Abstract
In order to discuss digital topological properties of a digital image (X,k), many recent papers have used the digital fundamental group and several digital topological invariants such as the k-linking number, the k-topological number, and so forth. Owing to some difficulties of an establishment of the multiplicative property of the digital fundamental group, a k-homotopic thinning method can be essentially used in calculating the digital fundamental group of a digital product with k-adjacency. More precisely, let \(\mathit{SC}_{k_{i}}^{n_{i},l_{i}}\) be a simple closed k i -curve with l i elements in \(\mathbf{Z}^{n_{i}},i\in\{1,2\}\) . For some k-adjacency of the digital product \(\mathit{SC}_{k_{1}}^{n_{1},l_{1}}\times\mathit{SC}_{k_{2}}^{n_{2},l_{2}}\subset\mathbf{Z}^{n_{1}+n_{2}}\) which is a torus-like set, proceeding with the k-homotopic thinning of \(\mathit{SC}_{k_{1}}^{n_{1},l_{1}}\times\mathit{SC}_{k_{2}}^{n_{2},l_{2}}\) , we obtain its k-homotopic thinning set denoted by DT k . Writing an algorithm for calculating the digital fundamental group of \(\mathit{SC}_{k_{1}}^{n_{1},l_{1}}\times\mathit {SC}_{k_{2}}^{n_{2},l_{2}}\) , we investigate the k-fundamental group of \((\mathit{SC}_{k_{1}}^{n_{1},l_{1}}\times\mathit{SC}_{k_{2}}^{n_{2},l_{2}},k)\) by the use of various properties of a digital covering (Z×Z,p 1×p 2,DT k ), a strong k-deformation retract, and algebraic topological tools. Finally, we find the pseudo-multiplicative property (contrary to the multiplicative property) of the digital fundamental group. This property can be used in classifying digital images from the view points of both digital k-homotopy theory and mathematical morphology.
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Han, SE. The k-Homotopic Thinning and a Torus-Like Digital Image in Z n . J Math Imaging Vis 31, 1–16 (2008). https://doi.org/10.1007/s10851-007-0061-2
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DOI: https://doi.org/10.1007/s10851-007-0061-2