Estimation of the complexity of a digital image from the viewpoint of fixed point theory

Remarks on the Preservation of the Almost Fixed Point Property Involving Several Types of Digitizations

“…To study the FPP or the AFPP for digital spaces from the viewpoint of digital topology, we first need to recall the k-adjacency relations of n-dimensional integer grids (see Equation (2)), a digital k-neighborhood, digital continuity, and so forth [2,14,17]. To study n-dimensional digital images, n ∈ N, as a generalization of the k-adjacency relations of Z n , n ∈ {1, 2, 3}, we will take the following approach [17] (see also [18]).…”

“…are k(m, n)-adjacent if at most m of their coordinates differ by ± 1, and all others coincide. According to the operator of Equation (1), the k(m, n)-adjacency relations of Z n , n ∈ N, are obtained [17] (see also [18]) as follows:…”

“…Then, it is obvious that each of the maps h and r is a 6-continuous map and the map g is an 18-continuous map (see Equations (7)- (9)). Hence, the composite r • h • g : (Y, 18) → (Y, 18) is an 18-continuous map. However, this composite does not have the AFPP of (Y, 18) (see the map r of Equation (9)).…”

“…For instance, on Z 3 , consider (Y := [−1, 1] 3 Z , 18 := k(2, 3)). Using the notion of 18-continuity of any self-map of (Y, 18) (see Proposition 1), we prove that the digital image (Y, 18) does not have the AFPP. To be precise, consider a self-map g of (Y, 18) in the following way: For…”

Topologies on Z n that Are Not Homeomorphic to the n-Dimensional Khalimsky Topological Space

“…More precisely, digital images can be considered as subsets of Z n with some structures, such as a digital adjacency (or the digital connectivity in the Rosenfeld model), the Khalimsky, the Marcus-Wyse, the H-topological, and the Alexandroff structures [10,11]. In particular, these structures play important roles in the fields of digital homotopy theory, fixed point theory, digital topological rough set theory, digital geometry, information theory, and so forth [12][13][14][15][16]. Thus, an intensive development of new topologies on Z n , which are different from the well-known topologies on Z n , can facilitate the study of pure and applied sciences including computer science.…”