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# Polyharmonic Smoothing Splines for Multi-Dimensional Signals with 1 ⁄ ||ω||τ-Like Spectra

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

### Proceedings of the Twenty-Ninth IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP'04), Montréal QC, Canada, May 17-21, 2004, pp. III-297-III-300.

Motivated by the fractal-like behavior of natural images, we propose a new smoothing technique that uses a regularization functional which is a fractional iterate of the Laplacian. This type of functional has previously been introduced by Duchon in the context of radial basis functions (RBFs) for the approximation of non-uniform data. Here, we introduce a new solution to Duchon's smoothing problem in multiple dimensions using non-separable fractional polyharmonic B-splines. The smoothing is performed in the Fourier domain by filtering, thereby making the algorithm fast enough for most multi-dimensional real-time applications.

@INPROCEEDINGS(http://bigwww.epfl.ch/publications/tirosh0401.html,
AUTHOR="Tirosh, S. and Van De Ville, D. and Unser, M.",
TITLE="Polyharmonic Smoothing Splines for Multi-Dimensional Signals with
$1/\|\omega\|^{\tau}$-Like Spectra",
BOOKTITLE="Proceedings of the Twenty-Ninth {IEEE} International
Conference on Acoustics, Speech, and Signal Processing
({ICASSP'04})",
YEAR="2004",
editor="",
volume="{III}",
series="",
pages="297--300",