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 \[ { | W ϕ i f ( α m β n , α m ) | : i ∈ { 1 , 2 , 3 } , m , n ∈ Z } \{\lvert \mathcal {W}_{\phi _i} f(\alpha ^{m}\beta n,\alpha ^{m}) \rvert : i\in \{1,2,3\}, m,n\in \mathbb {Z}\} \] 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.