[Paper Review] Homotopy relations for digital images

Laurence Boxer, P. Christopher Staecker|arXiv (Cornell University)|Sep 22, 2015

Digital Image Processing Techniques8 references3 citations

TL;DR

This paper introduces three new homotopy relations—homotopy similarity, same long homotopy type, and same real homotopy type—for digital images, extending traditional digital homotopy equivalence to compare bounded and unbounded images. Unlike standard homotopy equivalence, these relations allow finite and infinite digital images to be considered similar if they share fundamental group structure and homotopic behavior, even when one has infinite diameter. The key contribution is that these relations preserve fundamental group isomorphism and are closed under wedges and Cartesian products, enabling homotopy-theoretic analysis of unbounded digital spaces.

ABSTRACT

We introduce three generalizations of homotopy equivalence in digital images, to allow us to express whether a finite and an infinite digital image are similar with respect to homotopy. We show that these three generalizations are not equivalent to ordinary homotopy equivalence, and give several examples. We show that, like homotopy equivalence, our three generalizations imply isomorphism of fundamental groups, and are preserved under wedges and Cartesian products.

Motivation & Objective

• To address the limitation in digital topology where bounded and unbounded digital images cannot be considered homotopy equivalent under standard definitions.
• To define new homotopy relations that allow bounded and unbounded digital images to be considered similar in terms of homotopy properties.
• To preserve fundamental group isomorphism and closure under topological constructions like wedges and Cartesian products.
• To generalize digital homotopy theory to include infinite-diameter images while maintaining algebraic invariants.

Proposed method

• Introduce three new relations: homotopy similarity, same long homotopy type, and same real homotopy type, each generalizing digital homotopy equivalence.
• Define homotopy similarity via nested sequences of subimages with increasing size, where each subimage is homotopy equivalent to a corresponding subimage in the other space.
• Define same long homotopy type using infinite chains of homotopy equivalences over increasing time intervals.
• Define same real homotopy type using continuous homotopies over real-valued time intervals, adapted to digital settings.
• Use digital continuity and path-based homotopies in the standard $c_u$-adjacency framework to ensure well-definedness.
• Prove closure under wedges and finite Cartesian products by constructing component-wise homotopies and using relaxed adjacencies.

Experimental results

Research questions

• RQ1 Can bounded and unbounded digital images be meaningfully considered homotopically similar, despite differing in diameter?
• RQ2 Do the proposed relations—homotopy similarity, same long homotopy type, and same real homotopy type—imply one another, or are they strictly distinct?
• RQ3 Are these new relations equivalent to standard digital homotopy equivalence when both images are finite?
• RQ4 Do these relations preserve fundamental group isomorphism, and do they respect topological constructions like wedges and products?
• RQ5 Can the new relations be used to compare digital images with infinite diameter while retaining algebraic invariants?

Key findings

• Homotopy similarity, same long homotopy type, and same real homotopy type are strictly more general than standard digital homotopy equivalence and are not equivalent to one another.
• All three relations imply isomorphism of fundamental groups, even when one image has infinite diameter.
• The relations are preserved under wedging of pointed digital images, meaning the homotopic resemblance is maintained under this construction.
• Finite Cartesian products preserve all three relations when relaxed adjacencies are used, ensuring compatibility with product structures.
• Same long homotopy type implies same real homotopy type, but the converse does not hold in general.
• The relations are not equivalent to standard homotopy equivalence when one image is infinite, showing a fundamental distinction in behavior between finite and infinite digital images.