KEYWORDS: Phase retrieval, Radon, Magnesium, Mathematics, Space operations, Molybdenum, Binary data, Wavelets, Current controlled current source, Direct methods
We answer a number of open problems concerning phase retrieval and phase retrieval by projections. In particular, one main theorem classifies phase retrieval by projections via collections of sequences of vectors allowing norm retrieval. Another key result computes the minimal number of vectors needed to add to a frame in order for it to possess the complement property and hence allow phase retrieval. In furthering this idea, in a third main theorem we show that when a collection of subspaces is one subspace short from allowing phase retrieval, then any partition of these subspaces spans two hyperplanes. We offer many more results in this area as well as provide a large number of examples showing the limitations of the theory.
The mathematical study of phase retrieval was started in 2006 in a landmark paper of Balan, Casazza and Edidin. This quickly became a heavily studied topic with implications for many areas of research in both applied mathematics and engineering. Recently there have been developments in a new area of study pertaining to phase retrieval given by projections. We give an extensive overview of the papers regarding projection phase retrieval.
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