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Phase retrieval of bandlimited functions for the wavelet transform

by R. Alaifari and F. Bartolucci and M. Wellershoff

(Report number 2020-58)

Abstract
We study the recovery of square-integrable signals from the absolute values of their wavelet transforms, also called wavelet phase retrieval. We present a new uniqueness result for wavelet phase retrieval. To be precise, we show that any wavelet with finitely many vanishing moments allows for the unique recovery of real-valued bandlimited signals up to global sign. Additionally, we present the first uniqueness result for sampled wavelet phase retrieval in which the underlying wavelets are allowed to be complex-valued and we present a uniqueness result for phase retrieval from sampled Cauchy wavelet transform measurements.

Keywords: Phase retrieval, Wavelet transform, Morlet wavelet, Cauchy wavelets

@Techreport{ABW20_931,
  author = {R. Alaifari and F. Bartolucci and M. Wellershoff},
  title = {Phase retrieval of bandlimited functions for the wavelet transform},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2020-58},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2020/2020-58.pdf },
  year = {2020}
}

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