One-Dimensional Phase Retrieval: Regularization, Box Relaxation and Uniqueness

TL;DR

This paper provides a new characterization of the phase retrieval problem, showing that common regularization methods may not restrict solutions, and introduces a box relaxation approach that enables unique recovery of binary signals and an efficient denoising algorithm.

Contribution

It introduces a novel characterization of phase retrieval, demonstrates the equivalence of box relaxation to binary constraints, and develops an efficient denoising method for binary signals.

Findings

Gradient regularization does not restrict solutions.

Box relaxation is equivalent to binary constraints.

Binary signals can be uniquely recovered under certain conditions.

Abstract

Recovering a signal from its Fourier magnitude is referred to as phase retrieval, which occurs in different fields of engineering and applied physics. This paper gives a new characterization of the phase retrieval problem. Particularly useful is the analysis revealing that the common gradient-based regularization does not restrict the set of solutions to a smaller set. Specifically focusing on binary signals, we show that a box relaxation is equivalent to the binary constraint for Fourier-types of phase retrieval. We further prove that binary signals can be recovered uniquely up to trivial ambiguities under certain conditions. Finally, we use the characterization theorem to develop an efficient denoising algorithm.