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## ‣ Uso de ôndulas na compressão de operadores; Using wavelets for operator compression

Fonte: Universidade do Minho
Publicador: Universidade do Minho

Tipo: Dissertação de Mestrado

Publicado em //2005
Português

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Tese de mestrado em Matemática Computacional.; A chamada “teoria das ôndulas” constitui uma área recente e muito activa da matemática, que tem vindo a ser utilizada com sucesso em diversos campos de investigação, tal como reconhecimento de padrões, análise de sinal, compressão de dados, física quântica, acústica, etc., abrangendo, portanto, áreas desde a matemática pura e aplicada até à física, computação e engenharia.
Uma questão importante nesta teoria é a aplicação de um operador, representado numa certa base de ôndulas (escolhida adequadamente), a uma função arbitrária. A representação do operador pode fazer-se de duas formas distintas: standard e não-standard.
O principal objectivo desta tese é o estudo e descrição pormenorizada das representações standard e não-standard de operadores em bases de ôndulas.
Mostraremos, em particular, como estas representações podem ser usadas na compressão de certas classes de operadores.
Serão construídas explicitamente as representações de alguns casos particulares de operadores e calculadas, numericamente, as respectivas taxas de compressão.
Finalmente, serão referidas algumas aplicações dessas representações; Wavelet theory constitutes a recent and very active area of mathematics which has been used successfully in several fields of research...

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## ‣ Position Coding

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/06/2007
Português

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A position coding pattern is an array of symbols in which subarrays of a
certain fixed size appear at most once. So, each subarray uniquely identifies a
location in the larger array, which means there is a bijection of some sort
from this set of subarrays to a set of coordinates. The key to Fly Pentop
Computer paper and other examples of position codes is a method to read the
subarray and then convert it to coordinates. Position coding makes use of ideas
from discrete mathematics and number theory. In this paper, we will describe
the underlying mathematics of two position codes, one being the Anoto code that
is the basis of "Fly paper". Then, we will present two new codes, one which
uses binary wavelets as part of the bijection.; Comment: 14 pages, 7 figures

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## ‣ Positive definite maps, representations and frames

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/11/2005
Português

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We present a unitary approach to the construction of representations and
intertwining operators. We apply it to the $C^*$-algebras, groups, Gabor type
unitary systems and wavelets. We give an application of our method to the
theory of frames, and we prove a general dilation theorem which is in turn
applied to specific cases, and we obtain in this way a dilation theorem for
wavelets.

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## ‣ Ergodic optimization of prevalent super-continuous functions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Given a dynamical system, we say that a performance function has property P
if its time averages along orbits are maximized at a periodic orbit. It is
conjectured by several authors that for sufficiently hyperbolic dynamical
systems, property P should be typical among sufficiently regular performance
functions. In this paper we address this problem using a probabilistic notion
of typicality that is suitable to infinite dimension: the concept of prevalence
as introduced by Hunt, Sauer, and Yorke. For the one-sided shift on two
symbols, we prove that property P is prevalent in spaces of functions with a
strong modulus of regularity. Our proof uses Haar wavelets to approximate the
ergodic optimization problem by a finite-dimensional one, which can be
conveniently restated as a maximum cycle mean problem on a de Bruijin graph.; Comment: 23 pages, 4 figures. Final version, to appear in International
Mathematics Research Notices (IMRN)

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## ‣ Multiwavelet density estimation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/06/2012
Português

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Accurate density estimation methodologies play an integral role in a variety
of scientific disciplines, with applications including simulation models,
decision support tools, and exploratory data analysis. In the past, histograms
and kernel density estimators have been the predominant tools of choice,
primarily due to their ease of use and mathematical simplicity. More recently,
the use of wavelets for density estimation has gained in popularity due to
their ability to approximate a large class of functions, including those with
localized, abrupt variations. However, a well-known attribute of wavelet bases
is that they can not be simultaneously symmetric, orthogonal, and compactly
supported. Multiwavelets-a more general, vector-valued, construction of
wavelets-overcome this disadvantage, making them natural choices for estimating
density functions, many of which exhibit local symmetries around features such
as a mode. We extend the methodology of wavelet density estimation to use
multiwavelet bases and illustrate several empirical results where multiwavelet
estimators outperform their wavelet counterparts at coarser resolution levels.; Comment: Preprint

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## ‣ Taylor Series as Wide-sense Biorthogonal Wavelet Decomposition

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/02/2015
Português

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#Mathematics - Classical Analysis and ODEs#Computer Science - Information Theory#Mathematical Physics

Pointwise-supported generalized wavelets are introduced, based on Dirac,
doublet and further derivatives of delta. A generalized biorthogonal analysis
leads to standard Taylor series and new Dual-Taylor series that may be
interpreted as Laurent Schwartz distributions. A Parseval-like identity is also
derived for Taylor series, showing that Taylor series support an energy
theorem. New representations for signals called derivagrams are introduced,
which are similar to spectrograms. This approach corroborates the impact of
wavelets in modern signal analysis.; Comment: 6 pages, 4 figures. conference: XXII Simposio Brasileiro de
Telecomunicacoes, SBrT'05, 2005, Campinas, SP, Brazil

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## ‣ Phase retrieval for the Cauchy wavelet transform

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

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We consider the phase retrieval problem in which one tries to reconstruct a
function from the modulus of its wavelet transform. We study the unicity and
stability of the reconstruction. In the case where the wavelets are Cauchy
wavelets, we prove that the modulus of the wavelet transform uniquely
determines the function up to a global phase. We show that the reconstruction
operator is continuous but not uniformly continuous. We describe how to
construct pairs of functions which are far away in $L^2$-norm but whose wavelet
transforms are very close, in modulus. The principle is to modulate the wavelet
transform of a fixed initial function by a phase which varies slowly in both
time and frequency. This construction seems to cover all the instabilities that
we observe in practice; we give a partial formal justification to this fact.
Finally, we describe an exact reconstruction algorithm and use it to
numerically confirm our analysis of the stability question.

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## ‣ Some recent advances on the RBF

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/07/2002
Português

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This paper aims to survey our recent work relating to the radial basis
function (RBF) and its applications to numerical PDEs. We introduced the kernel
RBF involving general pre-wavelets and scale-orthogonal wavelets RBF. A
dimension-independent RBF error bound was also conjectured. The centrosymmetric
structure of RBF interpolation matrix under symmetric sample knots was pointed
out. On the other hand, we introduced the boundary knot method via nonsingular
general solution and dual reciprocity principle and the boundary particle
method via multiple reciprocity principle. By using the Green integral we
developed a domain-type Hermite RBF scheme called the modified Kansa method,
which significantly reduces calculation errors around boundary. To circumvent
the Gibbs phenomenon, the least square RBF collocation scheme was presented.
All above discretization schemes are meshfree, symmetric, spectral convergent,
integration-free and mathematically very simple. The numerical validations are
also briefly presented.

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## ‣ Orthonormal dilations of non-tight frames

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 20/04/2010
Português

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We establish dilation theorems for non-tight frames with additional
structure, i.e., frames generated by unitary groups of operators and projective
unitary representations. This generalizes previous dilation results for
Parseval frames due to Han and Larson and Gabardo and Han. We also extend the
dilation theorem for Parseval wavelets, due to Dutkay, Han, Picioroaga, and Sun
, by identifying the optimal class of frame wavelets for which dilation into an
orthonormal wavelet is possible.

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## ‣ Stable and robust sampling strategies for compressive imaging

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Computer Science - Computer Vision and Pattern Recognition#Computer Science - Information Theory#Mathematics - Numerical Analysis#94A08, 68U10, 65D18, 92C55

In many signal processing applications, one wishes to acquire images that are
sparse in transform domains such as spatial finite differences or wavelets
using frequency domain samples. For such applications, overwhelming empirical
evidence suggests that superior image reconstruction can be obtained through
variable density sampling strategies that concentrate on lower frequencies. The
wavelet and Fourier transform domains are not incoherent because low-order
wavelets and low-order frequencies are correlated, so compressive sensing
theory does not immediately imply sampling strategies and reconstruction
guarantees. In this paper we turn to a more refined notion of coherence -- the
so-called local coherence -- measuring for each sensing vector separately how
correlated it is to the sparsity basis. For Fourier measurements and Haar
wavelet sparsity, the local coherence can be controlled and bounded explicitly,
so for matrices comprised of frequencies sampled from a suitable inverse square
power-law density, we can prove the restricted isometry property with
near-optimal embedding dimensions. Consequently, the variable-density sampling
strategy we provide allows for image reconstructions that are stable to
sparsity defects and robust to measurement noise. Our results cover both
reconstruction by $\ell_1$-minimization and by total variation minimization.
The local coherence framework developed in this paper should be of independent
interest in sparse recovery problems more generally...

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## ‣ Square Roots of -1 in Real Clifford Algebras

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Rings and Algebras#Mathematics - Complex Variables#15A66 (Primary) 11E88, 42A38, 30G35 (Secondary)

It is well known that Clifford (geometric) algebra offers a geometric
interpretation for square roots of -1 in the form of blades that square to
minus 1. This extends to a geometric interpretation of quaternions as the side
face bivectors of a unit cube. Systematic research has been done [32] on the
biquaternion roots of -1, abandoning the restriction to blades. Biquaternions
are isomorphic to the Clifford (geometric) algebra $Cl(3,0)$ of $\mathbb{R}^3$.
Further research on general algebras $Cl(p,q)$ has explicitly derived the
geometric roots of -1 for $p+q \leq 4$ [17]. The current research abandons this
dimension limit and uses the Clifford algebra to matrix algebra isomorphisms in
order to algebraically characterize the continuous manifolds of square roots of
-1 found in the different types of Clifford algebras, depending on the type of
associated ring ($\mathbb{R}$, $\mathbb{H}$, $\mathbb{R}^2$, $\mathbb{H}^2$, or
$\mathbb{C}$). At the end of the paper explicit computer generated tables of
representative square roots of -1 are given for all Clifford algebras with
$n=5,7$, and $s=3 \, (mod 4)$ with the associated ring $\mathbb{C}$. This
includes, e.g., $Cl(0,5)$ important in Clifford analysis, and $Cl(4,1)$ which
in applications is at the foundation of conformal geometric algebra. All these
roots of -1 are immediately useful in the construction of new types of
geometric Clifford Fourier transformations.; Comment: 31 pages...

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## ‣ Unitary systems and wavelet sets

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/04/2006
Português

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We give a detailed description of the local commutant approach to wavelet
theory using operator algebraic methods. We include a new result on
interpolation pairs of wavelet sets: Every pair in the generalized Journe
family of wavelet sets is an interpolation pair. In the last section, we give a
discussion of several open problems on wavelets and frame-wavelets.; Comment: This is a semi-expository article which was written by the author for
the proceedings of the 4th International Conference on Wavelet Analysis and
its Applications, Macau, China, December 2005 [WAA2005], as a keynote
speaker; 30p

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## ‣ Linear Stable Sampling Rate: Optimality of 2D Wavelet Reconstructions from Fourier Measurements

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

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In this paper we analyze two-dimensional wavelet reconstructions from Fourier
samples within the framework of generalized sampling. For this, we consider
both separable compactly-supported wavelets and boundary wavelets. We prove
that the number of samples that must be acquired to ensure a stable and
accurate reconstruction scales linearly with the number of reconstructing
wavelet functions. We also provide numerical experiments that corroborate our
theoretical results.

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## ‣ A hypergeometric basis for the Alpert multiresolution analysis

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Classical Analysis and ODEs#Mathematics - Functional Analysis#Mathematics - Numerical Analysis

We construct an explicit orthonormal basis of piecewise ${}_{i+1}F_{i}$
hypergeometric polynomials for the Alpert multiresolution analysis. The Fourier
transform of each basis function is written in terms of ${}_2F_3$
hypergeometric functions. Moreover, the entries in the matrix equation
connecting the wavelets with the scaling functions are shown to be balanced
${}_4 F_3$ hypergeometric functions evaluated at $1$, which allows to compute
them recursively via three-term recurrence relations.
The above results lead to a variety of new interesting identities and
orthogonality relations reminiscent to classical identities of higher-order
hypergeometric functions and orthogonality relations of Wigner $6j$-symbols.

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## ‣ The local trace function of shift invariant subspaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 13/02/2003
Português

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We define the local trace function for subspaces of $\ltworn$ which are
invariant under integer translation. Our trace function contains the dimension
function and the spectral function defined by Bownik and Rzeszotnik and
completely characterizes the given translation invariant subspace. It has
properties such as positivity, additivity, monotony and some form of
continuity. It behaves nicely under dilations and modulations. We use the local
trace function to deduce, using short and simple arguments, some fundamental
facts about wavelets such as the characterizing equations, the equality between
the dimension function and the multiplicity function and some new relations
between scaling functions and wavelets.

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## ‣ Prime Coset Sum: A Systematic Method for Designing Multi-D Wavelet Filter Banks with Fast Algorithms

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/07/2014
Português

Relevância na Pesquisa

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As constructing multi-D wavelets remains a challenging problem, we propose a
new method called prime coset sum to construct multi-D wavelets. Our method
provides a systematic way to construct multi-D non-separable wavelet filter
banks from two 1-D lowpass filters, with one of whom being interpolatory. Our
method has many important features including the following: 1) it works for any
spatial dimension, and any prime scalar dilation, 2) the vanishing moments of
the multi-D wavelet filter banks are guaranteed by certain properties of the
initial 1-D lowpass filters, and furthermore, 3) the resulting multi-D wavelet
filter banks are associated with fast algorithms that are faster than the
existing fast tensor product algorithms.

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## ‣ Reproducing subgroups of Sp(2,R). Part II: admissible vectors

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/12/2012
Português

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In part I we introduced the class ${\mathcal E}_2$ of Lie subgroups of
$Sp(2,\R)$ and obtained a classification up to conjugation (Theorem 1.1). Here,
we determine for which of these groups the restriction of the metaplectic
representation gives rise to a reproducing formula. In all the positive cases
we characterize the admissible vectors with a generalized Calder\'on equation.
They include products of 1D-wavelets, directional wavelets, shearlets, and many
new examples.; Comment: 37 pages, 3 figures

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## ‣ On generic frequency decomposition. Part 1: vectorial decomposition

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/04/2008
Português

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The famous Fourier theorem states that, under some restrictions, any periodic
function (or real world signal) can be obtained as a sum of sinusoids, and
hence, a technique exists for decomposing a signal into its sinusoidal
components. From this theory an entire branch of research has flourished: from
the Short-Time or Windowed Fourier Transform to the Wavelets, the Frames, and
lately the Generic Frequency Analysis. The aim of this paper is to take the
Frequency Analysis a step further. It will be shown that keeping the same
reconstruction algorithm as the Fourier Theorem but changing to a new computing
method for the analysis phase allows the generalization of the Fourier Theorem
to a large class of nonorthogonal bases. New methods and algorithms can be
employed in function decomposition on such generic bases. It will be shown that
these algorithms are a generalization of the Fourier analysis, i.e. they are
reduced to the familiar Fourier tools when using orthogonal bases. The
differences between this tool and the wavelets and frames theories will be
discussed. Examples of analysis and reconstruction of functions using the given
algorithms and nonorthogonal bases will be given. In this first part the focus
will be on vectorial decomposition...

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## ‣ Asymptotic Analysis of Inpainting via Universal Shearlet Systems

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

Relevância na Pesquisa

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Recently introduced inpainting algorithms using a combination of applied
harmonic analysis and compressed sensing have turned out to be very successful.
One key ingredient is a carefully chosen representation system which provides
(optimally) sparse approximations of the original image. Due to the common
assumption that images are typically governed by anisotropic features,
directional representation systems have often been utilized. One prominent
example of this class are shearlets, which have the additional benefitallowing
faithful implementations. Numerical results show that shearlets significantly
outperform wavelets in inpainting tasks. One of those software packages,
www.shearlab.org, even offers the flexibility of usingdifferent parameter for
each scale, which is not yet covered by shearlet theory.
In this paper, we first introduce universal shearlet systems which are
associated with an arbitrary scaling sequence, thereby modeling the previously
mentioned flexibility. In addition, this novel construction allows for a smooth
transition between wavelets and shearlets and therefore enables us to analyze
them in a uniform fashion. For a large class of such scaling sequences, we
first prove that the associated universal shearlet systems form band-limited
Parseval frames for $L^2(\mathbb{R}^2)$ consisting of Schwartz functions.
Secondly...

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## ‣ Wavelet filter functions, the matrix completion problem, and projective modules over $C(\mathbb T^n)$

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Functional Analysis#Mathematics - Classical Analysis and ODEs#Mathematics - Operator Algebras#46L99 (primary), 42C40, 46H25 (secondary)

We discuss how one can use certain filters from signal processing to describe
isomorphisms between certain projective $C(\mathbb T^n)$-modules. Conversely,
we show how cancellation properties for finitely generated projective modules
over $C(\mathbb T^n)$ can often be used to prove the existence of continuous
high pass filters, of the kind needed for multivariate wavelets, corresponding
to a given continuous low-pass filter. However, we also give an example of a
continuous low-pass filter for which it is impossible to find corresponding
continuous high-pass filters. In this way we give another approach to the
solution of the matrix completion problem for filters of the kind arising in
wavelet theory.; Comment: 21 pages, various local improvements

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