Strobl22

Applied Harmonic Analysis and Friends

June 19th - 25th 2022

Strobl, AUSTRIA

"Phase retrieval of bandlimited functions for the wavelet transform"

Bartolucci, Francesca

The term phase retrieval indicates a large class of problems in which one seeks to recover a signal from phaseless measurements. We study the recovery of square-integrable functions from the absolute values of their wavelet transforms, also called wavelet phase retrieval. A fundamental issue consists of understanding for which choice of the mother wavelet $\psi$ and for which choice of the subspace $\mathcal{M}\subseteq L^2(\mathbb{R})$, every $f\in\mathcal{M}$ is determined up to a global phase by its wavelet coefficients $|\mathcal{W}_{\psi}f(b,a)|$, $b\in\mathbb{R}$, $a\in (0,+\infty)$. This is a famously difficult problem that has only been solved in very few cases, either by considering the setting in which both the signals and the mother wavelets are real-valued, or in which both are analytic signals. We derive a new uniqueness result for wavelet phase retrieval which guarantees the unique recovery of real-valued bandlimited signals, up to a global sign, even in the case when the mother wavelet is complex-valued. In particular, we prove the first uniqueness result for wavelet phase retrieval from samples in which the mother wavelets are allowed to be complex-valued.
http://univie.ac.at/projektservice-mathematik/e/talks/Bartolucci_2022-02_Strobl22_Bartolucci.pdf

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