2023/02/16 by Alaifari, Rima, Bartolucci, Francesca, Wellershoff, Matthias
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2302.08129
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 \| Wϕi f(αmβn,αm) |: i∈\1,2,3\, m,n∈ℤ\ for every α>1,β>0 with βln(α)≤ 4π/(1+4p), p>0, when the three wavelets ϕi are suitable linear combinations of the Poisson wavelet Pp of order p and its Hilbert transform \mathscrHPp. 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.