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 multiwavelet frame coefficients (Formula Presented) for every α > 1, β > 0 with β ln(α) ≤ 4π/(1 + 4p), p > 0, when the three wavelets φi are suitable linear combinations of the Poisson wavelet P_p of order p and its Hilbert transform H P_p. For complex-valued signals we find that this is not possible for any choice of the parameters α > 1, β > 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.