"The uniqueness problem in Gabor phase retrieval"Liehr, LukasThe uniqueness problem in Gabor phase retrieval concerns the determination of a square-integrable function from samples of the absolute value of its Gabor transform. We show that uniqueness fails if the Gabor transform is sampled on lattices of arbitrary density. This stands in fundamental contrast to ordinary sampling theory. Building on this non-uniqueness statement, we propose a potential solution: sampling on specific perturbations of a lattice ensures uniqueness. The latter yields the first uniqueness result through sampling on point configurations that exhibit finite density. In addition, we demonstrate that lattice-based uniqueness is achievable in certain restricted function spaces or if an ordinary lattice gets replaced by a so-called square-root lattice. |
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