Injectivity of Gabor phase retrieval from lattice measurements

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 Gr\"ochenig, Romero and St\"ockler, 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 M\"untz-type theorems as well as sampling theorems in shift-invariant spaces.Comment: 18 page

Generalized sampling: from shift-invariant to U-invariant spaces

The aim of this article is to derive a sampling theory in U-invariant subspaces of a separable Hilbert space ℋ where U denotes a unitary operator defined on ℋ. To this end, we use some special dual frames for L2(0, 1), and the fact that any U-invariant subspace with stable generator is the image of L2(0, 1) by means of a bounded invertible operator. The used mathematical technique mimics some previous sampling work for shift-invariant subspaces of L2(ℝ). Thus, sampling frame expansions in U-invariant spaces are obtained. In order to generalize convolution systems and deal with the time-jitter error in this new setting we consider a continuous group of unitary operators which includes the operator U

Invariant Subspaces of the Harmonic Dirichlet Space with Large Co-Dimension

In this paper, we comment on the complexity of the invariant subspaces (under the bilateral Dirichlet shift f → ζf) of the harmonic Dirichlet space D. Using the sampling theory of Seip and some work on invariant subspaces of Bergman spaces, we will give examples of invariant subspaces F ⊂ D with dim(F/ζF) = n, n ∈ N ∪ {∞}. We will also generalize this to the Dirichlet classes Dα, 0 \u3cα\u3c ∞, as well as the Besov classes Bα p ,

On Some Sampling-Related Frames in U-Invariant Spaces

This paper is concerned with the characterization as frames of some sequences in -invariant spaces of a separable Hilbert space H where U denotes an unitary operator defined on H ; besides, the dual frames having the same form are also found. This general setting includes, in particular, shift-invariant or modulation-invariant subspaces in L2(R) , where these frames are intimately related to the generalized sampling problem. We also deal with some related perturbation problems. In doing so, we need the unitary operator to belong to a continuous group of unitary operators

Sampling and Reconstruction of Spatial Signals

Digital processing of signals f may start from sampling on a discrete set Γ, f →(f(ϒη))ϒηεΓ. The sampling theory is one of the most basic and fascinating topics in applied mathematics and in engineering sciences. The most well known form is the uniform sampling theorem for band-limited/wavelet signals, that gives a framework for converting analog signals into sequences of numbers. Over the past decade, the sampling theory has undergone a strong revival and the standard sampling paradigm is extended to non-bandlimited signals including signals in reproducing kernel spaces (RKSs), signals with finite rate of innovation (FRI) and sparse signals, and to nontraditional sampling methods, such as phaseless sampling. In this dissertation, we first consider the sampling and Galerkin reconstruction in a reproducing kernel space. The fidelity measure of perceptual signals, such as acoustic and visual signals, might not be well measured by least squares. In the first part of this dissertation, we introduce a fidelity measure depending on a given sampling scheme and propose a Galerkin method in Banach space setting for signal reconstruction. We show that the proposed Galerkin method provides a quasi-optimal approximation, and the corresponding Galerkin equations could be solved by an iterative approximation-projection algorithm in a reproducing kernel subspace of Lp. A spatially distributed network contains a large amount of agents with limited sensing, data processing, and communication capabilities. Recent technological advances have opened up possibilities to deploy spatially distributed networks for signal sampling and reconstruction. We introduce a graph structure for a distributed sampling and reconstruction system by coupling agents in a spatially distributed network with innovative positions of signals. We split a distributed sampling and reconstruction system into a family of overlapping smaller subsystems, and we show that the stability of the sensing matrix holds if and only if its quasi-restrictions to those subsystems have l_2 uniform stability. This new stability criterion could be pivotal for the design of a robust distributed sampling and reconstruction system against supplement, replacement and impairment of agents, as we only need to check the uniform stability of affected subsystems. We also propose an exponentially convergent distributed algorithm for signal reconstruction, that provides a suboptimal approximation to the original signal in the presence of bounded sampling noises. Phase retrieval (Phaseless Sampling and Reconstruction) arises in various fields of science and engineering. It consists of reconstructing a signal of interest from its magnitude measurements. Sampling in shift-invariant spaces is a realistic model for signals with smooth spectrum. We consider phaseless sampling and reconstruction of real-valued signals in a shift-invariant space from their magnitude measurements on the whole Euclidean space and from their phaseless samples taken on a discrete set with finite sampling density. We find an equivalence between nonseparability of signals in a shift-invariant space and their phase retrievability with phaseless samples taken on the whole Euclidean space. We also introduce an undirected graph to a signal and use connectivity of the graph to characterize the nonseparability of high-dimensional signals. Under the local complement property assumption on a shift-invariant space, we find a discrete set with finite sampling density such that signals in shift-invariant spaces, that are determined by their magnitude measurements on the whole Euclidean space, can be reconstructed in a stable way from their phaseless samples taken on that discrete set. We also propose a reconstruction algorithm which provides a suboptimal approximation to the original signal when its noisy phaseless samples are available only

Sampling and Reconstruction of Signals in a Reproducing Kernel Subspace of Lp(Rd)L^p({\Bbb R}^d)

In this paper, we consider sampling and reconstruction of signals in a reproducing kernel subspace of L^p(\Rd), 1\le p\le \infty, associated with an idempotent integral operator whose kernel has certain off-diagonal decay and regularity. The space of $p$-integrable non-uniform splines and the shift-invariant spaces generated by finitely many localized functions are our model examples of such reproducing kernel subspaces of L^p(\Rd). We show that a signal in such reproducing kernel subspaces can be reconstructed in a stable way from its samples taken on a relatively-separated set with sufficiently small gap. We also study the exponential convergence, consistency, and the asymptotic pointwise error estimate of the iterative approximation-projection algorithm and the iterative frame algorithm for reconstructing a signal in those reproducing kernel spaces from its samples with sufficiently small gap

A matrix pencil approach to the existence of compactly supported reconstruction functions in average sampling

The aim of this work is to solve a question raised for average sampling in shift-invariant spaces by using the well-known matrix pencil theory. In many common situations in sampling theory, the available data are samples of some convolution operator acting on the function itself: this leads to the problem of average sampling, also known as generalized sampling. In this paper we deal with the existence of a sampling formula involving these samples and having reconstruction functions with compact support. Thus, low computational complexity is involved and truncation errors are avoided. In practice, it is accomplished by means of a FIR filter bank. An answer is given in the light of the generalized sampling theory by using the oversampling technique: more samples than strictly necessary are used. The original problem reduces to finding a polynomial left inverse of a polynomial matrix intimately related to the sampling problem which, for a suitable choice of the sampling period, becomes a matrix pencil. This matrix pencil approach allows us to obtain a practical method for computing the compactly supported reconstruction functions for the important case where the oversampling rate is minimum. Moreover, the optimality of the obtained solution is established