Connectivity calculus of fractal polyhedrons

PATTERN RECOGNITION

卷:48

Article

Molina-Abril, Helena¹  Real, Pedro¹  Nakamura, Akira²  Klette, Reinhard³ 

[1] Univ Seville, Seville, Spain

[2] Hiroshima Univ, Hiroshima 730, Japan

[3] Univ Auckland, Auckland 1, New Zealand

关键词: Connectivity;    Cycles;    Topological analysis;    Tunnels;    Directed graphs;    Betti number;    Fractal set;    Menger sponge;    Sierpinski pyramid;   

DOI  :  10.1016/j.patcog.2014.05.016

来源: Elsevier

【 摘 要 】

The paper analyzes the connectivity information (more precisely, numbers of tunnels and their homological (co)cycle classification) of fractal polyhedra. Homology chain contractions and its combinatorial counterparts, called homological spanning forest (HSF), are presented here as an useful topological tool, which codifies such information and provides an hierarchical directed graph-based representation of the initial polyhedra. The Menger sponge and the Sierpinski pyramid are presented as examples of these computational algebraic topological techniques and results focussing on the number of tunnels for any level of recursion are given. Experiments, performed on synthetic and real image data, demonstrate the applicability of the obtained results. The techniques introduced here are tailored to self-similar discrete sets and exploit homology notions from a representational point of view. Nevertheless, the underlying concepts apply to general cell complexes and digital images and are suitable for progressing in the computation of advanced algebraic topological information of 3-dimensional objects. (C) 2014 Elsevier Ltd. All rights reserved.

【 授权许可】

Free