Homological Computation Using Spanning Trees

  • H. Molina-Abril
  • P. Real
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5856)

Abstract

We introduce here a new \(\mathbb{F}_2\) homology computation algorithm based on a generalization of the spanning tree technique on a finite 3-dimensional cell complex K embedded in ℝ3. We demonstrate that the complexity of this algorithm is linear in the number of cells. In fact, this process computes an algebraic map φ over K, called homology gradient vector field (HGVF), from which it is possible to infer in a straightforward manner homological information like Euler characteristic, relative homology groups, representative cycles for homology generators, topological skeletons, Reeb graphs, cohomology algebra, higher (co)homology operations, etc. This process can be generalized to others coefficients, including the integers, and to higher dimension.

Keywords

Cell complex chain homotopy digital volume homology gradient vector field tree spanning tree 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • H. Molina-Abril
    • 1
    • 2
  • P. Real
    • 1
  1. 1.Departamento de Matematica Aplicada IUniversidad de Sevilla 
  2. 2.Faculty of Informatics, PRIP GroupVienna University of Technology 

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