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## ‣ Multidimensional wavelets

Colthurst, Thomas
Fonte: Massachusetts Institute of Technology Publicador: Massachusetts Institute of Technology
Tipo: Tese de Doutorado Formato: 82 p.
Português
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by Thoams Colthurst.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1997.; Includes bibliographical references (p. 78-82).

## ‣ Coxeter Groups, Wavelets, Multiresolution and Sampling

Dobrescu, M.; Olafsson, G.
Tipo: Artigo de Revista Científica
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In this short note we discuss the interplay between finite Coxeter groups and construction of wavelet sets, generalized multiresolution analysis and sampling.; Comment: Will appear in Contemporary Mathematics

## ‣ Image Separation using Wavelets and Shearlets

Kutyniok, Gitta; Lim, Wang-Q
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
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In this paper, we present an image separation method for separating images into point- and curvelike parts by employing a combined dictionary consisting of wavelets and compactly supported shearlets utilizing the fact that they sparsely represent point and curvilinear singularities, respectively. Our methodology is based on the very recently introduced mathematical theory of geometric separation, which shows that highly precise separation of the morphologically distinct features of points and curves can be achieved by $\ell^1$ minimization. Finally, we present some experimental results showing the effectiveness of our algorithm, in particular, the ability to accurately separate points from curves even if the curvature is relatively large due to the excellent localization property of compactly supported shearlets.

## ‣ Wavelets in function spaces on cellular domains

Scharf, Benjamin
Tipo: Artigo de Revista Científica
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Nowadays the theory and application of wavelet decompositions plays an important role not only for the study of function spaces (of Lebesgue, Hardy, Sobolev, Besov, Triebel-Lizorkin type) but also for its applications in signal and numerical analysis, partial differential equations and image processing. In this context it it a hard problem to construct wavelet bases for suitable function spaces on domains, e. g. the unit cube. A big step in this direction are the contributions of Hans Triebel from 2006 to 2008 where he constructed Riesz bases for classes of Besov- and Triebel-Lizorkin spaces on domains, starting with Daubechies wavelets. But there was a problem coming from the method: He had to exclude a big number of function spaces, in particular a large class of classical Sobolev spaces. The main goal of this thesis is a construction of Riesz bases of wavelet systems also for the exceptional cases using a modification of the function spaces - the so-called reinforced function spaces.; Comment: 109 pages, my thesis for becoming a Dr. rer. nat. at Friedrich-Schiller-Universitaet Jena

## ‣ Independent Component Analysis by Wavelets

Barbedor, Pascal
Tipo: Artigo de Revista Científica
Português
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We propose an ICA contrast based on the density estimation of the observed signal and its marginals by means of wavelets. The risk of the associated moment estimator is linked with approximation properties in Besov spaces. It is shown to converge faster than the at least expected minimax rate carried over from the underlying density estimations. Numerical simulations performed on some common types of densities yield very competitive results, with a high sensitivity to small departures from independence.; Comment: 22 pages

## ‣ Radix Representations, Self-Affine Tiles, and Multivariable Wavelets

Curry, Eva
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We investigate the connection between radix representations for Z^n and self-affine tilings of R^n. We apply our results to show that Haar-like multivariable wavelets exist for all dilation matrices that are sufficie; Comment: 8 pages, 1 figure

## ‣ Invariant polytopes of linear operators with applications to regularity of wavelets and of subdivisions

Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We generalize the recent invariant polytope algorithm for computing the joint spectral radius and extend it to a wider class of matrix sets. This, in particular, makes the algorithm applicable to sets of matrices that have finitely many spectrum maximizing products. A criterion of convergence of the algorithm is proved. As an application we solve two challenging computational open problems. First we find the regularity of the Butterfly subdivision scheme for various parameters $\omega$. In the "most regular" case $\omega = \frac{1}{16}$, we prove that the limit function has H\"older exponent $2$ and its derivative is "almost Lipschitz" with logarithmic factor $2$. Second we compute the H\"older exponent of Daubechies wavelets of high order.; Comment: 36 pages

## ‣ Wavelets, Multiplier spaces and application to Schr\"{o}dinger type operators with non-smooth potentials

Li, Pengtao; Yang, Qixiang; Zhu, Yueping
Tipo: Artigo de Revista Científica
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In this paper, we employ Meyer wavelets to characterize multiplier spaces between Sobolev spaces without using capacity. Further, we introduce logarithmic Morrey spaces $M^{t,\tau}_{r,p}(\mathbb{R}^{n})$ to establish the inclusion relation between Morrey spaces and multiplier spaces. By wavelet characterization and fractal skills, we construct a counterexample to show that the scope of the index $\tau$ of $M^{t,\tau}_{r,p}(\mathbb{R}^{n})$ is sharp. As an application, we consider a Schr\"odinger type operator with potentials in $M^{t,\tau}_{r,p}(\mathbb{R}^{n})$.

## ‣ Wavelets and Hilbert modules

Wood, Peter John
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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A Hilbert module is a generalisation of a Hilbert space for which the inner product takes its values in a C*-algebra instead of the complex numbers. We use the bracket product to construct some Hilbert modules over a group C*-algebra which is generated by the group of translations associated with a wavelet. We shall investigate bracket products and their Fourier transform in the space of square integrable functions in Euclidean space. We will also show that some wavelets are associated with Hilbert modules over the space of essentially bounded functions over higher dimensional tori.; Comment: To appear in the Journal of Fourier Analysis and Applications

## ‣ MRA Super-wavelets

Bildea, Stefan; Dutkay, Dorin Ervin; Picioroaga, Gabriel
Tipo: Artigo de Revista Científica
Português
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We construct a multiresolution theory for spaces bigger then L^2(R). For a good choice of the dilation and translation operators on these larger spaces, it is possible to build singly generated wavelet bases, thus obtaining examples of "super-wavelets"; Comment: v2

## ‣ Oversampling generates super-wavelets

Dutkay, Dorin Ervin; Jorgensen, Palle E. T.
Tipo: Artigo de Revista Científica
Português
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We show that the second oversampling theorem for affine systems generates super-wavelets. These are frames generated by an affine structure on the space $L^2(\br)\oplus...\oplus L^2(\br)$.; Comment: 9 pages, AMS-LaTeX. v2: The introduction has been expanded and a number of revisions have been made. Revised for Proceedings of the American Mathematical Society

## ‣ Quantization Opportunities for Polyharmonic Subdivision Wavelets Applied to Astronomical Images

Kounchev, Ognyan; Kalaglarsky, Damyan
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We continue the study of a new family of multivariate wavelets which are obtained by "polyharmonic subdivision". We provide the results of experiments considering the distribution of the wavelet coefficients for the Lena image and for astronomical images. The main purpose of this investigation is to find a clue for proper quantization algorithms.; Comment: 10 pages, 12 figures

## ‣ Wavelet Electrodynamics II: Atomic Composition of Electromagnetic Waves

Kaiser, Gerald
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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The representation of solutions of Maxwell's equations as superpositions of scalar wavelets with vector coefficients developed earlier is generalized to wavelets with polarization, which are matrix-valued. The construction proceeds in four stages: (1) A Hilbert space H of solutions is considered, based on a conformally invariant inner product. (2) The analytic-signal transform extends solutions from real space-time to a complex space-time domain T (double tube). The evaluation map E_z, which sends any solution F=B+iE in H to the value F(z) at z\in T, is bounded. The electromagnetic wavelets are defined as the adjoints the \Psi_z=E_z^*. (3) The eight real parameters z=x+iy\in T are given a complete physical interpretation: x\in R^4 is interpreted as a space-time point about which \Psi_z is focussed, and the timelike vector y gives its scale and velocity. Thus wavelets parameterized by the set of {\sl Euclidean} points (real space, imaginary time) have stationary centers, and the others are Doppler-shifted versions of the former. All the wavelets can be obtained from a single "mother wavelet" by conformal transformations. (4) A resolution of unity is established in H, giving a representation of solutions as "atomic compositions" of wavelets parameterized by z\in E. This yields a constructive method for generating solutions with initial data specified locally in space and by scale. Other representations...

## ‣ Wavelets and Triebel type oscillation spaces

Li, Pengtao; Yang, Qixiang; Zheng, Bentuo
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We apply wavelets to identify the Triebel type oscillation spaces with the known Triebel-Lizorkin-Morrey spaces $\dot{F}^{\gamma_1,\gamma_2}_{p,q}(\mathbb{R}^{n})$. Then we establish a characterization of $\dot{F}^{\gamma_1,\gamma_2}_{p,q}(\mathbb{R}^{n})$ via the fractional heat semigroup. Moreover, we prove the continuity of Calder\'on-Zygmund operators on these spaces. The results of this paper also provide necessary tools for the study of well-posedness of Navier-Stokes equations.

## ‣ Cuntz-Krieger algebras and wavelets on fractals

Marcolli, Matilde; Paolucci, Anna Maria
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We consider representations of Cuntz--Krieger algebras on the Hilbert space of square integrable functions on the limit set, identified with a Cantor set in the unit interval. We use these representations and the associated Perron-Frobenius and Ruelle operators to construct families of wavelets on these Cantor sets.; Comment: 37 pages, LaTeX

## ‣ Splines and Wavelets on Geophysically Relevant Manifolds

Pesenson, Isaac
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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Analysis on the unit sphere $\mathbb{S}^{2}$ found many applications in seismology, weather prediction, astrophysics, signal analysis, crystallography, computer vision, computerized tomography, neuroscience, and statistics. In the last two decades, the importance of these and other applications triggered the development of various tools such as splines and wavelet bases suitable for the unit spheres $\mathbb{S}^{2}$, $\>\>\mathbb{S}^{3}$ and the rotation group $SO(3)$. Present paper is a summary of some of results of the author and his collaborators on generalized (average) variational splines and localized frames (wavelets) on compact Riemannian manifolds. The results are illustrated by applications to Radon-type transforms on $\mathbb{S}^{d}$ and $SO(3)$.; Comment: The final publication is available at http://www.springerlink.com

## ‣ Equations For Frame Wavelets In $L^2(\R^2)$

Dai, Xingde
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We establish system of equations for single function normalized tight frame wavelets with compact supports associated with $2\times 2$ expansive integral matrices in $L^2(\R^2)$.; Comment: 27 pages, 3 figures, 1 table

## ‣ On the Hilbert transform of wavelets

Chaudhury, Kunal Narayan; Unser, Michael
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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A wavelet is a localized function having a prescribed number of vanishing moments. In this correspondence, we provide precise arguments as to why the Hilbert transform of a wavelet is again a wavelet. In particular, we provide sharp estimates of the localization, vanishing moments, and smoothness of the transformed wavelet. We work in the general setting of non-compactly supported wavelets. Our main result is that, in the presence of some minimal smoothness and decay, the Hilbert transform of a wavelet is again as smooth and oscillating as the original wavelet, whereas its localization is controlled by the number of vanishing moments of the original wavelet. We motivate our results using concrete examples.; Comment: Appears in IEEE Transactions on Signal Processing, vol. 59, no. 4, pp. 1890-1894, 2011

## ‣ Super-wavelets versus poly-Bergman spaces

Abreu, Luis Daniel
Tipo: Artigo de Revista Científica
Motivated by potential applications in multiplexing and by recent results on Gabor analysis with Hermite windows due to Gr\"{o}chenig and Lyubarskii, we investigate vector-valued wavelet transforms and vector-valued wavelet frames, which constitute special cases of super-wavelets, with a particular attention to the case when the analyzing wavelet vector is related to Fourier transforms of Laguerre functions. We construct an isometric isomorphism between $L^{2}(\mathbb{R}^{+},\mathbf{C}^{n})$ and poly-Bergman spaces, with a view to relate the sampling sequences in the poly-Bergman spaces to the wavelet frames and super-frames with the windows $\Phi_{n}$. One of the applications of the theory is a proof that $b\ln a<2\pi (n+1)$ is a necessary condition for the (scalar) wavelet frame associated to the $\Phi_{n}$ to exist. This seems to be the first known result of this type outside the setting of analytic functions (the case $n=0$, which has been completely studied by Seip in 1993).; Comment: Preliminar version; 19 pages
While the Clifford (geometric) algebra Fourier Transform (CFT) is global, we introduce here the local Clifford (geometric) algebra (GA) wavelet concept. We show how for $n=2,3 (\mod 4)$ continuous $Cl_n$-valued admissible wavelets can be constructed using the similitude group $SIM(n)$. We strictly aim for real geometric interpretation, and replace the imaginary unit $i \in \C$ therefore with a GA blade squaring to $-1$. Consequences due to non-commutativity arise. We express the admissibility condition in terms of a $Cl_{n}$ CFT and then derive a set of important properties such as dilation, translation and rotation covariance, a reproducing kernel, and show how to invert the Clifford wavelet transform. As an explicit example, we introduce Clifford Gabor wavelets. We further invent a generalized Clifford wavelet uncertainty principle. Extensions of CFTs and Clifford wavelets to $Cl_{0,n'}, n' = 1,2 (\mod 4)$ appear straight forward. Keywords: Clifford geometric algebra, Clifford wavelet transform, multidimensional wavelets, continuous wavelets, similitude group.; Comment: 8 pages