Finite topology as applied to image analysis
Computer Vision, Graphics, and Image Processing
A topological approach to digital topology
American Mathematical Monthly
Topological connectedness and 8-connectedness in digital pictures
CVGIP: Image Understanding
Computer Vision and Image Understanding
Comparability graphs and digital topology
Computer Vision and Image Understanding
Topological adjacency relations on Z,n
Theoretical Computer Science
Digital Topologies Revisited: An Approach Based on the Topological Point-Neighbourhood
DGCI '97 Proceedings of the 7th International Workshop on Discrete Geometry for Computer Imagery
Image Analysis and Mathematical Morphology
Image Analysis and Mathematical Morphology
Journal of Mathematical Imaging and Vision
Compatible topologies on graphs: an application to graph isomorphism problem complexity
Theoretical Computer Science
Digital Surfaces and Boundaries in Khalimsky Spaces
Journal of Mathematical Imaging and Vision
Multidimensional Size Functions for Shape Comparison
Journal of Mathematical Imaging and Vision
A quotient-universal digital topology
Theoretical Computer Science
Convenient Closure Operators on $\mathbb Z^2$
IWCIA '09 Proceedings of the 13th International Workshop on Combinatorial Image Analysis
Jordan curve theorems with respect to certain pretopologies on Z²
DGCI'09 Proceedings of the 15th IAPR international conference on Discrete geometry for computer imagery
A jordan curve theorem in the digital plane
IWCIA'11 Proceedings of the 14th international conference on Combinatorial image analysis
Curves, hypersurfaces, and good pairs of adjacency relations
IWCIA'04 Proceedings of the 10th international conference on Combinatorial Image Analysis
Computer assisted location of the lower limb mechanical axis
ITIB'12 Proceedings of the Third international conference on Information Technologies in Biomedicine
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We show that there are only two topologies in Z2 and five topologies in Z3 whose connected sets are connected in the intuitive sense. Both topologies for Z2 are well known (e.g., one is presented in D. Marcus, F. Wyse et al., Amer. Math. Monthly 77 (1979) 1119, and the second in E. Khalimsky et al., Topology and its Applications 36 (1990) 1) and found applications in computer graphics and computer vision (e.g., A. Rosenfeld, Amer. Math. Monthly 77 (1979) pp. 621, and T.Y. Kong et al., Amer. Math. Monthly 98 (1991) 901). Two of the five topologies for Z3 are products of the topologies known from Z1 and Z2. The remaining three topologies for Z3 are also generated from the two topologies for Z2 however, they are not product topologies.