Totally positive refinable functions with general dilation M

We construct a new class of approximating functions that are M-refinable and provide shape preserving approximations. The refinable functions in the class are smooth, compactly supported, centrally symmetric and totally positive. Moreover, their refinable masks are associated with convergent subdivision schemes. The presence of one or more shape parameters gives a great flexibility in the applications. Some examples for dilation M=4and M=5are also given

Ternary shape-preserving subdivision schemes

We analyze the shape-preserving properties of ternary subdivision schemes generated by bell-shaped masks. We prove that any bell-shaped mask, satisfying the basic sum rules, gives rise to a convergent monotonicity preserving subdivision scheme, but convexity preservation is not guaranteed. We show that to reach convexity preservation the first order divided difference scheme needs to be bell-shaped, too. Finally, we show that ternary subdivision schemes associated with certain refinable functions with dilation 3 have shape-preserving properties of higher order

Positivity of refinable functions defined by nonnegative finite masks

AbstractLet (a(j)|j=0,1,…,N) with a(0),a(N)≠0 be a given nonnegative mask. Assume that the subdivision scheme with this mask is convergent. Let the associated refinable function be ϕ. So the support of ϕ is contained in [0,N]. Melkman conjectured in 1997 that unless the scheme is interpolatory and N>2 the refinable function ϕ is positive on (0,N). In the present paper we confirm this conjecture. A lower bound of ϕ on [2−m,N−2−m] is also given

The correlation between the convergence of subdivision processes and solvability of refinement equations.

We consider a univariate two-scale difference equation, which is studied in approximation theory, curve design and wavelets theory. This paper analysis the correlation between the existence of smooth compactly supported solutions of this equation and the convergence of the corresponding cascade algorithm/subdivision scheme. We introduce a criterion that expresses this correlation in terms of mask of the equation. It was shown that the convergence of subdivision scheme depends on values that the mask takes at the points of its generalized cycles. In this paper we show that the criterion is sharp in the sense that an arbitrary generalized cycle causes the divergence of a suitable subdivision scheme. To do this we construct a general method to produce divergent subdivision schemes having smooth refinable functions. The criterion therefore establishes a complete classification of divergent subdivision schemes

Vector Subdivision Schemes for Arbitrary Matrix Masks

Employing a matrix mask, a vector subdivision scheme is a fast iterative averaging algorithm to compute refinable vector functions for wavelet methods in numerical PDEs and to produce smooth curves in CAGD. In sharp contrast to the well-studied scalar subdivision schemes, vector subdivision schemes are much less well understood, e.g., Lagrange and (generalized) Hermite subdivision schemes are the only studied vector subdivision schemes in the literature. Because many wavelets used in numerical PDEs are derived from refinable vector functions whose matrix masks are not from Hermite subdivision schemes, it is necessary to introduce and study vector subdivision schemes for any general matrix masks in order to compute wavelets and refinable vector functions efficiently. For a general matrix mask, we show that there is only one meaningful way of defining a vector subdivision scheme. Motivated by vector cascade algorithms and recent study on Hermite subdivision schemes, we shall define a vector subdivision scheme for any arbitrary matrix mask and then we prove that the convergence of the newly defined vector subdivision scheme is equivalent to the convergence of its associated vector cascade algorithm. We also study convergence rates of vector subdivision schemes. The results of this paper not only bridge the gaps and establish intrinsic links between vector subdivision schemes and vector cascade algorithms but also strengthen and generalize current known results on Lagrange and (generalized) Hermite subdivision schemes. Several examples are provided to illustrate the results in this paper on various types of vector subdivision schemes with convergence rates

Combinatorial Properties of Multivariate Subdivision Scheme with Nonnegative Masks

Unterteilungsalgorithmen liefern wichtige Techniken zur schnellen Erzeugung von Kurven und Oberflächen. Diese spielen auch eine zentrale Rolle in Wavelets. Ein Unterteilungsalgorithmus ist durch eine Maske definiert. Es ist bekannt, dass die Konvergenz dieser Algorithmen per gemeinsamen Spektralradius charakterisiert werden kann, der durch endlich viele Matrizen definiert ist. Allerdings ist die Berechnung des gemeinsamen Spektralradius im allgemeinen sehr schwierig. Unser Ziel ist es im multivariaten Fall einfach zu überprüufende Kriterien zu finden, die hinreichend und notwendig für die Konvergenz dieser Algorithmen sind. Die Einfachheit der Kriterien bedeutet, dass sich die Kriterien in polynomialer Zeit bzgl. der Masken, z.B. die Größe des Trägers von Masken, nachprüfen lassen. Nach einem einleitenden Kapitel 1 und einem grundlegenden Kapitel 2 konzentrieren wir uns daher in drei Schritten auf die Klasse der multivariaten Subdivisions-Schemata mit nichtnegativen Masken. Die Dissertation ist folgendermaßen aufgebaut: Wir beginnen zunächst in Kapitel 3 und 4 mit einer Demonstration des Zusammenhangs zwischen der Konvergenz des Subdivisions-Schemas und einiger Abbildungen für Gitter. Danach geben wir ein neues hinreichendes und notwendiges Konvergenzkriterium für nichtnegative Subdivisions-Schemata an. Theorem 3.3.1 stellt den zentralen Beitrag dieses Kapitels dar. Darauffolgend betrachten wir in Kapitel 5 und 6, dass die Konvergenz eines nichtnegativen Subdivisions-Schemas nicht von den Werten der Maske abhängt, sondern lediglich von ihrem Träger. Wir geben die unterschiedlichen Eigenschaften zwischen inneren Punkten und Randpunkten auf ihrem Träger mit Hilfe der weiterer notwendiger Konvergenzbedingung an. Dabei stellt sich heraus, dass der Zusammenhang der Matrix A eine einfache und adäquate Bedingung ist, um diese Eigenschaften zu garantieren. Im letzten Kapitel leiten wir nun einfach und schnell zu berechnende hinreichende Konvergenzbedingungen für multivariate Subdivisions-Schemata mit nichtnegativer Maske her, sofern der Träger spezielle Eigenschaften besitzt. Dabei nutzen wir obige Resultate

Extended Hermite Subdivision Schemes

International audienceSubdivision schemes are efficient tools for building curves and surfaces. For vector subdivision schemes, it is not so straightforward to prove more than the Hölder regularity of the limit function. On the other hand, Hermite subdivision schemes produce function vectors that consist of derivatives of a certain function, so that the notion of convergence automatically includes regularity of the limit. In this paper, we establish an equivalence betweena spectral condition and operator factorizations, then we study how such schemes with smooth limit functions can be extended into ones with higher regularity. We conclude by pointing out this new approach applied to cardinal splines

Optimising Spatial and Tonal Data for PDE-based Inpainting

Some recent methods for lossy signal and image compression store only a few selected pixels and fill in the missing structures by inpainting with a partial differential equation (PDE). Suitable operators include the Laplacian, the biharmonic operator, and edge-enhancing anisotropic diffusion (EED). The quality of such approaches depends substantially on the selection of the data that is kept. Optimising this data in the domain and codomain gives rise to challenging mathematical problems that shall be addressed in our work. In the 1D case, we prove results that provide insights into the difficulty of this problem, and we give evidence that a splitting into spatial and tonal (i.e. function value) optimisation does hardly deteriorate the results. In the 2D setting, we present generic algorithms that achieve a high reconstruction quality even if the specified data is very sparse. To optimise the spatial data, we use a probabilistic sparsification, followed by a nonlocal pixel exchange that avoids getting trapped in bad local optima. After this spatial optimisation we perform a tonal optimisation that modifies the function values in order to reduce the global reconstruction error. For homogeneous diffusion inpainting, this comes down to a least squares problem for which we prove that it has a unique solution. We demonstrate that it can be found efficiently with a gradient descent approach that is accelerated with fast explicit diffusion (FED) cycles. Our framework allows to specify the desired density of the inpainting mask a priori. Moreover, is more generic than other data optimisation approaches for the sparse inpainting problem, since it can also be extended to nonlinear inpainting operators such as EED. This is exploited to achieve reconstructions with state-of-the-art quality. We also give an extensive literature survey on PDE-based image compression methods