Undersampled Phase Retrieval With Outliers

Abstract

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 1D Monte Carlo tests and 2D 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.

Analysis-form sparse phase retrieval using variable-splitting

“…These and other outliers are not well-modeled by the quadratic data fit terms used in noisy forms of conventional sparse recovery methods. Instead, this work employs the 1-norm data fit term used in robust regression and in recent phase retrieval work [18], [19]:…”

“…For instance, a wide range of recent methods introduce image sparsity as a prior model. These methods include Fienupstyle alternating projections [7], [8], [9], semidefinite programming via "matrix lifting" [10], [11], [12], [13], [14], [15], generalized approximate message passing [16], greedy pursuit [17], and variable splitting [18]. However, with the exception of semidefinite programming, these methods are restricted to synthesis-form or canonical sparsity.…”

“…This paper describes an extension to the variable splitting approach that applies analysis-form sparse regularization in the form of a total variation (TV) penalty to the phase retrieval problem, without such extreme memory requirements. In addition, because imaging at high resolution is generally constrained to very low signal-to-noise ratio (SNR) data, this work uses the 1-norm data fit term employed in both variable splitting [18] and semidefinite programming [19].…”

Sparse representation utilizing tight frame for phase retrieval

“…In the image PR field, the image regularization, such as l 1 regularization [13,14], is focused by researchers. They often formulate the non-convex l 1 minimization problem and solve the problem by alternating directions method of multipliers (ADMM) [15], which can obtain a suboptimal solution to the non-convex problem.…”

“…Due to the nonconvexity of the objective function, the optimal solution to the corresponding problem is difficult to obtain. Nevertheless, ADMM technique, which can obtain a satisfied solution to the PR problem [13,14], is utilized in this paper.…”