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Self-Similarity: From Fractals to Splines

M. Unser

Invited talk, proceedings of the International Conference on Algorithms for Approximation V (A4A5'05), Chester, United Kingdom, July 18-22, 2005, pp. 13.



In this talk, we will show how the concept of self-similarity can be used as a bridge for connecting splines and fractals. Our starting point is the identification of the class of differential operators L that are both shift- and scale-invariant. This results in a family of generalized fractional derivatives indexed by two parameters. We specify the corresponding L-splines, which yields an extended class of fractional splines. The operator L also defines an energy measure, which can be used as an regularization functional for fitting the noisy samples of a signal. We show that, when the grid is uniform, the corresponding smoothing spline estimator is a cardinal fractional spline that can be computed efficiently by means of an FFT-based filtering algorithm. Using Gelfand's theory of generalized stochastic processes, we then prove that the above fractional derivatives act as the whitening operators of a class of self-similar processes that includes fractional Brownian motion. Thanks to this result, we show that the fractional smoothing spline algorithm can be used to obtain the minimum mean square error (MMSE) estimation of a self-similar process at any location, given a series of noisy measurements at the integers. This proves that the fractional splines are the optimal function spaces for estimating fractal-like processes; it also provides the optimal regularization parameters.

This is joint work with Thierry Blu.


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