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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 non-stationary stochastic processes that are statistically self-similar (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 (FFT-based) 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.


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


[1] M. Unser, T. Blu, "Fractional Splines and Wavelets," SIAM Review, vol. 42, no. 1, pp. 43-67, March 2000.

[2] M. Unser, T. Blu, "Self-Similarity: Part I—Splines and Operators," IEEE Transactions on Signal Processing, in press.

[3] T. Blu, M. Unser, "Self-Similarity: Part II—Optimal Estimation of Fractal Processes," IEEE Transactions on Signal Processing, in press.

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