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.

Denna avhandling fokuserar på visuell analys av rumsliga strukturer inom komplexa datamängder. Det primära målet är att extrahera meningsfulla egenskaper från sådana data och etablera jämförelser mellan dessa egenskaper för att stödja centrala visuella analysuppgifter, såsom spårning, jämförelse och ensembleanalys, skräddarsydda för specifika tillämpningsområden inom vetenskap och teknik. För att nå detta mål ligger fokus på att anpassa och utöka metoder från topologisk dataanalys (TDA) och integrera dem i visuella utforskningsmiljöer.

Detta arbete behandlar data från två olika vetenskapliga tillämpningsområden. För det första funktionell MRI (fMRI) data, där syftet är att extrahera patientspecifika neurala aktiveringsregioner och spåra deras dynamik över tid. En stor utmaning i samband med fMRI-analys är att data är i sig brusiga, eftersom en komplicerad blandning av flera bruskällor ofta förorenar den verkliga signalen i en fMRI-skanning.

Den andra tillämpningen handlar om granulära material, som är samlingar av diskreta partiklar såsom grus, sand eller pulver. Dessa partikelmängder beskrivs som dynamiska rumsliga grafer som representerar kraftnätverk. Dessa grafer har naturligtvis en flerskalig natur, eftersom lokala interaktioner på partikelnivå formar globala mönster. Huvudmålet är att förstå samspelet mellan storskaliga fenomen i granulära material, såsom störningar, mekaniskt beteende och dynamik, och dessa lokala interaktioner, vilket är ett aktivt forskningsområde.

TDA är en kraftfull metod för att hantera sådana utmaningar i datamängder och har framgångsrikt tillämpats i många vetenskapliga tillämpningar. Den utnyttjar principer från algebraisk topologi och beräkningsgeometri för att extrahera fler-skaliga egenskaper som är robusta mot brus och har stor potential för förenkling, abstraktion och sammanfattning av komplexa data.

Det centrala bidraget i detta arbete är utveckling och implementering av TDA och visualiseringsmetoder inom ett skräddarsytt visuellt analysramverk för att stödja domänforskaren för explorativa analyser av dynamiska komplexa data.

Mer specifikt inkluderar avhandlingen en omfattande undersökning av befintliga topologiska deskriptorer för skalär fältjämförelse, etablering av en taxonomi av metoder och integrering av den i en interaktiv visuell litteraturläsare för intuitiv utforskning. Byggande på denna grund utvecklades nya metoder för att extrahera, representera och analysera strukturella och dynamiska mönster i hjärnaktivitetsdata och kraftnätverk i granulära material. Dessa metoder utnyttjar sammanslagningsträd, flerskalig segmentering och cykelextraktionstekniker för att avslöja samband över rumsliga och tidsmässiga skalor. Dessutom utformades effektiva ramverk för att spåra och visualisera dynamiska funktioner för att stödja interaktiv utforskning och underlätta domänspecifik tolkning.

For millennia humans have interacted with a range of chemical substances that are an inte-gral part of human history and culture. The chemical characteristics of the three similar-looking chemical species psilocybin, nicotine, and caffeine have remarkably different biological effects. The narratives of these chemical characters are compelling—their role traverses the unobservable submicro-scopic world, influences our biological essence, and impacts societal values and sci-entific judgement. Visual storytelling through modern display technologies offers an educa-tional method to communicate the nature and effect of these chemical actors. Biological information that transcends multi-directionally and constantly from submicroscopic through to macroscopic processes can be made acces-sible and meaningful to students and the pub-lic. In doing so, links are forged between scientific knowledge and the manner we view these chemical forms from a human and socie-tal context. In attempting to capture these complexities, the aim of this chapter is to con-ceptualise and design a visual story to commu-nicate the effect of psilocybin, nicotine, and caffeine on humans for public engagement. Whilst remarkably similar in chemical struc-ture, these chemical species have dramatically different psychological and physiological effects on the human body. At the same time, human interaction with these substances is intertwined with historical associations, cul-tural norms, as well as perceived and enacted taboos.

Merge trees, a type of topological descriptor, serve to identify and summarize the topological characteristics associated with scalar fields. They present a great potential for the analysis and visualization of time-varying data. First, they give compressed and topology-preserving representations of data instances. Second, their comparisons provide a basis for studying the relations among data instances, such as their distributions, clusters, outliers, and periodicities. A number of comparative measures have been developed for merge trees. However, these measures are often computationally expensive since they implicitly consider all possible correspondences between critical points of the merge trees. In this paper, we perform geometry-aware comparisons of merge trees. The main idea is to decouple the computation of a comparative measure into two steps: a labeling step that generates a correspondence between the critical points of two merge trees, and a comparison step that computes distances between a pair of labeled merge trees by encoding them as matrices. We show that our approach is general, computationally efficient, and practically useful. Our general framework makes it possible to integrate geometric information of the data domain in the labeling process. At the same time, it reduces the computational complexity since not all possible correspondences have to be considered. We demonstrate via experiments that such geometry-aware merge tree comparisons help to detect transitions, clusters, and periodicities of a time-varying dataset, as well as to diagnose and highlight the topological changes between adjacent data instances.

We present a method for detecting patterns in time-varying functional magnetic resonance imaging (fMRI) data based on topological analysis. The oxygenated blood flow measured by fMRI is widely used as an indicator of brain activity. The signal is, however, prone to noise from various sources. Random brain activity, physiological noise, and noise from the scanner can reach a strength comparable to the signal itself. Thus, extracting the underlying signal is a challenging process typically approached by applying statistical methods. The goal of this work is to investigate the possibilities of recovering information from the signal using topological feature vectors directly based on the raw signal without medical domain priors. We utilize merge trees to define a robust feature vector capturing key features within a time step of fMRI data. We demonstrate how such a concise feature vector representation can be utilized for exploring the temporal development of brain activations, connectivity between these activations, and their relation to cognitive tasks.

In topological data analysis and visualization, topological descriptors such as persistence diagrams, merge trees, contour trees, Reeb graphs, and Morse–Smale complexes play an essential role in capturing the shape of scalar field data. We present a state-of-the-art report on scalar field comparison using topological descriptors. We provide a taxonomy of existing approaches based on visualization tasks associated with three categories of data: single fields, time-varying fields, and ensembles. These tasks include symmetry detection, periodicity detection, key event/feature detection, feature tracking, clustering, and structure statistics. Our main contributions include the formulation of a set of desirable mathematical and computational properties of comparative measures, and the classification of visualization tasks and applications that are enabled by these measures.