Lagrangian analysis of vector and tensor fields: Algorithmic foundations and applications in medical imaging and computational fluid dynamics

Both vector and tensor fields are important mathematical tools used to describe the physics of many phenomena in science and engineering. Effective vector and tensor field visualization techniques are therefore needed to interpret and analyze the corresponding data and achieve new insight into the considered problem. This dissertation is concerned with the extraction of important structural properties from vector and tensor datasets. Specifically, we present a unified approach for the characterization of distinguished manifolds that form the skeleton of vector and tensor fields and play a key role in understanding their properties. This dissertation makes several important contributions in this context. First, we propose a versatile model of geometric structure applicable to vector and tensor fields across a wide range of applications. Next, we present an application of this model to the robust characterization of edges in both symmetric second-order and higher-order tensor fields. Then, we introduce a space-varying time-scale concept that improves upon the structure characterization capabilities of existing methods. In addition, we propose a new model that combines generalized notions of Eulerian and Lagrangian perspectives for the structural analysis of multi-field datasets. Overall, we present a set of methods that apply a generalized notion of Lagrangian description, originally developed in fluid dynamics, to the analysis of vector and tensor fields. We evaluate them and study their performance in the context of problems spanning computational fluid dynamics and medical imaging applications