Subdivisions with infinitely supported mask

AbstractIn this paper we investigate the convergence of subdivision schemes associated with masks being polynomially decay sequences. Two-scale vector refinement equations are the formφ(x)=∑α∈Za(α)φ(2x-α),x∈R,where the vector of functions φ=(φ1,…,φr)T is in (L2(R))r and a≕(a(α))α∈Z is polynomially decay sequence of r×r matrices called refinement mask. Associated with the mask a is a linear operator on (L2(R))r given byQaf(x)≔∑α∈Za(α)f(2x-α),x∈R,f=(f1,…,fr)T∈(L2(R))r.By using same methods in [B. Han, R. Q. Jia, Characterization of Riesz bases of wavelets generated from multiresolution analysis, manuscript]; [B. Han, Refinable functions and cascade algorithms in weighted spaces with infinitely supported masks, manuscript]; [R.Q. Jia, Q.T. Jiang, Z.W. Shen, Convergence of cascade algorithms associated with nonhomogeneous refinement equations, Proc. Amer. Math. Soc. 129 (2001) 415–427]; [R.Q. Jia, Convergence of vector subdivision schemes and construction of biorthogonal multiple wavelets, in: Advances in Wavelet, Hong Kong,1997, Springer, Singapore, 1998, pp. 199–227], a characterization of convergence of the sequences (Qanf)n=1,2,… in the L2-norm is given, which extends the main results in [R.Q. Jia, S.D. Riemenschneider, D.X. Zhou, Vector subdivision schemes and multiple wavelets, Math. Comp. 67 (1998) 1533–1563] on convergence of the subdivision schemes associated with a finitely supported mask to the case in which mask a is polynomially decay sequence. As an application, we also obtain a characterization of smoothness of solutions of the refinement equation mentioned above for the case r=1

Smoothness of multivariate refinable functions with infinitely supported masks

AbstractIn this paper, we investigate the smoothness of multivariate refinable functions with infinitely supported masks and an isotropic dilation matrix. Using some methods as in [R.Q. Jia, Characterization of smoothness of multivariate refinable functions in Sobolev spaces, Trans. Amer. Math. Soc. 351 (1999) 4089–4112], we characterize the optimal smoothness of multivariate refinable functions with polynomially decaying masks and an isotropic dilation matrix. Our characterizations extend some of the main results of the above mentioned paper with finitely supported masks to the case in which masks are infinitely supported

Benchmark of FEM, Waveguide and FDTD Algorithms for Rigorous Mask Simulation

An extremely fast time-harmonic finite element solver developed for the transmission analysis of photonic crystals was applied to mask simulation problems. The applicability was proven by examining a set of typical problems and by a benchmarking against two established methods (FDTD and a differential method) and an analytical example. The new finite element approach was up to 100 times faster than the competing approaches for moderate target accuracies, and it was the only method which allowed to reach high target accuracies.Comment: 12 pages, 8 figures (see original publication for images with a better resolution

Bell-shaped nonstationary refinable ripplets

We study the approximation properties of the class of nonstationary refinable ripplets introduced in \cite{GP08}. These functions are solution of an infinite set of nonstationary refinable equations and are defined through sequences of scaling masks that have an explicit expression. Moreover, they are variation-diminishing and highly localized in the scale-time plane, properties that make them particularly attractive in applications. Here, we prove that they enjoy Strang-Fix conditions and convolution and differentiation rules and that they are bell-shaped. Then, we construct the corresponding minimally supported nonstationary prewavelets and give an iterative algorithm to evaluate the prewavelet masks. Finally, we give a procedure to construct the associated nonstationary biorthogonal bases and filters to be used in efficient decomposition and reconstruction algorithms. As an example, we calculate the prewavelet masks and the nonstationary biorthogonal filter pairs corresponding to the $C^2$ nonstationary scaling functions in the class and construct the corresponding prewavelets and biorthogonal bases. A simple test showing their good performances in the analysis of a spike-like signal is also presented. Keywords: total positivity, variation-dimishing, refinable ripplet, bell-shaped function, nonstationary prewavelet, nonstationary biorthogonal basisComment: 30 pages, 10 figure

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

Elliptic scaling functions as compactly supported multivariate analogs of the B-splines

In the paper, we present a family of multivariate compactly supported scaling functions, which we call as elliptic scaling functions. The elliptic scaling functions are the convolution of elliptic splines, which correspond to homogeneous elliptic differential operators, with distributions. The elliptic scaling functions satisfy refinement relations with real isotropic dilation matrices. The elliptic scaling functions satisfy most of the properties of the univariate cardinal B-splines: compact support, refinement relation, partition of unity, total positivity, order of approximation, convolution relation, Riesz basis formation (under a restriction on the mask), etc. The algebraic polynomials contained in the span of integer shifts of any elliptic scaling function belong to the null-space of a homogeneous elliptic differential operator. Similarly to the properties of the B-splines under differentiation, it is possible to define elliptic (not necessarily differential) operators such that the elliptic scaling functions satisfy relations with these operators. In particular, the elliptic scaling functions can be considered as a composition of segments, where the function inside a segment, like a polynomial in the case of the B-splines, vanishes under the action of the introduced operator.Comment: To appear in IJWMI

Semi-sharp creases on subdivision curves and surfaces

We explore a method for generalising Pixar semi-sharp creases from the univariate cubic case to arbitrary degree subdivision curves. Our approach is based on solving simple matrix equations. The resulting schemes allow for greater flexibility over existing methods, via control vectors. We demonstrate our results on several high-degree univariate examples and explore analogous methods for subdivision surfacesThis work was supported by the Engineering and Physical Sciences Research Council [EP/H030115/1].This is the author accepted manuscript and will be under embargo until the 23rd of August 2015. The final version has been published in Computer Graphics Forum here: http://onlinelibrary.wiley.com/doi/10.1111/cgf.12447/abstract