Book contents
- Frontmatter
- Contents
- List of contributors
- Preface
- 1 Characterization and construction of radial basis functions
- 2 Approximation and interpolation with radial functions
- 3 Representing and analyzing scattered data on spheres
- 4 A survey on L2-approximation orders from shift-invariant spaces
- 5 Introduction to shift-invariant spaces. Linear independence
- 6 Theory and algorithms for nonuniform spline wavelets
- 7 Applied and computational aspects of nonlinear wavelet approximation
- 8 Subdivision, multiresolution and the construction of scalable algorithms in computer graphics
- 9 Mathematical methods in reverse engineering
- Index
6 - Theory and algorithms for nonuniform spline wavelets
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- List of contributors
- Preface
- 1 Characterization and construction of radial basis functions
- 2 Approximation and interpolation with radial functions
- 3 Representing and analyzing scattered data on spheres
- 4 A survey on L2-approximation orders from shift-invariant spaces
- 5 Introduction to shift-invariant spaces. Linear independence
- 6 Theory and algorithms for nonuniform spline wavelets
- 7 Applied and computational aspects of nonlinear wavelet approximation
- 8 Subdivision, multiresolution and the construction of scalable algorithms in computer graphics
- 9 Mathematical methods in reverse engineering
- Index
Summary
Abstract
We investigate mutually orthogonal spline wavelet spaces on nonuniform partitions of a bounded interval, addressing the existence, uniqueness and construction of bases of minimally supported spline wavelets. The relevant algorithms for decomposition and reconstruction are considered as well as some stability-related questions. In addition, we briefly review the bivariate case for tensor products and arbitrary triangulations. We conclude this chapter with a discussion of some special cases.
Introduction
Splines have become the standard mathematical tool for representing smooth shapes in computer graphics and geometric modeling. Wavelets have been introduced more recently, but are by now well established both in mathematics and in applied sciences like signal processing and numerical analysis. The two concepts are closely related as splines provide some of the most important examples of wavelets. Although there is an extensive literature on cardinal spline wavelets (spline wavelets with uniform knot spacing), see Chui (1992), relatively little has been published about spline wavelets on arbitrary, nonuniform knots, which form the subject of this chapter. These kinds of wavelets, however, are needed for performing operations like decomposition, reconstruction and thresholding on splines given on a nonuniform knot vector, which typically occur in practical applications.
The flexibility of splines in modeling is due to good approximation properties, useful geometric interpretations of the B-spline coefficients, and simple algorithms for adding and removing knots. Full advantage of these capabilities can only be taken on general nonuniform knot vectors, where also multiple knots are allowed.
- Type
- Chapter
- Information
- Multivariate Approximation and Applications , pp. 152 - 187Publisher: Cambridge University PressPrint publication year: 2001
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