Unique wavelet sign retrieval from samples without bandlimiting

Author:

Alaifari Rima,Bartolucci Francesca,Wellershoff Matthias

Abstract

We study the problem of recovering a signal from magnitudes of its wavelet frame coefficients when the analyzing wavelet is real-valued. We show that every real-valued signal can be uniquely recovered, up to global sign, from its multiwavelet frame coefficients \[ { | W ϕ i f ( α m β n , α m ) | : i { 1 , 2 , 3 } , m , n Z } \{\lvert \mathcal {W}_{\phi _i} f(\alpha ^{m}\beta n,\alpha ^{m}) \rvert : i\in \{1,2,3\}, m,n\in \mathbb {Z}\} \] for every α > 1 , β > 0 \alpha >1,\beta >0 with β ln ( α ) 4 π / ( 1 + 4 p ) \beta \ln (\alpha )\leq 4\pi /(1+4p) , p > 0 p>0 , when the three wavelets ϕ i \phi _i are suitable linear combinations of the Poisson wavelet P p P_p of order p p and its Hilbert transform H P p \mathscr {H}P_p . For complex-valued signals we find that this is not possible for any choice of the parameters α > 1 , β > 0 \alpha >1,\beta >0 , and for any window. In contrast to the existing literature on wavelet sign retrieval, our uniqueness results do not require any bandlimiting constraints or other a priori knowledge on the real-valued signals to guarantee their unique recovery from the absolute values of their wavelet coefficients.

Funder

Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung

Publisher

American Mathematical Society (AMS)

Reference19 articles.

1. Phase retrieval of bandlimited functions for the wavelet transform;Alaifari, Rima;Appl. Comput. Harmon. Anal.,2023

2. Reconstructing real-valued functions from unsigned coefficients with respect to wavelet and other frames;Alaifari, Rima;J. Fourier Anal. Appl.,2017

3. Phase retrieval in the general setting of continuous frames for Banach spaces;Alaifari, Rima;SIAM J. Math. Anal.,2017

4. On signal reconstruction without phase;Balan, Radu;Appl. Comput. Harmon. Anal.,2006

5. Phase retrieval in infinite-dimensional Hilbert spaces;Cahill, Jameson;Trans. Amer. Math. Soc. Ser. B,2016

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