An undersampled phase retrieval algorithm via gradient iteration

This work addresses the issue of undersampled phase retrieval using the gradient framework and proximal regularization theorem. It is formulated as an optimization problem in terms of least absolute shrinkage and selection operator (LASSO) form with $(l_{2}+P_{1})$ norms minimization in the case of sparse incident signals. Then, inspired by the compressive phase retrieval via majorization-minimization technique (C-PRIME) algorithm, a gradient-based PRIME algorithm is proposed to solve a quadratic approximation of the original problem. Moreover, we also proved that the C-PRIME method can be regarded as a special case of the proposed algorithm. As demonstrated by simulation results, both the magnitude and phase recovery abilities of the proposed algorithm are excellent. Furthermore, the experimental results also show the mean square error (MSE) performance of the proposed algorithm versus iterative step.