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Topological landscapes: A terrain metaphor for scientific data
 IEEE Transactions on Visualization and Computer Graphics
"... Abstract—Scientific visualization and illustration tools are designed to help people understand the structure and complexity of scientific data with images that are as informative and intuitive as possible. In this context the use of metaphors plays an important role since they make complex informat ..."
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Abstract—Scientific visualization and illustration tools are designed to help people understand the structure and complexity of scientific data with images that are as informative and intuitive as possible. In this context the use of metaphors plays an important role since they make complex information easily accessible by using commonly known concepts. In this paper we propose a new metaphor, called “Topological Landscapes, ” which facilitates understanding the topological structure of scalar functions. The basic idea is to construct a terrain with the same topology as a given dataset and to display the terrain as an easily understood representation of the actual input data. In this projection from an ndimensional scalar function to a twodimensional (2D) model we preserve function values of critical points, the persistence (function span) of topological features, and one possible additional metric property (in our examples volume). By displaying this topologically equivalent landscape together with the original data we harness the natural human proficiency in understanding terrain topography and make complex topological information easily accessible.
D.: Structuring feature space: A nonparametric method for volumetric transfer function generation
 IEEE Trans. Vis. Comput. Graph
"... Fig. 1. An application of nonparametric clustering to the value vs. value gradient magnitude feature space using the CT visible woman feet dataset. Abstract — The use of multidimensional transfer functions for direct volume rendering has been shown to be an effective means of extracting materials ..."
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Fig. 1. An application of nonparametric clustering to the value vs. value gradient magnitude feature space using the CT visible woman feet dataset. Abstract — The use of multidimensional transfer functions for direct volume rendering has been shown to be an effective means of extracting materials and their boundaries for both scalar and multivariate data. The most common multidimensional transfer function consists of a twodimensional (2D) histogram with axes representing a subset of the feature space (e.g., value vs. value gradient magnitude), with each entry in the 2D histogram being the number of voxels at a given feature space pair. Users then assign color and opacity to the voxel distributions within the given feature space through the use of interactive widgets (e.g., box, circular, triangular selection). Unfortunately, such tools lead users through a trialanderror approach as they assess which data values within the feature space map to a given area of interest within the volumetric space. In this work, we propose the addition of nonparametric clustering within the transfer function feature space in order to extract patterns and guide transfer function generation. We apply a nonparametric kernel density estimation to group voxels of similar features within the 2D histogram. These groups are then binned and colored based on their estimated density, and the user may interactively grow and shrink the binned regions to explore feature boundaries and extract regions of interest. We also extend this scheme to temporal volumetric data in which time steps of 2D histograms are composited into a histogram volume. A threedimensional (3D) density estimation is then applied, and users can explore regions within the feature space across time without adjusting the transfer function at each time step. Our work
Efficient Algorithms for Computing Reeb Graphs
"... The Reeb graph tracks topology changes in level sets of a scalar function and finds applications in scientific visualization and geometric modeling. We describe an algorithm that constructs the Reeb graph of a Morse function defined on a 3manifold. Our algorithm maintains connected components of th ..."
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Cited by 16 (2 self)
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The Reeb graph tracks topology changes in level sets of a scalar function and finds applications in scientific visualization and geometric modeling. We describe an algorithm that constructs the Reeb graph of a Morse function defined on a 3manifold. Our algorithm maintains connected components of the two dimensional levels sets as a dynamic graph and constructs the Reeb graph in O(n log n+n log g(log log g) 3) time, where n is the number of triangles in the tetrahedral mesh representing the 3manifold and g is the maximum genus over all level sets of the function. We extend this algorithm to construct Reeb graphs of dmanifolds in O(n log n(log log n) 3) time, where n is the number of triangles in the simplicial complex that represents the dmanifold. Our result is a significant improvement over the previously known O(n 2) algorithm. Finally, we present experimental results of our implementation and demonstrate that, in practice, our algorithm for 3manifolds performs better than what the theoretical bound suggests.
Loop Surgery for Volumetric Meshes: Reeb Graphs Reduced to Contour Trees
"... Fig. 1. The Reeb graph of a pressure stress function on the volumetric mesh of a brake disk is shown at several scales of hypervolumebased simplification. At the finest resolution of this dataset (3.5 million tetrahedra), our approach computes the Reeb graph in 7.8 seconds while the fastest previou ..."
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Cited by 13 (6 self)
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Fig. 1. The Reeb graph of a pressure stress function on the volumetric mesh of a brake disk is shown at several scales of hypervolumebased simplification. At the finest resolution of this dataset (3.5 million tetrahedra), our approach computes the Reeb graph in 7.8 seconds while the fastest previous techniques [19, 12] do not produce a result. Abstract—This paper introduces an efficient algorithm for computing the Reeb graph of a scalar function f defined on a volumetric mesh M in R3. We introduce a procedure called loop surgery that transforms M into a mesh M ′ by a sequence of cuts and guarantees the Reeb graph of f (M′) to be loop free. Therefore, loop surgery reduces Reeb graph computation to the simpler problem of computing a contour tree, for which wellknown algorithms exist that are theoretically efficient (O(nlogn)) and fast in practice. Inverse cuts reconstruct the loops removed at the beginning. The time complexity of our algorithm is that of a contour tree computation plus a loop surgery overhead, which depends on the number of handles of the mesh. Our systematic experiments confirm that for reallife volumetric data, this overhead is comparable to the computation of the contour tree, demonstrating virtually linear scalability on meshes ranging from 70 thousand to 3.5 million tetrahedra. Performance numbers show that our algorithm, although restricted to volumetric data, has an average speedup factor of 6,500 over the previous fastest techniques, handling larger and more complex datasets. We demonstrate the versatility of our approach by extending fast topologically clean isosurface extraction to nonsimply connected domains. We apply this technique in the context of pressure analysis for mechanical design. In this case, our technique produces results in matter of seconds even for the largest models. For the same models, previous Reeb graph techniques do not produce a result. Index Terms—Reeb graph, scalar field topology, isosurfaces, topological simplification. 1
Interactive exploration and analysis of large scale turbulent combustion using topologybased data segmentation
 IEEE Transactions on Visualization and Computer Graphics
, 2011
"... Abstract—Largescale simulations are increasingly being used to study complex scientific and engineering phenomena. As a result, advanced visualization and data analysis are also becoming an integral part of the scientific process. Often, a key step in extracting insight from these large simulations ..."
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Cited by 12 (7 self)
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Abstract—Largescale simulations are increasingly being used to study complex scientific and engineering phenomena. As a result, advanced visualization and data analysis are also becoming an integral part of the scientific process. Often, a key step in extracting insight from these large simulations involves the definition, extraction, and evaluation of features in the space and time coordinates of the solution. However, in many applications these features involve a range of parameters and decisions that will affect the quality and direction of the analysis. Examples include particular level sets of a specific scalar field, or local inequalities between derived quantities. A critical step in the analysis is to understand how these arbitrary parameters/decisions impact the statistical properties of the features, since such a characterization will help to evaluate the conclusions of the analysis as a whole. We present a new topological framework that in a single pass extracts and encodes entire families of possible features definitions as well as their statistical properties. For each time step we construct a hierarchical merge tree a highly compact, yet flexible feature representation. While this data structure is more than two orders of magnitude smaller than the raw simulation data it allows us to extract a set of feature for any given parameter selection in a postprocessing step. Furthermore, we augment the trees with additional attributes making it possible to gather a large number of useful global, local, as well as conditional statistic that would otherwise be extremely difficult to compile. We also use this representation to create tracking graphs that describe the temporal evolution of the features over time. Our system provides a linkedview interface to explore the timeevolution of the graph interactively alongside the segmentation, thus making it possible to perform extensive data analysis in a very efficient manner. We demonstrate our framework
Efficient OutputSensitive Construction of Reeb Graphs
"... The Reeb graph tracks topology changes in level sets of a scalar function and finds applications in scientific visualization and geometric modeling. This paper describes a nearoptimal twostep algorithm that constructs the Reeb graph of a Morse function defined over manifolds in any dimension. The ..."
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Cited by 9 (4 self)
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The Reeb graph tracks topology changes in level sets of a scalar function and finds applications in scientific visualization and geometric modeling. This paper describes a nearoptimal twostep algorithm that constructs the Reeb graph of a Morse function defined over manifolds in any dimension. The algorithm first identifies the critical points of the input manifold, and then connects these critical points in the second step to obtain the Reeb graph. A simplification mechanism based on topological persistence aids in the removal of noise and unimportant features. A radial layout scheme results in a featuredirected drawing of the Reeb graph. Experimental results demonstrate the efficiency of the Reeb graph construction in practice and its applications.
FeatureBased Statistical Analysis of Combustion Simulation Data
"... Fig. 1. Our framework provides a natural and intuitive workflow for the exploration of global trends in featurebased statistics. By efficiently encoding hierarchical metadata in a preprocessing step, interactive data exploration of the equivalent of one terabyte of simulation data is performed o ..."
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Cited by 9 (4 self)
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Fig. 1. Our framework provides a natural and intuitive workflow for the exploration of global trends in featurebased statistics. By efficiently encoding hierarchical metadata in a preprocessing step, interactive data exploration of the equivalent of one terabyte of simulation data is performed on a commodity desktop. Abstract — We present a new framework for featurebased statistical analysis of largescale scientific data and demonstrate its effectiveness by analyzing features from Direct Numerical Simulations (DNS) of turbulent combustion. Turbulent flows are ubiquitous and account for transport and mixing processes in combustion, astrophysics, fusion, and climate modeling among other disciplines. They are also characterized by coherent structure or organized motion, i.e. nonlocal entities whose geometrical features can directly impact molecular mixing and reactive processes. While traditional multipoint statistics provide correlative information, they lack nonlocal structural information, and hence, fail to provide mechanistic causality information between organized fluid motion and mixing and reactive processes. Hence, it is of great interest to capture and track flow features and their statistics together with their correlation with relevant scalar quantities, e.g. temperature or species concentrations. In our approach we encode the set of all possible flow features by precomputing merge trees augmented with attributes, such as statistical moments of various scalar fields, e.g. temperature, as well as lengthscales computed via spectral analysis. The computation is performed in an efficient streaming manner in a preprocessing step and results in a collection of metadata that is orders of magnitude smaller than the original simulation data. This metadata is sufficient to support a fully flexible and interactive analysis
Applying Manifold Learning to Plotting Approximate Contour Trees
"... A contour tree is a powerful tool for delineating the topological evolution of isosurfaces of a singlevalued function, and thus has been frequently used as a means of extracting features from volumes and their timevarying behaviors. Several sophisticated algorithms have been proposed for construct ..."
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Cited by 7 (0 self)
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A contour tree is a powerful tool for delineating the topological evolution of isosurfaces of a singlevalued function, and thus has been frequently used as a means of extracting features from volumes and their timevarying behaviors. Several sophisticated algorithms have been proposed for constructing contour trees while they often complicate the software implementation especially for higherdimensional cases such as timevarying volumes. This paper presents a simple yet effective approach to plotting in 3D space, approximate contour trees from a set of scattered samples embedded in the highdimensional space. Our main idea is to take advantage of manifold learning so that we can elongate the distribution of highdimensional data samples to embed it into a lowdimensional space while respecting its local proximity of sample points. The contribution of this paper lies in the introduction of new distance metrics to manifold learning, which allows us to reformulate existing algorithms as a variant of currently available dimensionality reduction scheme. Efficient reduction of data sizes together with segmentation capability is also developed to equip our approach with a coarsetofine analysis even for largescale datasets. Examples are provided to demonstrate that our proposed scheme can successfully traverse the features of volumes and their temporal behaviors through the constructed contour trees.
Symmetry in Scalar Field Topology
"... Fig. 1. Symmetric patterns identified using contour trees in electron microscopy data of RuBisCO molecule in complex with RuBisCO large subunit methyltransferase (EMDB 1734). (a) Volume rendering of the molecule highlighting repeating structures in the scalar field. (b) Four different types of regio ..."
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Fig. 1. Symmetric patterns identified using contour trees in electron microscopy data of RuBisCO molecule in complex with RuBisCO large subunit methyltransferase (EMDB 1734). (a) Volume rendering of the molecule highlighting repeating structures in the scalar field. (b) Four different types of regions, indicative of the different subunits in the molecule, identified by the symmetry detection algorithm shown in cyan, magenta, brown, and violet. Regions with the same color are symmetric with respect to the scalar field distribution. (c) Subtrees of the contour tree are classified into different groups based on similarity. Subtrees belonging to a common group are shown with the same color and the corresponding regions are identified to be symmetric. Abstract — Study of symmetric or repeating patterns in scalar fields is important in scientific data analysis because it gives deep insights into the properties of the underlying phenomenon. Though geometric symmetry has been well studied within areas like shape processing, identifying symmetry in scalar fields has remained largely unexplored due to the high computational cost of the associated algorithms. We propose a computationally efficient algorithm for detecting symmetric patterns in a scalar field distribution by analysing the topology of level sets of the scalar field. Our algorithm computes the contour tree of a given scalar field and identifies subtrees that are similar. We define a robust similarity measure for comparing subtrees of the contour tree and use it to group similar subtrees together. Regions of the domain corresponding to subtrees that belong to a common group are extracted and reported to be symmetric. Identifying symmetry in scalar fields finds applications in visualization, data exploration, and feature detection. We describe two applications in detail: symmetryaware transfer function design and symmetryaware isosurface extraction. Index Terms—Scalar field symmetry, contour tree, similarity measure, persistence, isosurface extraction, transfer function design. 1
Direct Interval Volume Visualization
, 2010
"... We extend direct volume rendering with a unified model for generalized isosurfaces, also called interval volumes, allowing a wider spectrum of visual classification. We generalize the concept of scaleinvariant opacity—typical for isosurface rendering— to semitransparent interval volumes. Scalein ..."
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Cited by 6 (2 self)
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We extend direct volume rendering with a unified model for generalized isosurfaces, also called interval volumes, allowing a wider spectrum of visual classification. We generalize the concept of scaleinvariant opacity—typical for isosurface rendering— to semitransparent interval volumes. Scaleinvariant rendering is independent of physical space dimensions and therefore directly facilitates the analysis of data characteristics. Our model represents sharp isosurfaces as limits of interval volumes and combines them with features of direct volume rendering. Our objective is accurate rendering, guaranteeing that all isosurfaces and interval volumes are visualized in a crackfree way with correct spatial ordering. We achieve simultaneous direct and interval volume rendering by extending preintegration and explicit peak finding with datadriven splitting of ray integration and hybrid computation in physical and data domains. Our algorithm is suitable for efficient parallel processing for interactive applications as demonstrated by our CUDA implementation.