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Optimal Wiener Filtering for fMRI Images with Polyharmonic Smoothing Splines

S. Tirosh, D. Van De Ville, M. Unser

Proceedings of the 2004 Annual Meeting of the Swiss Society of Biomedical Engineering (SSBE'04), Zürich ZH, Swiss Confederation, September 2-3, 2004, poster no. 5.

Motivated by the fractal-like behavior of fMRI images [1] (and other images as well [2]), we propose a smoothing technique which uses a regularization functional that is a fractional iterate of the Laplacian.

This type of functional was introduced by Duchon in the context of radial basis functions (RBFs). We solve it using non-separable fractional polyharmonic B-splines [3].

We show a way of choosing the order of differentiation s, and prove that our algorithm is equivalent to the optimal discretization of the continuous-time Wiener filter for fractal-like signals (with a O(|ω|-s) spectral decay).


  1. E. Zarahn, G.K. Aguirre, M. D′Esposito, "Empirical Analyses of BOLD fMRI Statistics," Neuroimage, vol. 5, no. 3, pp. 179-197, April 1997.

  2. A.P. Pentland, "Fractal-Based Description of Natural Scenes," IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 6, no. 6, pp. 661-674, November 1984.

  3. C. Rabut, "Elementary m-Harmonic Cardinal B-Splines," Numerical Algorithms, vol. 2, no. 1, pp. 39-62, February 1992.

AUTHOR="Tirosh, S. and Van De Ville, D. and Unser, M.",
TITLE="Optimal {W}iener Filtering for {fMRI} Images with Polyharmonic
        Smoothing Splines",
BOOKTITLE="2004 Annual Meeting of the Swiss Society of Biomedical
        Engineering ({SSBE'04})",
address="Z{\"{u}}rich ZH, Swiss Confederation",
month="September 2-3,",
note="Poster no.\ 5")

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