Undersampled Phase Retrieval with Outliers

TL;DR

This paper introduces a versatile framework for phase retrieval from undersampled, outlier-corrupted data, employing multiple initializations, convex surrogates, and ADMM optimization, with a focus on robustness and generality.

Contribution

It presents a novel, flexible approach that incorporates a Laplace noise model and supports analysis-form sparsity priors, improving robustness to outliers in phase retrieval.

Findings

Enhanced support recovery with Laplace noise model

Reduced mean squared error in outlier scenarios

Demonstrated effectiveness through 1D and 2D simulations

Abstract

We propose a general framework for reconstructing transform-sparse images from undersampled (squared)-magnitude data corrupted with outliers. This framework is implemented using a multi-layered approach, combining multiple initializations (to address the nonconvexity of the phase retrieval problem), repeated minimization of a convex majorizer (surrogate for a nonconvex objective function), and iterative optimization using the alternating directions method of multipliers. Exploiting the generality of this framework, we investigate using a Laplace measurement noise model better adapted to outliers present in the data than the conventional Gaussian noise model. Using simulations, we explore the sensitivity of the method to both the regularization and penalty parameters. We include 1D Monte Carlo and 2D image reconstruction comparisons with alternative phase retrieval algorithms. The results suggest the proposed method, with the Laplace noise model, both increases the likelihood of correct support recovery and reduces the mean squared error from measurements containing outliers. We also describe exciting extensions made possible by the generality of the proposed framework, including regularization using analysis-form sparsity priors that are incompatible with many existing approaches.