TY - JOUR
T1 - Phase retrieval from power spectra of masked signals
AU - Bandeira, Afonso S.
AU - Chen, Yutong
AU - Mixon, Dustin G.
N1 - Funding Information:
A.S.B. was supported by AFOSR Grant No. FA9550-12-1-0317 and CMUC/FCT project PTDC/MAT/114394/2009 COMPETE/FEDER; Y.C. was supported by DARPA Grant No. N66001-13-1-4052 and AFOSR Grant No. FA9550-13-1-0076; and D.G.M. was supported by NSF Grant No. DMS-1321779. The views expressed in this article are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense or the U.S. Government.
Publisher Copyright:
© The authors 2014. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
PY - 2014/6/1
Y1 - 2014/6/1
N2 - In diffraction imaging, one is tasked with reconstructing a signal from its power spectrum. To resolve the ambiguity in this inverse problem, one might invoke prior knowledge about the signal, but phase retrieval algorithms in this vein have found limited success. One alternative is to create redundancy in the measurement process by illuminating the signal multiple times, distorting the signal each time with a different mask. Despite several recent advances in phase retrieval, the community has yet to construct an ensemble of masks which uniquely determines all signals and admits an efficient reconstruction algorithm. In this paper, we leverage the recently proposed polarization method to construct such an ensemble. First, we construct four explicit masks which enable polarization recovery of any signal with non-vanishing Fourier transform, and then we construct Θ(log M) random masks which, with high probability, simultaneously enable polarization recovery of any signal whatsoever. We also present numerical simulations to illustrate the stability of the polarization method in this setting. In comparison to a state-of-the-art phase retrieval algorithm known as PhaseLift, we find that polarization is much faster with comparable stability.
AB - In diffraction imaging, one is tasked with reconstructing a signal from its power spectrum. To resolve the ambiguity in this inverse problem, one might invoke prior knowledge about the signal, but phase retrieval algorithms in this vein have found limited success. One alternative is to create redundancy in the measurement process by illuminating the signal multiple times, distorting the signal each time with a different mask. Despite several recent advances in phase retrieval, the community has yet to construct an ensemble of masks which uniquely determines all signals and admits an efficient reconstruction algorithm. In this paper, we leverage the recently proposed polarization method to construct such an ensemble. First, we construct four explicit masks which enable polarization recovery of any signal with non-vanishing Fourier transform, and then we construct Θ(log M) random masks which, with high probability, simultaneously enable polarization recovery of any signal whatsoever. We also present numerical simulations to illustrate the stability of the polarization method in this setting. In comparison to a state-of-the-art phase retrieval algorithm known as PhaseLift, we find that polarization is much faster with comparable stability.
KW - Angular synchronization
KW - Diffraction imaging
KW - Phase retrieval
KW - Polarization
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U2 - 10.1093/imaiai/iau002
DO - 10.1093/imaiai/iau002
M3 - Article
AN - SCOPUS:84920051134
SN - 2049-8772
VL - 3
SP - 83
EP - 102
JO - Information and Inference
JF - Information and Inference
IS - 2
ER -