Abstract: A myriad of physical phenomena, such as fluid flows, magnetic fields, and population dynamics are described by vector fields. More often than not, vector fields are complex and their analysis is challenging. Vector field topology is a powerful analysis technique that consists of identifying the most essential structure of a vector field. Its topological features include critical points and separatrices, which segment the domain into regions of coherent flow behavior, and provide a sparse and semantically meaningful representation of the underlying data. However, a broad adoption of this formidable technique has been hampered by the lack of open source software implementing it. The Visualization Toolkit (VTK) now contains the filter vtkVectorFieldTopology that extracts the topological skeleton of 2D and 3D vector fields. We provide a hands-on tutorial on how to compute the topology of a vector field, both in vtkpython and ParaView, and discuss a tracking algorithm to explore the evolution of these features temporally.

Abstract: The basics of topological data analysis including persistent homology will be introduced. We will also assist participants in installing TDA software.

Abstract: This session gives an overview of the Topology ToolKit (TTK), an open-source library for topological data analysis and visualization. After introducing the main concepts behind TTK's usage, we will demo how to use it with simple examples in ParaView and Python. Then, we will present TTK's capabilities, going from simple critical point extraction to more advanced features (Reeb graphs, Morse-Smale complexes, distances and barycenters of persistence diagrams, etc) as well as TTK's current capabilities for processing point cloud data. This tutorial will focus on a beginnerÃ¢â‚¬â„¢s introduction to topological data analysis as it is used in practice. In particular, instead of focusing on theoretical aspects and algorithmic details, this tutorial focuses on how topological methods can be used to provide concise yet meaningful analysis of complex data.

In the process, we will provide links to numerous online resources containing further tutorials, documentations, examples and exercises that can be found online: https://topology-tool-kit.github.io/. If you have any questions with TTK, please don't hesitate to send us a message at ttk-users@googlegroups.com.

Abstract: Topological inference is the problem of identifying likely spaces parametrizing a given data set. Persistent (co)homology is one tool from TDA to approach this problem: the topological features captured in persistence diagrams --- number of cluster, presence of cycles, etc --- constrain the likely spaces underlying the data. That said, going from persistence computations to data parametrization is hard. This tutorial introduces a framework to tackle this problem; specifically, by leveraging the fact that the cohomology functor is representable via maps to Eilenberg-MacLane spaces (Brown representability theorem). We will show how this leads to algorithms that turn persistent cohomology computations into topology preserving multiscale maps from data to appropriate spaces. The tutorial will present both the theory and applications of these ideas.

https://github.com/ctralie/DREiMac

Abstract: Common information visualizations, e.g., scatterplots, suffer from overdraw even with modest amounts of data. Several techniques exist to reduce this overdraw, e.g., changing visual encodings or subsampling data. However, most guidance on their use remains largely rules-of-thumb. By applying Topological Data Analysis (TDA) to the problem, we have developed techniques that are mathematically robust, correspond to human perception and cognition, and are surprisingly effective at selecting effective visualizations of data. This tutorial will introduce participants to techniques for resolving these issues on three common information visualizations, namely scatterplots, line charts, and graph visualizations. The solutions to these problems are a mix of optimization interfaces and mechanisms for interactively exploring data, all using Topological Data Analysis.

Abstract: The mapper algorithm is a popular tool from topological data analysis for extracting topological summaries of high-dimensional datasets. This tutorial provides an overview of the mapper algorithm and walks through a few open-source tools for visualizing and analyzing high-dimensional point cloud datasets via the mapper algorithm. It will include several application examples to demonstrate the analytical workflow. The tutorial aims to help the attendees quickly get started with applying the mapper algorithm to the visual explorations of high-dimensional data.

Abstract: This talk will detail PNNLâ€™s efforts towards making hypergraph visualization practical for and accessible to a broad audience of users, e.g., data scientists, mathematicians, and analysts. While many hypergraph visualization tools and approaches exist as research prototypes, few are created with broad applicability and composability in mind. Our first hypergraph visualization software integrated directly with our open-source python package, hypernetx and produced static Euler visualizations for use in exploratory data analysis, presentations, and publications. Following this, we implemented an interactive version of this tool as a Jupyter widget using the D3.js force directed algorithm. Our efforts now are focused on providing a consistent API for multiple alternative interactive hypergraph visualizations, including specialized visualizations for hypergraphs with given node orders. Additionally, our software builds on Plotly Dash to allow other developers to integrate these visualizations more easily into complex, custom applications.

Abstract: Local maxima and minima, or extremal events, in experimental time series can be used as a coarse summary to characterize data. However, the discrete sampling in recording experimental measurements suggests uncertainty on the true timing of extrema during the experiment. This in turn gives uncertainty in the timing order of extrema within the time series. Motivated by applications in genomic time series and biological network analysis, we construct a weighted directed acyclic graph (DAG) called an extremal event DAG using techniques from persistent homology that is robust to measurement noise. Furthermore, we define a distance between extremal event DAGs based on the edit distance between strings. We prove several properties including local stability for the extremal event DAG distance with respect to pairwise L_\infty distances between functions in the time series data. Lastly, we provide algorithms, publicly free software, and implementations on extremal event DAG construction and comparison.

Abstract: In my talk, I will present how the TDA Mapper algorithm is applied to quantify ecological states of the Upper Mississippi River System (UMRS). This analysis/quantification was performed globally (to the entire UMRS water quality dataset) and locally (to pool specific water quality dataset). Additionally, I will discuss whether or not there is a shift in regime/state over the last 30 years.

Abstract: Hexahedral (hex-) meshes are preferred to achieve accurate and efficient simulations involving the solving of non-linear PDEs for many critical applications. Despite numerous efforts, producing high quality hexahedral meshes for arbitrary models remains a challenge. In the meantime, each pure hex mesh has a structure, called the base complex, that partitions the mesh (and thus the volume) into a stack of larger hexahedral blocks. This coarse representation suggests an efficient and multi-resolution hex-mesh generation and modification framework. In addition, the complexity of this structure affects a few subsequent tasks performed on the corresponding hex-mesh. However, there lacks a comprehensive metric and intuitive visualization to describe the complexity of this hex-mesh structure. To address that, I will introduce a first complexity metric for hex-mesh structure along with its effective visualization. I will then apply this metric and the structure visualization for the comparison of different hex-mesh generation and simplification methods to demonstrate its effectiveness. In the second half of the talk, I will present two techniques that we developed for the simplification and optimization of hex-mesh structures. I will provide sample results of the two techniques to demonstrate their robustness and effectiveness.

Abstract: Enabling insight into large and complex spatial datasets is a central theme in scientific data visualization. Visualizing large spatial data sets requires efficient data representations as well as powerful analysis algorithms for data transformation. In our work, we focus on spatial data representing continuous phenomena, such as scalar fields (terrains, volume data sets, etc.), on point clouds and on unstructured meshes connecting them. This talk concentrates on the first two steps in the data visualization pipeline, concerning compact and scalable representations for large-size spatial data sets, and data transformation methods based on topological data analysis. Specifically, we focus on a new approach to mesh data representation based on a modular decomposition, the stellar decomposition, and present data structures for meshes and simplicial complexes based on it as well as open-source libraries implementing them. We discuss topological methods we have developed for data segmentation, simplification and noise removal, based on discrete Morse theory. We present an approach to the analysis of multivariate fields, based on a combinatorial matching of the simplices discretizing a multivariate field, which preserves its multidimensional persistent homology. Our applications of these techniques in environmental science are discussed as well as some research challenges.

Abstract: We present a spacetime meshing framework that simplifies, scales, and delivers various feature-tracking algorithms for scientific data. The key of this framework is the simplicial spacetime meshing scheme that generalizes both regular and unstructured spatial meshes to spacetime while tessellating spacetime mesh elements into simplices. The benefits of using simplicial spacetime meshes include (1) reducing ambiguity cases for feature extraction and tracking, (2) simplifying the handling of degeneracies using symbolic perturbations, and (3) enabling scalable and parallel processing. The use of simplicial spacetime meshing simplifies and improves the implementation of several feature-tracking algorithms for critical points, quantum vortices, and isosurfaces. We implemented the simplicial spacetime meshing framework as open-source software, namely the feature tracking kit (FTK), which provides end-users with VTK/ParaView filters, Python bindings, a command-line interface, and programming interfaces for feature-tracking applications. We demonstrate use cases and scalability studies through both synthetic data and scientific applications, including tokamak, fluid dynamics, and superconductivity simulations.

Abstract: Persistent homology has emerged as a powerful tool for understanding complex data. However, interpretation and use of persistent shape statistics often hinges on a (theoretically) intractable inverse problem: selection of "good" cycle representatives in homology. In this talk, we'll introduce/recall the basic notions of persistent homology, as well as some recent scientific applications. We'll discuss what makes cycle representatives "good" in this context, and explain recent empirical results that show good representatives can almost always be found by "guessing." We'll then explore some beautiful and interesting properties of optimal generators observed empirically, in real-world data.

Abstract: Periodic nets have a long and involved history in crystallography and more generally the chemical and materials science community. A fairly new idea for their analysis and construction centers around viewing embedded graphs on special (minimal) periodic surfaces as graphs in R3. Recently, this approach has led to a novel connection between the mapping class group of a surface, a purely topological object, and entangled nets in three-dimensional Euclidean space, bringing new topological techniques to bear on a project that is inspired primarily by its applications to chemistry and materials science. In this talk, I would like to introduce the basics behind the approach and investigate the promising new idea of analysing the resulting structures and their relationship with the mapping class group using methods from computational topology, in particular, persistent homology.

Abstract: As collections of data grow in size, it is increasingly important to have efficient means of analyzing large data sets. In this project, we use Mapper, a tool that summarizes data into a graph, to discover an underlying structure relating the shapes of more than 3,300 Passiflora leaves from 40 different species. We choose to study leaves of the Passiflora genus in particular due to their extraordinary diversity of shape. As the Mapper graph has a structure, or "shape" of its own, we think of it as a "shape of shapes" that provides information on the interplay between the developmental processes determining leaf shape within a single plant and the evolutionary processes between species. In particular, we examine the interactions between leaf species and both leaf age and leaf area by constructing a Mapper graph for each measure. For each node in the resulting graphs, we then compute the average leaf shape to obtain a graph structure that reveals how morphometric differences between species relate to the developmental changes that must occur for those shapes to be realized. We additionally present a new Mapper visualization in Javascript which incorporates both tooltips that display the average leaf shape per node as well as supporting pie chart nodes, useful for visualization of categorical data such as Passiflora species.

Abstract: Persistence diagrams have been widely used to quantify the underlying features of filtered topological spaces in data visualization. In many applications, computing distances between diagrams is essential; however, computing these distances has been challenging due to the computational cost. In this paper, we propose a persistence diagram hashing framework that learns a binary code representation of persistence diagrams, which allows for fast computation of distances. This framework is built upon a generative adversarial network (GAN) with a diagram distance loss function to steer the learning process. Instead of using standard representations, we hash diagrams into binary codes, which have natural advantages in large-scale tasks. The training of this model is domain-oblivious in that it can be computed purely from synthetic, randomly created diagrams. As a consequence, our proposed method is directly applicable to various datasets without the need for retraining the model. These binary codes, when compared using fast Hamming distance, better maintain topological similarity properties between datasets than other vectorized representations. To evaluate this method, we apply our framework to the problem of diagram clustering and we compare the quality and performance of our approach to the state-of-the-art. In addition, we show the scalability of our approach on a dataset with 10k persistence diagrams, which is not possible with current techniques. Moreover, our experimental results demonstrate that our method is significantly faster with the potential of less memory usage, while retaining comparable or better quality comparisons.

Abstract: Understanding cluster structure is a basic component of exploratory data analysis. The framework of density-based clustering is a popular choice for exploratory data analysis, as it makes few assumptions about the distribution of the data. There are efficient algorithms, including some incorporating ideas from TDA, and much is known about the theoretical properties of these algorithms. In practice, using a clustering algorithm inevitably requires choosing one or more parameters, and this choice often has a large effect on the output. We explain how one can use persistence vineyards, and their conceptual offspring, the "prominence vineyard", to visualize how the output of a density-based clustering algorithm changes as one varies the parameters. This provides an intuitive guide for informed parameter selection, and also reveals multi-scale clustering structure that varies with distance scale or density threshold.

Abstract: Convolutional neural networks are powerful tools for the analysis of large amounts of data. Once trained, these models are simply series of linear and non-linear functions that transform inputs through a high-dimensional feature space. Many interpretability methods attempt to understand the decision process of a model by looking at the hidden layer representations produced by each transformation of an input, known as â€œactivationsâ€. Topological data analysis techniques are particularly effective at representing the complexities of hidden layer activations because of their ability to compactly summarize complex, high-dimensional data while still preserving relevant global structures. This talk will highlight recent work using topological summaries of activations and Mapper visualizations to understand how datasets of images are represented in the feature space of a trained model.

Abstract: 3D symmetric tensor fields have many applications in solid mechanics. Topological structures of these fields can provide additional understanding of the underlying physics. Computationally, a 3D linear tensor field is defined on a tetrahedral mesh. With numerical extraction of topological features such as degenerate curves, we can analyze and verify the number of such features in a linear symmetric tensor field. In particular, we present results on: (1) how many degenerate curves, a core constituent of tensor field topology (2) how many transition points, which occur between a pair of positive and negative indexed degenerate points and (3) possible bifurcations when a small perturbation is introduced into the field.