28 research outputs found
Highly Symmetric Multiple Bi-Frames for Curve and Surface Multiresolution Processing
Wavelets and wavelet frames are important and useful mathematical tools in numerous applications, such as signal and image processing, and numerical analysis. Recently, the theory of wavelet frames plays an essential role in signal processing, image processing, sampling theory, and harmonic analysis. However, multiwavelets and multiple frames are more flexible and have more freedom in their construction which can provide more desired properties than the scalar case, such as short compact support, orthogonality, high approximation order, and symmetry. These properties are useful in several applications, such as curve and surface noise-removing as studied in this dissertation. Thus, the study of multiwavelets and multiple frames construction has more advantages for many applications.
Recently, the construction of highly symmetric bi-frames for curve and surface multiresolution processing has been investigated. The 6-fold symmetric bi-frames, which lead to highly symmetric analysis and synthesis bi-frame algorithms, have been introduced. Moreover, these multiple bi-frame algorithms play an important role on curve and surface multiresolution processing. This dissertation is an extension of the study of construction of univariate biorthogonal wavelet frames (bi-frames for short) or dual wavelet frames with each framelet being symmetric in the scalar case. We will expand the study of biorthogonal wavelets and bi-frames construction from the scalar case to the vector case to construct biorthogonal multiwavelets and multiple bi-frames in one-dimension. In addition, we will extend the study of highly symmetric bi-frames for triangle surface multiresolution processing from the scalar case to the vector case.
More precisely, the objective of this research is to construct highly symmetric biorthogonal multiwavelets and multiple bi-frames in one and two dimensions for curve and surface multiresolution processing. It runs in parallel with the scalar case. We mainly present the methods of constructing biorthogonal multiwavelets and multiple bi-frames in both dimensions by using the idea of lifting scheme. On the whole, we discuss several topics include a brief introduction and discussion of multiwavelets theory, multiresolution analysis, scalar wavelet frames, multiple frames, and the lifting scheme. Then, we present and discuss some results of one-dimensional biorthogonal multiwavelets and multiple bi-frames for curve multiresolution processing with uniform symmetry: type I and type II along with biorthogonality, sum rule orders, vanishing moments, and uniform symmetry for both types. In addition, we present and discuss some results of two-dimensional biorthogonal multiwavelets and multiple bi-frames and the multiresolution algorithms for surface multiresolution processing. Finally, we show experimental results on curve and surface noise-removing by applying our multiple bi-frame algorithms
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 für eine riesige Klasse von Operatorgleichungen, einschließlich
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 (abhängig von der Regularität der Wavelets) äquivalent zu
Glättungsnormen wie Besov- oder Sobolev-Normen.
- Für 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 führen 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 maßgeblichen Ergebnisse lassen sich für lineare, symmetrische, elliptische Operatorgleichungen
erzielen. Es existiert auch eine Verallgemeinerung für nichtlineare elliptische
Gleichungen. Hier verbirgt sich jedoch eine ernste Schwierigkeit: Jeder numerische Algorithmus
für 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 beschränkten 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
Gröbner bases and wavelet design
AbstractIn this paper, we detail the use of symbolic methods in order to solve some advanced design problems arising in signal processing. Our interest lies especially in the construction of wavelet filters for which the usual spectral factorization approach (used for example to construct the well-known Daubechies filters) is not applicable. In these problems, we show how the design equations can be written as multivariate polynomial systems of equations and accordingly how Gröbner algorithms offer an effective way to obtain solutions in some of these cases
On Dimension Extension of a Class of Iterative Equations
This investigation aims at studying some special properties (convergence, polynomial preservation order, and orthogonal symmetry) of a class of r-dimension iterative equations, whose state variables are described by the following nonlinear iterative equation: ϕn(x)=T(ϕn−1(x)):=∑j=0mHjϕn−1(2x−k). The obtained results in this paper are complementary to some published results. As an application, we construct orthogonal symmetric multiwavelet with additional vanishing moments. Two examples are also arranged to demonstrate the correctness and effectiveness of the main results