Ten Good Reasons for Using Spline Wavelets

The purpose of this note is to highlight some of the unique properties of spline wavelets. These wavelets can be classified in four categories: othogonal (Battle-Lemarié), semi-orthogonal (e.g., B-spline), shift-orthogonal, and biorthogonal (Cohen-DaubechiesFeauveau). Unlike most other wavelet bases, splines have explicit formulae in both the time and frequency domain, which greatly facilitates their manipulation. They allow for a progressive transition between the two extreme cases of a multiresolution: Haar's piecewise constant representation (spline of degree zero) versus Shannon's bandlimited model (which corresponds to a spline of infinite order). Spline wavelets are extremely regular and usually symmetric or anti-symmetric. They can be designed to have compact support and to achieve optimal time-frequency localization (B-spline wavelets). The underlying scaling functions are the B-splines, which are the shortest and most regular scaling functions of order L. Finally, splines have the best approximation properties among all known wavelets of a given order L. In other words, they are the best for approximating smooth functions.