The k-Homotopic Thinning and a Torus-Like Digital Image in Z ⁿ

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.