Signal processing: denoising, statistical estimators and wavelets.

This research was carried out in collaboration with Dr. Umberto Amato while visiting Istituto per Applicazioni della Matematica of Naples, Italy.

The research was concerned with the problem of the wavelet transform and its applications to one-dimensional data (signals) and two-dimensional data (images). One of the considered subtopics was data denoising by means of Tikhonov regularization of the wavelet coefficients. Compression capabilities of wavelets reveal useful not only in picking the significant part of information contained in a signal, but also in suppressing the noise affecting the signal, without degrading it too much. A regularization technique has been devised, which, together with the Generalized Cross Validation criterion (GCVC) for choosing the proper amount of smoothing, gives rise to a very simple completely objective method for smoothing data. Comparisons, both theoretical and experimental, were made with the denoising techniques, developed by Donoho and Johnstone in a statistical framework, on some test problems from the literature and on a true problem coming from interferometry.
The effectiveness of wavelet regularization as a denoising device is mainly due to the fact that the true signal is compressed by the wavelet transform in a "few" coefficients, so that most of the coefficients in the wavelet expansion of the noisy signal are due to the noise contribution. When the signal is periodic, the mentioned effect is achieved by using the periodic wavelet transform. For non periodic signals however, such a compression is no longer obtained due to the boundary discontinuities so that the regularized solution is affected by artificially induced oscillations. To avoid this difficulty a wavelet transform for non periodic functions on an interval has been used, based on a newly developed method by Cohen, Daubechies, Jawerth and Vial. Numerical experiments on one-dimensional and two-dimensional data confirmed the removal of boundary effects.
A modified variant of GCVC for the case of correlated noise affecting the data has been proposed.
The regularization method has been generalized to Besov spaces and to more general regularizing functionals. It has been proved in particular that other techniques for smoothing data (such as various statistical estimators) can be viewed as solutions of regularization problems.

References from list of works related to this topic: papers [56] - [60], [62] - [65].