10,837 research outputs found
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
let , , be a countable family of closed, co-compact
subgroups of a second countable locally compact abelian group and study
systems of the form with generators in and with each
being 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 technical
local integrability condition (-LIC) we characterize when GTI systems
constitute tight and dual frames that yield reproducing formulas for .
This generalizes results on generalized shift invariant systems, where each
is assumed to be countable and each is a uniform lattice in
, to the case of uncountably many generators and (not necessarily discrete)
closed, co-compact subgroups. Furthermore, even in the case of uniform lattices
, our characterizations improve known results since the class of GTI
systems satisfying the -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 for . In
addition, we will see that the admissibility conditions for the continuous and
discrete wavelet and Gabor transform in are special cases
of the same general characterizing equations.Comment: Minor changes (v2). To appear in Trans. Amer. Math. So
Optimal Window and Lattice in Gabor Transform Application to Audio Analysis
This article deals with the use of optimal lattice and optimal window in
Discrete Gabor Transform computation. In the case of a generalized Gaussian
window, extending earlier contributions, we introduce an additional local
window adaptation technique for non-stationary signals. We illustrate our
approach and the earlier one by addressing three time-frequency analysis
problems to show the improvements achieved by the use of optimal lattice and
window: close frequencies distinction, frequency estimation and SNR estimation.
The results are presented, when possible, with real world audio signals
Fast learning algorithm for Gabor transformation, A
Includes bibliographical references.An adaptive learning approach for the computation of the coefficients of the generalized nonorthogonal 2-D Gabor transform representation is introduced in this correspondence. The algorithm uses a recursive least squares (RLS) type algorithm. The aim is to achieve minimum mean squared error for the reconstructed image from the set of the Gabor coefficients. The proposed RLS learning offers better accuracy and faster convergence behavior when compared with the least mean squares (LMS)-based algorithms. Applications of this scheme in image data reduction are also demonstrated
Spreading function representation of operators and Gabor multiplier approximation
Modification of signals in the time-frequency domain are used in many applications. However, the modification is often restricted to be purely multiplicative. In this paper, it is shown that, in the continuous case, a quite general class of operators can be represented by a twisted convolution in the short-time Fourier transform domain. The discrete case of Gabor transforms turns out to be more intricate. A similar representation will however be derived by means of a special form for the operator's spreading function (twisted spline type function). The connection between STFT- and Gabor-multipliers, their spreading function and the twisted convolution representation will be investigated. A precise characterization of the best approximation and its existence is given for both cases. Finally, the concept of Gabor multipliers is generalized to better approximate ''overspread'' operators
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 , we will construct an uncountable
family of lattices such that each pairing of
with some 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 such that each pairing
leaves the frame bounds invariant. Therefore, we are tempted to speak about
"Gabor Frame Sets of Invariance".Comment: To appear in "Journal of Pseudo-Differential Operators and
Applications
Multidimensional Gabor-Like Filters Derived from Gaussian Functions on Logarithmic Frequency Axes
A novel wavelet-like function is presented that makes it convenient to create
filter banks given mainly two parameters that influence the focus area and the
filter count. This is accomplished by computing the inverse Fourier transform
of Gaussian functions on logarithmic frequency axes in the frequency domain.
The resulting filters are similar to Gabor filters and represent oriented brief
signal oscillations of different sizes. The wavelet-like function can be
thought of as a generalized Log-Gabor filter that is multidimensional, always
uses Gaussian functions on logarithmic frequency axes, and innately includes
low-pass filters from Gaussian functions located at the frequency domain
origin
Gabor Frame Decomposition of Evolution Operators and Applications
We compute the Gabor matrix for Schr\"odinger-type evolution operators.
Precisely, we analyze the Heat Equation, already presented in
\cite{2012arXiv1209.0945C}, giving the exact expression of the Gabor matrix
which leads to better numerical evaluations. Then, using asymptotic integration
techniques, we obtain an upper bound for the Gabor matrix in one-dimension for
the generalized Heat Equation, new in the literature. Using Maple software, we
show numeric representations of the coefficients' decay. Finally, we show the
super-exponential decay of the coefficients of the Gabor matrix for the
Harmonic Repulsor, together with some numerical evaluations. This work is the
natural prosecution of the ideas presented in \cite{2012arXiv1209.0945C} and
\cite{MR2502369}.Comment: 29 pages, 7 figure
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