Stable Phase Retrieval Using Low-Redundancy Frames of Polynomials

In many applications, measurements of a signal consist of the magnitudes of linear functionals while the phase information of these functionals is unavailable. Examples of these type of measurements occur in optics, quantum mechanics, speech recognition, and x-ray crystallography. The main topic of this thesis is the recovery of the phase information of a signal using a small number of these magnitude measurements. This is called phase retrieval. We provide a choice of 4d − 4 magnitude measurements that uniquely determines any d dimensional signal, up to a unimodular constant. Then we provide a choice of 6d − 3 magnitude measurements that admits a stable polynomial time algorithm to recover the signal under the influence of noise. We also explore the behavior of pathological signals in this algorithm, as well as the mean squared error. Finally, we show that if the signal is known to be s sparse, then we only need a suitable choice of O(s log d/s) such measurements for the stable algorithm to successfully recover the signal.

Name thesis:degree_name

Doctor of Philosophy

Level thesis:degree_level

Doctoral

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Houston

Year dc:date.issued

2015

Author dc:creator

• Hammen, Nathaniel 1984-

Advisor dc:contributor.advisor

• Bodmann, Bernhard G.

Committee members dc:contributor.committeemember

• Mixon, Dustin G.
• Papadakis, Emanuel I.
• Azencott, Robert

• Functional Analysis
• Numerical Analysis

Statement dc:rights

• The author of this work is the copyright owner. UH Libraries and the Texas Digital Library have their permission to store and provide access to this work. UH Libraries has secured permission to reproduce any and all previously published materials contained in the work. Further transmission, reproduction, or presentation of this work is prohibited except with permission of the author(s).

Language dc:language.iso

eng

Handle dc:identifier.uri

https://hdl.handle.net/10657/4883

OAI identifier oai:identifier

oai:uh-ir.tdl.org:10657/4883