position information of the illuminated object,and its extensive applications can be found in many science and technology areas,like crystallography,astronomy,and optical imaging.However,phase retrieval is a typical ill-posed mathematical problem.There is no unique one-to-one mapping from one-dimensional signal vector to the magnitude values of its Fourier transform.Although the uniqueness of such a mapping theoretically exists for most of higher-dimensional signals,the high-precision recovered result still cannot be obtained numerically due to the error in measurement and computation and the non-convexity of solution model.To address the non-uniqueness issue mentioned above,many researchers have proposed various methods to improve the property of Fourier measurement matrix by manufacturing complex masks installed in the optical path or constructing a structured lighting setup to modulate the illuminating beam.But the methods are complicated in technical realization and suffers higher cost in economy.Alternatively,this thesis attempts to tackle the non-uniqueness problem with a software approach.First,the Transport of Intensity Equation(TIE)is employed to restrict the solution in a smaller range or to compress it in the lower-dimensional TIE solution space.Second,Fourier measurement matrices are properly subdivided to form multiple well-posed phase-retrieval problems each for a portion of the signals,which is justified similarly to the approaches for sparse signal recovery.Finally,on the basis of less variables,an appropriate initial solution and well-posed measurement matrices,the convex optimization algorithms like PhaseCut are utilized to obtain the recovered phase values with high-precision.Mainly,this thesis presents several research works as follows:(1)After a brief description about the physical theory on diffraction of coherent light and its application scenarios,the theoretic foundation and mathematical models for traditional phase retrieval methods like the iteration algorithm and TIE method are presented.Following this,the two iteration algorithms,GS algorithm and HIO algorithm,are implemented in MATLAB,and the simulation results are discussed,as a comparative reference for the approaches proposed in this research.(2)Several relaxed convex optimization models and their solution algorithms for phase retrieval problem are presented,and the reason that leads to their failure in solving the problem of the original Fourier measurement based phase-retrieval is investigated.Inspired from the success of the approach for sparse signal recovery,a fractional phase retrieval approach like 0-1 mask,which recovers the signal separately with multiple portions,is proposed and its realization in the framework of convex optimization is provided.The experiment shows that it can recover the signals exactly.(3)From the under-determination property of TIE,two algorithms,the optimization on the variables of its solution space and the convex optimization based on the signal compression with the basis vectors in TIE solution space,are proposed.Combined with traditional convex optimization for the original Fourier measurement,the two-step method can provide better recovery results.This method well resolves the ill-posedness problem of Fourier measurement without additional hardware equipment.