## Optimal discrete Morse functions for 2-manifolds (2003)

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Venue: | Computational Geometry: Theory and Applications |

Citations: | 17 - 5 self |

### BibTeX

@ARTICLE{Lewiner03optimaldiscrete,

author = {Thomas Lewiner and Helio Lopes and Geovan Tavares},

title = {Optimal discrete Morse functions for 2-manifolds},

journal = {Computational Geometry: Theory and Applications},

year = {2003},

volume = {26},

pages = {2003}

}

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### Abstract

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.

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Citation Context ... on T . (4) Define the discrete Morse function on the complement of T . First step: Construction on a face-spanning tree. The face-spanning tree T can be constructed by any of the standard algorithms =-=[22]. In-=- particular, we can use some mesh compression’s strategies. For example, Fig. 6 shows a spanning tree constructed by the Edgebreaker’s compression algorithm [17]. Second step: Addition of one edge... |

583 |
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Citation Context ... (G. Tavares). 0925-7721/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0925-7721(03)00014-2s222 T. Lewiner et al. / Computational Geometry 26 (2003) 221–233 Morse th=-=eory [19]-=- is a fundamental tool for investigating the topology of smooth manifolds. Particularly for computer graphics, many applications have been induced [8,14,21] from the smooth case. Morse proved that the... |

247 | Topological persistence and simplification
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Citation Context ...ion as cycles). For the reader interested only in computing the Betti numbers, a classical algorithm is described in [5]. We would like to suggest [4] for a basic introduction to homology theory, and =-=[7] for a -=-concise presentation of Z2 homology. Let K be a cell complex. A p-chain c (p) is a subset of p-cells in K, c (p) = � cσ .σ (p) . σ (p) ∈Ks224 T. Lewiner et al. / Computational Geometry 26 (2003... |

157 | Morse theory for cell complexes
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Citation Context ...ny continuous function must have a maximum and a minimum. Morse theory provides a significant refinement of this observation. Forman’s discrete Morse theory. Recent insights in Morse theory by Forma=-=n [10,11]-=- extend several aspects of this fundamental tool to discrete structures. Its combinatorial aspect allows computation completely independent of a geometric realization: the algorithm does not require a... |

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Citation Context ...s for the Betti numbers (in the presence of torsion, Z2 homology “counts” some torsion as cycles). For the reader interested only in computing the Betti numbers, a classical algorithm is described=-= in [5]-=-. We would like to suggest [4] for a basic introduction to homology theory, and [7] for a concise presentation of Z2 homology. Let K be a cell complex. A p-chain c (p) is a subset of p-cells in K, c (... |

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Citation Context ...tational Geometry 26 (2003) 221–233 Morse theory [19] is a fundamental tool for investigating the topology of smooth manifolds. Particularly for computer graphics, many applications have been induce=-=d [8,14,21]-=- from the smooth case. Morse proved that the topology of a manifold is very closely related to the critical points of a real smooth map defined on it (i.e., the points where the gradient vanishes). Th... |

55 | Computational topology
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Citation Context ...onal Topology. Computational Geometry and Object Modeling. 1 Introduction Figure 1: An optimal discrete Morse function on a Möbius strip, with 2 critical cells. Applications of computational topology =-=[6, 23]-=- in computational science and engineering are many and growing. These include meshing, morphing, feature extraction, data compression, surface coding and more, in areas such as computer graphics, soli... |

53 | Hierarchical Morse complexes for piecewise linear 2– manifolds
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Citation Context ...tational Geometry 26 (2003) 221–233 Morse theory [19] is a fundamental tool for investigating the topology of smooth manifolds. Particularly for computer graphics, many applications have been induce=-=d [8,14,21]-=- from the smooth case. Morse proved that the topology of a manifold is very closely related to the critical points of a real smooth map defined on it (i.e., the points where the gradient vanishes). Th... |

51 |
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Citation Context ...to represent solid models: it is a consistent collection of cells (vertices, edges, faces, ...). Fig. 2 gives an example of such a structure. A complete introduction to cell complexes can be found in =-=[18]. More fo-=-rmally, a cell α (p) of dimension p is a set homeomorphic to the open p-ball {x ∈ R p : �x� < 1}. When the dimension p of the cell is obvious, we will simply denote α instead of α (p) . A cel... |

49 |
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(Show Context)
Citation Context ...n: the algorithm does not require any coordinate or floating-point calculation. Forman proves several results and provides many applications of his theory [12,13], and new ones have appeared recently =-=[2]-=-. Once a Morse function has been defined on a smooth manifold, then informations about its topology can be deduced from its critical points. Similarly to the smooth case, Forman proved that the topolo... |

34 | A user guide to discrete Morse theory
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Citation Context ...pletely independent of a geometric realization: the algorithm does not require any coordinate or floating-point calculation. Forman proves several results and provides many applications of his theory =-=[12,13]-=-, and new ones have appeared recently [2]. Once a Morse function has been defined on a smooth manifold, then informations about its topology can be deduced from its critical points. Similarly to the s... |

27 | Morse theory for implicit surface modeling
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Citation Context ...tational Geometry 26 (2003) 221–233 Morse theory [19] is a fundamental tool for investigating the topology of smooth manifolds. Particularly for computer graphics, many applications have been induce=-=d [8,14,21]-=- from the smooth case. Morse proved that the topology of a manifold is very closely related to the critical points of a real smooth map defined on it (i.e., the points where the gradient vanishes). Th... |

20 | Discrete Morse functions from lexicographic orders
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Citation Context ...f the cases (see Section 5). Prior work. As far as we know, there have been no results yet in computing explicitly a discrete Morse function with optimality requirements. The work of Babson and Hersh =-=[1]-=- gives a construction and some interpretation of Forman functions on cell complexes out of lexicographic orders. The construction of such lexicographic orders is not mentioned in [1] and the optimalit... |

19 |
Computational topology
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Citation Context ...nal Topology. Computational Geometry and Object Modeling. Figure 1: An optimal discrete Morse function on a Möbius strip, with 2 critical cells. 1 Introduction Applications of computational topology =-=[6, 23]-=- in computational science and engineering are many and growing. These include meshing, morphing, feature extraction, data compression, surface coding and more, in areas such as computer graphics, soli... |

16 | Edgebreaker: a simple compression for surface with handles
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(Show Context)
Citation Context ... by any of the standard algorithms [22]. In particular, we can use some mesh compression’s strategies. For example, Fig. 6 shows a spanning tree constructed by the Edgebreaker’s compression algori=-=thm [17]-=-. Second step: Addition of one edge of the boundary. We test whether the manifold has a boundary during the first step. If we found a boundary edge, we add it to T . This edge will be a loop in the du... |

13 |
F.H.: Basic Concepts of Algebraic Topology
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(Show Context)
Citation Context ...e presence of torsion, Z2 homology “counts” some torsion as cycles). For the reader interested only in computing the Betti numbers, a classical algorithm is described in [5]. We would like to sugg=-=est [4] for a -=-basic introduction to homology theory, and [7] for a concise presentation of Z2 homology. Let K be a cell complex. A p-chain c (p) is a subset of p-cells in K, c (p) = � cσ .σ (p) . σ (p) ∈Ks22... |

6 |
A computationally intractable problem on simplicial complexes
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- 1996
(Show Context)
Citation Context ...ger. Question: Does it exist a discrete Morse function on K with at most n critical cells? The Morse optimality problem reduces, in the general case, to the Collapsibility problem, which is proven in =-=[9]-=- to be a MAX-SNP hard problem, i.e., an NP-hard problem for which any polynomial approximation algorithm can lead to a result arbitrary far from the optimum: Collapsibility problem Instance: A pair (K... |

6 |
Algorithm to build and unbuild 2 and 3 dimensional manifolds
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- 1996
(Show Context)
Citation Context ... discrete Morse function f defined on it, the cell complex L built out of the only critical cells of f is homotopy equivalent to K. This corresponds in the smooth case to the handlebody decomposition =-=[16]-=-. 6. Future works We introduced a scheme for constructing discrete Morse function on finite cell complexes of dimension 2. This construction is linear in time in all cases, and is proven to be optimal... |

6 |
Morse Theory, number 51
- Milnor
- 1963
(Show Context)
Citation Context ...al medicine and astrophysics. In applications to computer graphics, being able to detect topological singularities helps designing more robust and efficient algorithms in time and space. Morse theory =-=[19]-=- is a fundamental tool for investigating the topology of smooth manifolds. Particularly for computer graphics, many applications have been induced [8, 21, 14] from the smooth case. Morse proved that t... |

5 | Some applications of combinatorial differential topology
- Forman
- 2001
(Show Context)
Citation Context ...pletely independent of a geometric realization: the algorithm does not require any coordinate or floating-point calculation. Forman proves several results and provides many applications of his theory =-=[12,13]-=-, and new ones have appeared recently [2]. Once a Morse function has been defined on a smooth manifold, then informations about its topology can be deduced from its critical points. Similarly to the s... |

5 | Visualizing Forman’s discrete vector field
- Lewiner, Lopes, et al.
- 2003
(Show Context)
Citation Context ...complexes of dimension 2. This construction is linear in time in all cases, and is proven to be optimal in the case of 2-manifolds. This algorithm has been extended for arbitrary finite dimensions in =-=[15]-=-, without proof of optimality. However, the experimental results showed our algorithm gave an optimal result in most of the cases. This opens the question of which conditions on the cell complex would... |

5 | Incremental construction properties in dimension two— shellability, extendable shellability and vertex decomposability
- Moriyama, Takeuchi
- 2000
(Show Context)
Citation Context ... With geometrical constraint. Morse function. For all the manifolds cases, the resulting function was optimal. For the non-manifolds complexes (in particular for the examples of Moriyama and Takeuchi =-=[20]-=-), the function had at most 4 redundant critical cells. The experimental results on a Pentium III, 550 MHz, confirm the linear complexity (Fig. 8(a)) and the independence of the complexity from topolo... |

3 |
Basic concepts of algebraic topology. Undergraduate text in Mathematics
- Croom
- 1978
(Show Context)
Citation Context ...e presence of torsion, Z2 homology “counts” some torsion as cycles). For the reader interested only in computing the Betti numbers, a classical algorithm is described in [5]. We would like to suggest =-=[4]-=- for a basic introduction to homology theory, and [7] for a concise presentation of Z2 homology. The chain group and the boundary operator Let K be a cell complex. A p–chain c (p) is a subset of p–cel... |

2 | Discrete Morse complexes
- Chari, Joswig
- 2002
(Show Context)
Citation Context ...an’s discrete Morse theory relates the topology of a cell complex to the critical cells of a discrete Morse function. For a complete introduction, see Forman’s presentations [12,13] and Chari’s =-=works [2,3]. -=-The focus of this paper is to provide an optimal construction of discrete Morse functions—optimal in the sense that the function has the minimum possible number of critical cells in each dimension. ... |

2 | A user guide to discrete Morse theory, preprint - Forman - 2001 |

1 |
O’Rourke (Eds.), Handlebook of Discrete Computational Geometry
- Vegter, topology, et al.
- 1997
(Show Context)
Citation Context ...003 Elsevier B.V. All rights reserved. Keywords: Morse theory; Forman theory; Computational topology; Computational geometry and object modeling 1. Introduction Applications of computational topology =-=[6,23]-=- in computational science and engineering are many and growing. These include meshing, morphing, feature extraction, data compression, surface coding and more, in areas such as computer graphics, soli... |

1 | Discrete Morse complexes, preprint - Chari, Joswig - 2001 |