Phase Retrieval Versus Phaseless Reconstruction

TL;DR

This paper clarifies the relationship between phase retrieval and phaseless reconstruction, proving their equivalence and exploring related concepts like projections and weak phase retrieval.

Contribution

It establishes the equivalence between phase retrieval and phaseless reconstruction, and differentiates weak phase retrieval from these concepts.

Findings

Phase retrieval is equivalent to phaseless reconstruction.

Phase retrieval by projections is equivalent to phaseless reconstruction by projections.

Weak phase retrieval is fundamentally different from phaseless reconstruction.

Abstract

In 2006, Balan/Casazza/Edidin \cite{BCE} introduced the frame theoretic study of phaseless reconstruction. Since then, this has turned into a very active area of research. Over the years, many people have replaced the term {\it phaseless reconstruction} with {\it phase retrieval}. Casazza then asked: {\it Are these really the same?} In this paper, we will show that phase retrieval is equivalent to phaseless reconstruction. We then show, more generally, that phase retrieval by projections is equivalent to phaseless reconstruction by projections. Finally, we study {\it weak phase retrieval} and discover that it is very different from phaseless reconstruction.