A PhD thesis @ NALAG

TW 2000_1

M. Jansen
Wavelet tresholding and noise reduction

Advisor: Adhemar Bultheel

Abstract

A wavelet transform decomposes data into a sparse, multiscale representation. This dissertation uses both features in wavelet based noise reduction algorithms.

Sparsity is the key to wavelet thresholding: coefficients with magnitude below a threshold are replaced by zero. After an introduction to wavelets and their applications, this text discusses the minimum risk threshold. This threshold minimizes the expected mean square error of the output. This error cannot be computed exactly when the uncorrupted data are unknown. We present procedure based on generalized cross validation (GCV) to estimate the optimal threshold. An asymptotic argument motivates this estimation method. To this end, we first study the asymptotic behavior of the minimum risk threshold. We compare these minimum risk and GCV thresholds with the well known universal threshold.

The multiresolution character of a wavelet decomposition allows for refinements of the general threshold scheme. Tree structured thresholding reduces false structures in the output. Scale dependent thresholds are necessary to deal with correlated noise or non-orthogonal wavelet transforms. The synthesis from a non-decimated wavelet transform has an additional smoothing effect. We also investigate noise reduction in the framework of integer wavelet transform.

The next part concentrates on images. An approximation theoretic argument learns that wavelets might not be the ultimate basis functions for image processing. Moreover, selecting the coefficients with large magnitude is a local approach. A Bayesian procedure could lead to a more structured coefficient selection, which better preserves edges in the output. The geometric prior model favors clusters of important coefficients.

The last part investigates the applicability of threshold algorithms for non-equidistant data and second generation wavelets. Experimental results indicate that instability of the wavelet transform hinder the classification of the coefficients according to their importance. We propose an algorithm to overcome this difficulty.