Results 1  10
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22
Computing Contour Trees in All Dimensions
, 1999
"... We show that contour trees can be computed in all dimensions by a simple algorithm that merges two trees. Our algorithm extends, simplifies, and improves work of Tarasov and Vyalyi and of van Kreveld et al. ..."
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Cited by 131 (8 self)
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We show that contour trees can be computed in all dimensions by a simple algorithm that merges two trees. Our algorithm extends, simplifies, and improves work of Tarasov and Vyalyi and of van Kreveld et al.
MorseSmale Complexes for Piecewise Linear 3Manifolds
, 2003
"... We define the MorseSmale complex of a Morse function over a 3manifold as the overlay of the descending and ascending manifolds of all critical points. In the generic case, its 3dimensional cells are shaped like crystals and are separated by quadrangular faces. In this paper, we give a combinatori ..."
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Cited by 105 (28 self)
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We define the MorseSmale complex of a Morse function over a 3manifold as the overlay of the descending and ascending manifolds of all critical points. In the generic case, its 3dimensional cells are shaped like crystals and are separated by quadrangular faces. In this paper, we give a combinatorial algorithm for constructing such complexes for piecewise linear data.
Topological Volume Skeletonization and its Application to Transfer Function Design
 Graphical Models
, 2004
"... Topological volume skeletonization is a novel approach for automating transfer function design in visualization by extracting the topological structure of a volume dataset. The skeletonization process yields a graph called a volume skeleton tree, which consists of volumetric critical points and thei ..."
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Cited by 35 (8 self)
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Topological volume skeletonization is a novel approach for automating transfer function design in visualization by extracting the topological structure of a volume dataset. The skeletonization process yields a graph called a volume skeleton tree, which consists of volumetric critical points and their connectivity. The resultant graph provides critical field values whose color and opacity are accentuated in the design of transfer functions for direct volume rendering. Visually pleasing results of volume visualization demonstrate the feasibility of the present approach. 1
Simple and Optimal OutputSensitive Construction of Contour Trees Using Monotone Paths
, 2004
"... Contour trees are used when highdimensional data are preprocessed for efficient extraction of isocontours for the purpose of visualization. So far, efficient algorithms for contour trees are based on processing the data in sorted order. We present a new algorithm that avoids sorting of the whole da ..."
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Cited by 19 (1 self)
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Contour trees are used when highdimensional data are preprocessed for efficient extraction of isocontours for the purpose of visualization. So far, efficient algorithms for contour trees are based on processing the data in sorted order. We present a new algorithm that avoids sorting of the whole dataset, but sorts only a subset of socalled componentcritical points. They form only a small fraction of the vertices in the dataset, for typical data that arise in practice. The algorithm is simple, achieves the optimal outputsensitive bound in running time, and works in any dimension. Our experiments show that the algorithm compares favorably with the previous best algorithm.
The Safari Interface for Visualizing Timedependent Volume Data Using Isosurfaces and Contour Spectra
, 2002
"... Wedes5FI e a geometricbast for thevis5076jU7T0 of timevarying volume data of one ors everal variables as they occur ins cientific and engineeringapplications We demonsjU75 a prototype interface for gridded data, extending the contours pectrum interface of Bajaj, PasTF67j and Schikor ..."
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Cited by 16 (4 self)
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Wedes5FI e a geometricbast for thevis5076jU7T0 of timevarying volume data of one ors everal variables as they occur ins cientific and engineeringapplications We demonsjU75 a prototype interface for gridded data, extending the contours pectrum interface of Bajaj, PasTF67j and Schikore to higherdimens6T0 and to topological properties that are not decomposmpj7 And we explore the datastaj6T9T Key words: Scientific vis950jU0TT7F computational topology,s implicial mes #Corres onding author.
Topological manipulation of isosurfaces
, 2004
"... In this thesis, I show how to use the topological information encoded in an abstraction called the contour tree to enable interactive manipulation of individual contour surfaces in an isosurface scene, using an interface called the flexible isosurface. Underpinning this interface are several improve ..."
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Cited by 15 (2 self)
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In this thesis, I show how to use the topological information encoded in an abstraction called the contour tree to enable interactive manipulation of individual contour surfaces in an isosurface scene, using an interface called the flexible isosurface. Underpinning this interface are several improvements and extensions to existing work on the contour tree. The first, and most critical, extension, is the path seed: a new method of generating seeds from the contour tree for isosurface extraction. The second extension is to compute geometric information called local spatial measures for contours and store this information in the contour tree. The third extension is to use local spatial measures to simplify both the contour tree and isosurface displays. This simplification can also be used for noise removal. Lastly, this thesis extends work with contour trees from simplicial meshes to arbitrary meshes, interpolants, and tessellation cases. ii Contents ii
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 10 (3 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
IsoContour Queries and Gradient Descent with Guaranteed Delivery in Sensor Networks
"... Abstract—We study the problem of datadriven routing and navigation in a distributed sensor network over a continuous scalar field. Specifically, we address the problem of searching for the collection of sensors with readings within a specified range. This is named the isocontour query problem. We ..."
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Cited by 5 (3 self)
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Abstract—We study the problem of datadriven routing and navigation in a distributed sensor network over a continuous scalar field. Specifically, we address the problem of searching for the collection of sensors with readings within a specified range. This is named the isocontour query problem. We develop a gradient based routing scheme such that from any query node, the query message follows the signal field gradient or derived quantities and successfully discovers all isocontours of interest. Due to the existence of local maxima and minima, the guaranteed delivery requires preprocessing of the signal field and the construction of a contour tree in a distributed fashion. Our approach has the following properties: (i) the gradient routing uses only local node information and its message complexity is close to optimal, as shown by simulations; (ii) the preprocessing message complexity is linear in the number of nodes and the storage requirement for each node is a small constant. The same preprocessing also facilitates route computation between any pair of nodes where the the route lies within any user supplied range of values. I.
Combinatorial Construction of MorseSmale Complexes for Data Analysis and Visualization
, 2008
"... Scientific data is becoming increasingly complex, and sophisticated techniques are required for its effective analysis and visualization. The MorseSmale complex is an efficient data structure that represents the complete gradient flow behavior of a scalar function, and can be used to identify, ord ..."
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Cited by 2 (0 self)
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Scientific data is becoming increasingly complex, and sophisticated techniques are required for its effective analysis and visualization. The MorseSmale complex is an efficient data structure that represents the complete gradient flow behavior of a scalar function, and can be used to identify, order, and selectively remove features. This dissertation presents two algorithms for constructing MorseSmale complexes in any dimensions. The first algorithm uses persistencebased simplification to remove excess topology from an artificially generated MorseSmale complex, with important topological features preserved. The second algorithm uses discrete Morse theory to generate an explicit representation of the discrete gradient flow of a scalar function, and uses this representation to compute the MorseSmale complex directly. This second method enables a divideandconquer strategy for handling large data, and is presented in a general framework that admits many common data formats, such as simplicial, gridded, and adaptive multiresolution (AMR) meshes. Practical considerations are also presented, such as data structures, proper handling of boundary conditions, strategies to accelerate cancellations, and a method to extract a betterquality representation of the topology. A realworld example is also included, where the algorithms and techniques presented in this dissertation are applied to extract the core structure of a porous solid.
A Hybrid Parallel Algorithm for Computing and Tracking Level Set Topology
"... Abstract—The contour tree is a topological abstraction of a scalar field that captures evolution in level set connectivity. It is an effective representation for visual exploration and analysis of scientific data. We describe a workefficient, output sensitive, and scalable parallel algorithm for co ..."
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Cited by 2 (2 self)
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Abstract—The contour tree is a topological abstraction of a scalar field that captures evolution in level set connectivity. It is an effective representation for visual exploration and analysis of scientific data. We describe a workefficient, output sensitive, and scalable parallel algorithm for computing the contour tree of a scalar field defined on a domain that is represented using either an unstructured mesh or a structured grid. A hybrid implementation of the algorithm using the GPU and multicore CPU can compute the contour tree of an input containing 16 million vertices in less than ten seconds with a speedup factor of upto 13. Experiments based on an implementation in a multicore CPU environment show nearlinear speedup for large data sets. I.