
Fractional Splines and Fractals
A MATLAB package is made available for computing the fractional smoothing spline estimator of a 1D signal, and for generating fBms (fractional Brownian motion).
Fractional Brownian motion processes, which were introduced by Mandelbrot in 1968, are nonstationary stochastic processes that are statistically selfsimilar (fractals). We have shown recently that fractional splines and fBms are intimately connected: they can be defined as solution of the same type of (fractional) differential equation. In the first case, the excitation is a deterministic stream of Dirac impulses (discrete signal), while in the second, it is white Gaussian noise. Thanks to this connection, we could prove that a smoothing spline estimator with an appropriate set of parameters (spline order or degree and regularization factor) produces the minimum mean squares error reconstruction of a fBm signal corrupted by additive noise.
The package provides a very efficient (FFTbased) implementation of this estimation procedure. It also includes a benchmark/demo for testing its optimality: a fBm signal is generated, corrupted by additive noise, and reconstructed from the noisy measurements with an oversampling factor of m.

Description
The available functions are:
 fsdemo.m: Demo/benchmark programme, described above.
 smoothspline.m: Function that returns samples of the smoothing spline for a given input sequence, regularization factor (lambda), spline order (gamma), and oversampling factor (m). See MATLAB help for details.
 fBmper.m: Generator of fractional (pseudo)Brownian motion.
 fractsplineautocorr.m: Function for frequency domain computation of fractional spline autocorrelation.

MATLAB Download

References
[1] M. Unser, T. Blu, "Fractional Splines and Wavelets," SIAM Review, vol. 42, no. 1, pp. 4367, March 2000.
[2] M. Unser, T. Blu, "SelfSimilarity: Part I—Splines and Operators," IEEE Transactions on Signal Processing, in press.
[3] T. Blu, M. Unser, "SelfSimilarity: Part II—Optimal Estimation of Fractal Processes," IEEE Transactions on Signal Processing, in press.

