Robust Compressive Phase Retrieval via Deep Generative Priors

This problem is known as phase retrieval and is encountered frequently in applications including X-ray crystallography [1, 2], astronomy [3], optics [4], tomography, microscopy, array imaging [5], acoustics [6], quantum mechanics [7] and ptychography [8], where it is extremely difficult or infeasible to measure phase information of signal while recording magnitude measurements is much easier. In its full generality, the inverse problem 1 is severely ill-posed due to its nonlinear and non-convex nature. Traditional approaches to overcome the ill posedness of phase retrieval generally falls into two categories. First approach is to introduce redundancy into measurement system, where we take more measurements than dimension of true signal x, i.e., m n usually in the form of oversampled Fourier transform [9], short-time Fourier transform [10], random Gaussian measurements [11], coded diffraction patterns using random masks or structured illuminations [12, 13], wavelet transform [14], and Gabor frames [15]. Second approach is to exploit some known knowledge about true signal x (prior information) such as sparsity [16, 17, 18] or non-negativity [19, 20].