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A Graph-with-Loop Structure for a Topological Representation of 3D Objects

Author | González Díaz, Rocío
Jiménez Rodríguez, María José Medrano Garfia, Belén Real Jurado, Pedro |

Department | Universidad de Sevilla. Departamento de Matemática Aplicada I |

Date | 2007 |

Published in | Computer Analysis of Images and Patterns (CAIP 2007), Lecture Notes in Computer Science, Vol. 4673, p. 506-513 |

Document type | Chapter of Book |

Abstract | Given a cell complex K whose geometric realization |K| is embedded in R 3 and a continuous function h: |K|→R (called the height function), we construct a graph G h (K) which is an extension of the Reeb graph R h (|K|). ... Given a cell complex K whose geometric realization |K| is embedded in R 3 and a continuous function h: |K|→R (called the height function), we construct a graph G h (K) which is an extension of the Reeb graph R h (|K|). More concretely, the graph G h (K) without loops is a subdivision of R h (|K|). The most important difference between the graphs G h (K) and R h (|K|) is that G h (K) preserves not only the number of connected components but also the number of “tunnels” (the homology generators of dimension 1) of K. The latter is not true in general for R h (|K|). Moreover, we construct a map ψ: G h (K)→K identifying representative cycles of the tunnels in K with the ones in G h (K) in the way that if e is a loop in G h (K), then ψ(e) is a cycle in K such that all the points in |ψ(e)| belong to the same level set in |K|. |

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http://dx.doi.org/10.1007/978-3-540-74272-2_63

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DOI:
http://dx.doi.org/10.1007/978-3-540-74272-2_63