RICAM Colloquium - Mathematical Aspects of STFT Phase Retrieval
Thursday, December 14, 15:30, SP 416-2
Mathematical Aspects of STFT Phase Retrieval
Short Time Fourier Transform (STFT) Phase retrieval refers to the problem of reconstructing a function from its spectrogram (e.g., the absolute values of its STFT). Such problems appear in several signal processing applications and most prominently in the reconstruction of ptychographic measurements in microscopic imaging. In practice one is interested in reliable and stable reconstruction algorithms that utilize a small number of spectrogram samples but it is not known to which extent this is always possible. In this talk I will present several mathematical results that partly elucidate this question. In a first part I will present results that characterize the noise stability of spectrogram measurements via a certain spectral quantity associated with the given spectrogram. In a second part we consider the question to which extent it is possible to reconstruct a function from discrete spectrogram samples. We show that such a reconstruction is never possible if the sampling set is a lattice and irrespective of the window function used in the STFT transform. On the positive side we construct alternative sampling sets that allow for unique reconstruction of each L2 function from its sampled spectrogram.These results have a direct impact on the design of signal acquisition methods in the context of ptychography. They also build on a surprisingly broad mix of techniques from harmonic analyis, functional analyis, complex analysis, and Riemannian geometry.