Construction of fractional spline wavelet bases
Michael A. Unser, Thierry Blu · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 1999
We extend Schoenberg's B-splines to all fractional degrees (alpha) > -1/2. These splines are constructed using linear combinations of the integer shifts of the power functions x(alpha ) (one-sided) or x(alpha )* (symmetric); in each case, they are (alpha) -Hoelder continuous for (alpha) > 0. They satisfy most of the properties of the traditional B-splines; in particular, the Biesz basis condition and the two-scale relation, which makes them suitable for the construction of new families of wavelet bases. What is especially interesting from a wavelet perspective is that the fractional B-splines have a fractional order of approximately ((alpha) + 1), while they reproduce the polynomials of degree [(alpha) ]. We show how they yield continuous-order generalization of the orthogonal Battle- Lemarie wavelets and of the semi-orthogonal B-spline wavelets. As (alpha) increases, these latter wavelets tend to be optimally localized in time and frequency in the sense specified by the uncertainty principle. The corresponding analysis wavelets also behave like fractional differentiators; they may therefore be used to whiten fractional Brownian motion processes.