In this thesis, we propose a theoretical as well as practical framework to combine geometric prior information to a statistical/probabilitstic methodology in the investigation of a denoising problem in its generic form together with its various applications in signal/image analysis.
We are able in the process, to investigate, understand and mitigate existing limitations of so-called nonlinear diffusion techniques (such as the Perona-Malik equation) from a probabilistic view point, and propose a new nonlinear denoising method that is based on a random walk whose transition probabilities are selected by the information of a two-sided gradient. This results in a piecewise constant filtered image and lifts the long-standing problem of an unknown evolution stopping time.
Our second contribution is in establishing a direct link between multi-resolution analysis techniques and so-called scale space analysis methods, which we in turn utilize to improve the performance of segmentation-optimized image analysis techniques. This is accomplished by using wavelets of higher order vanishing moments, specifically, we achieve a reduction in the typical "blocky" artifacts and a better preservation of texture information.
Our third and final contribution is to propose a drastically different approach by isolating statistically independent components in a signal, which we later use as a basis for discrimination against noise, or potentially as plain features. This is related to the well known independent component analysis ( ICA ), for which we first propose Jensen-Rényi divergence as an information- theoretic criterion. In addition, we propose a Rényi mutual divergence as a better criterion to separate mixed signals along with a non-parametric estimation technique for such a measure for 1-D problems.
A particle system simulation method is on our future plan of work and is currently ongoing to further investigate the stochastic properties of our diffusion framework.
