Efficiently Storing Well-Composed Polyhedral Complexes Computed Over 3D Binary Images

  • Rocio Gonzalez-Diaz
  • Maria-Jose Jimenez
  • Belen Medrano

DOI: 10.1007/s10851-017-0722-8

Cite this article as:
Gonzalez-Diaz, R., Jimenez, MJ. & Medrano, B. J Math Imaging Vis (2017). doi:10.1007/s10851-017-0722-8


A 3D binary image I can be naturally represented by a combinatorial-algebraic structure called cubical complex and denoted by Q(I), whose basic building blocks are vertices, edges, square faces and cubes. In Gonzalez-Diaz et al. (Discret Appl Math 183:59–77, 2015), we presented a method to “locally repair” Q(I) to obtain a polyhedral complex P(I) (whose basic building blocks are vertices, edges, specific polygons and polyhedra), homotopy equivalent to Q(I), satisfying that its boundary surface is a 2D manifold. P(I) is called a well-composed polyhedral complex over the pictureI. Besides, we developed a new codification system for P(I), encoding geometric information of the cells of P(I) under the form of a 3D grayscale image, and the boundary face relations of the cells of P(I) under the form of a set of structuring elements. In this paper, we build upon (Gonzalez-Diaz et al. 2015) and prove that, to retrieve topological and geometric information of P(I), it is enough to store just one 3D point per polyhedron and hence neither grayscale image nor set of structuring elements are needed. From this “minimal” codification of P(I), we finally present a method to compute the 2-cells in the boundary surface of P(I).


3D binary image Well composedness Cubical complex Well-composed polyhedral complex 

Funding information

Funder NameGrant NumberFunding Note
  • MTM2015-67072-P

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Applied Mathematics I, School of Computer EngineeringUniversity of SevilleSevilleSpain

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