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23
Wavelets on Irregular Point Sets
 Phil. Trans. R. Soc. Lond. A
, 1999
"... this article we review techniques for building and analyzing wavelets on irregular point sets in one and two dimensions. We discuss current results both on the practical and theoretical side. In particular we focus on subdivision schemes and commutation rules. Several examples are included. ..."
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Cited by 49 (0 self)
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this article we review techniques for building and analyzing wavelets on irregular point sets in one and two dimensions. We discuss current results both on the practical and theoretical side. In particular we focus on subdivision schemes and commutation rules. Several examples are included.
A Multiresolution Framework for Variational Subdivision
, 1998
"... Subdivision is a powerful paradigm for the generation of curves and surfaces. It is easy to implement, computationally efficient, and useful in a variety of applications because of its intimate connection with multiresolution analysis. An important task in computer graphics and geometric modeling is ..."
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Cited by 45 (0 self)
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Subdivision is a powerful paradigm for the generation of curves and surfaces. It is easy to implement, computationally efficient, and useful in a variety of applications because of its intimate connection with multiresolution analysis. An important task in computer graphics and geometric modeling is the construction of curves that interpolate a given set of points and minimize a fairness functional (variational design). In the context of subdivision, fairing leads to special schemes requiring the solution of a banded linear system at every subdivision step. We present several examples of such schemes including one that reproduces nonuniform interpolating cubic splines. Expressing the construction in terms of certain elementary operations we are able to embed variational subdivision in the lifting framework, a powerful technique to construct wavelet filter banks given a subdivision scheme. This allows us to extend the traditional lifting scheme for FIR filters to a certain class of IIR filters. Consequently we show how to build variationally optimal curves and associated, stable wavelets in a straightforward fashion. The algorithms to perform the corresponding decomposition and reconstruction transformations are easy to implement and efficient enough for interactive applications.
Regularity Of Irregular Subdivision
, 1998
"... . We study the smoothness of the limit function for one dimensional unequally spaced interpolating subdivision schemes. The new grid points introduced at every level can lie in irregularly spaced locations between old, adjacent grid points and not only midway as is usually the case. For the natural ..."
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Cited by 34 (5 self)
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. We study the smoothness of the limit function for one dimensional unequally spaced interpolating subdivision schemes. The new grid points introduced at every level can lie in irregularly spaced locations between old, adjacent grid points and not only midway as is usually the case. For the natural generalization of the four point scheme introduced by Dubuc and Dyn, Levin, and Gregory, we show that, under some geometric restrictions, the limit function is always C 1 ; under slightly stronger restrictions we show that the limit function is almost C 2 , the same regularity as in the regularly spaced case. 1. Introduction Subdivision is a powerful mechanism for the construction of smooth curves and surfaces. The main idea behind subdivision is to iterate upsampling and local averaging to build complex geometrical shapes. Originally such schemes were studied in the context of corner cutting [13, 5] as well as for building piecewise polynomial curves, e.g., the de Casteljau algorithm f...
Analysis of quasiuniform subdivision
, 2003
"... We study the smoothness of quasiuniform bivariate subdivision. A quasiuniform bivariate scheme consists of different uniform rules on each side of the yaxis, far enough from the axis, some different rules near the yaxis, and is uniform in the ydirection. For schemes that generate polynomials up ..."
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Cited by 19 (1 self)
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We study the smoothness of quasiuniform bivariate subdivision. A quasiuniform bivariate scheme consists of different uniform rules on each side of the yaxis, far enough from the axis, some different rules near the yaxis, and is uniform in the ydirection. For schemes that generate polynomials up to degree m, we derive a sufficient condition for C m continuity of the limit function, which is simple enough to be used in practice. It amounts to showing that the joint spectral radius of a certain pair of matrices has to be less than 2 −m. We also relate the Hölder exponent of the mth order derivatives to that joint spectral radius. The main tool is an extension of existing analysis techniques for uniform subdivision schemes, although a different proof is required for the quasiuniform case. The same idea is also applicable to the analysis of quasiuniform subdivision processes in higher dimension. Along with the analysis we present a ‘tri–quad ’ scheme, which is combined of a scheme on a triangular grid on the half plane x<0 and a scheme on a square grid on the other half plane x>0 and special rules near the yaxis. Using the new analysis tools it is shown that the tri–quad scheme is globally C².
Piecewise uniform subdivision schemes
 Mathematical Methods for Curves and Surfaces
, 1995
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A piecewise polynomial approach to analyzing interpolatory subdivision
, 2010
"... The fourpoint interpolatory subdivision scheme of Dubuc and its generalizations to irregularly spaced data studied by Warren and by Daubechies, Guskov, and Sweldens are based on fitting cubic polynomials locally. In this paper we analyze the convergence of the scheme by viewing the limit function a ..."
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Cited by 5 (5 self)
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The fourpoint interpolatory subdivision scheme of Dubuc and its generalizations to irregularly spaced data studied by Warren and by Daubechies, Guskov, and Sweldens are based on fitting cubic polynomials locally. In this paper we analyze the convergence of the scheme by viewing the limit function as the limit of piecewise cubic functions arising from the scheme. This allows us to recover the regularity results of Daubechies et al. in a simpler way and to obtain the approximation order of the scheme and its first derivative.
Nonuniform BSpline subdivision using refine and smooth
 In IMA Conference on the Mathematics of Surfaces
, 2007
"... Abstract. Subdivision surfaces would be useful in a greater number of applications if an arbitrarydegree, nonuniform scheme existed that was a generalisation of NURBS. As a step towards building such a scheme, we investigate nonuniform analogues of the LaneRiesenfeld ‘refine and smooth ’ subdivi ..."
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Cited by 4 (1 self)
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Abstract. Subdivision surfaces would be useful in a greater number of applications if an arbitrarydegree, nonuniform scheme existed that was a generalisation of NURBS. As a step towards building such a scheme, we investigate nonuniform analogues of the LaneRiesenfeld ‘refine and smooth ’ subdivision paradigm. We show that the assumptions made in constructing such an analogue are critical, and conclude that Schaefer’s global knot insertion algorithm is the most promising route for further investigation in this area. 1
Regular algebraic curve segments (III)—Applications
 in data fitting, Computer Aided Geom. Design
, 2000
"... In this paper (part three of the trilogy) we use low degree G 1 and G 2 continuous regular algebraic spline curves de ned within parallelograms, to interpolate an ordered set of data points in the plane. We explicitly characterize curve families whose members have the required interpolating properti ..."
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Cited by 4 (2 self)
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In this paper (part three of the trilogy) we use low degree G 1 and G 2 continuous regular algebraic spline curves de ned within parallelograms, to interpolate an ordered set of data points in the plane. We explicitly characterize curve families whose members have the required interpolating properties and possess a minimal number of in ection points. The regular algebraic spline curves considered here have many attractive features: They are easy to construct. There exist convenient geometric control handles to locally modify the shape of the curve. The error of the approximation is controllable. Since the spline curve isalways inside the parallelogram, the error of the t is bounded by the size of the parallelogram. The spline curve can be rapidly displayed, even though the algebraic curve segments are implicitly de ned. Key words: Algebraic curve; tensor product; polygonal chain, parallelogram. 1