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## ‣ Multidimensional wavelets

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 82 p.

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by Thoams Colthurst.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1997.; Includes bibliographical references (p. 78-82).

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## ‣ Coxeter Groups, Wavelets, Multiresolution and Sampling

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/10/2007
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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

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## ‣ Image Separation using Wavelets and Shearlets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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.

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## ‣ Wavelets in function spaces on cellular domains

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/02/2013
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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

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## ‣ Independent Component Analysis by Wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/02/2010
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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/02/2015
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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/01/2013
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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
$M^{t,\tau}_{r,p}(\mathbb{R}^{n})$.

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## ‣ Wavelets and Hilbert modules

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/10/2004
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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

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## ‣ MRA Super-wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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## ‣ Oversampling generates super-wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/04/2012
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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 20/08/2001
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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...

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## ‣ Wavelets and Triebel type oscillation spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/01/2014
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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.

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## ‣ Cuntz-Krieger algebras and wavelets on fractals

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/08/2009
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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

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## ‣ Splines and Wavelets on Geophysically Relevant Manifolds

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/03/2014
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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/09/2015
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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

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## ‣ On the Hilbert transform of wavelets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/07/2011
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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

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## ‣ Super-wavelets versus poly-Bergman spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/09/2009
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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

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## ‣ Clifford (Geometric) Algebra Wavelet Transform

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/06/2013
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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

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