[Paper Review] Determining the Number of Holes of a 2D Digital Component is Easy

Li M. Chen|arXiv (Cornell University)|Nov 16, 2012

Digital Image Processing Techniques2 references3 citations

TL;DR

This paper presents a simple, efficient formula to compute the number of holes in a 2D digital connected component: h = 1 + (|C₄| − |C₂|)/4, where C₄ and C₂ are sets of corner points with four and two direct neighbors, respectively. The method leverages digital topology and Euler characteristic principles, enabling exact hole counting via basic point counting without complex algorithms.

ABSTRACT

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.

Motivation & Objective

• To derive a simple, exact formula for determining the number of holes in a 2D digital connected component.
• To provide a computationally efficient alternative to complex topological algorithms for hole detection in digital images.
• To establish a direct link between corner point configurations (C₂ and C₄) and topological genus in 2D digital space.
• To validate the formula using both 3D embedding and curve-based topological reasoning.

Proposed method

• Embed the 2D component S into 3D space as a double S₁ ∪ S₂ at z=1 and z=2 planes to form a solid with genus g = h.
• Use the genus formula g = 1 + (|M₅| + 2|M₆| − |M₃|)/8, where M₃, M₅, M₆ are sets of points with 3, 5, and 6 neighbors in the 3D surface.
• Relate 2D corner points to 3D neighbor counts: |M₃| = 2|C₂| and |M₅| = 2|C₄|, with |M₆| = 0 for non-pathological boundaries.
• Derive the hole formula h = 1 + (|C₄| − |C₂|)/4 through substitution and simplification of the genus formula.
• Provide a second proof using the curve theorem: CP₂ = CP₄ + 4 for simple closed curves, summing over outer and inner boundaries.
• Validate the formula with concrete 2D examples, counting C₂ and C₄ points and computing h via the formula.

Experimental results

Research questions

• RQ1 Can the number of holes in a 2D digital component be computed using only local corner point information?
• RQ2 Is there a simple, exact formula for genus (number of holes) in 2D digital topology that avoids complex topological algorithms?
• RQ3 How do the counts of corner points with two and four direct neighbors relate to the topological genus of a 2D component?
• RQ4 Can the genus of a 2D component be derived from a 3D embedding of its double surface?

Key findings

• The formula h = 1 + (|C₄| − |C₂|)/4 correctly computes the number of holes in a 2D digital component with no pathological boundary cases.
• For the first example, |C₂| = 8 and |C₄| = 4, yielding h = 1 + (4 − 8)/4 = 0, correctly identifying no holes.
• For the second example, |C₂| = 6 and |C₄| = 6, yielding h = 1 + (6 − 6)/4 = 1, correctly identifying one hole.
• Each additional hole increases |C₄| by 4 relative to |C₂|, confirming the linear relationship in the formula.
• The formula is derived from both 3D embedding and curve-based topology, confirming consistency across methods.
• The method requires only counting corner points with 2 or 4 direct neighbors, making it computationally efficient and easy to implement.