Determining the Number of Holes of a 2D Digital Component is Easy

The number of holes in a connected component in 2D images is a basic invariant. In this note, a simple formula was proven using our previous results in digital topology (Chen 2004, Chen and Rong (2010). The new is: $h =1+ (|C_4|-|C_2|)/4$, where h is the number of holes, and $C_i$ indicate the set of corner points having $i$ direct adjacent points in the component.

💡 Research Summary

The paper addresses a fundamental problem in binary image analysis: determining the number of holes (topological genus) of a connected component in a two‑dimensional digital image. While the hole count is a basic invariant used in shape analysis, character recognition, medical imaging, and many other fields, traditional approaches rely on either the Euler characteristic (χ = V − E + F) or on multi‑pass labeling and boundary‑following algorithms. These methods typically require a full scan of the image, construction of adjacency graphs, or repeated traversals of the component’s interior, which can be computationally expensive for large‑scale or real‑time applications.

Building on earlier work in digital topology (Chen 2004; Chen & Rong 2010), the authors propose a remarkably simple closed‑form expression for the hole count: \