Non-negativity constraints in the one-dimensional discrete-time phase retrieval problem

TL;DR

This paper investigates whether non-negativity constraints can ensure unique solutions in one-dimensional discrete-time phase retrieval problems, concluding that such constraints are generally insufficient to guarantee uniqueness due to prevalent ambiguities.

Contribution

The paper provides a theoretical analysis showing non-negativity alone does not resolve ambiguities in 1D phase retrieval, highlighting the limitations of common assumptions.

Findings

Non-negativity is not sufficient for uniqueness in 1D phase retrieval.

Ambiguities are common and not rare exceptions.

Theoretical characterization of ambiguities in the problem.

Abstract

Phase retrieval problems occur in a width range of applications in physics and engineering such as crystallography, astronomy, and laser optics. Common to all of them is the recovery of an unknown signal from the intensity of its Fourier transform. Because of the well-known ambiguousness of these problems, the determination of the original signal is generally challenging. Although there are many approaches in the literature to incorporate the assumption of non-negativity of the solution into numerical algorithms, theoretical considerations about the solvability with this constraint occur rarely. In this paper, we consider the one-dimensional discrete-time setting and investigate whether the usually applied a priori non-negativity can overcame the ambiguousness of the phase retrieval problem or not. We show that the assumed non-negativity of the solution is usually not a sufficient a priori condition to ensure uniqueness in one-dimensional phase retrieval. More precisely, using an appropriate characterization of the occurring ambiguities, we show that neither the uniqueness nor the ambiguousness are rare exceptions.