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70
Finding the homology of submanifolds with high confidence from random samples
, 2004
"... Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high dimensional spaces. We consider the case where data is drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to “learn ” the hom ..."
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Cited by 177 (6 self)
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Recently there has been a lot of interest in geometrically motivated approaches to data analysis in high dimensional spaces. We consider the case where data is drawn from sampling a probability distribution that has support on or near a submanifold of Euclidean space. We show how to “learn ” the homology of the submanifold with high confidence. We discuss an algorithm to do this and provide learning-theoretic complexity bounds. Our bounds are obtained in terms of a condition number that limits the curvature and nearness to self-intersection of the submanifold. We are also able to treat the situation where the data is “noisy ” and lies near rather than on the submanifold in question.
Computing a Canonical Polygonal Schema of an Orientable Triangulated Surface
, 2001
"... A closed orientable surface of genus g can be obtained by appropriate identication of pairs of edges of a 4ggon (the polygonal schema). The identied edges form 2g loops on the surface, that are disjoint except for their common end-point. These loops are generators of both the fundamental group and t ..."
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Cited by 66 (3 self)
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A closed orientable surface of genus g can be obtained by appropriate identication of pairs of edges of a 4ggon (the polygonal schema). The identied edges form 2g loops on the surface, that are disjoint except for their common end-point. These loops are generators of both the fundamental group and the homology group of the surface. The inverse problem is concerned with nding a set of 2g loops on a triangulated surface, such that cutting the surface along these loops yields a (canonical) polygonal schema. We present two optimal algorithms for this inverse problem. Both algorithms have been implemented using the CGAL polyhedron data structure.
Optimally cutting a surface into a disk
- Discrete & Computational Geometry
, 2002
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Computational Topology: Ambient Isotopic Approximation of 2-Manifolds
- THEORETICAL COMPUTER SCIENCE
, 2001
"... A fundamental issue in theoretical computer science is that of establishing unambiguous formal criteria for algorithmic output. This paper does so within the domain of computer-aided geometric modeling. For practical geometric modeling algorithms, it is often desirable to create piecewise linear app ..."
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Cited by 40 (19 self)
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A fundamental issue in theoretical computer science is that of establishing unambiguous formal criteria for algorithmic output. This paper does so within the domain of computer-aided geometric modeling. For practical geometric modeling algorithms, it is often desirable to create piecewise linear approximations to compact manifolds embedded in and it is usually desirable for these two representations to be "topologically equivalent". Though this has traditionally been taken to mean that the two representations are homeomorphic, such a notion of equivalence suffers from a variety of technical and philosophical difficulties; we adopt the stronger notion of ambient isotopy. It is shown here, that for any C², compact, 2-manifold without boundary, which is embedded in R³, there exists a piecewise linear ambient isotopic approximation. Furthermore, this isotopy has compact support, with specific bounds upon the size of this compact neighborhood. These bounds may be useful for practical application in computer graphics and engineering design simulations. The proof given relies upon properties of the medial axis, which is explained in this paper.
Shape understanding by contour driven retiling
- THE VISUAL COMPUTER
, 2003
"... Given a triangle mesh representing a closed manifold surface of arbitrary genus, a method is proposed to automatically extract the Reeb graph of the manifold with respect to the height function. The method is based on a slicing strategy that traces contours while inserting them directly in the mesh ..."
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Cited by 30 (7 self)
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Given a triangle mesh representing a closed manifold surface of arbitrary genus, a method is proposed to automatically extract the Reeb graph of the manifold with respect to the height function. The method is based on a slicing strategy that traces contours while inserting them directly in the mesh as constraints. Critical areas, which identify isolated and non-isolated critical points of the surface, are recognized and coded in the extended Reeb graph (ERG). The remeshing strategy guarantees that topological features are correctly maintained in the graph, and the tiling of ERG nodes reproduces the original shape at a minimal, but topologically correct, geometric level.
Optimal discrete Morse functions for 2-manifolds
- 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 24 (6 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 2-manifolds, 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.
Morse Theory for Filtrations and Efficient Computation of Persistent Homology
"... We introduce an efficient preprocessing algorithm to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups. The technique is based on an extension of combinatorial Morse theory from complexes to filtrations. ..."
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Cited by 23 (8 self)
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We introduce an efficient preprocessing algorithm to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups. The technique is based on an extension of combinatorial Morse theory from complexes to filtrations.
Applications of Forman’s discrete Morse theory to topology visualization and mesh compression
- IEEE Trans. Vis. Comput. Graph
"... Abstract. Morse theory is a powerful tool for investigating the topology of smooth manifolds. It has been widely used by the computational topology, computer graphics and geometric modeling communities to devise topology based algorithms and data structures. Forman introduced a discrete version of t ..."
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Cited by 20 (3 self)
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Abstract. Morse theory is a powerful tool for investigating the topology of smooth manifolds. It has been widely used by the computational topology, computer graphics and geometric modeling communities to devise topology based algorithms and data structures. Forman introduced a discrete version of this theory, which is purely combinatorial. This work aims to build, visualize and apply the basic elements of Forman’s discrete Morse theory. It intends to use some of those concepts to visually study the topology of an object. As a basis, an algorithmic construction of optimal Forman’s discrete gradient vector fields is provided. This construction is then used to topologically analyze mesh compression schemes, such as Edgebreaker and Grow&Fold. In particular, this paper proves that the complexity class of the strategy optimization of Grow&Fold is MAX–SNP hard.
Computing Homology Groups of Simplicial Complexes in R³
, 1998
"... Recent developments in analyzing molecular structures and representing solid models using simplicial complexes have further enhanced the need for computing structural information about simplicial complexes in R 3 . This paper develops basic techniques required to manipulate and analyze structures ..."
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Cited by 17 (0 self)
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Recent developments in analyzing molecular structures and representing solid models using simplicial complexes have further enhanced the need for computing structural information about simplicial complexes in R 3 . This paper develops basic techniques required to manipulate and analyze structures of complexes in R 3 . A new approach to analyze simplicial complexes in Euclidean 3-space R 3 is described. First, methods from topology are used to analyze triangulated 3-manifolds in R 3 . Then it is shown that these methods can, in fact, be applied to arbitrary simplicial complexes in R 3 after (simulating) the process of thickening a complex to a 3-manifold homotopic to it. As a consequence considerable structural information about the complex can be determined and certain discrete problems solved as well. For example, it is shown how to determine the homology groups, as well as concrete representations of their generators, for a given complex in R 3 . Keywords. Topology, homot...
