Phase retrieval using iterative Fourier transform and convex optimization algorithm

Phase is an inherent characteristic of any wave field. Statistics show that greater than 25% of the information is encoded in the amplitude term and 75% of the information is in the phase term. The technique of phase retrieval means acquire phase by computation using magnitude measurements and provides data information for holography display, 3D field reconstruction, X-ray crystallography, diffraction imaging, astronomical imaging and many other applications. Mathematically, solving phase retrieval problem is an inverse problem taking the physical and computation constraints. Some recent algorithms use the principle of compressive sensing, such as PhaseLift, PhaseCut and compressive phase retrieval etc. they formulate phase retrieval problems as one of finding the rank-one solution to a system of linear matrix equations and make the overall algorithm a convex program over nxn matrices. However, by "lifting" a vector problem to a matrix one, these methods lead to a much higher computational cost as a result. Furthermore, they only use intensity measurements but few physical constraints. In the paper, a new algorithm is proposed that combines above convex optimization methods with a well known iterative Fourier transform algorithm (IFTA). The IFTA iterates between the object domain and spectral domain to reinforce the physical information and reaches convergence quickly which has been proved in many applications such as compute-generated-hologram (CGH). Herein the output phase of the IFTA is treated as the initial guess of convex optimization methods, and then the reconstructed phase is numerically computed by using modified TFOCS. Simulation results show that the combined algorithm increases the likelihood of successful recovery as well as improves the precision of solution.