Self-Similar Vector Fields

Abstract

We propose statistically self-similar and rotation-invariant models for vector fields, study some of the more significant properties of these models, and suggest algorithms and methods for reconstructing vector fields from numerical observations, using the same notions of self-similarity and invariance that give rise to our stochastic models. We illustrate the efficacy of the proposed schemes by applying them to the problems of denoising synthetic flow phantoms and enhancing flow-sensitive magnetic resonance imaging (MRI) of blood flow in the aorta. In constructing our models and devising our applied schemes and algorithms, we rely on two fundamental notions. The first of these, referred to as "innovation modelling" in the thesis, is the principle —applicable both analytically and synthetically— of reducing complex phenomena to combinations of simple independent components or "innovations". The second fundamental idea is that of "invariance", which indicates that in the absence of any distinguishing factor, two equally valid models or solutions should be given equal consideration