Stable Phase Retrieval Using Low-Redundancy Frames of Polynomials

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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.

Functional Analysis, Numerical Analysis
Portions of this document appear in: Bodmann, Bernhard G., and Nathaniel Hammen. "Stable phase retrieval with low-redundancy frames." Advances in computational mathematics 41, no. 2 (2015): 317-331. And in: Bodmann, Bernhard G., and Nathaniel Hammen. "Algorithms and error bounds for noisy phase retrieval with low-redundancy frames." Applied and Computational Harmonic Analysis 43, no. 3 (2017): 482-503. And in: Bodmann, Bernhard G., and Nathaniel Hammen. "Error bounds for noisy compressive phase retrieval." In 2015 International Conference on Sampling Theory and Applications (SampTA), pp. 342-346. IEEE, 2015.