Title: Topological characterisation of 3D digital image based on topological and Betti numbers

Authors: Yibo Zhao

Addresses: School of Data Science and Computer, Shandong Women's University, Jinan 250300, Shandong, China

Abstract: The Euler-Poincaré characteristic (EPC) is recognised as a key topological parameter in digital topology, commonly used to describe object connectivity and derive various quantities and functions. Its calculation is closely linked to Betti numbers, which represent the number of tunnels, cavities, and components. A new algorithm has been proposed for calculating the number of tunnels in a 3 × 3 × 3 neighbourhood of a point in three-dimensional (3D) discrete space, where the number of tunnels is determined as the topological number T₆ (p) minus 1. The correctness of the algorithm has been proven, and the resulting property of the number of tunnels has been demonstrated. Additionally, the correctness of the Euler-Poincaré property calculated by Jiang's algorithm has been validated.

Keywords: Euler-Poincaré characteristics; EPCs; topological numbers; tunnels; component.

DOI: 10.1504/IJDSDE.2025.148520

International Journal of Dynamical Systems and Differential Equations, 2025 Vol.14 No.3, pp.267 - 279

Received: 07 Mar 2025
Accepted: 22 Apr 2025
Published online: 10 Sep 2025 *

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