Phase retrieval of sparse signals using optimization transfer and ADMM

We propose a reconstruction method for the phase retrieval problem prevalent in optics, crystallography, and other imaging applications. Our approach uses signal sparsity to provide robust reconstruction, even in the presence of outliers. Our method is multi-layered, involving multiple random initial conditions, convex majorization, variable splitting, and alternating directions method of multipliers (ADMM)-based implementation. Monte Carlo simulations demonstrate that our algorithm can correctly and robustly detect sparse signals from full and undersampled sets of squared-magnitude-only measurements, corrupted by additive noise or outliers.