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Computing Persistent Homology
 Discrete Comput. Geom
"... We show that the persistent homology of a filtered d dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enabl ..."
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Cited by 148 (20 self)
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We show that the persistent homology of a filtered d dimensional simplicial complex is simply the standard homology of a particular graded module over a polynomial ring. Our analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields. It also enables us to derive a natural algorithm for computing persistent homology of spaces in arbitrary dimension over any field. This results generalizes and extends the previously known algorithm that was restricted to subcomplexes of S and Z2 coefficients. Finally, our study implies the lack of a simple classification over nonfields. Instead, we give an algorithm for computing individual persistent homology groups over an arbitrary PIDs in any dimension.
Topology and Data
, 2008
"... An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that ..."
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Cited by 119 (4 self)
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An important feature of modern science and engineering is that data of various kinds is being produced at an unprecedented rate. This is so in part because of new experimental methods, and in part because of the increase in the availability of high powered computing technology. It is also clear that the nature of the data
Fast construction of the VietorisRips complex
 Computer and Graphics
, 2010
"... (To appear in Computer & Graphics) The VietorisRips complex characterizes the topology of a point set. This complex is popular in topological data analysis as its construction extends easily to higher dimensions. We formulate a twophase approach for its construction that separates geometry fro ..."
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Cited by 20 (3 self)
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(To appear in Computer & Graphics) The VietorisRips complex characterizes the topology of a point set. This complex is popular in topological data analysis as its construction extends easily to higher dimensions. We formulate a twophase approach for its construction that separates geometry from topology. We survey methods for the first phase, give three algorithms for the second phase, implement all algorithms, and present experimental results. Our software can also be used for constructing any clique complex, such as the weak witness complex. 1
The tidy set: A minimal simplicial set for computing homology of clique complexes
 In Proc. ACM Symposium of Computational Geometry
, 2010
"... We introduce the tidy set, a minimal simplicial set that captures the topology of a simplicial complex. The tidy set is particularly effective for computing the homology of clique complexes. This family of complexes include the VietorisRips complex and the weak witness complex, methods that are pop ..."
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Cited by 17 (1 self)
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We introduce the tidy set, a minimal simplicial set that captures the topology of a simplicial complex. The tidy set is particularly effective for computing the homology of clique complexes. This family of complexes include the VietorisRips complex and the weak witness complex, methods that are popular in topological data analysis. The key feature of our approach is that it skips constructing the clique complex. We give algorithms for constructing tidy sets, implement them, and present experiments. Our preliminary results show that tidy sets are orders of magnitude smaller than clique complexes, giving us a homology engine with small memory requirements.
Computing Multidimensional Persistence
"... The theory of multidimensional persistence captures the topology of a multifiltration – a multiparameter family of increasing spaces. Multifiltrations arise naturally in the topological analysis of scientific data. In this paper, we give a polynomial time algorithm for computing multidimensional pe ..."
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Cited by 12 (1 self)
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The theory of multidimensional persistence captures the topology of a multifiltration – a multiparameter family of increasing spaces. Multifiltrations arise naturally in the topological analysis of scientific data. In this paper, we give a polynomial time algorithm for computing multidimensional persistence.
Topology based selection and curation of level sets
 IN TOPOINVIS 2007, ACCEPTED
, 2007
"... The selection of appropriate level sets for the quantitative visualization of three dimensional imaging or simulation data is a problem that is both fundamental and essential. The selected level set needs to satisfy several topological and geometric constraints to be useful for subsequent quantita ..."
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Cited by 9 (4 self)
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The selection of appropriate level sets for the quantitative visualization of three dimensional imaging or simulation data is a problem that is both fundamental and essential. The selected level set needs to satisfy several topological and geometric constraints to be useful for subsequent quantitative processing and visualization. For an initial selection of an isosurface, guided by contour tree data structures, we detect the topological features by computing stable and unstable manifolds of the critical points of the distance function induced by the isosurface. We further enhance the description of these features by associating geometric attributes with them. We then rank the attributed features and provide a handle to them for curation of the topological anomalies.
The Ring of Algebraic Functions on Persistence Bar Codes
, 2012
"... Persistent homology ([3], [13]) is a fundamental tool in the area of computational topology. It can be used to infer topological structure in data sets (see [1], [4]), but variations on the method can be applied to study aspects of the shape of point clouds which are not overtly topological ([5], [8 ..."
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Cited by 8 (2 self)
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Persistent homology ([3], [13]) is a fundamental tool in the area of computational topology. It can be used to infer topological structure in data sets (see [1], [4]), but variations on the method can be applied to study aspects of the shape of point clouds which are not overtly topological ([5], [8]). The
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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Cited by 7 (3 self)
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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