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Toward Parallel Computation of Dense Homotopy Skeletons for nD Digital Objects

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Combinatorial Image Analysis (IWCIA 2017)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 10256))

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Abstract

An appropriate generalization of the classical notion of abstract cell complex, called primal-dual abstract cell complex (pACC for short) is the combinatorial notion used here for modeling and analyzing the topology of nD digital objects and images. Let \(D\subset I\) be a set of n-xels (ROI) and I be a n-dimensional digital image. We design a theoretical parallel algorithm for constructing a topologically meaningful asymmetric pACC HSF(D), called Homological Spanning Forest of D (HSF of D, for short) starting from a canonical symmetric pACC associated to I and based on the application of elementary homotopy operations to activate the pACC processing units. From this HSF-graph representation of D, it is possible to derive complete homology and homotopy information of it. The preprocessing procedure of computing HSF(I) is thoroughly discussed. In this way, a significant advance in understanding how the efficient HSF framework for parallel topological computation of 2D digital images developed in [2] can be generalized to higher dimension is made.

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References

  1. Berciano, A., Molina-Abril, H., Real, P.: Searching high order invariants in computer imagery. Appl. Algebra Eng. Commun. Comput. 23(1–2), 17–28 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Díaz-del-Río, F., Real, P., Onchis, D.: A parallel homological spanning forest framework for 2D topological image analysis. Pattern Recogn. Lett. 83, 49–58 (2016)

    Article  Google Scholar 

  3. Harker, S., Mischaikow, K., Mrozek, M., Nanda, V., Wagner, H., Juda, M., Dlotko, P.: The efficiency of a homology algorithm based on discrete Morse theory and coreductions. Image-A: Appl. Math. Image Eng. 1(1), 41–48 (2010)

    Google Scholar 

  4. Floriani, L., Fugacci, U., Iuricich, F.: Homological shape analysis through discrete morse theory. In: Breuß, M., Bruckstein, A., Maragos, P., Wuhrer, S. (eds.) Perspectives in Shape Analysis. MV, pp. 187–209. Springer, Cham (2016). doi:10.1007/978-3-319-24726-7_9

    Chapter  Google Scholar 

  5. González-Díaz, R., Real, P.: On the cohomology of 3D digital images. Discrete Appl. Math. 147(2), 245–263 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hopcroft, J., Tarjan, R.: Algorithm 447: efficient algorithms for graph manipulation. Commun. ACM 16(6), 372–378 (1973)

    Article  Google Scholar 

  7. Hurewicz, W.: Homology and homotopy theory. In Proceedings of the International Mathematical Congress of 1950, p. 344. University of Toronto Press (1952)

    Google Scholar 

  8. Klette, R.: Cell complexes through time. In International Symposium on Optical Science and Technology. International Society for Optics and Photonics, pp. 134–145 (2000)

    Google Scholar 

  9. Kovalevsky, V.: Finite topology as applied to image analysis. Comput. Vis. Graph. Image Process. 46, 141–161 (1989)

    Article  Google Scholar 

  10. Molina-Abril, H., Real, P., Nakamura, A., Klette, R.: Connectivity calculus of fractal polyhedrons. Pattern Recogn. 48(4), 1150–1160 (2015)

    Article  Google Scholar 

  11. Molina-Abril, H., Real, P.: Homological spanning forest framework for 2D image analysis. Ann. Math. Artif. Intell. 64(4), 385–409 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Molina-Abril, H., Real, P.: Homological optimality in Discrete Morse Theory through chain homotopies. Pattern Recogn. Lett. 11, 1501–1506 (2012)

    Article  Google Scholar 

  13. Pavlidis, T.: Algorithms for Graphics and Image Processing. Springer Science and Business Media, Heidelberg (1977)

    MATH  Google Scholar 

  14. Real, P., Molina-Abril, H., Gonzalez-Lorenzo, A., Bac, A., Mari, J.L.: Searching combinatorial optimality using graph-based homology information. Appl. Algebra Eng. Comm. Comp. 26(1–2), 103–120 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Real, P.: Homological perturbation theory and associativity. Homology, Homotopy Appl. 2(1), 51–88 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Real, P.: An algorithm computing homotopy groups. Math. Comput. Simul. 42(4–6), 461–465 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Veblen, O.: Analisis Situs, vol. 5. A.M.S. Publications, Providence (1931)

    MATH  Google Scholar 

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Acknowledgments

This work has been supported by the Spanish research projects (supported by the Ministerio de Economía y Competitividad and FEDER funds) COFNET (Event-based Cognitive Visual and Auditory Sensory Fusion, TEC2016-77785-P) and TOP4COG (Topological Recognition of 4D Digital Images via HSF model, MTM2016-81030-P (AEI/FEDER,UE)). The last co-author gratefully acknowledges the support of the Austrian Science Fund FWF-P27516.

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Authors and Affiliations

  1. Institute of Mathematics, University of Seville, Avda. Reina Mercedes s/n, 41012, Seville, Spain

    Pedro Real

  2. Computer Architecture and Technology Department, University of Seville, Avda. Reina Mercedes s/n, 41012, Seville, Spain

    Fernando Diaz-del-Rio

  3. Faculty of Mathematics, Univ. of Vienna, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Darian Onchis

  4. Faculty of Mathematics and Computer Science, West Univ. of Timisoara, Bulevardul Vasile Parvan 4, 300223, Timisoara, Romania

    Darian Onchis

Authors
  1. Pedro Real
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  2. Fernando Diaz-del-Rio
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  3. Darian Onchis
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Corresponding author

Correspondence to Fernando Diaz-del-Rio .

Editor information

Editors and Affiliations

  1. SUNY Buffalo State, Buffalo, New York, USA

    Valentin E. Brimkov

  2. Department of Applied Professional Studies, SUNY Fredonia, Fredonia, New York, USA

    Reneta P. Barneva

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Real, P., Diaz-del-Rio, F., Onchis, D. (2017). Toward Parallel Computation of Dense Homotopy Skeletons for nD Digital Objects. In: Brimkov, V., Barneva, R. (eds) Combinatorial Image Analysis. IWCIA 2017. Lecture Notes in Computer Science(), vol 10256. Springer, Cham. https://doi.org/10.1007/978-3-319-59108-7_12

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  • DOI: https://doi.org/10.1007/978-3-319-59108-7_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-59107-0

  • Online ISBN: 978-3-319-59108-7

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