Unique wavelet sign retrieval from samples without bandlimiting

TL;DR

This paper demonstrates that real-valued signals can be uniquely reconstructed from wavelet magnitude samples without bandlimiting constraints, using specific wavelet combinations, while complex signals cannot.

Contribution

It introduces a novel wavelet sign retrieval method that guarantees unique recovery of real signals without prior bandlimiting assumptions, unlike previous approaches.

Findings

Unique recovery of real signals from wavelet magnitudes without bandlimiting.

No such guarantee exists for complex signals with the same method.

The method uses specific linear combinations of Poisson wavelets and their Hilbert transforms.

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 \[ \{\lvert \mathcal{W}_{\phi_i} f(\alpha^{m}\beta n,\alpha^{m}) \rvert: i\in\{1,2,3\}, m,n\in\mathbb{Z}\} \] for every $\alpha>1,\beta>0$ with $\beta\ln(\alpha)\leq 4\pi/(1+4p)$, $p>0$, when the three wavelets $\phi_i$ are suitable linear combinations of the Poisson wavelet $P_p$ of order $p$ and its Hilbert transform $\mathscr{H}P_p$. For complex-valued signals we find that this is not possible for any choice of the parameters $\alpha>1,\beta>0$, 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.