Matrix methods in visualization

Abstract

The theory of matrices has a long history that began over 4000 years ago. It took a while until matrices were studied systematically in the context of linear algebra. While these results from the 18th and 19th century were mainly characterized by theoretical thoughts, the modern use of matrices is usually linked to computational aspects. This aspect made the theory of matrices extremely useful for applied sciences, such as computer graphics and visualization, and paved the way for innovative matrix methods. The overall goal of this thesis is to integrate such matrix methods into the field of data analysis and visualization, where emphasis is placed on matrix decompositions. In this context, the following four concepts are addressed: the examination of linear structures and matrix formulations, the utilization of matrix formulations and matrix methods, the customization of matrix methods for visualization, and the augmentation of visualization techniques. These four conceptual steps characterize a sequential process that is used throughout the chapters of this thesis. With a main focus on data-driven methods that reveal time evolutionary and statistical patterns, the contents of the chapters refer to different fields of application. Chapter 2 demonstrates applications of Dynamic Mode Decomposition in the context of visual computing, and Chapter 3 addresses the challenges of uncertainty propagation and visualization. In contrast, Chapters 4 and 5 present methods in the context of structural analysis (solid mechanics) and smoothed particle hydrodynamics (fluid mechanics). The overall content of this thesis demonstrates the versatile, effective use of matrices for visual computing.