Phase Retrieval via Smoothing Projected Gradient Method

Phase retrieval is a kind of ill-posed inverse problem, which is present in various applications, such as optics, astronomical imaging, and X-ray crystallography. Mathematically this inverse problem consists on recovering an unknown signal x epsilon R-n/C-n from a set of absolute square projections y(k) =vertical bar < a(k), x >vertical bar(2) , k = 1, . . ., m, where a(k) are the sampling vectors. However, the square absolute function is in general non-convex and non-differentiable, which are desired properties in order to solve the problem, when traditional convex optimization algorithms are used. Therefore, this paper introduces a special differentiable function, known as smoothing function, in order to solve the phase retrieval problem by using the smoothing projected gradient (SPG) method. Moreover, to accelerate the convergence of this algorithm, this paper uses a nonlinear conjugate gradient method applied to the smoothing function as the search direction. Simulation results are provided to validate its efficiency on existing algorithms for phase retrieval. It is shown that compared with recently developed algorithms, the proposed method is able to accelerate the convergence.