Phase Retrieval Versus Phaseless Reconstruction

In 2006, Balan, Casazza, and Edidin [1] 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 phaseless reconstruction with phase retrieval. At a meeting in 2012, Casazza asked: 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 in the real case.