Gabor Frames Research Articles

Projected Gabor deconvolution

Gabor deconvolution consists of approximating a nonstationary problem as several stationary subproblems via the Gabor frame, solving each subproblem independently, and then recombining/projecting the subsolutions into an approximate solution to the original nonstationary problem. The approximations, however, cause inherent instability due to systematic errors that prevent the algorithm from converging to the true solution even for noise-free cases. Furthermore, the method will be time consuming when a nonlinear optimization is used for shaping the reflectivity structure. An alternative projected Gabor deconvolution is considered, which is based on the Gabor expansion. However, in contrast to the conventional form, in the new method, first the problem is projected into the time domain, and then an inversion is performed to obtain the final reflectivity. Compared with the Gabor deconvolution, the projected alternative (1) exhibits an improved convergence property, (2) is more efficient for sparse deconvolution because only a single optimization is required to be solved, and (3) is more flexible for incorporating prior information about noise and reflectivity structure via a least-squares method. Numerical tests using simulated and field data are presented showing that the new method generates more accurate and stable reflectivity models compared with the Gabor deconvolution.

Reproducing formulas for generalized translation invariant systems on locally compact abelian groups

In this paper we connect the well-established discrete frame theory of generalized shift invariant systems to a continuous frame theory. To do so, we letΓj\Gamma _{\!j},j∈Jj \in J, be a countable family of closed, co-compact subgroups of a second countable locally compact abelian groupGGand study systems of the form⋃j∈J{gj,p(⋅−γ)}γ∈Γj,p∈Pj\bigcup _{j \in J}\{ g_{j,p}(\cdot - \gamma )\}_{\gamma \in \Gamma _{\!j}, p \in P_j}with generatorsgj,pg_{j,p}inL2(G)L^2(G)and with eachPjP_jbeing a countable or an uncountable index set. We refer to systems of this form as generalized translation invariant (GTI) systems. Many of the familiar transforms, e.g., the wavelet, shearlet and Gabor transform, both their discrete and continuous variants, are GTI systems. Under a technicalα\alphalocal integrability condition (α\alpha-LIC) we characterize when GTI systems constitute tight and dual frames that yield reproducing formulas forL2(G)L^2(G). This generalizes results on generalized shift invariant systems, where eachPjP_jis assumed to be countable and eachΓj\Gamma _{\!j}is a uniform lattice inGG, to the case of uncountably many generators and (not necessarily discrete) closed, co-compact subgroups. Furthermore, even in the case of uniform latticesΓj\Gamma _{\! j}, our characterizations improve known results since the class of GTI systems satisfying theα\alpha-LIC is strictly larger than the class of GTI systems satisfying the previously used local integrability condition. As an application of our characterization results, we obtain new characterizations of translation invariant continuous frames and Gabor frames forL2(G)L^2(G). In addition, we will see that the admissibility conditions for the continuous and discrete wavelet and Gabor transform inL2(Rn)L^2(\mathbb {R}^n)are special cases of the same general characterizing equations.

Gabor frame sets of invariance: a Hamiltonian approach to Gabor frame deformations.

In this work we study families of pairs of window functions and lattices which lead to Gabor frames which all possess the same frame bounds. To be more precise, for every generalized Gaussian g, we will construct an uncountable family of lattices lbrace Lambda _tau rbrace such that each pairing of g with some Lambda _tau yields a Gabor frame, and all pairings yield the same frame bounds. On the other hand, for each lattice we will find a countable family of generalized Gaussians lbrace g_i rbrace such that each pairing leaves the frame bounds invariant. Therefore, we are tempted to speak about Gabor Frame Sets of Invariance.

Gabor frames and asymptotic behavior of Schwartz distributions

We obtain characterizations of asymptotic properties of Schwartz distribution by using Gabor frames. Our characterizations are indeed Tauberian theorems for shift asymptotics (S-asymptotics) in terms of short-time Fourier transforms with respect to windows generating Gabor frames. For it, we show that the Gabor coefficient operator provides (topological) isomorphisms of the spaces of tempered distributions S'(Rd) and distributions of exponential type K'1(Rd) onto their images.

THE GABOR FRAME AS A DISCRETIZATION FOR THE 2D TRANSVERSE-ELECTRIC SCATTERING-PROBLEM DOMAIN INTEGRAL EQUATION

We apply the Gabor frame as a projection method to numerically solve a 2D ransverse-electric-polarized domain-integral equation for a homogeneous medium. Since the Gabor frame is spatially as well as spectrally very well convergent, it is convenient to use for solving a domain integral equation. The mixed spatial and spectral nature of the Gabor frame creates a natural and fast way to Fourier transform a function. In the spectral domain we employ a coordinate scaling to smoothen the branchcut found in the Green function. We have developed algorithms to perform multiplication and convolution efficiently, scaling as O(N log N ) on the number of Gabor coefficients, yielding an overall algorithm that also scales as O(N log N ).

On Homotopy Continuation for Speech Restoration.

In this paper, a homotopy-based method is employed for the recovery of speech recordings from missing or corrupted samples taken in a noisy environment. The model for the acquisition device is a compressed sensing scenario using Gabor frames. To recover an approximation of the speech file, we used the basis pursuit denoising method with the homotopy continuation algorithm. We tested the proposed method with various speech recordings.

Weaving frames

We study an intriguing question in frame theory we call Frames that is partially motivated by preprocessing of Gabor frames. Two frames $\{\varphi_i\}_{i\in I}$ and $\{\psi_i \}_{i\in I}$ for a Hilbert space ${\mathbb H}$ are woven if there are constants $0<A \le B $ so that for every subset $\sigma \subset I$, the family $\{\varphi_i\}_{i\in \sigma} \cup \{\psi_i\}_{i\in \sigma^c}$ is a frame for $\mathbb H$ with frame bounds $A,B$. Fundamental properties of woven frames are developed and key differences between weaving Riesz bases and weaving frames are considered. In particular, it is shown that a Riesz basis cannot be woven with a redundant frame. We also introduce an apparently weaker form of weaving but show that it is equivalent to weaving. Weaving frames has potential applications in wireless sensor networks that require distributed processing under different frames, as well as preprocessing of signals using Gabor frames.

On Partition of Unities Generated by Entire Functions and Gabor Frames in $$ L^2({\mathbb R}^d) $$ L 2 ( R d ) and $$\ell ^2({\mathbb Z}^d)$$ ℓ 2 ( Z d )

We characterize the entire functions P of d variables, $$d\ge 2,$$ for which the $${\mathbb Z}^d$$ -translates of $$P\chi _{[0,N]^d}$$ satisfy the partition of unity for some $$N\in \mathbb N.$$ In contrast to the one-dimensional case, these entire functions are not necessarily periodic. In the case where P is a trigonometric polynomial, we characterize the maximal smoothness of $$P\chi _{[0,N]^d},$$ as well as the function that achieves it. A number of especially attractive constructions are achieved, e.g., of trigonometric polynomials leading to any desired (finite) regularity for a fixed support size. As an application we obtain easy constructions of matrix-generated Gabor frames in $$ L^2({\mathbb R}^d) ,$$ with small support and high smoothness. By sampling this yields dual pairs of finite Gabor frames in $$\ell ^2({\mathbb Z}^d).$$

Gabor frames for quasicrystals, K-theory, and twisted gap labeling

We study the connection between Gabor frames for quasicrystals, the topology of the hull of a quasicrystal Λ, and the K-theory of an associated twisted groupoid algebra. In particular, we construct a finitely generated projective module over this algebra, and any multiwindow Gabor frame for Λ can be used to construct an idempotent representing this module in K-theory. For lattice subsets in dimension two, this allows us to prove a twisted version of Bellissard's gap labeling theorem.

Gabor frames with trigonometric spline dual windows

A dual Gabor window pair has two functions that can reconstruct any function in [Formula: see text] using certain systems of their modulated and translated forms. Few explicit examples of dual Gabor window pairs are known. This paper constructs explicit examples with trigonometric form in one dimension as well as higher dimensions. Also, in the discrete time domain, the trigonometric form allows us to evaluate the Gabor coefficients efficiently using the Discrete Fourier Transform. The windows have fixed support and arbitrary smoothness.

Gabor frames of Gaussian beams for the Schrödinger equation

The present paper is devoted to the semiclassical analysis of linear Schrödinger equations from a Gabor frame perspective. We consider (time-dependent) smooth Hamiltonians with at most quadratic growth. Then we construct higher order parametrices for the corresponding Schrödinger equations by means of ħ -Gabor frames, as recently defined by M. de Gosson, and we provide precise L 2 -estimates of their accuracy, in terms of the Planck constant ħ . Nonlinear parametrices, in the spirit of the nonlinear approximation, are also presented. Numerical experiments are exhibited to compare our results with the early literature.

Weak Gabor bi-frames on periodic subsets of the real line

In this paper, we introduce the concept of weak Gabor bi-frame (WGBF) in a general closed subspace [Formula: see text] of [Formula: see text]. It is a generalization of Gabor bi-frame, and is new even if [Formula: see text]. A WGBF for [Formula: see text] contains all information of [Formula: see text] to some extent. Let [Formula: see text], [Formula: see text], and [Formula: see text] be an [Formula: see text]-periodic subset of [Formula: see text] with positive measure. This paper is devoted to characterizing WGBFs for [Formula: see text] of the form [Formula: see text] It is well-known that, if [Formula: see text], the projections of Gabor frames for [Formula: see text] onto [Formula: see text] cannot cover all Gabor frames for [Formula: see text]. This paper presents a Zak transform-domain and a time-domain characterization of WGBFs for [Formula: see text]. These characterizations are new even if [Formula: see text]. Some examples are also provided to illustrate the generality of our theory.

Density and duality theorems for regular Gabor frames

We investigate Gabor frames on locally compact abelian groups with time–frequency shifts along non-separable, closed subgroups of the phase space. Density theorems in Gabor analysis state necessary conditions for a Gabor system to be a frame or a Riesz basis, formulated only in terms of the index subgroup. In the classical results the subgroup is assumed to be discrete. We prove density theorems for general closed subgroups of the phase space, where the necessary conditions are given in terms of the “size” of the subgroup. From these density results we are able to extend the classical Wexler–Raz biorthogonal relations and the duality principle in Gabor analysis to Gabor systems with time–frequency shifts along non-separable, closed subgroups of the phase space. Even in the euclidean setting, our results are new.

Discrete coherent states for higher Landau levels

We consider the quantum dynamics of a charged particle evolving under the action of a constant homogeneous magnetic field, with emphasis on the discrete subgroups of the Heisenberg group (in the Euclidean case) and of the SL(2,R) group (in the Hyperbolic case). We investigate completeness properties of discrete coherent states associated with higher order Euclidean and hyperbolic Landau levels, partially extending classic results of Perelomov and of Bargmann, Butera, Girardello and Klauder. In the Euclidean case, our results follow from identifying the completeness problem with known results from the theory of Gabor frames. The results for the hyperbolic setting follow by using a combination of methods from coherent states, time-scale analysis and the theory of Fuchsian groups and their associated automorphic forms.

Phase Retrieval from Gabor Measurements

Compressed sensing investigates the recovery of sparse signals from linear measurements. But often, in a wide range of applications, one is given only the absolute values (squared) of the linear measurements. Recovering such signals (not necessarily sparse) is known as the phase retrieval problem. We consider this problem in the case when the measurements are time-frequency shifts of a suitably chosen generator, i.e. coming from a Gabor frame. We prove an easily checkable injectivity condition for recovery of any signal from all $N^2$ time-frequency shifts, and for recovery of sparse signals, when only some of those measurements are given.

Some Results on the Lattice Parameters of Quaternionic Gabor Frames

Gabor frames play a vital role not only in modern harmonic analysis but also in several fields of applied mathematics, for instances, detection of chirps, or image processing. In this work we present a non-trivial generalization of Gabor frames to the quaternionic case and give new density results. The key tool is the two-sided windowed quaternionic Fourier transform (WQFT). As in the complex case, we want to write the WQFT as an inner product between a quaternion-valued signal and shifts and modulates of a real-valued window function. We demonstrate a Heisenberg uncertainty principle and for the results regarding the density, we employ the quaternionic Zak transform to obtain necessary and sufficient conditions to ensure that a quaternionic Gabor system is a quaternionic Gabor frame. We conclude with a proof that the Gabor conjecture does not hold true in the quaternionic case.

Random Access Protocols With Collision Resolution in a Noncoherent Setting

Wireless systems are increasingly used for Machine-Type Communication (MTC), where the users sporadically send very short messages. In such a setting, the overhead imposed by channel estimation is substantial, thereby demanding noncoherent communication. In this paper we consider a noncoherent setup in which users randomly access the medium to send short messages to a common receiver. We propose a transmission scheme based on Gabor frames, where each user has a dedicated codebook of M possible codewords, while the codebook simultaneously serves as an ID for the user. The scheme is used as a basis for a simple protocol for collision resolution.

Relating tribological stimuli to somatosensory electroencephalographic responses.

The present study deals with the extraction of neural correlates evoked by tactile stimulation of the human fingertip. A reciprocal sliding procedure was performed using a home-built tribometer while simultaneously electroencephalographic (EEG) data from the somatosensory cortex was recorded. The tactile stimuli were delivered by a sliding block with equidistant, perpendicular ridges. The experiments were designed and performed in a fully passive way to prevent attentional locked influences from the subjects. In order to improve the signal-to-noise ratio (SNR) of event related single-trials (ERPs), nonlocal means in addition to 2D-anisotropic denoising schemes based on tight Gabor frames were applied. This novel approach allowed for an easier extraction of ERP alternations. A negative correlation between the latency of the P100 component of the resulting brain responses and the intensity of the underlying lateral forces was found. These findings lead to the conclusion that an increasing stimulus intensity results in a decreasing latency of the brain responses.

Gabor systems and almost periodic functions

Inspired by results of Kim and Ron, given a Gabor frame in L2(R), we determine a non-countable generalized frame for the non-separable space AP2(R) of the Besicovic almost periodic functions. Gabor type frames for suitable separable subspaces of AP2(R) are constructed. We show furthermore that Bessel-type estimates hold for the AP norm with respect to a countable Gabor system using suitable almost periodic norms of sequences.

Dual Gramian analysis: Duality principle and unitary extension principle

Abstract. Dual Gramian analysis is one of the fundamental tools developed in a series of papers [37, 40, 38, 39, 42] for studying frames. Using dual Gramian analysis, the frame operator can be represented as a family of matrices composed of the Fourier transforms of the generators of (generalized) shiftinvariant systems, which allows us to characterize most properties of frames and tight frames in terms of their generators. Such a characterization is applied in the above-mentioned papers to two widely used frame systems, namely Gabor and wavelet frame systems. Among many results, we mention here the discovery of the duality principle for Gabor frames [40] and the unitary extension principle for wavelet frames [38]. This paper aims at establishing the dual Gramian analysis for frames in a general Hilbert space and subsequently characterizing the frame properties of a given system using the dual Gramian matrix generated by its elements. Consequently, many interesting results can be obtained for frames in Hilbert spaces, e.g., estimates of the frame bounds in terms of the frame elements and the duality principle. Moreover, this new characterization provides new insights into the unitary extension principle in [38], e.g., the connection between the unitary extension principle and the duality principle in a weak sense. One application of such a connection is a simplification of the construction of multivariate tight wavelet frames from a given refinable mask. In contrast to the existing methods that require completing a unitary matrix with trigonometric polynomial entries from a given row, our method greatly simplifies the tight wavelet frame construction by converting it to a constant matrix completion problem. To illustrate its simplicity, the proposed construction scheme is used to construct a few examples of multivariate tight wavelet frames from box splines with certain desired properties, e.g., compact support, symmetry or anti-symmetry.