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On the Composition of Digitally Continuous Multivalued Functions

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Abstract

In Escribano et al. (Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 4992. Springer, Berlin, pp. 81–92, 2008), Escribano et al. (Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 5810. Springer, Berlin, pp. 275–287, 2009) and Escribano et al. (J Math Imaging Vis 42:76–91, 2012), a notion of continuity in digital spaces, which extends the usual notion of digital continuity, was introduced. In Escribano et al. (Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 4992. Springer, Berlin, pp. 81–92, 2008), it was claimed that the composition of two digitally continuous multivalued functions is a digitally continuous multivalued function. However, this result is only correct in the following situation: If \(X\subset {{\mathrm{{\mathbb {Z}}}}}^m , Y\subset {{\mathrm{{\mathbb {Z}}}}}^n , Z\subset {{\mathrm{{\mathbb {Z}}}}}^p \), and \(F:X\leadsto Y\) and \(G:Y\leadsto Z\) are, respectively, a \((k,k')\)-continuous multivalued function and a \((k',k'')\)-continuous multivalued function, with \(k=3^m-1\), then \(GF:X\leadsto Z\) is a \((k,k'')\)-continuous multivalued function. In this paper, we give a proof of this result. On the other hand, in Escribano et al. (Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 5810. Springer, Berlin, pp. 275–287, 2009) and Escribano et al. (J Math Imaging Vis 42:76-91, 2012), the continuity of the composition of continuous multivalued functions is used in some proofs. Therefore, those proofs are only valid for \(k=8\). We give in this paper new proofs for the case \(k=4\) not requiring this property, with the exception of a result on the sequential deletion of 4-simple points which we state here as Conjecture 1.

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References

  1. Boxer, L.: Digitally continuous functions. Pattern Recognit. Lett. 15, 833–839 (1994)

    Article  MATH  Google Scholar 

  2. Boxer, L.: A classical construction for the digital fundamental group. J. Math. Imaging Vis. 10, 51–62 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  3. Boxer, L.: Properties of digital homotopy. J. Math. Imaging Vis. 22, 19–26 (2005)

    Article  MathSciNet  Google Scholar 

  4. Boxer, L.: Homotopy properties of sphere-like digital images. J. Math. Imaging Vis. 24, 167–175 (2006)

    Article  MathSciNet  Google Scholar 

  5. Boxer, L.: Remarks on digitally continuous multivalued functions. J. Adv. Math. 9(1), 1755–1762 (2014)

    Google Scholar 

  6. Escribano, C., Giraldo, A., Sastre, M.A.: Digitally continuous multivalued functions. In: Coeurjolly, D., Sivignon, I., Tougne, L., Dupont, F. (eds.) Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 4992, pp. 81–92. Springer (2008)

  7. Escribano, C., Giraldo, A., Sastre, M.A.: Thinning algorithms as multivalued \({\cal N}\)-retractions. In: Brlek, S., Provençal, X., Reutenauer, C. (eds.) Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 5810, pp. 275–287. Springer (2009)

  8. Escribano, C., Giraldo, A., Sastre, M.A.: Digitally continuous multivalued functions, morphological operations and thinning algorithms. J. Math. Imaging Vis. 42, 76–91 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Escribano, C., Giraldo, A., Sastre, M.A.: Characterization of the deletion of (26,6)-simple points as multivalued \(({\cal N},26)\)-retractions. Dis. Appl. Math. 183, 31–41 (2014)

  10. Gale, D.: The game of Hex and the Brouwer fixed-point theorem. Am. Math. Mon. 86, 818–827 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  11. Giraldo, A., Gross, A., Latecki, L.J.: Digitizations preserving shape. Pattern Recognit. 32, 365–376 (1999)

    Article  Google Scholar 

  12. Giraldo, A., Sanjurjo, J.M.R.: Density and finiteness. A discrete approach to shape. Topol. Appl. 76, 61–77 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  13. Han, X., Xu, C., Prince, J.: A topology preserving level set method for geometric deformable models. IEEE Trans. Pattern Anal. Mach. Intell. 25(6), 755–768 (2003)

    Article  Google Scholar 

  14. Klette, R., Rosenfeld, A.: Digital Geometry. Elsevier, Amsterdam (2004)

    MATH  Google Scholar 

  15. Kong, T.Y., Rosenfeld, A.: Digital topology: introduction and survey. Comput. Vis. Graph. Image Process. 48, 357–393 (1989)

    Article  Google Scholar 

  16. Kong, T.Y., Rosenfeld, A. (eds.) Topological Algorithms for Digital Image Processing, Elsevier, Amsterdam (1996)

  17. Kovalevsky, V.: A new concept for digital geometry. In: Ying-Lie, O., et al. (eds.) Shape in Picture. Proceedings of the NATO Advanced Research Workshop, Driebergen, The Netherlands, 1992. Computer and Systems Sciences, vol. 126. Springer-Verlag (1994)

  18. Rosenfeld, A.: Digital topology. Am. Math. Mon. 86(8), 621–630 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rosenfeld, A.: Continuous functions in digital pictures. Pattern Recognit. Lett. 4, 177–184 (1986)

    Article  MATH  Google Scholar 

  20. Ségonne, F.: Active contours under topology control-genus preserving level sets. Int. J. Comput. Vis. 79(2), 107–117 (2008)

    Article  Google Scholar 

  21. Tsaur, R., Smyth, M.B.: “Continuous” multifunctions in discrete spaces with applications to fixed point theory. In: Bertrand, G., et al. (eds.) Digital and image geometry: advanced lectures. Lecture Notes in Computer Science, vol. 2243, pp. 75–88. Springer-Verlag, New York (2001)

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Acknowledgments

The authors thank the referees for helpful comments and suggestions which have helped to improve the final version of the paper. Antonio Giraldo has been supported by Ministerio de Ciencia e Innovación (MICINN MTM2012-30719).

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  1. Departamento de Matemática Aplicada a las Tecnologías de la Información y las Comunicaciones, E.T.S. Ingenieros Informáticos, Universidad Politécnica, Campus de Montegancedo, Boadilla del Monte, 28660, Madrid, Spain

    Antonio Giraldo & María Asunción Sastre

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  1. Antonio Giraldo
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  2. María Asunción Sastre
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Correspondence to Antonio Giraldo.

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Giraldo, A., Sastre, M.A. On the Composition of Digitally Continuous Multivalued Functions. J Math Imaging Vis 53, 196–209 (2015). https://doi.org/10.1007/s10851-015-0570-3

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