Simple pairs of points in digital spaces. Topology-preserving transformations of digital spaces by contracting simple pairs of points

Transformations of digital spaces preserving local and global topology play an important role in thinning, skeletonization and simplification of digital images. In the present paper, we introduce and study contractions of simple pair of points based on the notions of a digital contractible space and contractible transformations of digital spaces. We show that the contraction of a simple pair of points preserves local and global topology of a digital space. Relying on the obtained results, we study properties if digital manifolds. In particular, we show that a digital n-manifold can be transformed to its compressed form with the minimal number of points by sequential contractions of simple pairs. Key Words: Graph, digital space, contraction, splitting, simple pair, homotopy, thinning

💡 Research Summary

The paper addresses a fundamental problem in digital image processing: how to simplify a digital space (a graph representation of pixels or voxels) while guaranteeing that both local and global topological properties are preserved. Existing thinning and skeletonization techniques rely on the removal of a single “simple point,” a vertex whose deletion does not alter the adjacency structure of its neighborhood. However, in many practical situations a single‑point deletion is insufficient for aggressive reduction, especially when the underlying structure is complex. To overcome this limitation the authors introduce the notion of a simple pair of points and a corresponding contraction operation.

A digital space is modeled as an undirected graph (G=(V,E)) where each vertex corresponds to a pixel/voxel and edges encode a chosen adjacency (e.g., 4‑ or 8‑adjacency in 2‑D, 6‑ or 26‑adjacency in 3‑D). A subgraph is called digitally contractible if it can be reduced to a single vertex by a sequence of simple‑point deletions; this mirrors the classical topological concept of a contractible space. Building on this, a simple pair ({p,q}) is defined as two adjacent vertices whose closed neighborhoods (N