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Denoising of Fractal-like images using Polyharmonic B-splines Start Demo
Author: Alex Prudencio Arispe
Notice Of Disclaimers: The information, opinions, data, and statements contained herein are not necessarily those of the Biomedical Imaging Group at the Swiss Federal Institute of Technology Lausanne (EPFL) and should not be interpreted, acted on or represented as such. Please contact the author of this algorithm if you have a specific question.
March 2005


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.


Implemented as a semester project by Alex Prudencio Arispe.


[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.

Applet instructions

• Choose an input image from the list.

• Add some white Gaussian noise of standard deviation sigma.

• You can chose to compute the optimal values of parameters lambda and s [1], or to set them.

• If "compute optimal values" is chosen, one can chose the computation method.

• Press "Run" to start the denoising process.

Toolbar in the image display

pointer_on Get the coordinates and value of a pixel.
info_on Get the maximum, minimum and the mean value of the image.
frame Open a new window containing the image.
zoomin_on Zoom out by a factor 2.
(N/A on Netscape)
zoomout_on Zoom in by a factor 2. (N/A on Netscape)
move Move the zoomed part of the image. (N/A on Netscape)



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