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

High-order balanced multiwavelets: theory, factorization, and design

This correspondence deals with multiwavelets, which are a recent generalization of wavelets in the context of time-varying filter banks and with their applications to signal processing and especially com- pression. By their inherent structure, multiwavelets are fit for processing multichannel signals. This is the main issue in which we will be interested here. The outline of the correspondence is as follows. First, we will review material on multiwavelets and their links with multifilter banks and, especially, time-varying filter banks. Then, we will have a close look at the problems encountered when using multiwavelets in applications, and we will propose new solutions for the design of multiwavelets filter banks by introducing the so-called balanced multiwavelets

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

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

Quaternion Matrices : Statistical Properties and Applications to Signal Processing and Wavelets

Similarly to how complex numbers provide a possible framework for extending scalar signal processing techniques to 2-channel signals, the 4-dimensional hypercomplex algebra of quaternions can be used to represent signals with 3 or 4 components. For a quaternion random vector to be suited for quaternion linear processing, it must be (second-order) proper. We consider the likelihood ratio test (LRT) for propriety, and compute the exact distribution for statistics of Box type, which include this LRT. Various approximate distributions are compared. The Wishart distribution of a quaternion sample covariance matrix is derived from first principles. Quaternions are isomorphic to an algebra of structured 4x4 real matrices. This mapping is our main tool, and suggests considering more general real matrix problems as a way of investigating quaternion linear algorithms. A quaternion vector autoregressive (VAR) time-series model is equivalent to a structured real VAR model. We show that generalised least squares (and Gaussian maximum likelihood) estimation of the parameters reduces to ordinary least squares, but only if the innovations are proper. A LRT is suggested to simultaneously test for quaternion structure in the regression coefficients and innovation covariance. Matrix-valued wavelets (MVWs) are generalised (multi)wavelets for vector-valued signals. Quaternion wavelets are equivalent to structured MVWs. Taking into account orthogonal similarity, all MVWs can be constructed from non-trivial MVWs. We show that there are no non-scalar non-trivial MVWs with short support [0,3]. Through symbolic computation we construct the families of shortest non-trivial 2x2 Daubechies MVWs and quaternion Daubechies wavelets.Open Acces

Wavelet-based numerical methods adaptive modelling of shallow water flows

Mesh adaptation techniques are commonly coupled with the numerical schemes in an attempt to improve the modelling efficiency and capturing of the different physical scales which are involved in the shallow water flow problems. This work designs an adaptive technique that avails from the wavelets theory for transforming the local single resolution information into multiresolution information in which these data information became accessible. The adaptivity of wavelets was first comprehensively tested via using an arbitrary function in which the spatial resolution adaptivity was achieved from the local solution itself and it was based on a single user-prescribed parameter. Secondly, the adaptive technique was combined with two standard numerical modelling schemes (i.e. finite volume and discontinuous Galerkin schemes) to produce two wavelet-based adaptive schemes. These schemes are designed for modelling one-dimensional shallow water flows and are referred to the Haar wavelets finite volume (HWFV) and multiwavelet discontinuous Galerkin (MWDG) schemes. Both adaptive schemes were systematically tested using hydraulic test cases. The results demonstrated that the proposed adaptive technique could serve as lucid foundation on which to construct holistic and smart adaptive schemes for simulating real shallow water flow