The B-spline has long been important in computer graphics for representing curves and surfaces, but it was recently realised that the recursive subdivision method of construction could be used to formulate B-splines in wavelet terms; this led to excellent new applications (see Stollnitz et al., 1996) in which curves and surfaces could be analysed or modified at any chosen scale, from local to global. In addition to proving results, we provide some exemplification of these things.

By introducing an equivalent definition of B-splines as an m-fold convolution of boxes we bring out an intimate connection with the Fourier transform. This in turn provides an alternative derivation of such formulae as the Cox–de Boor relations.

The last two sections are designated as appendices, optional follow-up to the main treatment. In Section 17.4 we derive wavelet-identifying formulae that hold for arbitrary size of control polygon, whilst Section 17.5 addresses mathematical aspects of the natural generalisation from curve to surface wavelets achievable by the subdivision system of Loop (1987). Multiresolution and editing examples are exhibited.

Splines from boxes

We begin with Bézier splines, to give background and to introduce some of the ideas behind splines, including that of convexity in Section 17.1.2. Moving on to the even more useful B-splines in Section 17.1.3, we present their definition as a convolution of box functions, excellent for the coming wavelet formulation.