Digital Hausdorff distance on a connected digital image

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.

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  • Boxer, L., Digitally continuous functions, Pattern Recognition Letters, 15 (1994), 833-839.
  • Boxer, L., A classical construction for the digital fundamental group, Journal of Mathematical Imaging and Vision, 10 (1999), 51-62.
  • Boxer, L., Multivalued functions in digital topology, Note di Matematica, 37(2) (2017), 61-76.
  • Boxer, L., Alternate product adjacencies in digital topology, Applied General Topology, 19(1) (2018), 21-53.
  • 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.
  • Boxer, L. Staecker, P. C., Connectivity preserving multivalued functions in digital topology, Journal of Mathematical Imaging and Vision, 55(3) (2016), 370-377.
  • Chartrand, G., Lesniak, L., Graphs & digraphs, 2nd ed., Wadsworth, Inc., Belmont, Ca, 1986.
  • 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.
  • 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.
  • Giraldo, A., Sastre, M., On the composition of digitally continuous multivalued functions, Journal of Mathematical Imaging and Vision, 53 (2) (2015), 196-209.
  • Han, S.-E., Non-product property of the digital fundamental group, Information Sciences, 171(1) (2005), 73-91.
  • Han, S.-E., Digital fundamental group and Euler characteristic of a connected sum of digital closed surfaces, Information Sciences, 177 (2007), 3314-3326.
  • 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.
  • 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
  • Herman, G.T., Oriented surfaces in digital spaces, CVGIP: Graphical Models and Image Processing, 55 (1993), 381-396.
  • Khalimsky, E., Motion, deformation and homotopy in finite spaces, In Proceedings IEEE International Conference on Systems, Man, and Cybernetics, (1987), 227-234.
  • Leader, S., On clusters in proximity spaces, Fund. Math., 47 (1959), 205-213.
  • Lu, J., Li, J., Yan, Z., Zhang, C., Zero-Shot Learning by Generating Pseudo Feature Representations, arXiv1703.06389v1, 2017, 1-9.
  • 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.
  • Munkres, J., Topology (2nd ed.), pp. 280-281, Prentice Hall, 1999.
  • Peters, J.F., personal communication.
  • 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.
  • 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.
  • 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.
  • Peters, J.F., Proximal Voronoï regions, convex polygons, & Leader uniform topology, Adv. Math.: S.J., 4 (1) (2015), 1-5, Zbl 1335.65032.
  • Rosenfeld, A., Continuous functions on digital images, Pattern Recognition Letters, 4 (1987), 177-184.
  • 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.