Phase Transitions in Phase Retrieval

Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


Consider a scenario in which an unknown signal is transformed by a known linear operator, and then the pointwise absolute value of the unknown output function is reported. This scenario appears in several applications, and the goal is to recover the unknown signal – this is called phase retrieval. Phase retrieval has been a popular subject of research in the last few years, both in determining whether complete information is available with a given linear operator and in finding efficient and stable phase retrieval algorithms in the cases where complete information is available. Interestingly, there are a few ways to measure information completeness, and each way appears to be governed by a phase transition of sorts. This chapter will survey the state of the art with some of these phase transitions, and identify a few open problems for further research.

Key words:

Phase retrieval Phase transition Informationally complete Full spark Almost injectivity Unit norm tight frames 



The author was supported by NSF Grant No. DMS-1321779. The views expressed in this chapter are those of the author and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the U.S. Government.


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsAir Force Institute of TechnologyWright-Patterson AFBUSA

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