Topological Structures and Operators for Bivariate Data Visualization

Abstract

Understanding complex phenomena in diverse scientific and engineering disciplines often relies on decoding the interplay among real-valued or scalar fields that are measured or computed over a spatial domain. This thesis explores the relationships between paired scalar fields, resulting in contributions that may be categorized under three themes: computation, generalization, and applicability. Specifically, it presents methods for efficient computation and exploration of topological structures of bivariate fields, extends univariate concepts to the bivariate fields, and applies these structures across various applications.

The first part of the thesis focuses on the efficient computation of fiber surfaces and the generalization of flexible isosurface computation to bivariate fields. Utilizing the bivariate counterpart of critical points, the Jacobi set, we present an output-sensitive algorithm tailored for fiber surface computation for fields defined on tetrahedral meshes. The algorithm facilitates flexible extraction of individual fiber surface components through interactive guided exploration within the proximity of Jacobi edges.

The second part introduces novel approaches for analyzing static bivariate fields using continuous scatterplots (CSPs) and fiber surfaces, with applications in electronic transition datasets. Two CSP-based operators are introduced: a range-driven CSP lens operator and a domain-driven CSP peel operator. The former enables direct queries on CSPs using mathematically defined lenses, leading to segmented CSPs seamlessly linked to the spatial domain. The latter computes peeled CSP layers for domain segments, revealing distinctive properties within each respective segment. Developed in collaboration with a theoretical chemist, these operators offer valuable insights into electron density fields, helping analyze donor-acceptor behaviors in molecular subgroups.

The third part generalizes the concept of tracks to bivariate fields and delves into the representation and analysis of physical phenomena evolving over time. A novel track-based representation is introduced, capturing the evolution of bivariate features. The method leverages image moments to capture CSP similarities and dissimilarities, and PCA to represent them in 2D space. Case studies demonstrate the applicability and effectiveness of the technique in studying excited-state dynamics in two molecular systems.