Topology-Driven Visual Analysis of Structures in Dynamic Spatial Data

This thesis focuses on the visual analysis of spatial structures within complex datasets. The primary goal is to extract meaningful features from such data and establish comparisons between these features to support core visual analysis tasks, such as tracking, comparison, and ensemble analysis, tailored to specific application domains in science and engineering. To reach this goal, the focus is to adapt and extend methods from topological data analysis (TDA) and integrate them in visual exploration environments. This work addresses data from two different scientific application domains. First functional MRI (fMRI) data, where the aim is to extract subject-specific neural activation regions and track their dynamics over time. A major challenge associated with fMRI analysis is that the data is inherently noisy, as a complicated mixture of multiple sources of noise often pollutes the true signal in an fMRI scan. The second application deals with granular materials, which are collections of discrete particles such as gravel, sand, or powder. These particle sets are described as dynamic spatial graphs representing force networks. These graphs naturally have a multiscale nature, as local particle-level interactions shape global patterns. The main goal is to understand the interplay between the large-scale phenomena in granular materials, such as jamming, mechanical behavior, and dynamics, and these local interactions, which is an active research area. TDA is a powerful approach for addressing such challenges in datasets and has successfully been applied to many scientific applications. It leverages principles from algebraic topology and computational geometry to extract multiscale features that are robust to noise and have great potential for simplification, abstraction, and summarization of complex data. The core contribution of this work is the development and implementation of TDA and visualization methods within a tailored visual analysis framework to support the domain scientist for explorative analyses of dynamic complex data. More specifically, the thesis includes a survey of existing topological descriptors for scalar field comparison, establishing a taxonomy of methods and integrating it into an interactive visual literature browser for intuitive exploration. Building on this foundation, novel approaches were developed to extract, represent, and analyze structural and dynamic patterns in the brain activity data and the force networks in granular materials. These methods leverage merge trees, multiscale segmentation, and cycle extraction techniques to reveal relationships across spatial and temporal scales. Furthermore, efficient frameworks for tracking and visualizing dynamic features were designed to support interactive exploration and facilitate domain-specific interpretation.