On The 2D Phase Retrieval Problem

TL;DR

This paper investigates the 2D phase retrieval problem, demonstrating that it can be reduced to a constrained 1D problem, and proves that a single additional constraint ensures unique solutions for almost all signals.

Contribution

It establishes a connection between 2D and 1D phase retrieval, showing that one extra constraint guarantees uniqueness in the 1D formulation for almost all signals.

Findings

A single additional constraint suffices for uniqueness in 1D phase retrieval.

2D phase retrieval can be formulated as a constrained 1D problem.

Analytical solutions are possible under these constraints with combinatorial complexity.

Abstract

The recovery of a signal from the magnitude of its Fourier transform, also known as phase retrieval, is of fundamental importance in many scientific fields. It is well known that due to the loss of Fourier phase the problem in 1D is ill-posed. Without further constraints, there is no unique solution to the problem. In contrast, uniqueness up to trivial ambiguities very often exists in higher dimensions, with mild constraints on the input. In this paper we focus on the 2D phase retrieval problem and provide insight into this uniqueness property by exploring the connection between the 2D and 1D formulations. In particular, we show that 2D phase retrieval can be cast as a 1D problem with additional constraints, which limit the solution space. We then prove that only one additional constraint is sufficient to reduce the many feasible solutions in the 1D setting to a unique solution for almost all signals. These results allow to obtain an analytical approach (with combinatorial complexity) to solve the 2D phase retrieval problem when it is unique.