Multivalued Functions in Digital Topology

We study several types of multivalued functions in digital topology

Digital shy maps

[EN] We study properties of shy maps in digital topology.Boxer, L. (2017). Digital shy maps. Applied General Topology. 18(1):143-152. doi:10.4995/agt.2017.6663.SWORD143152181C. Berge, Graphs and Hypergraphs, 2nd edition, North-Holland, Amsterdam, 1976. https://doi.org/10.1016/0167-8655(94)90012-4Boxer, L. (1994). Digitally continuous functions. Pattern Recognition Letters, 15(8), 833-839. doi:10.1016/0167-8655(94)90012-4L. Boxer, A classical construction for the digital fundamental group, Pattern Recognition Letters 10 (1999), 51-62. https://doi.org/10.1007/s10851-005-4780-y https://doi.org/10.1007/s10851-006-9698-5Boxer, L. (2005). Properties of Digital Homotopy. Journal of Mathematical Imaging and Vision, 22(1), 19-26. doi:10.1007/s10851-005-4780-yBoxer, L. (2006). Digital Products, Wedges, and Covering Spaces. Journal of Mathematical Imaging and Vision, 25(2), 159-171. doi:10.1007/s10851-006-9698-5L. Boxer, Remarks on digitally continuous multivalued functions, Journal of Advances in Mathematics 9, no. 1 (2014), 1755-1762.L. Boxer and I. Karaca, Fundamental groups for digital products, Advances and Applications in Mathematical Sciences 11, no. 4 (2012), 161-180.Boxer, L., & Staecker, P. C. (2016). Connectivity Preserving Multivalued Functions in Digital Topology. Journal of Mathematical Imaging and Vision, 55(3), 370-377. doi:10.1007/s10851-015-0625-5Escribano, C., Giraldo, A., & Sastre, M. A. (s. f.). Digitally Continuous Multivalued Functions. Lecture Notes in Computer Science, 81-92. doi:10.1007/978-3-540-79126-3_9Escribano, C., Giraldo, A., & Sastre, M. A. (2011). Digitally Continuous Multivalued Functions, Morphological Operations and Thinning Algorithms. Journal of Mathematical Imaging and Vision, 42(1), 76-91. doi:10.1007/s10851-011-0277-zGiraldo, A., & Sastre, M. A. (2015). On the Composition of Digitally Continuous Multivalued Functions. Journal of Mathematical Imaging and Vision, 53(2), 196-209. doi:10.1007/s10851-015-0570-3HAN, S. (2005). Non-product property of the digital fundamental group. Information Sciences, 171(1-3), 73-91. doi:10.1016/j.ins.2004.03.018V. A. Kovalevsky, A new concept for digital geometry, shape in picture, Springer, New York (1994). https://doi.org/10.1016/0167-8655(86)90017-6Rosenfeld, A. (1986). ‘Continuous’ functions on digital pictures. Pattern Recognition Letters, 4(3), 177-184. doi:10.1016/0167-8655(86)90017-6Tsaur, R., & Smyth, M. B. (2001). «Continuous» Multifunctions in Discrete Spaces with Applications to Fixed Point Theory. Lecture Notes in Computer Science, 75-88. doi:10.1007/3-540-45576-0_

Digitally Continuous Multivalued Functions, Morphological Operations and Thinning Algorithms

In a recent paper (Escribano et al. in Discrete Geometry for Computer Imagery 2008. Lecture Notes in Computer Science, vol. 4992, pp. 81–92, 2008) we have introduced a notion of continuity in digital spaces which extends the usual notion of digital continuity. Our approach, which uses multivalued functions, provides a better framework to define topological notions, like retractions, in a far more realistic way than by using just single-valued digitally continuous functions. In this work we develop properties of this family of continuous functions, now concentrating on morphological operations and thinning algorithms. We show that our notion of continuity provides a suitable framework for the basic operations in mathematical morphology: erosion, dilation, closing, and opening. On the other hand, concerning thinning algorithms, we give conditions under which the existence of a retraction F:X⟶X∖D guarantees that D is deletable. The converse is not true, in general, although it is in certain particular important cases which are at the basis of many thinning algorithms