Injectivity of Gabor phase retrieval from lattice measurements
Philipp Grohs, Lukas Liehr
Abstract
We establish novel uniqueness results for the Gabor phase retrieval problem: if G:L2(R)→L2(R2) denotes the Gabor transform then every f∈L4[−c2,c2] is determined up to a global phase by the values |Gf(x,ω)| where (x,ω) are points on the lattice b−1Z×(2c)−1Z and b>0 is an arbitrary positive constant. This for the first time shows that compactly-supported, complex-valued functions can be uniquely reconstructed from lattice samples of their spectrogram. Moreover, by making use of recent developments related to sampling in shift-invariant spaces by Gröchenig, Romero and Stöckler, we prove analogous uniqueness results for functions in shift-invariant spaces with Gaussian generator. Generalizations to nonuniform sampling are also presented. Finally, we compare our results to the situation where the considered signals are assumed to be real-valued.