Topological Data Analysis in Visualization: Theory, Applications, Software

Abstract: Topology is the mathematical study of the connectivity of space. This study is often based on topological invariants, which help establish equivalence between spaces or help distinguish between them. This lecture will provide a gentle introduction to topology with a focus on topological invariants that are amenable to computation. After a quick introduction to homoemorphism and manifolds, we will study invariants such as the Euler Characteristic, Betti numbers, and homology groups. Next, we will study two applications to illustrate the use of the invariants — the surface classification theorem and critical point classification via lower link.

Biography: Vijay Natarajan is a Professor in the Department of Computer Science and Automation at Indian Institute of Science, Bangalore. He received his Ph.D. in computer science from Duke University and holds the B.E. degree in computer science and M.Sc. degree in mathematics from BITS Pilani. His research interests include scientific visualization, computational geometry, and computational topology. In current work, he is developing topological methods for time-varying and multi-field data visualization, and studying applications in biology, material science, and climate science.

Abstract: During this lecture we will introduce elements of discrete topology such as simplicial complexes and illustrate how they can be used to give a geometric representation of point cloud data or metric spaces. Persistent homology, a fundamental tool in Topological Data Analysis (TDA) will then be defined. Persistent homology encodes homological properties of a sequence of simplicial complexes, revealing shape characteristics of the data. Several ways have been proposed in TDA to summarise persistent homology in a way which can then be used for data analysis; we will review some of them (persistence diagrams, barcodes, landscapes, stable ranks) with an emphasis on their robustness to perturbations of the data.

Biography: Martina Scolamiero is an Assistant Professor in Mathematics with specialization in Geometry and Mathematical Statistics in Artificial Intelligence. Her research interests are in the areas of applied and computational topology. She works on defining topological invariants that are suitable for data analysis, understanding their statistical properties, and their applicability in machine learning. She is also interested in applications of topological methods to neuroscience and psychiatry.

Abstract: In this lecture, I will discuss practical aspects of how topology is used for understanding the shape of data represented as a point cloud in some space. Some key ideas such as thickening, filtrations, and simplicial complexes will be described. I will then introduce three types of complexes that can be constructed as a result of this thickening, namely, Vietoris-Rips, Čech, and alpha complexes. We will learn about the relationship between them and their advantages and disadvantages. In the end, we will also take up some concrete simple examples of point cloud data and discuss how we can compute it's topological inavriants through persistent homology.

Biography: Talha Bin Masood is an Assistant Professor at Linköping University in Sweden. He received his Ph.D. in Computer Science from the Indian Institute of Science, Bangalore. After that, he worked as post doctoral researcher at Linköping University from 2018 to 2022. His research interests include scientific visualization, computational geometry, computational topology, and their applications to various scientific domains.

Abstract: Scalar field topology (SFT) deals with scalar functions defined on topological spaces, commonly referred to as scalar fields. Since scalar fields are very common data representations, SFT has successfully been applied in many research areas to characterize features. To this end, SFT provides several data abstractions, such as critical points, contours, merge/contour trees, Reeb graphs, ascending/descending manifolds, Morse-Smale complexes, fibers, and so forth. These abstractions capture the inherent structure of the input data, and due to their generality, they can describe a plethora of semantic features, including pressure minima in climate data, boundaries between mixing fluids in CFD simulations, dark matter halos in cosmology, and bonds in chemistry. This talk introduces these abstractions and demonstrates their application in a variety of research applications.

Biography: Jonas Lukasczyk is a Staff Scientist at TU Kaiserslautern. He received his Ph.D. degree from the Visual Information Analysis Group, Technische Universitat Kaiserslautern, Germany, where he also studied applied computer science and mathematics. His recent work focuses on topology-based characterization of features and their evolution in large-scale simulations. He is an active contributor to one of the flagship software libraries for topological analysis of scientific data within Visualization domain called Topology Toolkit (TTK).

Abstract: This lecture will continue the discussion on topological descriptors for scalar fields with a focus on the Morse-Smale complex. Ideas from Morse theory of smooth scalar functions can be transported to piecewise-linear functions and applied towards the study of scientific data. The Morse-Smale (MS) complex represents a partition of the domain of a scalar field into regions that exhibit uniform gradient flow behavior. The domain is partitioned into cells, each of which is defined by a pair of critical points of the scalar field. In this lecture, we will introduce the MS complex, study some characteristic properties, and outline an algorithm for computing the complex. The practical utility of the MS complex depends on the existence of methods for topological simplification that help identify and remove noise. Finally, we will describe some applications of the MS complex.

Biography: Vijay Natarajan is a Professor in the Department of Computer Science and Automation at Indian Institute of Science, Bangalore. He received his Ph.D. in computer science from Duke University and holds the B.E. degree in computer science and M.Sc. degree in mathematics from BITS Pilani. His research interests include scientific visualization, computational geometry, and computational topology. In current work, he is developing topological methods for time-varying and multi-field data visualization, and studying applications in biology, material science, and climate science.

Abstract:In this lecture, I will discuss two topological concepts for the analysis of vector and tensor-valued fields. I will start with the limit-set topology assuming a continuous field as introduced into visualization by Helman and Hesselink, focussing on two- and three-dimensional fields. As an alternative, we take a brief look at the discrete vector field topology, which is based on Forman's discrete vector fields. Then we will discuss some approaches that have been considered for simplifying vector fields. Finally, I will talk about some challenges and open problems in vector/tensor field visualization.

Biography: Ingrid Hotz received her M.S. degree in theoretical Physics from the Ludwig Maximilian University in Munich Germany and the PhD degree from the Computer Science Department at the University of Kaiserslautern, Germany. During 2003-2006 she worked as a postdoctoral researcher at the Institute for Data Analysis and Visualization (IDAV) at the University of California. Then she was the leader of a research group at the Zuse Institute in Berlin Germany during 2006-2013. From 2013 to 2015 she was the head of the scientific visualization group at the German Aerospace Center (DLR). Since 2015 she is a Professor in Scientific Visualization at the Linköping University in Scientific Visualization and has an affiliation with the Center for Medical Image Science and Visualization (CMIV) in Linköping. The main focus of her research lies in the area of data analysis and scientific visualization, ranging from basic research questions to effective solutions to visualization problems in applications including flow analysis, engineering and physics, medical applications, and mechanical engineering ranging from small- to large-scale simulations.

Abstract: Insights from data are highly influenced by the shape perceived in the data. However, two problems exist. First, data are often high dimensional, making their shape difficult to visualize. Second, once visualized, they often suffer from scalability and readability issues, even with modest amounts of data. By applying topology-based descriptors to these problems, approaches can utilize the shape for presenting and interacting with data in ways that are mathematically robust and correspond to human perception and cognition. This talk will discuss the applications of topology-based descriptors, namely persistent homology, contour trees, and mapper, in several commonly used visualization types, including scatterplots, line charts, node-link diagrams, and dimension reduction.

Biography: Paul Rosen is an Associate Professor in the Kahlert School of Computing and the Scientific Computing and Imaging Institute at the University of Utah. He received his Ph.D. from Purdue University in 2010. Subsequently, he was a Research Assistant Professor at the University of Utah from 2010 to 2015. He was then an Assistant and Associate Professor at the University of South Florida from 2015 to 2022. He has been the co-author of over 80 papers, 6 having received best paper awards or honorable mentions in the areas of computer graphics, data visualization, and topological data analysis. His research interests lie at the intersection of scientific and information visualization, where he utilizes a mix of human-centered design and geometry- and topology-based approaches to improve the efficacy of visualization tools. His research has been supported by the National Institutes of Health, the National Radio Astronomy Observatory, the Defense Intelligence Agency, and National Science Foundation grants, including an NSF CAREER Award in 2019. He is also one of the General Chairs for IEEE VIS 2024, which will be held in Tampa, Florida.

Abstract: Many scientific measurements and/or simulations lead to scalar fields which are real valued functions measured over an area, or a volume. They can be static or dynamic i.e., measured across time, or ensembles i.e., dependent on various parameters and initial conditions. Topological structures provide a combinatorial, abstract, and succinct representation of scalar fields with proven methods to simplify these fields. With availability of such structures which capture varying degrees of information, with or without geometric context, comparison becomes one of the key operations in any data analysis pipeline to gain deeper understanding of the scientific phenomena with appropriate focus and context. Such an understanding might not be possible with just analyzing the individual fields or their corresponding topological structures. In this talk we introduce comparative analysis of topological structures, talk about theoretical and practical considerations involved in designing of such comparative methods, differentiate between global and local methods, and finally show use-cases where the comparison forms the foundation of multiple applications like symmetry detection, periodicity detection, temporal summarization, and feature tracking.

Biography: Raghavendra Sridharamurthy is a postdoctoral researcher at the Scientific Computing and Imaging Institute (SCI), University of Utah. He completed his PhD and master's in computer science from the Indian Institute of Science (IISc), Bengaluru. His research interests include scientific visualization, computational topology, topological data analysis, and their applications.