#### 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].