Digitally continuous multivalued functions

We introduce in this paper a notion of continuity in digital spaces which extends the usual notion of digital continuity. Our approach uses multivalued maps. We show how the multivalued approach 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 particular, we characterize the deletion of simple points, one of the most important processing operations in digital topology, as a particular kind of retraction

Some consequences of restrictions on digitally continuous functions

We study the consequences of some restrictions on digitally continuous functions. One of our results modifies easily to yield an analogous result for topological spaces

Some Consequences of Restrictions on Digitally Continuous Functions

We study the consequences of some restrictions on digitally continuous functions. One of our results modifies easily to yield an analogous result for topological spaces.Comment: arXiv admin note: substantial text overlap with arXiv:2106.0601

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

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_

Digital fixed points, approximate fixed points, and universal functions

[EN] A. Rosenfeld introduced the notion of a digitally continuous function between digital images, and showed that although digital images need not have fixed point properties analogous to those of the Euclidean spaces modeled by the images, there often are approximate fixed point properties of such images. In the current paper, we obtain additional results concerning fixed points and approximate fixed points of digitally continuous functions. Among these are several results concerning the relationship between universal functions and the approximate fixed point property (AFPP).Boxer, L.; Ege, O.; Karaca, I.; Lopez, J.; Louwsma, J. (2016). Digital fixed points, approximate fixed points, and universal functions. Applied General Topology. 17(2):159-172. doi:10.4995/agt.2016.4704.SWORD15917217