Special Analysis seminar: Stable phase retrieval using three mask frames of polynomials with oversampling

Tuesday, December 6, 2016 - 9:30am
Math Sci 110
Nathaniel Hammen
(Air Force Institute of Technology)

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 talk is the recovery of the phase information of a signal using a small number of these magnitude measurements. This recovery 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 show that even though at least 4 masks are normally required in the structured illumination model, this set of measurements may be thought of as a 3 mask oversampled structured illumination model. We also explore the behavior of pathological signals in this algorithm. 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.

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