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تسجيل مستخدم جديدStability estimates for phase retrieval from discrete Gabor measurements
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نشر من قبل
Matthias Wellershoff
تاريخ النشر
2019
مجال البحث
الهندسة المعلوماتية
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English
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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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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
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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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