Fractional Splines, Wavelet Bases, and Applications

The purpose of this presentation is to describe a recent family of basis functions—the fractional B-splines—which appear to be intimately connected to fractional calculus. Among other properties, we show that they are the convolution kernels that link the discrete (finite differences) and continuous (derivatives) fractional differentiation operators. We also provide simple closed forms for the fractional derivatives of these splines. The fractional B-splines satisfy a fundamental two-scale relation. Consequently, they can be used as building blocks for constructing a variety of orthogonal and semi-orthogonal wavelet bases of $L _{ 2 }$ ; these are indexed by a continuous order parameter γ = α + 1, where α is the (fractional) degree of the spline. We show that the corresponding wavelets behave like multiscale differentiation operators of fractional order γ. This is in contrast with classical wavelets whose differentiation order is constrained to be an integer. We also briefly discuss some recent applications in medical and seismic imaging.