Stability Estimates for Phase Retrieval from Discrete Gabor Measurements

Phase retrieval refers to the problem of recovering some signal (which is often modelled as an element of a Hilbert space) from phaseless measurements. It has been shown that in the deterministic setting phase retrieval from frame coefficients is always unstable in infinite-dimensional Hilbert spaces (Cahill et al. in Trans Am Math Soc Ser B 3(3):63–76, 2016) and possibly severely ill-conditioned in finite-dimensional Hilbert spaces (Cahill et al. in Trans Am Math Soc Ser B 3(3):63–76, 2016). Recently, it has also been shown that phase retrieval from measurements induced by the Gabor transform with Gaussian window function is stable under a more relaxed semi-global phase recovery regime based on atoll functions (Alaifari in Found Comput Math 19(4):869–900, 2019). In finite dimensions, we present first evidence that this semi-global reconstruction regime allows one to do phase retrieval from measurements of bandlimited signals induced by the discrete Gabor transform in such a way that the corresponding stability constant only scales like a low order polynomial in the space dimension. To this end, we utilise reconstruction formulae which have become common tools in recent years (Bojarovska and Flinth in J Fourier Anal Appl 22(3):542–567, 2016; Eldar et al. in IEEE Signal Process Lett 22(5):638–642, 2014; Li et al. in IEEE Signal Process Lett 24(4):372–376, 2017; Nawab et al. in IEEE Trans Acoust Speech Signal Process 31(4):986–998, 1983).

Inverting spectrogram measurements via aliased Wigner distribution deconvolution and angular synchronization

We propose a two-step approach for reconstructing a signal $\textbf x\in \mathbb{C}^d$ from subsampled discrete short-time Fourier transform magnitude (spectogram) measurements: first, we use an aliased Wigner distribution deconvolution approach to solve for a portion of the rank-one matrix $\widehat{\textbf{x}}\widehat{\textbf{x}}^{*}.$ Secondly, we use angular synchronization to solve for $\widehat{\textbf{x}}$ (and then for $\textbf{x}$ by Fourier inversion). Using this method, we produce two new efficient phase retrieval algorithms that perform well numerically in comparison to standard approaches and also prove two theorems; one which guarantees the recovery of discrete, bandlimited signals $\textbf{x}\in \mathbb{C}^{d}$ from fewer than $d$ short-time Fourier transform magnitude measurements and another which establishes a new class of deterministic coded diffraction pattern measurements which are guaranteed to allow efficient and noise robust recovery.