An algebraic characterization of injectivity in phase retrieval

Aldo Conca; Dan Edidin; Milena Hering; Cynthia Vinzant

doi:10.1016/j.acha.2014.06.005

An algebraic characterization of injectivity in phase retrieval

Aldo Conca, Dan Edidin, Milena Hering, Cynthia Vinzant

School of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

A complex frame is a collection of vectors that span C^Mand define measurements, called intensity measurements, on vectors in C^M. In purely mathematical terms, the problem of phase retrieval is to recover a complex vector from its intensity measurements, namely the modulus of its inner product with these frame vectors. We show that any vector is uniquely determined (up to a global phase factor) from
4M−4 generic measurements. To prove this, we identify the set of frames defining non-injective measurements with the projection of a real variety and bound its dimension.

Original language	English
Pages (from-to)	346-356
Number of pages	11
Journal	Applied and Computational Harmonic Analysis
Volume	38
Early online date	30 Jun 2014
DOIs	https://doi.org/10.1016/j.acha.2014.06.005
Publication status	Published - Mar 2015

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1312.0158.pdSubmitted manuscript, 208 KB

Milena Hering
- School of Mathematics - Reader
Person: Academic: Research Active

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title = "An algebraic characterization of injectivity in phase retrieval",

abstract = "A complex frame is a collection of vectors that span CM and define measurements, called intensity measurements, on vectors in CM. In purely mathematical terms, the problem of phase retrieval is to recover a complex vector from its intensity measurements, namely the modulus of its inner product with these frame vectors. We show that any vector is uniquely determined (up to a global phase factor) from 4M−4 generic measurements. To prove this, we identify the set of frames defining non-injective measurements with the projection of a real variety and bound its dimension.",

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AU - Edidin, Dan

AU - Hering, Milena

AU - Vinzant, Cynthia

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AB - A complex frame is a collection of vectors that span CM and define measurements, called intensity measurements, on vectors in CM. In purely mathematical terms, the problem of phase retrieval is to recover a complex vector from its intensity measurements, namely the modulus of its inner product with these frame vectors. We show that any vector is uniquely determined (up to a global phase factor) from 4M−4 generic measurements. To prove this, we identify the set of frames defining non-injective measurements with the projection of a real variety and bound its dimension.

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DO - 10.1016/j.acha.2014.06.005

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JO - Applied and Computational Harmonic Analysis

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An algebraic characterization of injectivity in phase retrieval

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Milena Hering

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