Topological numbers and their use to characterize simple points for 2D binary images

TL;DR

This paper adapts topological numbers to 2D binary images to characterize simple points, comparing their effectiveness with other methods and analyzing the limits of thinning algorithms in topology preservation.

Contribution

It introduces a simplified approach using topological numbers for 2D images and compares it with existing methods for simple point characterization.

Findings

Topological numbers effectively characterize simple points in 2D images.

Comparison shows advantages over Hilditch and Yokoi methods.

Identifies the maximum local configurations for topology-preserving thinning.

Abstract

In this paper, we adapt the two topological numbers, which have been proposed to efficiently characterize simple points in specific neighborhoods for 3D binary images, to the case of 2D binary images. Unlike the 3D case, we only use a single neighborhood to define these two topological numbers for the 2D case. Then, we characterize simple points either by using the two topological numbers or by a single topological number linked to another one condition. We compare the characterization of simple points by topological numbers with two other ones based on Hilditch crossing number and Yokoi number. We also highlight the number of possible configurations corresponding to a simple point, which also represents the maximum limit of local configurations that a thinning algorithm operating by parallel deletion of simple (individual) points may delete while preserving topology (limit usually not reachable, depending on the deletion strategy).