ﻻ يوجد ملخص باللغة العربية
We study the phase reconstruction of signals $f$ belonging to complex Gaussian shift-invariant spaces $V^infty(varphi)$ from spectrogram measurements $|mathcal{G}f(X)|$ where $mathcal{G}$ is the Gabor transform and $X subseteq mathbb{R}^2$. An explicit reconstruction formula will demonstrate that such signals can be recovered from measurements located on parallel lines in the time-frequency plane by means of a Riesz basis expansion. Moreover, connectedness assumptions on $|f|$ result in stability estimates in the situation where one aims to reconstruct $f$ on compact intervals. Driven by a recent observation that signals in Gaussian shift-invariant spaces are determined by lattice measurements [Grohs, P., Liehr, L., Injectivity of Gabor phase retrieval from lattice measurements, arXiv:2008.07238] we prove a sampling result on the stable approximation from finitely many spectrogram samples. The resulting algorithm provides a non-iterative, provably stable and convergent approximation technique. In addition, it constitutes a method of approximating signals in function spaces beyond $V^infty(varphi)$, such as Paley-Wiener spaces.
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 funct
We study the expressive power of deep ReLU neural networks for approximating functions in dilated shift-invariant spaces, which are widely used in signal processing, image processing, communications and so on. Approximation error bounds are estimated
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
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
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