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

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

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

Analysis and Processing of Irregularly Distributed Point Clouds

We address critical issues arising in the practical implementation of processing real point cloud data that exhibits irregularities. We develop an adaptive algorithm based on Learning Theory for processing point clouds from a stationary sensor that standard algorithms have difficulty approximating. Moreover, we build the theory of distribution-dependent subdivision schemes targeted at representing curves and surfaces with gaps in the data. The algorithms analyze aggregate quantities of the point cloud over subdomains and predict these quantities at the finer level from the ones at the coarser level

Multigrid methods and stationary subdivisions

Multigridmethods are fast iterative solvers for sparse large ill-conditioned linear systems of equations derived, for instance, via discretization of PDEs in fluid dynamics, electrostatics and continuummechanics problems. Subdivision schemes are simple iterative algorithms for generation of smooth curves and surfaces with applications in 3D computer graphics and animation industry. This thesis presents the first definition and analysis of subdivision based multigrid methods. The main goal is to improve the convergence rate and the computational cost of multigrid taking advantage of the reproduction and regularity properties of underlying subdivision. The analysis focuses on the grid transfer operators appearing at the coarse grid correction step in the multigrid procedure. The convergence of multigrid is expressed in terms of algebraic properties of the trigonometric polynomial associated to the grid transfer operator. We interpreter the coarse-to-fine grid transfer operator as one step of subdivision. We reformulate the algebraic properties ensuring multigrid convergence in terms of regularity and generation properties of subdivision. The theoretical analysis is supported by numerical experiments for both algebraic and geometric multigrid. The numerical tests with the bivariate anisotropic Laplacian ask for subdivision schemes with anisotropic dilation. We construct a family of interpolatory subdivision schemes with such dilation which are optimal in terms of the size of the support versus their polynomial generation properties. The numerical tests confirmthe validity of our theoretical analysis