[Paper Review] Some enumerations of binary digital images

P. Christopher Staecker|arXiv (Cornell University)|Feb 22, 2015

Digital Image Processing Techniques4 references3 citations

TL;DR

This paper presents exhaustive enumerations of connected binary digital images in ℤ² under 4- and 8-adjacency up to isomorphism, classifying them by rigidity, irreducibility, and pointed irreducibility. Using computer-assisted brute-force search with Sage, it identifies that non-rigid irreducible images are rare and conjectures they are precisely the cycle graphs Cₙ for n > 4.

ABSTRACT

The topology of digital images has been studied much in recent years, but no attempt has been made to exhaustively catalog the structure of binary images of small numbers of points. We produce enumerations of several classes of digital images up to isomorphism and decide which among them are homotopy equivalent to one another. Noting some patterns in the results, we make some conjectures about digital images which are irreducible but not rigid.

Motivation & Objective

• To systematically enumerate all connected binary digital images in ℤ² with 4- and 8-adjacency up to isomorphism.
• To classify these images by homotopy-theoretic properties: rigidity, irreducibility, and pointed irreducibility.
• To identify patterns in non-rigid irreducible images and formulate conjectures about their structure.
• To provide machine-readable data and visualizations for future research in digital topology.

Proposed method

• Conducting exhaustive computer searches using Sage to enumerate all connected simple graphs up to isomorphism for small point counts.
• Applying graph isomorphism checks to classify images under 4-adjacency (up to 12 points) and 8-adjacency (up to 9 points).
• Using homotopy equivalence definitions: a map is homotopic to identity in one step if it is continuous and fixes the image structure.
• Implementing algorithms to test whether the identity map is homotopic to a non-surjective map in one step, determining reducibility.
• Verifying rigidity by checking whether only the identity map is homotopic to itself.
• Proving that every 4-adjacency image is isomorphic to an 8-adjacency image via a linear transformation f(x,y) = (x+y, x−y).

Experimental results

Research questions

• RQ1 Which connected digital images in ℤ² with 4-adjacency are irreducible but not rigid?
• RQ2 Are there non-rigid irreducible images beyond cycle graphs Cₙ for n > 4 under 4-adjacency?
• RQ3 How do the counts of rigid, irreducible, and pointed irreducible images grow with the number of points under 4- and 8-adjacency?
• RQ4 Is every 4-adjacency image isomorphic to an 8-adjacency image, and if so, via what transformation?
• RQ5 What structural properties characterize non-rigid irreducible digital images in ℤ²?

Key findings

• For 4-adjacency, only cycle graphs Cₙ with n > 4 are non-rigid and irreducible, suggesting they are the only such images.
• For 8-adjacency, the same pattern holds: non-rigid irreducible images are conjectured to be isomorphic only to Cₙ for n > 4.
• The number of 8-adjacency images exceeds that of 4-adjacency images for n ≥ 3 due to additional diagonal adjacencies.
• The identity map is homotopic to a non-surjection in one step if and only if the image is reducible, per Lemma 2.9 of [4].
• The transformation f(x,y) = (x+y, x−y) establishes an isomorphism between any 4-adjacency image and an 8-adjacency image in ℤ².
• The 13-point 4-adjacency image in Figure 2(a) is reducible but pointed irreducible, confirming that pointed irreducibility does not imply irreducibility.