Topological Classification of points in $Z^2$ by using Topological Numbers for $2$D discrete binary images

TL;DR

The paper develops a local topological framework for 2D discrete binary images by using topological numbers defined on the $N^*_8(x)$ neighborhood to classify points into six categories: isolated, interior, simple, curve, and junctions with 3 or 4 curves. It proves a local criterion—$x$ is an $n$-simple point iff $T_n(x,X)=1$ and $T_{ar n}(x,\overline{X})=1$—and enumerates all $2^8=256$ local configurations into six classes with explicit counts. The core contribution is the precise six-class topological classification based on $(T_n,T_{ar n})$ pairs, enabling topology-preserving point deletion strategies and targeted feature extraction. This framework supports applications in curve skeletonization and the analysis of structural features in domains like fingerprints and medical imaging. All mathematical notation is given in $LaTeX$-style format within $...$ delimiters.

Abstract

In this paper, we propose a topological classification of points for 2D discrete binary images. This classification is based on the values of the calculus of topological numbers. Six classes of points are proposed: isolated point, interior point, simple point, curve point, point of intersection of 3 curves, point of intersection of 4 curves. The number of configurations of each class is also given.

Figures (8)

• Figure 1: (a) A $2$D binary image, (b) a corresponding mapping to $\mathcal{Z}^2$.
• Figure 2: (a) $N_4(x)$, (b) $N_8(x)$, (c) $4$-neighbors of $x$, (d) $8$-neighbors of $x$.
• Figure 3: (a) $x$ is $4$-simple for $X$ and is not $8$-simple for $X$, (b) $x$ is $4$-simple for $X$ and $8$-simple for $X$, (c) $x$ is $8$-simple for $X$ and is not $4$-simple for $X$.
• Figure 4: Number of configurations for the different values of topological numbers and according to the values $(n,\overline{n})$. In the last three columns, some examples of configurations are given for each possibility of values for the pair ($T_n(x,X), T_{\overline{n}}(x,\overline{X})$) by specifying if there are any configuration verifying these pairs only for $n=4$, both for $n=4$ and $n=8$, or only for $n=8$.
• Figure 5: The point $x_1$ is a $4$-simple point (Fig. \ref{['fig:table_nombres_topos_2D']} (g)), $x_2$ is the $4$-interior point (Fig. \ref{['fig:table_nombres_topos_2D']} (e)), $x_3$ is a $4$-isolated point (Fig. \ref{['fig:table_nombres_topos_2D']} (a)), $x_4$ is a $4$-curve point (Fig. \ref{['fig:table_nombres_topos_2D']} (j)), $x_5$ is the $(4,4)$-curves junction point (Fig. \ref{['fig:table_nombres_topos_2D']} (p)), $x_6$ is a $(4,3)$-curves junction point (Fig. \ref{['fig:table_nombres_topos_2D']} (m)).