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Register a new user$L^2$-stability analysis for Gabor phase retrieval
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We consider the problem of reconstructing the missing phase information from spectrogram data $|mathcal{G} f|,$ with $$ mathcal{G}f(x,y)=int_mathbb{R} f(t) e^{-pi(t-x)^2}e^{-2pi i t y}dt, $$ the Gabor transform of a signal $fin L^2(mathbb{R})$. More specifically, we are interested in domains $Omegasubseteq mathbb{R}^2$, which allow for stable local reconstruction, that is $$ |mathcal{G}g| approx |mathcal{G}f| quad text{in} ~Omega quadLongrightarrow quad exists tauinmathbb{T}:quad mathcal{G}g approx taumathcal{G}f quad text{in} ~Omega. $$ In recent work [P. Grohs and M. Rathmair. Stable Gabor Phase Retrieval and Spectral Clustering. Comm. Pure Appl. Math. (2019)] and [P. Grohs and M. Rathmair. Stable Gabor phase retrieval for multivariate functions. J. Eur. Math. Soc. (2021)] we established a characterization of the stability of this phase retrieval problem in terms of the connectedness of the observed measurements. The main downside of the aforementioned results is that the similarity of two spectrograms is measured w.r.t. a first order weighted Sobolev norm. In this article we remove this flaw and essentially show that the Sobolev norm may be replaced by the $L^2-$norm. Using this result allows us to show that it suffices to sample the spectrogram on suitable discrete sampling sets -- a property of crucial importance for practical applications.
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In recent work [P. Grohs and M. Rathmair. Stable Gabor Phase Retrieval and Spectral Clustering. Communications on Pure and Applied Mathematics (2018)] the instabilities of the Gabor phase retrieval problem, i.e., the problem of reconstructing a function $f$ from its spectrogram $|mathcal{G}f|$, where $$ mathcal{G}f(x,y)=int_{mathbb{R}^d} f(t) e^{-pi|t-x|^2} e^{-2pi i tcdot y} dt, quad x,yin mathbb{R}^d, $$ have been completely classified in terms of the disconnectedness of the spectrogram. These findings, however, were crucially restricted to the onedimensional case ($d=1$) and therefore not relevant for many practical applications. In the present paper we not only generalize the aforementioned results to the multivariate case but also significantly improve on them. Our new results have comprehensive implications in various applications such as ptychography, a highly popular method in coherent diffraction imaging.
The problem of reconstructing a function from the magnitudes of its frame coefficients has recently been shown to be never uniformly stable in infinite-dimensional spaces [5]. This result also holds for frames that are possibly continuous [2]. On the other hand, the problem is always stable in finite-dimensional settings. A prominent example of such a phase retrieval problem is the recovery of a signal from the modulus of its Gabor transform. In this paper, we study Gabor phase retrieval and ask how the stability degrades on a natural family of finite-dimensional subspaces of the signal domain $L^2(mathbb{R})$. We prove that the stability constant scales at least quadratically exponentially in the dimension of the subspaces. Our construction also shows that typical priors such as sparsity or smoothness promoting penalties do not constitute regularization terms for phase retrieval.
We establish novel uniqueness results for the Gabor phase retrieval problem: If $mathcal{G} : L^2(mathbb{R}) to L^2(mathbb{R}^2)$ denotes the Gabor transform then every $f in L^4[-c/2,c/2]$ is determined up to a global phase by the values $|mathcal{G}f(x,omega)|$ where $(x,omega)$ are points on the lattice $mathbb{Z} times (2c)^{-1}mathbb{Z}$. This for the first time shows that compactly supported 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 Grochenig, Romero and Stockler, we prove analogous uniqueness results for functions in general shift-invariant spaces with Gaussian generator. Generalizations to nonuniform lattices are also presented. The results are based on a combination of certain Muntz-type theorems as well as sampling theorems in shift-invariant spaces.
We consider the recovery of square-integrable signals from discrete, equidistant samples of their Gabor transform magnitude and show that, in general, signals can not be recovered from such samples. In particular, we show that for any lattice, one can construct functions in $L^2(mathbb{R})$ which do not agree up to global phase but whose Gabor transform magnitudes sampled on the lattice agree. These functions can be constructed to be either real-valued or complex-valued and have good concentration in both time and frequency.
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 [7] and possibly severely ill-conditioned in finite-dimensional Hilbert spaces [7]. 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 [1]. 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 [6,12,18,20].
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