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## ‣ Smoothed Affine Wigner Transform

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/07/2010
Português

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We study a generalization of Husimi function in the context of wavelets. This
leads to a nonnegative density on phase-space for which we compute the
evolution equation corresponding to a Schr\"Aodinger equation.

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## ‣ On spatial and temporal multilevel dynamics and scaling effects in epileptic seizures

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Quantitative Biology - Neurons and Cognition#Mathematics - Dynamical Systems#Nonlinear Sciences - Chaotic Dynamics#Nonlinear Sciences - Pattern Formation and Solitons#Physics - Medical Physics

Epileptic seizures are one of the most well-known dysfunctions of the nervous
system. During a seizure, a highly synchronized behavior of neural activity is
observed that can cause symptoms ranging from mild sensual malfunctions to the
complete loss of body control. In this paper, we aim to contribute towards a
better understanding of the dynamical systems phenomena that cause seizures.
Based on data analysis and modelling, seizure dynamics can be identified to
possess multiple spatial scales and on each spatial scale also multiple time
scales. At each scale, we reach several novel insights. On the smallest spatial
scale we consider single model neurons and investigate early-warning signs of
spiking. This introduces the theory of critical transitions to excitable
systems. For clusters of neurons (or neuronal regions) we use patient data and
find oscillatory behavior and new scaling laws near the seizure onset. These
scalings lead to substantiate the conjecture obtained from mean-field models
that a Hopf bifurcation could be involved near seizure onset. On the largest
spatial scale we introduce a measure based on phase-locking intervals and
wavelets into seizure modelling. It is used to resolve synchronization between
different regions in the brain and identifies time-shifted scaling laws at
different wavelet scales. We also compare our wavelet-based multiscale approach
with maximum linear cross-correlation and mean-phase coherence measures.; Comment: 24 pages...

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## ‣ A wavelet-based approximation of fractional Brownian motion with a parallel algorithm

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We construct a wavelet-based almost sure uniform approximation of fractional
Brownian motion (fBm) B_t^(H), t in [0, 1], of Hurst index H in (0, 1). Our
results show that by Haar wavelets which merely have one vanishing moment, an
almost sure uniform expansion of fBm of H in (0, 1) can be established. The
convergence rate of our approximation is derived. We also describe a parallel
algorithm that generates sample paths of an fBm efficiently.; Comment: 20 pages. J. of Applied Probability, to appear in March 2014

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## ‣ Noise Covariance Properties in Dual-Tree Wavelet Decompositions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/08/2011
Português

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Dual-tree wavelet decompositions have recently gained much popularity, mainly
due to their ability to provide an accurate directional analysis of images
combined with a reduced redundancy. When the decomposition of a random process
is performed -- which occurs in particular when an additive noise is corrupting
the signal to be analyzed -- it is useful to characterize the statistical
properties of the dual-tree wavelet coefficients of this process. As dual-tree
decompositions constitute overcomplete frame expansions, correlation structures
are introduced among the coefficients, even when a white noise is analyzed. In
this paper, we show that it is possible to provide an accurate description of
the covariance properties of the dual-tree coefficients of a wide-sense
stationary process. The expressions of the (cross-)covariance sequences of the
coefficients are derived in the one and two-dimensional cases. Asymptotic
results are also provided, allowing to predict the behaviour of the
second-order moments for large lag values or at coarse resolution. In addition,
the cross-correlations between the primal and dual wavelets, which play a
primary role in our theoretical analysis, are calculated for a number of
classical wavelet families. Simulation results are finally provided to validate
these results.

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## ‣ Realizations of infinite products, Ruelle operators and wavelet filters

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Using the notions and tools from realization in the sense of systems theory,
we establish an explicit and new realization formula for families of infinite
products of rational matrix-functions of a single complex variable. Our
realizations of these resulting infinite products have the following four
features: 1) Our infinite product realizations are functions defined in an
infinite-dimensional complex domain. 2) Starting with a realization of a single
rational matrix-function $M$, we show that a resulting infinite product
realization obtained from $M$ takes the form of an (infinite-dimensional)
Toeplitz operator with a symbol that is a reflection of the initial realization
for $M$. 3) Starting with a subclass of rational matrix functions, including
scalar-valued corresponding to low-pass wavelet filters, we obtain the
corresponding infinite products that realize the Fourier transforms of
generators of $\mathbf L_2(\mathbb R)$ wavelets. 4) We use both the
realizations for $M$ and the corresponding infinite product to produce a matrix
representation of the Ruelle-transfer operators used in wavelet theory. By
matrix representation we refer to the slanted (and sparse) matrix which
realizes the Ruelle-transfer operator under consideration.; Comment: corrected version

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## ‣ Adaptive density estimation under dependence

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Assume that $(X_t)_{t\in\Z}$ is a real valued time series admitting a common
marginal density $f$ with respect to Lebesgue's measure. Donoho {\it et al.}
(1996) propose a near-minimax method based on thresholding wavelets to estimate
$f$ on a compact set in an independent and identically distributed setting. The
aim of the present work is to extend these results to general weak dependent
contexts. Weak dependence assumptions are expressed as decreasing bounds of
covariance terms and are detailed for different examples. The threshold levels
in estimators $\widehat f_n$ depend on weak dependence properties of the
sequence $(X_t)_{t\in\Z}$ through the constant. If these properties are
unknown, we propose cross-validation procedures to get new estimators. These
procedures are illustrated via simulations of dynamical systems and non causal
infinite moving averages. We also discuss the efficiency of our estimators with
respect to the decrease of covariances bounds.

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## ‣ Some equations relating multiwavelets and multiscaling functions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/11/2005
Português

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The local trace function introduced in \cite{Dut} is used to derive equations
that relate multiwavelets and multiscaling functions in the context of a
generalized multiresolution analysis, without appealing to filters. A
construction of normalized tight frame wavelets is given. Particular instances
of the construction include normalized tight frame and orthonormal wavelet
sets.

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## ‣ An application of interpolating scaling functions to wave packet propagation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/08/2003
Português

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#Physics - Atomic Physics#Mathematics - Numerical Analysis#Physics - Computational Physics#Quantum Physics

Wave packet propagation in the basis of interpolating scaling functions (ISF)
is studied. The ISF are well known in the multiresolution analysis based on
spline biorthogonal wavelets. The ISF form a cardinal basis set corresponding
to an equidistantly spaced grid. They have compact support of the size
determined by the underlying interpolating polynomial that is used to generate
ISF. In this basis the potential energy matrix is diagonal and the kinetic
energy matrix is sparse and, in the 1D case, has a band-diagonal structure. An
important feature of the basis is that matrix elements of a Hamiltonian are
exactly computed by means of simple algebraic transformations efficiently
implemented numerically. Therefore the number of grid points and the order of
the underlying interpolating polynomial can easily be varied allowing one to
approach the accuracy of pseudospectral methods in a regular manner, similar to
high order finite difference methods. The results of numerical simulations of
an H+H_2 collinear collision show that the ISF provide one with an accurate and
efficient representation for use in the wave packet propagation method.; Comment: plain Latex, 11 pages, 4 figures attached in the JPEG format

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## ‣ Multiresolution Analysis Based on Coalescence Hidden-variable FIF

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/01/2012
Português

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In the present paper, multiresolution analysis arising from Coalescence
Hidden-variable Fractal Interpolation Functions (CHFIFs) is accomplished. The
availability of a larger set of free variables and constrained variables with
CHFIF in multiresolution analysis based on CHFIFs provides more control in
reconstruction of functions in L2(\mathbb{R})than that provided by
multiresolution analysis based only on Affine Fractal Interpolation Functions
(AFIFs). In our approach, the vector space of CHFIFs is introduced, its
dimension is determined and Riesz bases of vector subspaces Vk, k \in
\mathbb{Z}, consisting of certain CHFIFs in L2(\mathbb{R}) \cap C0(\mathbb{R})
are constructed. As a special case, for the vector space of CHFIFs of dimension
4, orthogonal bases for the vector subspaces Vk, k \in \mathbb{Z}, are
explicitly constructed and, using these bases, compactly supported continuous
orthonormal wavelets are generated.; Comment: 19 Pages, 3 Figures

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## ‣ Wavelet methods in multi-conjugate adaptive optics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/02/2013
Português

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#Mathematics - Numerical Analysis#Astrophysics - Instrumentation and Methods for Astrophysics#45Q05, 65R32, 65T60, 85-08, 85A35

The next generation ground-based telescopes rely heavily on adaptive optics
for overcoming the limitation of atmospheric turbulence. In the future adaptive
optics modalities, like multi-conjugate adaptive optics (MCAO), atmospheric
tomography is the major mathematical and computational challenge. In this
severely ill-posed problem a fast and stable reconstruction algorithm is needed
that can take into account many real-life phenomena of telescope imaging. We
introduce a novel reconstruction method for the atmospheric tomography problem
and demonstrate its performance and flexibility in the context of MCAO. Our
method is based on using locality properties of compactly supported wavelets,
both in the spatial and frequency domain. The reconstruction in the atmospheric
tomography problem is obtained by solving the Bayesian MAP estimator with a
conjugate gradient based algorithm. An accelerated algorithm with
preconditioning is also introduced. Numerical performance is demonstrated on
the official end-to-end simulation tool OCTOPUS of European Southern
Observatory.

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## ‣ Adaptive estimation of an additive regression function from weakly dependent data

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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A $d$-dimensional nonparametric additive regression model with dependent
observations is considered. Using the marginal integration technique and
wavelets methodology, we develop a new adaptive estimator for a component of
the additive regression function. Its asymptotic properties are investigated
via the minimax approach under the $\mathbb{L}_2$ risk over Besov balls. We
prove that it attains a sharp rate of convergence which turns to be the one
obtained in the $\iid$ case for the standard univariate regression estimation
problem.; Comment: Substantial improvement of the estimator and the main theorem

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## ‣ Average Interpolating Wavelets on Point Clouds and Graphs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/10/2011
Português

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#Mathematics - Functional Analysis#Computer Science - Information Theory#Statistics - Machine Learning

We introduce a new wavelet transform suitable for analyzing functions on
point clouds and graphs. Our construction is based on a generalization of the
average interpolating refinement scheme of Donoho. The most important
ingredient of the original scheme that needs to be altered is the choice of the
interpolant. Here, we define the interpolant as the minimizer of a smoothness
functional, namely a generalization of the Laplacian energy, subject to the
averaging constraints. In the continuous setting, we derive a formula for the
optimal solution in terms of the poly-harmonic Green's function. The form of
this solution is used to motivate our construction in the setting of graphs and
point clouds. We highlight the empirical convergence of our refinement scheme
and the potential applications of the resulting wavelet transform through
experiments on a number of data stets.

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## ‣ From Weyl-Heisenberg Frames to Infinite Quadratic Forms

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/07/2005
Português

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Let $a$, $b$ be two fixed positive constants. A function $g\in L^2({\mathbb
R})$ is called a \textit{mother Weyl-Heisenberg frame wavelet} for $(a,b)$ if
$g$ generates a frame for $L^2({\mathbb R})$ under modulates by $b$ and
translates by $a$, i.e., $\{e^{imbt}g(t-na\}_{m,n\in\mathbb{Z}}$ is a frame for
$L^2(\mathbb{R})$. In this paper, we establish a connection between mother
Weyl-Heisenberg frame wavelets of certain special forms and certain strongly
positive definite quadratic forms of infinite dimension. Some examples of
application in matrix algebra are provided.

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## ‣ Weyl-Heisenberg Frame Wavelets with Basic Supports

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 29/07/2005
Português

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Let $a$, $b$ be two fixed non-zero constants. A measurable set $E\subset
\mathbb{R}$ is called a Weyl-Heisenberg frame set for $(a, b)$ if the function
$g=\chi_{E}$ generates a Weyl-Heisenberg frame for $L^2(\mathbb{R})$ under
modulates by $b$ and translates by $a$, i.e.,
$\{e^{imbt}g(t-na\}_{m,n\in\mathbb{Z}}$ is a frame for $L^2(\mathbb{R})$. It is
an open question on how to characterize all frame sets for a given pair $(a,b)$
in general. In the case that $a=2\pi$ and $b=1$, a result due to Casazza and
Kalton shows that the condition that the set
$F=\bigcup_{j=1}^{k}([0,2\pi)+2n_{j}\pi)$ (where $\{n_{1}

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## ‣ Nonparametric methods for volatility density estimation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/10/2009
Português

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#Statistics - Methodology#Mathematics - Statistics Theory#Quantitative Finance - Statistical Finance#62G07, 62G08, 62M07, 62P20, 91G70

Stochastic volatility modelling of financial processes has become
increasingly popular. The proposed models usually contain a stationary
volatility process. We will motivate and review several nonparametric methods
for estimation of the density of the volatility process. Both models based on
discretely sampled continuous time processes and discrete time models will be
discussed.
The key insight for the analysis is a transformation of the volatility
density estimation problem to a deconvolution model for which standard methods
exist. Three type of nonparametric density estimators are reviewed: the
Fourier-type deconvolution kernel density estimator, a wavelet deconvolution
density estimator and a penalized projection estimator. The performance of
these estimators will be compared. Key words: stochastic volatility models,
deconvolution, density estimation, kernel estimator, wavelets, minimum contrast
estimation, mixing

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## ‣ On orthogonal $p$-adic wavelet bases

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/12/2013
Português

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A variety of different orthogonal wavelet bases has been found in L_2(R) for
the last three decades. It appeared that similar constructions also exist for
functions defined on some other algebraic structures, such as the Cantor and
Vilenkin groups and local fields of positive characteristic. In the present
paper we show that the situation is quite different for the field of $p$-adic
numbers. Namely, it is proved that any orthogonal wavelet basis consisting of
band-limited (periodic) functions is a modification of Haar basis. This is a
little bit unexpected because from the wavelet theory point of view, the
additive group of $p$-adic numbers looks very similar to the Vilenkin group
where analogs of the Daubechies wavelets (and even band-limited ones) do exist.
We note that all $p$-adic wavelet bases and frames appeared in the literature
consist of Schwartz-Bruhat functions (i.e., band-limited and compactly
supported ones).

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## ‣ Wavelet methods for shape perception in electro-sensing

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/10/2013
Português

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This paper aims at presenting a new approach to the electro-sensing problem
using wavelets. It provides an efficient algorithm for recognizing the shape of
a target from micro-electrical impedance measurements. Stability and resolution
capabilities of the proposed algorithm are quantified in numerical simulations.

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## ‣ From cardinal spline wavelet bases to highly coherent dictionaries

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/03/2008
Português

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Wavelet families arise by scaling and translations of a prototype function,
called the {\em {mother wavelet}}. The construction of wavelet bases for
cardinal spline spaces is generally carried out within the multi-resolution
analysis scheme. Thus, the usual way of increasing the dimension of the
multi-resolution subspaces is by augmenting the scaling factor. We show here
that, when working on a compact interval, the identical effect can be achieved
without changing the wavelet scale but reducing the translation parameter. By
such a procedure we generate a redundant frame, called a {\em{dictionary}},
spanning the same spaces as a wavelet basis but with wavelets of broader
support. We characterise the correlation of the dictionary elements by
measuring their `coherence' and produce examples illustrating the relevance of
highly coherent dictionaries to problems of sparse signal representation.

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## ‣ Wavelet Galerkin method for fractional elliptic differential equations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/05/2014
Português

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Under the guidance of the general theory developed for classical partial
differential equations (PDEs), we investigate the Riesz bases of wavelets in
the spaces where fractional PDEs usually work, and their applications in
numerically solving fractional elliptic differential equations (FEDEs). The
technique issues are solved and the detailed algorithm descriptions are
provided. Compared with the ordinary Galerkin methods, the wavelet Galerkin
method we propose for FEDEs has the striking benefit of efficiency, since the
condition numbers of the corresponding stiffness matrixes are small and
uniformly bounded; and the Toeplitz structure of the matrix still can be used
to reduce cost. Numerical results and comparison with the ordinary Galerkin
methods are presented to demonstrate the advantages of the wavelet Galerkin
method we provide.; Comment: 20 pages, 0 figures

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## ‣ Sampling, splines and frames on compact manifolds

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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 the Shannon-type sampling, 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: Will appear in International Journal on Geomathematics (GEM). arXiv
admin note: substantial text overlap with arXiv:1403.0963

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