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Fractal Models and Spline-Based Processing of Vector Signals

Investigators: Pouya Dehghani Tafti

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. As our second objective, we aim at developing spline-based mathematical tools and algorithms for the analysis and processing of vectorial medical datasets.

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's), 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 experimental 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 quantities and motion, the study of which can advance our understanding of physiological processes, as well as lead to better diagnostics and treatment, for instance in the context of cardiovascular diseases.

Main Contributions

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 account of 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.

Collaborations: Prof. Michael Unser

Period: 2007-ongoing

Funding: Swiss National Science Foundation

Major Publications

[1] 

P.D. Tafti, D. Van De Ville, M. Unser, "Invariances, Laplacian-Like Wavelet Bases, and the Whitening of Fractal Processes," IEEE Transactions on Image Processing, in press.

[2] 

P.D. Tafti, D. Van De Ville, M. Unser, "Innovation Modelling and Wavelet Analysis of Fractal Processes in Bio-Imaging," Proceedings of the Fifth IEEE International Symposium on Biomedical Imaging: From Nano to Macro (ISBI'08), Paris, France, May 14-17, 2008, pp. 1501-1504.

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