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Item Rendering and Extracting Extremal Features in 3D Fields(The Eurographics Association and John Wiley & Sons Ltd., 2018) Kindlmann, Gordon L.; Chiw, Charisee; Huynh, Tri; Gyulassy, Attila; Reppy, John; Bremer, Peer-Timo; Jeffrey Heer and Heike Leitte and Timo RopinskiVisualizing and extracting three-dimensional features is important for many computational science applications, each with their own feature definitions and data types. While some are simple to state and implement (e.g. isosurfaces), others require more complicated mathematics (e.g. multiple derivatives, curvature, eigenvectors, etc.). Correctly implementing mathematical definitions is difficult, so experimenting with new features requires substantial investments. Furthermore, traditional interpolants rarely support the necessary derivatives, and approximations can reduce numerical stability. Our new approach directly translates mathematical notation into practical visualization and feature extraction, with minimal mental and implementation overhead. Using a mathematically expressive domain-specific language, Diderot, we compute direct volume renderings and particlebased feature samplings for a range of mathematical features. Non-expert users can experiment with feature definitions without any exposure to meshes, interpolants, derivative computation, etc. We demonstrate high-quality results on notoriously difficult features, such as ridges and vortex cores, using working code simple enough to be presented in its entirety.Item Stability of Dissipation Elements: A Case Study in Combustion(The Eurographics Association and John Wiley and Sons Ltd., 2014) Gyulassy, Attila; Bremer, Peer-Timo; Grout, Ray; Kolla, Hemanth; Chen, Jacqueline; Pascucci, Valerio; H. Carr, P. Rheingans, and H. SchumannRecently, dissipation elements have been gaining popularity as a mechanism for measurement of fundamental properties of turbulent flow, such as turbulence length scales and zonal partitioning. Dissipation elements segment a domain according to the source and destination of streamlines in the gradient flow field of a scalar function f :M!R. They have traditionally been computed by numerically integrating streamlines from the center of each voxel in the positive and negative gradient directions, and grouping those voxels whose streamlines terminate at the same extremal pair. We show that the same structures map well to combinatorial topology concepts developed recently in the visualization community. Namely, dissipation elements correspond to sets of cells of the Morse- Smale complex. The topology-based formulation enables a more exploratory analysis of the nature of dissipation elements, in particular, in understanding their stability with respect to small scale variations. We present two examples from combustion science that raise significant questions about the role of small scale perturbation and indeed the definition of dissipation elements themselves.Item Exploring High-Dimensional Structure via Axis-Aligned Decomposition of Linear Projections(The Eurographics Association and John Wiley & Sons Ltd., 2018) Thiagarajan, Jayaraman J.; Liu, Shusen; Ramamurthy, Karthikeyan Natesan; Bremer, Peer-Timo; Jeffrey Heer and Heike Leitte and Timo RopinskiTwo-dimensional embeddings remain the dominant approach to visualize high dimensional data. The choice of embeddings ranges from highly non-linear ones, which can capture complex relationships but are difficult to interpret quantitatively, to axis-aligned projections, which are easy to interpret but are limited to bivariate relationships. Linear project can be considered as a compromise between complexity and interpretability, as they allow explicit axes labels, yet provide significantly more degrees of freedom compared to axis-aligned projections. Nevertheless, interpreting the axes directions, which are often linear combinations of many non-trivial components, remains difficult. To address this problem we introduce a structure aware decomposition of (multiple) linear projections into sparse sets of axis-aligned projections, which jointly capture all information of the original linear ones. In particular, we use tools from Dempster-Shafer theory to formally define how relevant a given axis-aligned project is to explain the neighborhood relations displayed in some linear projection. Furthermore, we introduce a new approach to discover a diverse set of high quality linear projections and show that in practice the information of k linear projections is often jointly encoded in ~ k axis-aligned plots. We have integrated these ideas into an interactive visualization system that allows users to jointly browse both linear projections and their axis-aligned representatives. Using a number of case studies we show how the resulting plots lead to more intuitive visualizations and new insights.Item A Quantized Boundary Representation of 2D Flows(The Eurographics Association and Blackwell Publishing Ltd., 2012) Levine, Joshua; Jadhav, Shreeraj; Bhatia, Harsh; Pascucci, Valerio; Bremer, Peer-Timo; S. Bruckner, S. Miksch, and H. PfisterAnalysis and visualization of complex vector fields remain major challenges when studying large scale simulation of physical phenomena. The primary reason is the gap between the concepts of smooth vector field theory and their computational realization. In practice, researchers must choose between either numerical techniques, with limited or no guarantees on how they preserve fundamental invariants, or discrete techniques which limit the precision at which the vector field can be represented. We propose a new representation of vector fields that combines the advantages of both approaches. In particular, we represent a subset of possible streamlines by storing their paths as they traverse the edges of a triangulation. Using only a finite set of streamlines creates a fully discrete version of a vector field that nevertheless approximates the smooth flow up to a user controlled error bound. The discrete nature of our representation enables us to directly compute and classify analogues of critical points, closed orbits, and other common topological structures. Further, by varying the number of divisions (quantizations) used per edge, we vary the resolution used to represent the field, allowing for controlled precision. This representation is compact in memory and supports standard vector field operations.Item Distortion-Guided Structure-Driven Interactive Exploration of High-Dimensional Data(The Eurographics Association and John Wiley and Sons Ltd., 2014) Liu, Shusen; Wang, Bei; Bremer, Peer-Timo; Pascucci, Valerio; H. Carr, P. Rheingans, and H. SchumannDimension reduction techniques are essential for feature selection and feature extraction of complex highdimensional data. These techniques, which construct low-dimensional representations of data, are typically geometrically motivated, computationally efficient and approximately preserve certain structural properties of the data. However, they are often used as black box solutions in data exploration and their results can be difficult to interpret. To assess the quality of these results, quality measures, such as co-ranking [LV09], have been proposed to quantify structural distortions that occur between high-dimensional and low-dimensional data representations. Such measures could be evaluated and visualized point-wise to further highlight erroneous regions [MLGH13]. In this work, we provide an interactive visualization framework for exploring high-dimensional data via its twodimensional embeddings obtained from dimension reduction, using a rich set of user interactions. We ask the following question: what new insights do we obtain regarding the structure of the data, with interactive manipulations of its embeddings in the visual space? We augment the two-dimensional embeddings with structural abstractions obtained from hierarchical clusterings, to help users navigate and manipulate subsets of the data. We use point-wise distortion measures to highlight interesting regions in the domain, and further to guide our selection of the appropriate level of clusterings that are aligned with the regions of interest. Under the static setting, point-wise distortions indicate the level of structural uncertainty within the embeddings. Under the dynamic setting, on-thefly updates of point-wise distortions due to data movement and data deletion reflect structural relations among different parts of the data, which may lead to new and valuable insights.Item Extracting Features from Time-Dependent Vector Fields Using Internal Reference Frames(The Eurographics Association and John Wiley and Sons Ltd., 2014) Bhatia, Harsh; Pascucci, Valerio; Kirby, Robert M.; Bremer, Peer-Timo; H. Carr, P. Rheingans, and H. SchumannExtracting features from complex, time-dependent flow fields remains a significant challenge despite substantial research efforts, especially because most flow features of interest are defined with respect to a given reference frame. Pathline-based techniques, such as the FTLE field, are complex to implement and resource intensive, whereas scalar transforms, such as l2, often produce artifacts and require somewhat arbitrary thresholds. Both approaches aim to analyze the flow in a more suitable frame, yet neither technique explicitly constructs one. This paper introduces a new data-driven technique to compute internal reference frames for large-scale complex flows. More general than uniformly moving frames, these frames can transform unsteady fields, which otherwise require substantial processing of resources, into a sequence of individual snapshots that can be analyzed using the large body of steady-flow analysis techniques. Our approach is simple, theoretically well-founded, and uses an embarrassingly parallel algorithm for structured as well as unstructured data. Using several case studies from fluid flow and turbulent combustion, we demonstrate that internal frames are distinguished, result in temporally coherent structures, and can extract well-known as well as notoriously elusive features one snapshot at a time.Item The Grassmannian Atlas: A General Framework for Exploring Linear Projections of High-Dimensional Data(The Eurographics Association and John Wiley & Sons Ltd., 2016) Liu, Shusen; Bremer, Peer-Timo; Jayaraman, Jayaraman Thiagarajan; Wang, Bei; Summa, Brian; Pascucci, Valerio; Kwan-Liu Ma and Giuseppe Santucci and Jarke van WijkLinear projections are one of the most common approaches to visualize high-dimensional data. Since the space of possible projections is large, existing systems usually select a small set of interesting projections by ranking a large set of candidate projections based on a chosen quality measure. However, while highly ranked projections can be informative, some lower ranked ones could offer important complementary information. Therefore, selection based on ranking may miss projections that are important to provide a global picture of the data. The proposed work fills this gap by presenting the Grassmannian Atlas, a framework that captures the global structures of quality measures in the space of all projections, which enables a systematic exploration of many complementary projections and provides new insights into the properties of existing quality measures.Item Interactive Investigation of Traffic Congestion on Fat-Tree Networks Using TREESCOPE(The Eurographics Association and John Wiley & Sons Ltd., 2018) Bhatia, Harsh; Jain, Nikhil; Bhatele, Abhinav; Livnat, Yarden; Domke, Jens; Pascucci, Valerio; Bremer, Peer-Timo; Jeffrey Heer and Heike Leitte and Timo RopinskiParallel simulation codes often suffer from performance bottlenecks due to network congestion, leaving millions of dollars of investments underutilized. Given a network topology, it is critical to understand how different applications, job placements, routing schemes, etc., are affected by and contribute to network congestion, especially for large and complex networks. Understanding and optimizing communication on large-scale networks is an active area of research. Domain experts often use exploratory tools to develop both intuitive and formal metrics for network health and performance. This paper presents TREESCOPE, an interactive, web-based visualization tool for exploring network traffic on large-scale fat-tree networks. TREESCOPE encodes the network topology using a tailored matrix-based representation and provides detailed visualization of all traffic in the network. We report on the design process of TREESCOPE, which has been received positively by network researchers as well as system administrators. Through case studies of real and simulated data, we demonstrate how TREESCOPE's visual design and interactive support for complex queries on network traffic can provide experts with new insights into the occurrences and causes of congestion in the network.Item Visual Exploration of High-Dimensional Data through Subspace Analysis and Dynamic Projections(The Eurographics Association and John Wiley & Sons Ltd., 2015) Liu, Shusen; Wang, Bei; Thiagarajan, Jayaraman J.; Bremer, Peer-Timo; Pascucci, Valerio; H. Carr, K.-L. Ma, and G. SantucciWe introduce a novel interactive framework for visualizing and exploring high-dimensional datasets based on subspace analysis and dynamic projections. We assume the high-dimensional dataset can be represented by a mixture of low-dimensional linear subspaces with mixed dimensions, and provide a method to reliably estimate the intrinsic dimension and linear basis of each subspace extracted from the subspace clustering. Subsequently, we use these bases to define unique 2D linear projections as viewpoints from which to visualize the data. To understand the relationships among the different projections and to discover hidden patterns, we connect these projections through dynamic projections that create smooth animated transitions between pairs of projections. We introduce the view transition graph, which provides flexible navigation among these projections to facilitate an intuitive exploration. Finally, we provide detailed comparisons with related systems, and use real-world examples to demonstrate the novelty and usability of our proposed framework.