264 research outputs found
Nonhomogeneous Wavelet Systems in High Dimensions
It is of interest to study a wavelet system with a minimum number of
generators. It has been showed by X. Dai, D. R. Larson, and D. M. Speegle in
[11] that for any real-valued expansive matrix M, a homogeneous
orthonormal M-wavelet basis can be generated by a single wavelet function. On
the other hand, it has been demonstrated in [21] that nonhomogeneous wavelet
systems, though much less studied in the literature, play a fundamental role in
wavelet analysis and naturally link many aspects of wavelet analysis together.
In this paper, we are interested in nonhomogeneous wavelet systems in high
dimensions with a minimum number of generators. As we shall see in this paper,
a nonhomogeneous wavelet system naturally leads to a homogeneous wavelet system
with almost all properties preserved. We also show that a nonredundant
nonhomogeneous wavelet system is naturally connected to refinable structures
and has a fixed number of wavelet generators. Consequently, it is often
impossible for a nonhomogeneous orthonormal wavelet basis to have a single
wavelet generator. However, for redundant nonhomogeneous wavelet systems, we
show that for any real-valued expansive matrix M, we can always
construct a nonhomogeneous smooth tight M-wavelet frame in with a
single wavelet generator whose Fourier transform is a compactly supported
function. Moreover, such nonhomogeneous tight wavelet frames are
associated with filter banks and can be modified to achieve directionality in
high dimensions. Our analysis of nonhomogeneous wavelet systems employs a
notion of frequency-based nonhomogeneous wavelet systems in the distribution
space. Such a notion allows us to separate the perfect reconstruction property
of a wavelet system from its stability in function spaces
Wavelets and their use
This review paper is intended to give a useful guide for those who want to
apply discrete wavelets in their practice. The notion of wavelets and their use
in practical computing and various applications are briefly described, but
rigorous proofs of mathematical statements are omitted, and the reader is just
referred to corresponding literature. The multiresolution analysis and fast
wavelet transform became a standard procedure for dealing with discrete
wavelets. The proper choice of a wavelet and use of nonstandard matrix
multiplication are often crucial for achievement of a goal. Analysis of various
functions with the help of wavelets allows to reveal fractal structures,
singularities etc. Wavelet transform of operator expressions helps solve some
equations. In practical applications one deals often with the discretized
functions, and the problem of stability of wavelet transform and corresponding
numerical algorithms becomes important. After discussing all these topics we
turn to practical applications of the wavelet machinery. They are so numerous
that we have to limit ourselves by some examples only. The authors would be
grateful for any comments which improve this review paper and move us closer to
the goal proclaimed in the first phrase of the abstract.Comment: 63 pages with 22 ps-figures, to be published in Physics-Uspekh
Extreme Value Analysis of Empirical Frame Coefficients and Implications for Denoising by Soft-Thresholding
Denoising by frame thresholding is one of the most basic and efficient
methods for recovering a discrete signal or image from data that are corrupted
by additive Gaussian white noise. The basic idea is to select a frame of
analyzing elements that separates the data in few large coefficients due to the
signal and many small coefficients mainly due to the noise \epsilon_n. Removing
all data coefficients being in magnitude below a certain threshold yields a
reconstruction of the original signal. In order to properly balance the amount
of noise to be removed and the relevant signal features to be kept, a precise
understanding of the statistical properties of thresholding is important. For
that purpose we derive the asymptotic distribution of max_{\omega \in \Omega_n}
|| for a wide class of redundant frames
(\phi_\omega^n: \omega \in \Omega_n}. Based on our theoretical results we give
a rationale for universal extreme value thresholding techniques yielding
asymptotically sharp confidence regions and smoothness estimates corresponding
to prescribed significance levels. The results cover many frames used in
imaging and signal recovery applications, such as redundant wavelet systems,
curvelet frames, or unions of bases. We show that `generically' a standard
Gumbel law results as it is known from the case of orthonormal wavelet bases.
However, for specific highly redundant frames other limiting laws may occur. We
indeed verify that the translation invariant wavelet transform shows a
different asymptotic behaviour.Comment: [Content: 39 pages, 4 figures] Note that in this version 4 we have
slightely changed the title of the paper and we have rewritten parts of the
introduction. Except for corrected typos the other parts of the paper are the
same as the original versions
Confidence bands in density estimation
Given a sample from some unknown continuous density
, we construct adaptive confidence bands that are
honest for all densities in a "generic" subset of the union of -H\"older
balls, , where is a fixed but arbitrary integer. The exceptional
("nongeneric") set of densities for which our results do not hold is shown to
be nowhere dense in the relevant H\"older-norm topologies. In the course of the
proofs we also obtain limit theorems for maxima of linear wavelet and kernel
density estimators, which are of independent interest.Comment: Published in at http://dx.doi.org/10.1214/09-AOS738 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Construction of interpolating and orthonormal multigenerators and multiwavelets on the interval
In den letzten Jahren haben sich Wavelets zu einem hochwertigen Hilfsmittel in der angewandten
Mathematik entwickelt. Eine Waveletbasis ist im Allgemeinen ein System von
Funktionen, das durch die Skalierung, Translation und Dilatation einer endlichen Menge
von Funktionen, den sogenannten Mutterwavelets, entsteht. Wavelets wurden sehr erfolgreich
in der digitalen Signal- und Bildanalyse, z. B. zur Datenkompression verwendet.
Ein weiteres wichtiges Anwendungsfeld ist die Analyse und die numerische Behandlung
von Operatorgleichungen. Insbesondere ist es gelungen, adaptive numerische Algorithmen
basierend auf Wavelets fĂŒr eine riesige Klasse von Operatorgleichungen, einschlieĂlich
Operatoren mit negativer Ordnung, zu entwickeln. Der Erfolg der Wavelet-
Algorithmen ergibt sich als Konsequenz der folgenden Fakten:
- Gewichtete Folgennormen von Wavelet-Expansionskoeffizienten sind in einem bestimmten
Bereich (abhÀngig von der RegularitÀt der Wavelets) Àquivalent zu
GlÀttungsnormen wie Besov- oder Sobolev-Normen.
- FĂŒr eine breite Klasse von Operatoren ist ihre Darstellung in Wavelet-Koordinaten
nahezu diagonal.
- Die verschwindenden Momente von Wavelets entfernen den glatten Teil einer Funktion
und fĂŒhren zu sehr effizienten Komprimierungsstrategien.
Diese Fakten können z. B. verwendet werden, um adaptive numerische Strategien mit
optimaler Konvergenzgeschwindigkeit zu konstruieren, in dem Sinne, dass diese Algorithmen
die Konvergenzordnung der besten N-Term-Approximationsschemata realisieren.
Die maĂgeblichen Ergebnisse lassen sich fĂŒr lineare, symmetrische, elliptische Operatorgleichungen
erzielen. Es existiert auch eine Verallgemeinerung fĂŒr nichtlineare elliptische
Gleichungen. Hier verbirgt sich jedoch eine ernste Schwierigkeit: Jeder numerische Algorithmus
fĂŒr diese Gleichungen erfordert die Auswertung eines nichtlinearen Funktionals,
welches auf eine Wavelet-Reihe angewendet wird. Obwohl einige sehr ausgefeilte Algorithmen
existieren, erweisen sie sich als ziemlich langsam in der Praxis. In neueren Studien
wurde gezeigt, dass dieses Problem durch sogenannte Interpolanten verbessert werden
kann. Dabei stellt sich heraus, dass die meisten bekannten Basen der Interpolanten
keine stabilen Basen in L2[a,b] bilden.
In der vorliegenden Arbeit leisten wir einen wesentlichen Beitrag zu diesem Problem
und konstruieren neue Familien von Interpolanten auf beschrÀnkten Gebieten, die nicht
nur interpolierend, sondern auch stabil in L2[a,b] sind. Da dies mit nur einem Generator
schwer (oder vielleicht sogar unmöglich) zu erreichen ist, werden wir mit Multigeneratoren
und Multiwavelets arbeiten.In recent years, wavelets have become a very powerful tools in applied
mathematics. In general,
a wavelet basis is a system of functions that is generated by scaling, translating and dilating a
finite set of functions, the so-called mother wavelets. Wavelets have been very successfully
applied in image/signal analysis, e.g., for denoising and compression purposes. Another
important field of applications is the analysis and the numerical treatment of operator
equations. In particular, it has been possible to design adaptive numerical algorithms based on
wavelets for a huge class of operator equations including operators of negative order. The
success of wavelet algorithms is an ultimative consequence of the following facts:
- Weighted sequence norms of wavelet expansion coefficients are equivalent in a certain
range (depending on the regularity of the wavelets) to smoothness norms such as Besov
or Sobolev norms.
- For a wide class of operators their representation in wavelet coordinates is nearly
diagonal.
-The vanishing moments of wavelets remove the smooth part of a function.
These facts can,
e.g., be used to construct adaptive numerical strategies that are guaranteed to
converge with optimal order, in the sense that these algorithms realize the convergence order
of best N-term approximation schemes. The most far-reaching results have been obtained for
linear, symmetric elliptic operator equations. Generalization to nonlinear elliptic equations also
exist. However, then one is faced with a serious bottleneck: every numerical algorithm for these
equations requires the evaluation of a nonlinear functional applied to a wavelet series.
Although some very sophisticated algorithms exist, they turn out to perform quite slowly in
practice. In recent studies, it has been shown that this problem can be ameliorated by means of
so called interpolants. However, then the problem occurs that most of the known bases of
interpolants do not form stable bases in L2[a,b].
In this PhD project, we intend to provide a significant
contribution to this problem. We want to
construct new families of interpolants on domains that are not only interpolating, but also
stable in L2[a,b]or even orthogonal. Since this is hard to achieve (or maybe even impossible)
with just one generator, we worked with multigenerators and multiwavelets
Noise Covariance Properties in Dual-Tree Wavelet Decompositions
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
High Frequency Asymptotics for Wavelet-Based Tests for Gaussianity and Isotropy on the Torus
We prove a CLT for skewness and kurtosis of the wavelets coefficients of a
stationary field on the torus. The results are in the framework of the
fixed-domain asymptotics, i.e. we refer to observations of a single field which
is sampled at higher and higher frequencies. We consider also studentized
statistics for the case of an unknown correlation structure. The results are
motivated by the analysis of cosmological data or high-frequency financial data
sets, with a particular interest towards testing for Gaussianity and isotropyComment: 33 pages, 3 figure
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