Undersampled Phase Retrieval With Outliers

This paper proposes a general framework for reconstructing sparse images from undersampled (squared)magnitude data corrupted with outliers and noise. This phase retrieval method uses a layered approach, combining repeated minimization of a convex majorizer (surrogate for a nonconvex objective function), and iterative optimization of that majorizer using a preconditioned variant of the alternating direction method of multipliers (ADMM). Since phase retrieval is nonconvex, this implementation uses multiple initial majorization vectors. The introduction of a robust 1-norm data fit term that is better adapted to outliers exploits the generality of this framework. The derivation also describes a normalization scheme for the regularization parameter and a known adaptive heuristic for the ADMM penalty parameter. Both 1-D Monte Carlo tests and 2-D image reconstruction simulations suggest the proposed framework, with the robust data fit term, reduces the reconstruction error for data corrupted with both outliers and additive noise, relative to competing algorithms having the same total computation.