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34
Saddle connectors  an approach to visualizing the topological skeleton of complex 3D vector fields
 IN PROC. IEEE VISUALIZATION
, 2003
"... One of the reasons that topological methods have a limited popularity for the visualization of complex 3D flow fields is the fact that such topological structures contain a number of separating stream surfaces. Since these stream surfaces tend to hide each other as well as other topological feature ..."
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Cited by 52 (17 self)
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One of the reasons that topological methods have a limited popularity for the visualization of complex 3D flow fields is the fact that such topological structures contain a number of separating stream surfaces. Since these stream surfaces tend to hide each other as well as other topological features, for complex 3D topologies the visualizations become cluttered and hardly interpretable. This paper proposes to use particular stream lines called saddle connectors instead of separating stream surfaces and to depict single surfaces only on user demand. We discuss properties and computational issues of saddle connectors and apply these methods to complex flow data. We show that the use of saddle connectors makes topological skeletons available as a valuable visualization tool even for topologically complex 3D flow data.
Simplicial Edge Representation of Protein Structures and Alpha Contact Potential with Confidence Measure
 Proteins
, 2003
"... Protein representation and potential function are two important ingredients for studying protein folding, equilibrium thermodynamics, and sequence design. We introduce a novel geometric representation of protein contact interactions using the edge simplices from the alpha shape of the protein struct ..."
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Cited by 22 (15 self)
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Protein representation and potential function are two important ingredients for studying protein folding, equilibrium thermodynamics, and sequence design. We introduce a novel geometric representation of protein contact interactions using the edge simplices from the alpha shape of the protein structure. This representation can eliminate implausible neighbors that are not in physical contact, and can avoid spurious contact between two residues when a third residue is between them. We developed statistical alpha contact potential using an oddsratio model. A studentized bootstrap method was then introduced to assess the 95% confidence intervals for each of the 210 propensity parameters. We found, with confidence, that there is significant longrange propensity (>30 residues apart) for hydrophobic interactions. We tested alpha contact potential for native structure discrimination using several sets of decoy structures, and found that it often performs comparably with atombased potentials requiring manymore parameters. We also show that accurate geometric representation is important, and that alpha contact potential has better performance than potential defined by cutoff distance between geometric centers of side chains. Hierarchical clustering of alpha contact potentials reveals natural grouping of residues. To explore the relationship between shape and physicochemical representations, we tested the minimum alphabet size necessary for native structure discrimination. We found that there is no significant difference in performance of discrimination when alphabet size varies from 7 to 20, if geometry is represented accurately by alpha simplicial edges. This result suggests that the geometry of packing plays an important role, but the specific residue types are often interch...
Isotopic implicit surface meshing
 In Symposium on Theory of computing
, 2004
"... This paper addresses the problem of piecewise linear approximation of implicit surfaces. We first give a criterion ensuring that the zeroset of a smooth function and the one of a piecewise linear approximation of it are isotopic. Then, we deduce from this criterion an implicit surface meshing algor ..."
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Cited by 21 (1 self)
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This paper addresses the problem of piecewise linear approximation of implicit surfaces. We first give a criterion ensuring that the zeroset of a smooth function and the one of a piecewise linear approximation of it are isotopic. Then, we deduce from this criterion an implicit surface meshing algorithm certifying that the output mesh is isotopic to the actual implicit surface. This is the first algorithm achieving this goal in a provably correct way. 1
Analysis of scalar fields over point cloud data
 In Proc. 20th ACMSIAM Sympos. Discrete Algorithms
, 2009
"... Given a realvalued function f defined over some metric space X, is it possible to recover some structural information about f from the sole information of its values at a finite set L ⊆ X of sample points, whose pairwise distances in X are given? We provide a positive answer to this question. More ..."
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Cited by 17 (5 self)
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Given a realvalued function f defined over some metric space X, is it possible to recover some structural information about f from the sole information of its values at a finite set L ⊆ X of sample points, whose pairwise distances in X are given? We provide a positive answer to this question. More precisely, taking advantage of recent advances on the front of stability for persistence diagrams, we introduce a novel algebraic construction, based on a pair of nested families of simplicial complexes built on top of the point cloud L, from which the persistence diagram of f can be faithfully approximated. We derive from this construction a series of algorithms for the analysis of scalar fields from point cloud data. These algorithms are simple and easy to implement, have reasonable complexities, and come with theoretical guarantees. To illustrate the generality of the approach, we present some experimental results obtained in various applications, ranging from clustering to sensor networks (see the electronic version of the paper for color pictures). 1
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
Optimal discrete Morse functions for 2manifolds
 Computational Geometry: Theory and Applications
, 2003
"... Morse theory is a powerful tool in its applications to computational topology, computer graphics and geometric modeling. It was originally formulated for smooth manifolds. Recently, Robin Forman formulated a version of this theory for discrete structures such as cell complexes. It opens up several c ..."
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Cited by 14 (4 self)
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Morse theory is a powerful tool in its applications to computational topology, computer graphics and geometric modeling. It was originally formulated for smooth manifolds. Recently, Robin Forman formulated a version of this theory for discrete structures such as cell complexes. It opens up several categories of interesting objects (particularly meshes) to applications of Morse theory. Once a Morse function has been defined on a manifold, then information about its topology can be deduced from its critical elements. The main objective of this paper is to introduce a linear algorithm to define optimal discrete Morse functions on discrete 2manifolds, where optimality entails having the least number of critical elements. The algorithm presented is also extended to general finite cell complexes of dimension at most 2, with no guarantee of optimality.
TerraStream: From elevation data to watershed hierarchies
 Proc. ACM Sympos. on Advances in Geographic Information Systems
"... We consider the problem of extracting a river network and a watershed hierarchy from a terrain given as a set of irregularly spaced points. We describe TerraStream, a “pipelined ” solution that consists of four main stages: construction of a digital elevation model (DEM), hydrological conditioning, ..."
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Cited by 11 (7 self)
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We consider the problem of extracting a river network and a watershed hierarchy from a terrain given as a set of irregularly spaced points. We describe TerraStream, a “pipelined ” solution that consists of four main stages: construction of a digital elevation model (DEM), hydrological conditioning, extraction of river networks, and construction of a watershed hierarchy. Our approach has several advantages over existing methods. First, we design and implement the pipeline so each stage is scalable to massive data sets; a single nonscalable stage would create a bottleneck and limit overall scalability. Second, we develop the algorithms in a general framework so that they work for both TIN and grid DEMs. TerraStream is flexible and allows users to choose from various models and parameters, yet our pipeline is designed to reduce (or eliminate) the need for manual intervention between stages. We have implemented TerraStream and present experimental results on real elevation point sets that show that our approach handles massive multigigabyte terrain data sets. For example, we can process a data set containing over 300 million points—over 20GB of raw data—in under 26 hours, where most of the time (76%) is spent in the initial CPUintensive DEM construction stage. 1
Stable Morse Decompositions for Piecewise Constant Vector Fields on Surfaces,” Computer Graphics Forum, to appear
, 2011
"... Abstract—In this paper, we introduce a new approach to computing a Morse decomposition of a vector field on a triangulated manifold surface. The basic idea is to convert the input vector field to a piecewise constant (PC) vector field, whose trajectories can be computed using simple geometric rules. ..."
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Cited by 10 (6 self)
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Abstract—In this paper, we introduce a new approach to computing a Morse decomposition of a vector field on a triangulated manifold surface. The basic idea is to convert the input vector field to a piecewise constant (PC) vector field, whose trajectories can be computed using simple geometric rules. To overcome the intrinsic difficulty in PC vector fields (in particular, discontinuity along mesh edges), we borrow results from the theory of differential inclusions. The input vector field and its PC variant have similar Morse decompositions. We introduce a robust and efficient algorithm to compute Morse decompositions of a PC vector field. Our approach provides subtriangle precision for Morse sets. In addition, we describe a Morse set classification framework which we use to color code the Morse sets in order to enhance the visualization. We demonstrate the benefits of our approach with three wellknown simulation data sets, for which our method has produced Morse decompositions that are similar to or finer than those obtained using existing techniques, and is over an order of magnitude faster. Index Terms—Morse decomposition, vector field topology. Ç 1
Stability of Critical Points with Interval Persistence
"... Scalar functions defined on a topological space Ω are at the core of many applications such as shape matching, visualization and physical simulations. Topological persistence is an approach to characterizing these functions. It measures how long topological structures in the sublevel sets {x ∈ Ω: f ..."
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Cited by 6 (3 self)
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Scalar functions defined on a topological space Ω are at the core of many applications such as shape matching, visualization and physical simulations. Topological persistence is an approach to characterizing these functions. It measures how long topological structures in the sublevel sets {x ∈ Ω: f(x) ≤ c} persist as c changes. Recently it was shown that the critical values defining a topological structure with relatively large persistence remain almost unaffected by small perturbations. This result suggests that topological persistence is a good measure for matching and comparing scalar functions. We extend these results to critical points in the domain by redefining persistence and critical points and replacing sublevel sets {x ∈ Ω: f(x) ≤ c} with interval sets {x ∈ Ω: a ≤ f(x) < b}. With these modifications we establish a stability result for critical points. This result is strengthened for maxima that can be used for matching two scalar functions.