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In an era of rapidly growing data sets, information reduction techniques such as extracting and highlighting characteristic features, are becoming increasingly important for efficient data analysis. Particularly relevant features of scalar fields are their critical points since they mark locations in the domain where a field's level set undergoes fundamental topological changes. There are well-established methods for locating and relating such points in a deterministic setting. However, many real-world phenomena studied in the computational sciences today are the result of a chaotic system that cannot be fully described by a single scalar field. Instead, the variability of such systems is typically captured with ensemble simulations, which generate a variety of possible outcomes of the simulated process. The topological analysis of such ensemble data sets, and uncertain data in general, is less well studied. In particular, there is no established definition for critical points of uncertain scalar fields. This thesis therefore aims to generalize the concept of critical points to uncertain scalar fields. While a deterministic field has a single set of critical points, each outcome of an uncertain scalar field has its own set of critical points. A first step towards finding an appropriate analog for critical points in uncertain data is to look at the distribution of all these critical points. In this work, different methods for analyzing this distribution are presented, which identify and track the likely locations of critical points over time, estimate their local occurrence probabilities, and eventually characterize their spatial uncertainty. A driving factor of winter weather in western Europe is the North Atlantic Oscillation (NAO), which is manifested by fluctuations in the sea level pressure difference between the Icelandic Low and the Azores High. Several methods have been developed to describe the strength of this oscillation. Some of them are based on certain assumptions, such as fixed positions of these two pressure systems. It is possible, however, that climate change will affect the locations of the main pressure variations and thus the validity of these descriptive methods. An alternative approach is based on the leading empirical orthogonal function (EOF) computed from the sea level pressure fields over the North Atlantic. The critical points of these fields indicate the actual locations of maximum pressure variations and can thus be used to assess how climate change affects these locations and to evaluate the validity of methods that use fixed locations to characterize the strength of the NAO. Because the climate is described by a chaotic system, such an analysis should incorporate the uncertain nature of climate predictions to produce statistically robust results. Extracting and tracking the positions of the maximum pressure variations that characterize the NAO therefore serves as a motivating practical application for the study of critical points in uncertain data in this work. Because uncertain data tend to be noisy, filtering is often required to separate relevant signals of variation from irrelevant fluctuations. A well-established method for extracting dominant signals from a time series of fields is to compute its empirical orthogonal functions (EOFs). In the first part of this thesis, this concept is extended to the analysis of spatiotemporal ensemble data sets to decompose their variation into modes describing the variation in the ensemble direction and modes describing the variation in the time direction. An application to different climate data sets revealed that, depending on the way an ensemble has been generated, temporal and ensemble-wise variations are not necessarily independent, making it difficult to separate these signals. Next, a computational pipeline for tracking likely locations of critical points in ensembles of scalar fields is presented. It computes leading EOFs on sliding time windows for all ensemble members, extracts regions where critical points can be expected from the resulting ensembles of EOFs for every time window, and finally tracks the barycenters of these regions over time. An application of this pipeline to sea level pressure fields over the North Atlantic revealed systematic shifts in the locations of the maximum pressure variations that characterize the NAO. These found shift were more pronounced for more extreme climate change scenarios. Existing methods for the identification of critical points in ensembles of scalar fields do not distinguish between uncertainties that are inherent in the analyzed system itself and those that are additionally introduced by using a finite sample of fields to capture these variations. In the next part of this thesis, two approaches for estimating the occurrence probabilities of critical points are presented that explicitly take into account and communicate to the viewer the additional uncertainties caused by estimating these probabilities from finite-sized ensembles. A comparison with existing works on synthetic data demonstrates the added value of the new approaches. The last part of this thesis is devoted to the question of how to characterize the spatial uncertainty of critical points. It provides a sound mathematical formulation of the problem of finding critical points with spatial uncertainty and computing their spatial distribution. This ultimately leads to the notion of uncertain critical points as a generalization of critical points to uncertain scalar fields. An analysis of the theoretical properties of these structures gave conditions under which well-interpretable results can be obtained and revealed interpretational difficulties when these conditions are not met. / In Zeiten immer größerer Datensätze gewinnen Techniken zur Informationsreduktion, etwa die Extraktion und Hervorhebung charakteristischer Merkmale, zunehmend an Bedeutung für eine effiziente Datenanalyse. Besonders relevante Merkmale von Skalarfeldern sind ihre kritischen Punkte, da sie Orte in der Domäne kennzeichnen, an denen sich die Topologie der Niveaumenge eines Feldes grundlegend verändert. Es existieren etablierte Methoden, um diese Punkte in deterministischen Feldern zu lokalisieren und sie miteinander in Beziehung zu setzen. Viele Alltagsphänomene, die heute untersucht werden, sind jedoch das Ergebnis chaotischer Systeme, die sich nicht vollständig durch ein einzelnes Skalarfeld beschreiben lassen. Stattdessen wird die Variabilität solcher Systeme mit Ensemblesimulationen erfasst, die eine Vielzahl möglicher Ergebnisse des simulierten Prozesses erzeugen. Die topologische Analyse solcher Ensemble-Datensätze und unsicherer Daten im Allgemeinen ist bisher weniger gut erforscht. Insbesondere gibt es noch keine etablierte Definition für die kritischen Punkte von unsicheren Skalarfeldern. In dieser Dissertation wird daher eine Verallgemeinerung des Konzepts kritischer Punkte auf unsichere Skalarfelder angestrebt. Während ein deterministisches Feld einen einzigen Satz kritischer Punkte hat, hat jede Realisierung eines unsicheren Skalarfeldes ihre eigenen kritischen Punkte. Ein erster Schritt, um ein geeignetes Analogon für kritische Punkte in unsicheren Daten zu finden, besteht darin, die Verteilung all dieser kritischen Punkte zu untersuchen. Zu diesem Zweck werden in dieser Arbeit verschiedene Methoden vorgestellt, die es ermöglichen, die wahrscheinlichen Orte kritischer Punkte zu identifizieren und über die Zeit zu verfolgen, die lokalen Wahrscheinlichkeiten für das Auftreten kritischer Punkte zu schätzen und schließlich die räumliche Unsicherheit von kritischen Punkten zu charakterisieren. Ein bestimmender Faktor für das Winterwetter in Westeuropa ist die Nordatlantische Oszillation (NAO), die sich in Schwankungen des Druckunterschieds auf Meereshöhe zwischen dem Islandtief und dem Azorenhoch äußert. Es existieren unterschiedliche Methoden, um die Stärke dieser Oszillation zu beschreiben, von denen einige auf bestimmten Annahmen beruhen, wie etwa der fixen Position der beiden Drucksysteme. Es ist jedoch möglich, dass der Klimawandel die Lage der Hauptdruckschwankungen und somit die Gültigkeit dieser Beschreibungsmethoden beeinträchtigt. Ein alternativer Ansatz basiert auf der führenden empirischen Orthogonalfunktion (EOF), welche aus den Druckfeldern auf Meereshöhe über dem Nordatlantik berechnet wird. Die kritischen Punkte dieses Feldes entsprechen den tatsächlichen Orten maximaler Druckschwankungen. Sie können daher verwendet werden, um die Auswirkungen des Klimawandels auf diese Orte zu bewerten und dadurch die Gültigkeit von Methoden, die feste Positionen zur Charakterisierung der Stärke der NAO verwenden, zu beurteilen. Da das Klima durch ein chaotisches System beschrieben wird, sollte eine solche Analyse die Unsicherheit von Klimavorhersagen berücksichtigen, um statistisch zuverlässige Ergebnisse zu erhalten. Die Extraktion und Verfolgung der für die NAO charakteristischen Positionen maximaler Druckschwankungen dient daher als motivierende praktische Anwendung für die Untersuchung kritischer Punkte in unsicheren Daten in dieser Arbeit. Da unsichere Daten oft verrauscht sind, ist meist zunächst eine Filterung erforderlich, um relevante Signale von irrelevanten Fluktuationen zu trennen. Ein etabliertes Konzept zur Extraktion dominanter Signale aus Zeitreihen von Skalarfeldern ist die empirische Orthogonalfunktionsanalyse (EOF-Analyse). Im ersten Teil dieser Arbeit wird dieses Konzept auf die Analyse von zeitabhängigen Ensemble-Datensätzen erweitert, um deren Variation in Moden zu zerlegen, die die jeweiligen Schwankungen in Ensemble- und Zeitrichtung beschreiben. Eine Anwendung auf verschiedene Klimadatensätze hat gezeigt, dass je nachdem, wie ein Ensemble generiert wurde, zeitliche und ensemblebezogene Variationen nicht zwangsläufig unabhängig sind, was eine Trennung dieser Signale erschwert. Im weiteren wird eine Berechnungspipeline zur Verfolgung der wahrscheinlichen Positionen kritischer Punkte in Ensemblen von Skalarfeldern vorgestellt. Sie berechnet zunächst die führenden EOFs auf gleitenden Zeitfenstern für jedes Ensemblemitglied, extrahiert dann aus den resultierenden Ensemblen von EOFs an jedem Zeitfenster Regionen, in denen kritische Punkte zu erwarten sind, und verfolgt schließlich die Baryzentren dieser Regionen über die Zeit. Die Anwendung dieser Pipeline auf die nordatlantischen Meeresspiegeldruckfelder hat eine systematische Verschiebungen der für die NAO charakteristischen Orte der maximalen Druckvariationen offenbart. Dabei führten extremere Klimawandelszenarien zu stärkeren Verschiebungen. Vorhandene Methoden zur Identifikation von kritischen Punkten in Ensemblen von Skalarfeldern unterscheiden nicht zwischen Unsicherheiten, die dem analysierten System selbst innewohnen, und solchen, die durch die Verwendung einer endlichen Stichprobe von Feldern zur Erfassung dieser Variationen zusätzlich verursacht werden. Im nächsten Teil dieser Arbeit werden daher zwei Ansätze zur Schätzung der Auftrittswahrscheinlichkeiten kritischer Punkte vorgestellt, die explizit auch die zusätzlichen Unsicherheiten berücksichtigen, die durch die Schätzung dieser Wahrscheinlichkeiten aus endlichen Ensemblen entstehen, und diese an den Betrachter kommunizieren. Der Mehrwert der neuen Verfahren wurde in einem Vergleich mit bestehenden Arbeiten auf synthetischen Daten demonstriert. Der letzte Teil dieser Arbeit ist der Frage gewidmet, wie sich die räumliche Unsicherheit kritischer Punkte charakterisieren lässt. Es wird eine fundierte mathematische Formulierung des Problems der Suche nach kritischen Punkten mit räumlicher Unsicherheit und der Berechnung ihrer räumlichen Verteilung erbracht. Das führt schließlich zum Begriff unsicherer kritischer Punkte als Verallgemeinerung von kritischen Punkten auf unsichere Skalarfelder. Eine Analyse der theoretischen Eigenschaften dieser Strukturen hat Bedingungen ergeben, unter denen einfach zu interpretierende Ergebnisse erzielt werden können, und offenbarte Interpretationsschwierigkeiten, die entstehen, wenn diese Bedingungen nicht erfüllt sind.

Direct Volume Rendering (DVR) is a visualization technique that has proved to be a very powerful tool in many scientific visualization applications. Diagnostic medical imaging is one domain where DVR could provide clear benefits in terms of unprecedented possibilities for analysis of complex cases and highly efficient work flow for certain routine examinations. The full potential of DVR in the clinical environment has not been reached, however, primarily due to limitations in conventional DVR methods and tools. This thesis presents methods addressing four major challenges for DVR in clinical use. The foundation of all methods is to incorporate the domain knowledge of the medical professional in the technical solutions. The first challenge is the very large data sets routinely produced in medical imaging today. To this end a multiresolution DVR pipeline is proposed, which dynamically prioritizes data according to the actual impact in the rendered image to be reviewed. Using this prioritization the system can reduce the data requirements throughout the pipeline and provide high performance and visual quality in any environment. Another problem addressed is how to achieve simple yet powerful interactive tissue classification in DVR. The methods presented define additional attributes that effectively captures readily available medical knowledge. The task of tissue detection is also important to solve in order to improve efficiency and consistency of diagnostic image review. Histogram-based techniques that exploit spatial relations in the data to achieve accurate and robust tissue detection are presented in this thesis. The final challenge is uncertainty visualization, which is very pertinent in clinical work for patient safety reasons. An animation method has been developed that automatically conveys feasible alternative renderings. The basis of this method is a probabilistic interpretation of the visualization parameters. Several clinically relevant evaluations of the developed techniques have been performed demonstrating their usefulness. Although there is a clear focus on DVR and medical imaging, most of the methods provide similar benefits also for other visualization techniques and application domains.

Ensemble simulation is a commonly used technique in operational forecasting of weather and floods. Multi-member ensemble output is usually large, multivariate, and challenging to interpret interactively. Forecast meteorologists and hydrologists are interested in understanding the uncertainties associated with the simulation; specifically variability between the ensemble members. The visualization of ensemble members is currently accomplished through spaghetti plots or hydrographs. To improve visualization techniques and tools for forecasters, we conducted a userstudy to evaluate the effectiveness of existing uncertainty visualization techniques on 1D and 2D synthetic datasets. We designed an uncertainty evaluation framework to enable easier design of such studies for scientific visualization. The techniques evaluated are errorbars, scaled size of glyphs, color-mapping on glyphs, and color-mapping of uncertainty on the data surface. Although we did not find a consistent order among the four techniques for all tasks, we found that the efficiency of techniques used highly depended on the tasks being performed. Errorbars consistently underperformed throughout the experiment. Scaling the size of glyphs and color-mapping of the surface performed reasonably well. With results from the user-study, we iteratively developed a tool named ‘Noodles’ to interactively explore the ensemble uncertainty in weather simulations. Uncertainty was quantified using standard deviation, inter-quartile range, width of the 95% confidence interval, and by bootstrapping the data. A coordinated view of ribbon and glyph-based uncertainty visualization, spaghetti plots, and data transect plots was provided to two meteorologists for expert evaluation. They found it useful in assessing uncertainty in the data, especially in finding outliers and avoiding the parametrizations leading to these outliers. Additionally, they could identify spatial regions with high uncertainty thereby determining poorly simulated storm environments and deriving physical interpretation of these model issues. We also describe uncertainty visualization capabilities developed for a tool named ‘FloodViz’ for visualization and analysis of flood simulation ensembles. Simple member and trend plots and composited inundation maps with uncertainty are described along with different types of glyph based uncertainty representations. We also provide feedback from a hydrologist using various features of the tool from an operational perspective.

Climate analysts use Coupled Model Intercomparison Project Phase 5 (CMIP5) simulations to make sense of models performance in predicting extreme events such as heavy precipitation. Similarly, weather analysts use numerical weather prediction models (NWP) to simulate weather conditions either by perturbing initial conditions or by changing multiple input parameterization schemes, e.g., cumulus and microphysics schemes. These simulations are used in operational weather forecasting and for studying the role of parameterization schemes in synoptic weather events like storms. This work addresses the need for visualizing the differences in both CMIP5 and NWP model output. This work proposes three glyph designs used for communicating CMIP5 model error. It also describes Ensemble Visual eXplorer tool that provides multiple ways of visualizing NWP model output and the related input parameter space. The proposed interactive dendrogram provides an effective way to relate multiple input parameterization schemes with spatial characteristics of model uncertainty features. The glyphs that were designed to communicate CMIP5 model error are extended to encode both parameterization schemes and graduated uncertainty, to provide related insights at specific locations such as storm center and the areas surrounding it. The work analyzes different ways of using glyphs to represent parametric uncertainty using visual variables such as color and size, in conjunction with Gestalt visual properties. It demonstrates the use of visual analytics in resolving some of the issues such as visual scalability. As part of this dissertation, we evaluated three glyph designs using average precipitation rate predicted by CMIP5 simulations, and Ensemble Visual eXplorer tool using WRF 1999 March 4th, North American storm track dataset.

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