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Adaptive Wiener filtering using polyharmonic wavelet packets

Florian Vaussard
Section Microtechnique, EPFL

Semester project
June 2006

Lots of images show a fractal-like behaviour, characterized by a power spectrum density following a decreasing exponential law. This provides a way to easily perform a Wiener filter, which reduces the problem to the estimation of only three parameters (the Hurst exponent H, the sigma0 coefficient, and the noise level sigma). In this semester project, the goal was to perform this filtering locally on the image. Indeed images are more likely to have a fractal-like behaviour locally than globally.

To do this, the wavelet packet transform is used. While the Fourier transform gives only a frequency information, the wavelet packet transform does a frequency / time decomposition allowing the localization of the estimation. The isotropic polyharmonic B-splines wavelets with quincunx subsampling are used because of their isotropic behaviour, more adapted to the isotropic behaviour of fractals.

The result gives a good estimator for the fractal parameters and the noise level. This allows to perform the local Wiener filtering with a small improvement compared to the global Wiener filter. But denoising using the isotropic polyharmonic B-splines is not as efficient as using more classical transformations.

a) Original image; b) Map with Hurst estimations;
c) Noisy image (SNR = -1.59dB); d) Filtered image (SNR = 10.7dB)