The Fractional Spline Wavelet Transform: Definition and Implementation

We define a new wavelet transform that is based on a recently defined family of scaling functions: the fractional B-splines. The interest of this family is that they interpolate between the integer degrees of polynomial B-splines and that they allow a fractional order of approximation. The orthogonal fractional spline wavelets essentially behave as a fractional differentiators. This property seems promising for the analysis of $1/f ^{ \alpha }$ ; noise that can be whitened by an appropriate choice of the degree of the spline transform. We present a practical FFT-based algorithm for the implementation of these fractional wavelet transforms, and give some examples of processing.