Phase retrieval of bandlimited functions for the wavelet transform

We study the problem of phase retrieval in which one aims to recover a function $f$ from the magnitude of its wavelet transform $|mathcal{W}_psi f|$. We consider bandlimited functions and derive new uniqueness results for phase retrieval, where the wavelet itself can be complex-valued. In particular, we prove the first uniqueness result for the case that the wavelet $psi$ has a finite number of vanishing moments. In addition, we establish the first result on unique reconstruction from samples of the wavelet transform magnitude when the wavelet coefficients are complex-valued