241 research outputs found
Approximation of dual Gabor frames, window decay, and wireless communications
We consider three problems for Gabor frames that have recently received much
attention. The first problem concerns the approximation of dual Gabor frames in
by finite-dimensional methods. Utilizing Wexler-Raz type duality
relations we derive a method to approximate the dual Gabor frame, that is much
simpler than previously proposed techniques. Furthermore it enables us to give
estimates for the approximation rate when the dimension of the finite model
approaches infinity. The second problem concerns the relation between the decay
of the window function and its dual . Based on results on
commutative Banach algebras and Laurent operators we derive a general condition
under which the dual inherits the decay properties of . The third
problem concerns the design of pulse shapes for orthogonal frequency division
multiplex (OFDM) systems for time- and frequency dispersive channels. In
particular, we provide a theoretical foundation for a recently proposed
algorithm to construct orthogonal transmission functions that are well
localized in the time-frequency plane
The development of the quaternion wavelet transform
The purpose of this article is to review what has been written on what other authors have called quaternion wavelet transforms (QWTs): there is no consensus about what these should look like and what their properties should be. We briefly explain what real continuous and discrete wavelet transforms and multiresolution analysis are and why complex wavelet transforms were introduced; we then go on to detail published approaches to QWTs and to analyse them. We conclude with our own analysis of what it is that should define a QWT as being truly quaternionic and why all but a few of the “QWTs” we have described do not fit our definition
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
Gabor expansion of an equivalent dipole antenna
The Gabor representation in the context of an aperture problem is an expansion of a radiated field in terms of a discrete set of linearly shifted and spatially rotated elementary Gaussian beams. The parameters that can be varied in this summation are the number of beams and corresponding beam widths. As the difficulty associated with the unique determination of the expansion coefficients was alleviated, the method has been successfully applied to one and two dimensional apertures. Although, the asymptotic evaluation of expansion functions has reduced the computational burden drasticallym it was at the expense of some loss in accuracy. The numerical experiments have established that the narrow beam algorithm with almost a priori predictability can be used in a variety of problems. Here, the Gabor representation has been applied to a narrow rectangular aperture illuminated with a sinusoidal field. The narrow aperture (height \u3e\u3e width) excited by sinusoidal field distribution approximates an equivalent dipole with a similar current distribution with only exception that aperture radiates into a half-space whereas \u27the dipole covers the entire space. Utilizing the narrow beam algorithm, once the expansion coefficients were determined, the radiated electric field potential in near, mid and far zones were evaluated. The criteria in determining the number of expansion coefficients was based on re-generation of the aperture field distribution with sufficient accuracy. It was observed that even though wide beam algorithm was applied, less number of terms resulted in erroneous side lobes and higher number of terms caused Gibbs phenomena in the region close to the aperture plane. The numerical evaluations are carried out for the half-wavelength high narrow aperture. Far zone numerical results of radiated potential utilizing Gabor expansion are compared to analytical expressions determined via Fourier transform. The unique application developed in this work in expressing the radiated field of an equivalent dipole antenna revealed that Gabor expansion can be a valuable tool in studying practical radiation and propagation problems
On quasi-greedy bases associated with unitary representations of countable groups
We consider the natural generating system for a cyclic subspace of a Hilbert space generated by a dual integrable unitary representation of a countable abelian group. We prove, under mild hypothesis, that whenever the generating system is a quasi-greedy basis it must also be an unconditional Riesz basis. A number of applications to Gabor systems and to general Vilenkin systems are considered. In particular, we show that any Gabor Schauder basis that also forms a quasi-greedy system in L2 is in fact a Riesz basis, and therefore satisfies the classical Balian-Low theorem
Generalized coorbit space theory and inhomogeneous function spaces of Besov-Lizorkin-Triebel type
Coorbit space theory is an abstract approach to function spaces and their
atomic decompositions. The original theory developed by Feichtinger and
Gr{\"o}chenig in the late 1980ies heavily uses integrable representations of
locally compact groups. Their theory covers, in particular, homogeneous
Besov-Lizorkin-Triebel spaces, modulation spaces, Bergman spaces, and the
recent shearlet spaces. However, inhomogeneous Besov-Lizorkin-Triebel spaces
cannot be covered by their group theoretical approach. Later it was recognized
by Fornasier and the first named author that one may replace coherent states
related to the group representation by more general abstract continuous frames.
In the first part of the present paper we significantly extend this abstract
generalized coorbit space theory to treat a wider variety of coorbit spaces. A
unified approach towards atomic decompositions and Banach frames with new
results for general coorbit spaces is presented. In the second part we apply
the abstract setting to a specific framework and study coorbits of what we call
Peetre spaces. They allow to recover inhomogeneous Besov-Lizorkin-Triebel
spaces of various types of interest as coorbits. We obtain several old and new
wavelet characterizations based on precise smoothness, decay, and vanishing
moment assumptions of the respective wavelet. As main examples we obtain
results for weighted spaces (Muckenhoupt, doubling), general 2-microlocal
spaces, Besov-Lizorkin-Triebel-Morrey spaces, spaces of dominating mixed
smoothness, and even mixtures of the mentioned ones. Due to the generality of
our approach, there are many more examples of interest where the abstract
coorbit space theory is applicable.Comment: to appear in Journal of Functional Analysi
Image segmentation in the wavelet domain using N-cut framework
We introduce a wavelet domain image segmentation algorithm based on Normalized Cut (NCut) framework in this thesis. By employing the NCut algorithm we solve the perceptual grouping problem of image segmentation which aims at the extraction of the global impression of an image. We capitalize on the reduced set of data to be processed and statistical features derived from the wavelet-transformed images to solve graph partitioning more efficiently than before. Five orientation histograms are computed to evaluate similarity/dissimilarity measure of local structure. We use properties of the wavelet transform filtering to capture edge information in vertical, horizontal and diagonal orientations. This approach allows for direct processing of compressed data and results in faster implementation of NCut framework than that in the spatial domain and also decent quality of segmentation of natural scene images
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