Summary

In this project, we follow two main objectives. The first is to provide statistical models of scalar and vector physical quantities that are hoped to be useful in a wide range of applications, including medical imaging. Our second objective is to develop mathematical and algorithmic tools appropriate for the analysis of such models and the processing of vectorial medical data.

Introduction

Stochastic fractals—random entities that exhibit special forms of invariance to coordinate transformations, including stochastic self-similarity—have been noted to provide interesting statistical models for a wide range of natural and man-made phenomena. We focus on a special class of such random entities known as fractional Brownian motions (fBm), and propose to extend this family in several important ways, in particular, by introducing a vector extension that allows a parametric tuning of the model to match the specific directional behaviour of the phenomenon under consideration.
In order to facilitate the study of such models and data they are meant to represent, new mathematical and algorithmic tools are proposed. These include a framework for the analysis of fractal processes using polyharmonic wavelets and vector splines and wavelets for the study of vector fractals.
The particular emphasis on vector models and signals is due to the fact that an increasing number of medical-imaging modalities now provide us with measurements of dynamic quantites and motion, the study of which can advance our understanding of physiological processes, as well as lead to better diagnostics and treatments, for instance in the context of cardiovascular diseases.

Main Contribution

During the course of this project, a rigorous and extendable mathematical characterization of scalar fractional Brownian random fields was conducted. An interesting link was established between this formulation and polyharmonic splines, posing the latter as the deterministic counterparts of these stochastic fractals in a precise sense.
We have also been working on a vector extension of fractional Brownian motion. This extension is formulated in a way that makes explicit its dependence on parameters that govern the directional comportment of the model. This quality can, in principle, allow one to take into account the specific physical properties of the data (such as those associated with incompressible fluid flow), resulting in new or improved schemes.
Finally, we have been investigating vector splines and wavelets suited to general vectorial signal processing and the study of our statistical models. Research on this subject is still ongoing and is hoped to lead to effective algorithms for vectorial data analysis.