Unique wavelet sign retrieval from samples without bandlimiting

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 multi-wavelet frame coefficients \[ \{\lvert \mathcal{W}_{ϕ_i} f(α^{m}βn,α^{m}) \rvert: i\in\{1,2,3\}, m,n\in\mathbb{Z}\} \] for every $\hat{I}\pm >1,{\hat{I}}^2>0$ with ${\hat{I}}^2\ln (\hat{I}\pm )\le 4\ddot{I}€/(1+4p)$, $p>0$, when the three wavelets $\ddot{I}{•}_i$ are suitable linear combinations of the Poisson wavelet $P_p$ of order $p$ and its Hilbert transform $\mathcal{H}{P}_p$. For complex-valued signals we find that this is not possible for any choice of the parameters $\hat{I}\pm >1,{\hat{I}}^2>0$, 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.

14 pages, 2 figures