Digital Hausdorff distance on a connected digital image

Abstract

A digital image X can be considered as a subset of Zⁿ together with an adjacency relation where Z is the set of the integers and n is a natural number. The aim of this study is to measure the closeness of two subsets of a connected digital image. To do this, we adapt the Hausdorff distance in the topological setting to its digital version. In this paper, we define a metric on a connected digital image by using the length of the shortest digital simple path. Then we use this metric to define the r-thickening of the subsets of a connected digital image and define the digital Hausdorff distance between them.

Keywords

• digital topology
• Hausdorff distance
• digital image

References

1. Boxer, L., Digitally continuous functions, Pattern Recognition Letters, 15 (1994), 833-839.
2. Boxer, L., A classical construction for the digital fundamental group, Journal of Mathematical Imaging and Vision, 10 (1999), 51-62.
3. Boxer, L., Multivalued functions in digital topology, Note di Matematica, 37(2) (2017), 61-76.
4. Boxer, L., Alternate product adjacencies in digital topology, Applied General Topology, 19(1) (2018), 21-53.
5. Boxer, L., Ege, O., Karaca, I., Lopez, J., Louwsma, J., Digital fixed points, approximate fixed points, and universal functions, Applied General Topology, 17(2) (2016), 159-172.
6. Boxer, L. Staecker, P. C., Connectivity preserving multivalued functions in digital topology, Journal of Mathematical Imaging and Vision, 55(3) (2016), 370-377.
7. Chartrand, G., Lesniak, L., Graphs & digraphs, 2nd ed., Wadsworth, Inc., Belmont, Ca, 1986.
8. Escribano, C., Giraldo, A., Sastre, M., Digitally continuous multivalued functions, in Discrete Geometry for Computer Imagery, Lecture Notes in Computer Science, v. 4992, 81-92, Springer, 2008.

9. Escribano, C., Giraldo, A., Sastre, M., Digitally continuous multi-valued functions, morphological thinning algorithms, Journal of Mathematical Imaging and Vision, 42(1) (2012), 76-91.
10. Giraldo, A., Sastre, M., On the composition of digitally continuous multivalued functions, Journal of Mathematical Imaging and Vision, 53 (2) (2015), 196-209.
11. Han, S.-E., Non-product property of the digital fundamental group, Information Sciences, 171(1) (2005), 73-91.
12. Han, S.-E., Digital fundamental group and Euler characteristic of a connected sum of digital closed surfaces, Information Sciences, 177 (2007), 3314-3326.
13. Han, S.-E., Estimation of the complexity of a digital image from the viewpoint of fixed point theory, Applied Mathematics and Computation, 347 (2019), 236-248.
14. Han, S.-E., Digital k-Contractibility of an n-Times Iterated Connected Sum of Simple Closed k-Surfaces and Almost Fixed Point Property, Mathematics, 8(345) (2020), 1-24, doi:10.3390/math8030345
15. Herman, G.T., Oriented surfaces in digital spaces, CVGIP: Graphical Models and Image Processing, 55 (1993), 381-396.
16. Khalimsky, E., Motion, deformation and homotopy in finite spaces, In Proceedings IEEE International Conference on Systems, Man, and Cybernetics, (1987), 227-234.
17. Leader, S., On clusters in proximity spaces, Fund. Math., 47 (1959), 205-213.
18. Lu, J., Li, J., Yan, Z., Zhang, C., Zero-Shot Learning by Generating Pseudo Feature Representations, arXiv1703.06389v1, 2017, 1-9.
19. Molina, M., Sánchez, J., Zero-Shot Learning with Partial Attributes, Intelligent Computing Systems. ISICS 2018. Communications in Computer and Information Science, vol 820, Springer Nature, Switzerland AG, (2018), 147--158, https://doi-org.uml.idm.oclc.org/10.1007/978-3-319-76261-6\_12.
20. Munkres, J., Topology (2nd ed.), pp. 280-281, Prentice Hall, 1999.
21. Peters, J.F., personal communication.
22. Peters, J.F., Computational geometry, topology and physics of digital images with applications. Shape complexes, optical vortex nerves and proximities, Springer Nature, Cham, Switzerland, 2020, xxv+440 pp., https://doi-org.uml.idm.oclc.org/10.1007/978-3-030-22192-8, Zbl07098311.
23. Peters, J.F., Topology of Digital Images. Visual Pattern Discovery in Proximity Spaces, Intelligent Systems Reference Library 134 (2014), Springer Heidelberg, xxvii+ 433pp, https://doi-org.uml.idm.oclc.org/10.1007/978-3-642-53845-2, Zentralblatt MATH Zbl 1295 68010.
24. Peters, J.F., Computational proximity. Excursions in the topology of digital images, Intelligent Systems Reference Library 102 (2016), Cham: Springer, xv + 411pp, https://doi-org.uml.idm.oclc.org/10.1007/978-3-319-30362-4, Zentralblatt MATH Zbl 1382.68008.
25. Peters, J.F., Proximal Voronoï regions, convex polygons, & Leader uniform topology, Adv. Math.: S.J., 4 (1) (2015), 1-5, Zbl 1335.65032.
26. Rosenfeld, A., Continuous functions on digital images, Pattern Recognition Letters, 4 (1987), 177-184.
27. Tsaur, R. and Smyth, M. B., "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 Springer, Berlin, Heidelberg, 2001.