Exponential Splines and Pseudo-Splines: Generation versus reproduction of exponential polynomials

Subdivision schemes are iterative methods for the design of smooth curves and surfaces. Any linear subdivision scheme can be identified by a sequence of Laurent polynomials, also called subdivision symbols, which describe the linear rules determining successive refinements of coarse initial meshes. One important property of subdivision schemes is their capability of exactly reproducing in the limit specific types of functions from which the data is sampled. Indeed, this property is linked to the approximation order of the scheme and to its regularity. When the capability of reproducing polynomials is required, it is possible to define a family of subdivision schemes that allows to meet various demands for balancing approximation order, regularity and support size. The members of this family are known in the literature with the name of pseudo-splines. In case reproduction of exponential polynomials instead of polynomials is requested, the resulting family turns out to be the non-stationary counterpart of the one of pseudo-splines, that we here call the family of exponential pseudo-splines. The goal of this work is to derive the explicit expressions of the subdivision symbols of exponential pseudo-splines and to study their symmetry properties as well as their convergence and regularity.Comment: 25 page

Analysis of uniform binary subdivision schemes for curve design

The paper analyses the convergence of sequences of control polygons produced by a binary subdivision scheme of the form .0,1,2,...kz,ikj,ifjbm0j1k12ifjam0j1k2if=∈+Σ==++Σ==+ The convergence of the control polygons to a Cu curve is analysed in terms of the convergence to zero of a derived scheme for the differences - . The analysis of the smoothness of the limit curve is reduced to kif the convergence analysis of "differentiated" schemes which correspond to divided differences of {/i ∈Z} with respect to the diadic parameteriz- kif ation = i/2kitk . The inverse process of "integration" provides schemes with limit curves having additional orders of smoothness

Recursive subdivision algorithms for curve and surface design

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.In this thesis, the author studies recursIve subdivision algorithms for curves and surfaces. Several subdivision algorithms are constructed and investigated. Some graphic examples are also presented. Inspired by the Chaikin's algorithm and the Catmull-Clark's algorithm, some non-uniform schemes, the non-uniform corner cutting scheme and the recursive subdivision algorithm for non-uniform B-spline curves, are constructed and analysed. The adapted parametrization is introduced to analyse these non-uniform algorithms. In order to solve the surface interpolation problem, the Dyn-Gregory-Levin's 4-point interpolatory scheme is generalized to surfaces and the 10-point interpolatory subdivision scheme for surfaces is formulated. The so-called Butterfly Scheme, which was firstly introduced by Dyn, Gregory Levin in 1988, is just a special case of the scheme. By studying the Cross-Differences of Directional Divided Differences, a matrix approach for analysing uniform subdivision algorithms for surfaces is established and the convergence of the 10-point scheme over both uniform and non-uniform triangular networks is studied. Another algorithm, the subdivision algorithm for uniform bi-quartic B-spline surfaces over arbitrary topology is introduced and investigated. This algorithm is a generalization of Doo-Sabin's and Catmull-Clark's algorithms. It produces uniform Bi-quartic B-spline patches over uniform data. By studying the local subdivision matrix, which is a circulant, the tangent plane and curvature properties of the limit surfaces at the so-called Extraordinary Points are studied in detail.The Chinese Educational Commission and The British Council (SBFSS/1987

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

A real algebra perspective on multivariate tight wavelet frames

Recent results from real algebraic geometry and the theory of polynomial optimization are related in a new framework to the existence question of multivariate tight wavelet frames whose generators have at least one vanishing moment. Namely, several equivalent formulations of the so-called Unitary Extension Principle (UEP) from [33] are interpreted in terms of hermitian sums of squares of certain nongenative trigonometric polynomials and in terms of semi-definite programming. The latter together with the results in [31, 35] answer affirmatively the long stading open question of the existence of such tight wavelet frames in dimesion d = 2; we also provide numerically efficient methods for checking their existence and actual construction in any dimension. We exhibit a class of counterexamples in dimension d = 3 showing that, in general, the UEP property is not sufficient for the existence of tight wavelet frames. On the other hand we provide stronger sufficient conditions for the existence of tight wavelet frames in dimension d ≥ 3 and illustrate our results by several examples

Annihilation operators for exponential spaces in subdivision

none3siWe investigate properties of differential and difference operators annihilating certain finite-dimensional spaces of exponential functions in two variables that are connected to the representation of real-valued trigonometric and hyperbolic functions. Although exponential functions appear in a variety of contexts, the motivation behind this technical note comes from considering subdivision schemes where annihilation operators play an important role. Indeed, subdivision schemes with the capability of preserving exponential functions can be used to obtain an exact description of surfaces parametrized in terms of trigonometric and hyperbolic functions, and annihilation operators are useful to automatically detect the frequencies of such functions.mixedConti C.; Lopez-Urena S.; Romani L.Conti C.; Lopez-Urena S.; Romani L