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62
Topologybased Simplification for Feature Extraction from 3D Scalar Fields
"... This paper describes a topological approach for simplifying continuous functions defined on volumetric domains. The MorseSmale complex provides a segmentation of the domain into monotonic regions having uniform gradient flow behavior. We present a combinatorial algorithm that simplifies the Morse ..."
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Cited by 35 (16 self)
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This paper describes a topological approach for simplifying continuous functions defined on volumetric domains. The MorseSmale complex provides a segmentation of the domain into monotonic regions having uniform gradient flow behavior. We present a combinatorial algorithm that simplifies the MorseSmale complex by repeated application of two atomic operations that removes pairs of critical points. The simplification procedure leaves important critical points untouched, and is therefore useful for extracting features. We present a visualization of the simplified topology.
A topological approach to simplification of threedimensional scalar functions
 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS (SPECIAL ISSUE IEEE VISUALIZATION
, 2006
"... This paper describes an efficient combinatorial method for simplification of topological features in a 3D scalar function. The MorseSmale complex, which provides a succinct representation of a function’s associated gradient flow field, is used to identify topological features and their significance ..."
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Cited by 23 (11 self)
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This paper describes an efficient combinatorial method for simplification of topological features in a 3D scalar function. The MorseSmale complex, which provides a succinct representation of a function’s associated gradient flow field, is used to identify topological features and their significance. The simplification process, guided by the MorseSmale complex, proceeds by repeatedly applying two atomic operations that each remove a pair of critical points from the complex. Efficient storage of the complex results in execution of these atomic operations at interactive rates. Visualization of the simplified complex shows that the simplification preserves significant topological features and removes small features and noise.
Efficient Computation of MorseSmale Complexes for ThreeDimensional Scalar Functions
, 2007
"... The MorseSmale complex is an efficient representation of the gradient behavior of a scalar function, and critical points paired by the complex identify topological features and their importance. We present an algorithm that constructs the MorseSmale complex in a series of sweeps through the data, ..."
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Cited by 23 (11 self)
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The MorseSmale complex is an efficient representation of the gradient behavior of a scalar function, and critical points paired by the complex identify topological features and their importance. We present an algorithm that constructs the MorseSmale complex in a series of sweeps through the data, identifying various components of the complex in a consistent manner. All components of the complex, both geometric and topological, are computed, providing a complete decomposition of the domain. Efficiency is maintained by representing the geometry of the complex in terms of point sets.
A Practical Approach to MorseSmale Complex Computation: Scalability and Generality
"... Abstract—The MorseSmale (MS) complex has proven to be a useful tool in extracting and visualizing features from scalarvalued data. However, efficient computation of the MS complex for large scale data remains a challenging problem. We describe a new algorithm and easily extensible framework for co ..."
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Cited by 16 (3 self)
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Abstract—The MorseSmale (MS) complex has proven to be a useful tool in extracting and visualizing features from scalarvalued data. However, efficient computation of the MS complex for large scale data remains a challenging problem. We describe a new algorithm and easily extensible framework for computing MS complexes for large scale data of any dimension where scalar values are given at the vertices of a closurefinite and weak topology (CW) complex, therefore enabling computation on a wide variety of meshes such as regular grids, simplicial meshes, and adaptive multiresolution (AMR) meshes. A new divideandconquer strategy allows for memoryefficient computation of the MS complex and simplification onthefly to control the size of the output. In addition to being able to handle various data formats, the framework supports implementationspecific optimizations, for example, for regular data. We present the complete characterization of critical point cancellations in all dimensions. This technique enables the topology based analysis of large data on offtheshelf computers. In particular we demonstrate the first full computation of the MS complex for a 1 billion/1024 3 node grid on a laptop computer with 2Gb memory. Index Terms—Topologybased analysis, MorseSmale complex, large scale data. 1
Simple homotopy type of Novikov complex and ζfunction of the gradient flow
, 1997
"... This paper is the second part of the author’s work on generic properties of Novikov complex (see [16], [13], [15], [14]) In these papers we proved, that for a C 0 generic gradient of a given Morse form ω the boundary operators in the Novikov complex are rational functions, and not merely power ser ..."
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Cited by 14 (1 self)
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This paper is the second part of the author’s work on generic properties of Novikov complex (see [16], [13], [15], [14]) In these papers we proved, that for a C 0 generic gradient of a given Morse form ω the boundary operators in the Novikov complex are rational functions, and not merely power series. In the present paper we study the twisted Lefschetz function ζL(−v), where ω is a Morse form, and v is a C 0 generic ωgradient. This ζfunction is defined with respect to the abelian covering ̂ M → M with the structure group H1(M). We prove that ζL(−v) equals to the torsion of a chain homotopy equivalence between Novikov Complex and the completed cellular chain complex of this covering of the manifold, and that this ζfunction belongs to the corresponding localization of the group ring of H1(M). The relations with SeibergWitten invariants of 3manifolds are discussed.
Sasakian geometry, hypersurface singularities, and Einstein
, 2005
"... This review article has grown out of notes for the three lectures the second author presented during the XXIVth Winter School of Geometry and Physics in Srni, Czech Republic, in January of 2004. Our purpose is twofold. We want give a brief introduction to some of the techniques we have developed ov ..."
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Cited by 12 (3 self)
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This review article has grown out of notes for the three lectures the second author presented during the XXIVth Winter School of Geometry and Physics in Srni, Czech Republic, in January of 2004. Our purpose is twofold. We want give a brief introduction to some of the techniques we have developed over the last 5 years
The comparison geometry of Ricci curvature, in Comparison geometry
 Fakultät für Mathematik, Universität
, 1997
"... Abstract. We survey comparison results that assume a bound on the manifold’s Ricci curvature. 1. ..."
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Cited by 10 (0 self)
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Abstract. We survey comparison results that assume a bound on the manifold’s Ricci curvature. 1.
The local linearization problem for smooth SL(n)actions
 Enseign. Math
, 1997
"... Abstract. This paper considers SL(n, R)actions on Euclidean space fixing the origin. We show that all C 1actions on R n are linearizable. We give C ∞actions of SL(2, R) on R 3 and of SL(3, R) on R 8 which are not linearizable. We classify the C 0actions of SL(n, R) on R n. Finally, the paper con ..."
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Cited by 9 (0 self)
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Abstract. This paper considers SL(n, R)actions on Euclidean space fixing the origin. We show that all C 1actions on R n are linearizable. We give C ∞actions of SL(2, R) on R 3 and of SL(3, R) on R 8 which are not linearizable. We classify the C 0actions of SL(n, R) on R n. Finally, the paper concludes with a study of the linearizability of SL(n, Z)actions. RÉSUMÉ. Dans cet article, on considère les actions de SL(n, R) sur l’espace euclidien qui fixent l’origine. On montre que les actions C1 sur Rn sont linéarisables. On donne des actions C ∞ de SL(2, R) sur R3 et de SL(3, R) sur R8 qui ne sont pas linéarisables. On classifie les actions C0 de SL(n, R) sur Rn. L’article s’achève par une étude de la linéarisabilité des actions de SL(n, Z). 1.