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.