Results 1  10
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27
Vines and vineyards by updating persistence in linear time
 In “Proc. 22nd
, 2006
"... Persistent homology is the mathematical core of recent work on shape, including reconstruction, recognition, and matching. Its pertinent information is encapsulated by a pairing of the critical values of a function, visualized by points forming a diagram in the plane. The original algorithm in [10] ..."
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Cited by 31 (9 self)
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Persistent homology is the mathematical core of recent work on shape, including reconstruction, recognition, and matching. Its pertinent information is encapsulated by a pairing of the critical values of a function, visualized by points forming a diagram in the plane. The original algorithm in [10] computes the pairs from an ordering of the simplices in a triangulation and takes worstcase time cubic in the number of simplices. The main result of this paper is an algorithm that maintains the pairing in worstcase linear time per transposition in the ordering. A sideeffect of the algorithm’s analysis is an elementary proof of the stability of persistence diagrams [7] in the special case of piecewiselinear functions. We use the algorithm to compute 1parameter families of diagrams which we apply to the study of protein folding trajectories. Categories and Subject Descriptors F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems—Geometrical problems and
Coarse and reliable geometric alignment for protein docking
 In “Proc. Pacific Sympos. Biocomput
, 2005
"... Abstract. We present an efficient algorithm for generating a small set of coarse alignments between interacting proteins using meaningful features on their surfaces. The proteins are treated as rigid bodies, but the results are more generally useful as the produced configurations can serve as input ..."
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Cited by 15 (5 self)
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Abstract. We present an efficient algorithm for generating a small set of coarse alignments between interacting proteins using meaningful features on their surfaces. The proteins are treated as rigid bodies, but the results are more generally useful as the produced configurations can serve as input to local improvement algorithms that allow for protein flexibility. We apply our algorithm to a diverse set of protein complexes from the Protein Data Bank, demonstrating the effectivity of our algorithm, both for bound and for unbound protein docking problems. 1.
QUANTIFYING HOMOLOGY CLASSES
, 2008
"... We develop a method for measuring homology classes. This involves three problems. First, we define the size of a homology class, using ideas from relative homology. Second, we define an optimal basis of a homology group to be the basis whose elements’ size have the minimal sum. We provide a greedy ..."
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Cited by 15 (4 self)
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We develop a method for measuring homology classes. This involves three problems. First, we define the size of a homology class, using ideas from relative homology. Second, we define an optimal basis of a homology group to be the basis whose elements’ size have the minimal sum. We provide a greedy algorithm to compute the optimal basis and measure classes in it. The algorithm runs in O(β^4 n³ log² n) time, where n is the size of the simplicial complex and β is the Betti number of the homology group. Third, we discuss different ways of localizing homology classes and prove some hardness results.
Analyzing and tracking burning structures in lean premixed hydrogen flames
 IEEE Transactions on Visualization and Computer Graphics
, 2010
"... Abstract—This paper presents topologybased methods to robustly extract, analyze, and track features defined as subsets of isosurfaces. First, we demonstrate how features identified by thresholding isosurfaces can be defined in terms of the Morse complex. Second, we present a specialized hierarchy t ..."
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Cited by 15 (12 self)
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Abstract—This paper presents topologybased methods to robustly extract, analyze, and track features defined as subsets of isosurfaces. First, we demonstrate how features identified by thresholding isosurfaces can be defined in terms of the Morse complex. Second, we present a specialized hierarchy that encodes the feature segmentation independent of the threshold while still providing a flexible multiresolution representation. Third, for a given parameter selection, we create detailed tracking graphs representing the complete evolution of all features in a combustion simulation over several hundred time steps. Finally, we discuss a user interface that correlates the tracking information with interactive rendering of the segmented isosurfaces enabling an indepth analysis of the temporal behavior. We demonstrate our approach by analyzing three numerical simulations of lean hydrogen flames subject to different levels of turbulence. Due to their unstable nature, lean flames burn in cells separated by locally extinguished regions. The number, area, and evolution over time of these cells provide important insights into the impact of turbulence on the combustion process. Utilizing the hierarchy, we can perform an extensive parameter study without reprocessing the data for each set of parameters. The resulting statistics enable scientists to select appropriate parameters and provide insight into the sensitivity of the results with respect to the choice of parameters. Our method allows for the first time to quantitatively correlate the turbulence of the burning process with the distribution of burning regions, properly segmented and selected. In particular, our analysis shows that counterintuitively stronger turbulence leads to larger cell structures, which burn more intensely than expected. This behavior suggests that flames could be stabilized under much leaner conditions than previously anticipated. Index Terms—Visualization, data analysis, topological data analysis, Morse complex, Reeb graph, feature detection, feature tracking, combustion simulations, burning regions. Ç
Quantifying homology classes II: Localization and stability
, 2007
"... Abstract. In the companion paper [7], we measured homology classes and computed the optimal homology basis. This paper addresses two related problems, namely, localization and stability. We localize a class with the cycle minimizing a certain objective function. We explore three different objective ..."
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Cited by 11 (3 self)
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Abstract. In the companion paper [7], we measured homology classes and computed the optimal homology basis. This paper addresses two related problems, namely, localization and stability. We localize a class with the cycle minimizing a certain objective function. We explore three different objective functions, namely, volume, diameter and radius. We show that it is NPhard to compute the smallest cycle using the former two. We also prove that the measurement defined in [7] is stable with regard to small changes of the geometry of the concerned space. 1.
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
Efficient OutputSensitive Construction of Reeb Graphs ⋆
"... Abstract. 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 dimen ..."
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Cited by 8 (4 self)
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Abstract. 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. 1
Feature tracking using Reeb graphs
 In Topologybased Methods in Visualization (Proc. TopoInVis 2009
, 2009
"... Abstract. Tracking features and exploring their temporal dynamics can aid scientists in identifying interesting time intervals in a simulation and serve as basis for performing quantitative analyses of temporal phenomena. In this paper, we develop a novel approach for tracking subsets of isosurfaces ..."
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Cited by 6 (4 self)
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Abstract. Tracking features and exploring their temporal dynamics can aid scientists in identifying interesting time intervals in a simulation and serve as basis for performing quantitative analyses of temporal phenomena. In this paper, we develop a novel approach for tracking subsets of isosurfaces, such as burning regions in simulated flames, which are defined as areas of high fuel consumption on a temperature isosurface. Tracking such regions as they merge and split over time can provide important insights into the impact of turbulence on the combustion process. However, the convoluted nature of the temperature isosurface and its rapid movement make this analysis particularly challenging. Our approach tracks burning regions by extracting a temperature isovolume from the fourdimensional spacetime temperature field. It then obtains isosurfaces for the original simulation time steps and labels individual connected “burning ” regions based on the local fuel consumption value. Based on this information, a boundary surface between burning and nonburning regions is constructed. The Reeb graph of this boundary surface is the tracking graph for burning regions.
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.
Design of data structures for mergeable trees
 In Proceedings of the 17th Annual ACMSIAM Symposium on Discrete Algorithms (SODA
, 2006
"... Motivated by an application in computational topology, we consider a novel variant of the problem of efficiently maintaining dynamic rooted trees. This variant requires merging two paths in a single operation. In contrast to the standard problem, in which only one tree arc changes at a time, a singl ..."
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Cited by 5 (1 self)
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Motivated by an application in computational topology, we consider a novel variant of the problem of efficiently maintaining dynamic rooted trees. This variant requires merging two paths in a single operation. In contrast to the standard problem, in which only one tree arc changes at a time, a single merge operation can change many arcs. In spite of this, we develop a data structure that supports merges on an nnode forest in O(log 2 n) amortized time and all other standard tree operations in O(log n) time (amortized, worstcase, or randomized depending on the underlying data structure). For the special case that occurs in the motivating application, in which arbitrary arc deletions (cuts) are not allowed, we give a data structure with an O(log n) time bound per operation. This is asymptotically optimal under certain assumptions. For the evenmore special case in which both cuts and parent queries are disallowed, we give an alternative O(log n)time solution that uses standard dynamic trees as a black box. This solution also applies to the motivating application. Our methods use previous work on dynamic trees in various ways, but the analysis of each algorithm requires novel ideas. We also investigate lower bounds for the problem under various assumptions. 1