Decomposing Cavities in Digital Volumes into Products of Cycles

  • Ainhoa Berciano
  • Helena Molina-Abril
  • Ana Pacheco
  • Paweł Pilarczyk
  • Pedro Real
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5810)


The homology of binary 3–dimensional digital images (digital volumes) provides concise algebraic description of their topology in terms of connected components, tunnels and cavities. Homology generators corresponding to these features are represented by nontrivial 0–cycles, 1–cycles and 2–cycles, respectively. In the framework of cubical representation of digital volumes with the topology that corresponds to the 26–connectivity between voxels, we introduce a method for algorithmic computation of a coproduct operation that can be used to decompose 2–cycles into products of 1–cycles (possibly trivial). This coproduct provides means of classifying different kinds of cavities; in particular, it allows to distinguish certain homotopically non-equivalent spaces that have isomorphic homology. We define this coproduct at the level of a cubical complex built directly upon voxels of the digital image, and we construct it by means of the classical Alexander-Whitney map on a simplicial subdivision of faces of the voxels.


homology cubical homology cubical set cell complex digital image cavity cycle Alexander Whitney diagonal chain homotopy algebraic gradient vector field 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Ainhoa Berciano
    • 1
  • Helena Molina-Abril
    • 2
  • Ana Pacheco
    • 2
  • Paweł Pilarczyk
    • 3
  • Pedro Real
    • 2
  1. 1.Departamento de Didactica de la Matematica y de las CC. ExperimentalesUniversidad del Pais Vasco-Euskal Herriko UnibertsitateaBilbao (Bizkaia)Spain
  2. 2.Departamento de Matematica Aplicada IUniversidad de SevillaAvenida Reina MercedesSpain
  3. 3.Centro de MatemáticaUniversidade do MinhoBragaPortugal

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