Abstract
We study connectivity preserving multivalued functions (Kovalevsky in A new concept for digital geometry, shape in picture, 1994) between digital images. This notion generalizes that of continuous multivalued functions (Escribano et al. in Discrete geometry for computer imagery, lecture notes in computer science, 2008; Escribano et al. in J Math Imaging Vis 42:76–91, 2012) studied mostly in the setting of the digital plane \({\mathbb {Z}}^2\). We show that connectivity preserving multivalued functions, like continuous multivalued functions, are appropriate models for digital morphological operations. Connectivity preservation, unlike continuity, is preserved by compositions, and generalizes easily to higher dimensions and arbitrary adjacency relations.
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References
Borsuk, K.: Theory of Retracts. Polish Scientific Publishers, Warsaw (1967)
Boxer, L.: Digitally continuous functions. Pattern Recogn. Lett. 15, 833–839 (1994)
Boxer, L.: A classical construction for the digital fundamental group. Pattern Recogn. Lett. 10, 51–62 (1999)
Boxer, L.: Properties of digital homotopy. J. Math. Imaging Vis. 22, 19–26 (2005)
Boxer, L.: Remarks on digitally continuous multivalued functions. J. Adv. Math. 9(1), 1755–1762 (2014)
Escribano, C., Giraldo, A., Sastre, M.: Digitally continuous multivalued functions. In: Discrete Geometry for Computer Imagery. Lecture Notes in Computer Science, vol. 4992, pp. 81–92. Springer, Berlin (2008)
Escribano, C., Giraldo, A., Sastre, M.: Digitally continuous multivalued functions, morphological operations and thinning algorithms. J. Math. Imaging Vis. 42, 76–91 (2012)
Giraldo, A., Sastre, M.: On the composition of digitally continuous multivalued functions. J. Math. Imaging Vis. 58, 196–209 (2015)
Kong, T.Y., Rosenfeld, A. (ed.): Topological Algorithms for Digital Image Processing. Elsevier, Amsterdam (1996)
Kovalevsky, V.A.: A New Concept for Digital Geometry, Shape in Picture. Springer, New York (1994)
Rosenfeld, A.: Digital topology. Am Math Mon 86, 621–630 (1979)
Rosenfeld, A.: ‘Continuous’ functions on digital images. Pattern Recogn. Lett. 4, 177–184 (1987)
Soille, P.: Morphological operators. In: Jhne, B., et al. (eds.) Signal Processing and Pattern Recognition. Handbook of Computer Vision and Applications, vol. 2, pp. 627–682. Academic Press, San Diego (1999)
Tsaur, R., Smyth, M.: “Continuous” multifunctions in discrete spaces with applications to fixed point theory. In: Bertrand, G., Imiya, A., Klette, R. (eds.), Digital and Image Geometry, Lecture Notes in Computer Science, vol. 2243, pp 151–162. Springer, Berlin (2001), doi:10.1007/3-540-45576-05
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We are grateful for the suggestions of the anonymous reviewers.
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Boxer, L., Staecker, P.C. Connectivity Preserving Multivalued Functions in Digital Topology. J Math Imaging Vis 55, 370–377 (2016). https://doi.org/10.1007/s10851-015-0625-5
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DOI: https://doi.org/10.1007/s10851-015-0625-5