**Overview**
This demonstration shows an implementation of a denoising
algorithm for fractal-like images using polyharmonic B-splines. The
algorithm is described in [1] [2].
Motivated by the fractal-like behavior of natural images, we
present 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.
Based on the signal model, we chose the order of the basis
functions, thus achieving a suitable tool for fractal-like signals. Using
statistical analysis, it can be proved that our algorithm is equivalent to
the optimal discretization of the continuous-time Wiener filter for
fractal-like signals, which is the best possible linear technique. We also
obtain the best regularization parameter, achieving a completely automatic
smoothing process.
**Contact**
Implemented as a semester project by
Alex Prudencio Arispe.
**References**
[1] S. Tirosh, D. Van De Ville, M. Unser, "
Polyharmonic Smoothing Splines for Multi-Dimensional Signals with 1 / ||w|| t -Like Spectra ,
" 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.
[2] S. Tirosh, D. Van De Ville, M. Unser, " Polyharmonic
Smoothing Splines and the Multi-Dimensional Wiener Filtering of
Fractal-like Signals," submitted, 2005. |