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Register a new userUniqueness of STFT phase retrieval for bandlimited functions
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We consider the problem of phase retrieval from magnitudes of short-time Fourier transform (STFT) measurements. It is well-known that signals are uniquely determined (up to global phase) by their STFT magnitude when the underlying window has an ambiguity function that is nowhere vanishing. It is less clear, however, what can be said in terms of unique phase-retrievability when the ambiguity function of the underlying window vanishes on some of the time-frequency plane. In this short note, we demonstrate that by considering signals in Paley-Wiener spaces, it is possible to prove new uniqueness results for STFT phase retrieval. Among those, we establish a first uniqueness theorem for STFT phase retrieval from magnitude-only samples in a real-valued setting.
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We study the problem of phase retrieval in which one aims to recover a function $f$ from the magnitude of its wavelet transform $|mathcal{W}_psi f|$. We consider bandlimited functions and derive new uniqueness results for phase retrieval, where the wavelet itself can be complex-valued. In particular, we prove the first uniqueness result for the case that the wavelet $psi$ has a finite number of vanishing moments. In addition, we establish the first result on unique reconstruction from samples of the wavelet transform magnitude when the wavelet coefficients are complex-valued
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.
In a variety of fields, in particular those involving imaging and optics, we often measure signals whose phase is missing or has been irremediably distorted. Phase retrieval attempts the recovery of the phase information of a signal from the magnitude of its Fourier transform to enable the reconstruction of the original signal. A fundamental question then is: Under which conditions can we uniquely recover the signal of interest from its measured magnitudes? In this paper, we assume the measured signal to be sparse. This is a natural assumption in many applications, such as X-ray crystallography, speckle imaging and blind channel estimation. In this work, we derive a sufficient condition for the uniqueness of the solution of the phase retrieval (PR) problem for both discrete and continuous domains, and for one and multi-dimensional domains. More precisely, we show that there is a strong connection between PR and the turnpike problem, a classic combinatorial problem. We also prove that the existence of collisions in the autocorrelation function of the signal may preclude the uniqueness of the solution of PR. Then, assuming the absence of collisions, we prove that the solution is almost surely unique on 1-dimensional domains. Finally, we extend this result to multi-dimensional signals by solving a set of 1-dimensional problems. We show that the solution of the multi-dimensional problem is unique when the autocorrelation function has no collisions, significantly improving upon a previously known result.
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.
We give some details about the stationary phase lemma. We first prove a special case where the high order terms are derived explicitly. Based on that, we prove a more general case by using Morse lemma.
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