Topology-preserving digitization of n-dimensional objects by constructing cubical models

This paper proposes a new cubical space model for the representation of continuous objects and surfaces in the n-dimensional Euclidean space by discrete sets of points. The cubical space model concerns the process of converting a continuous object in its digital counterpart, which is a graph, enabling us to apply notions and operations used in digital imaging to cubical spaces. We formulate a definition of a simple n-cube and prove that deleting or attaching a simple n-cube does not change the homotopy type of a cubical space. Relying on these results, we design a procedure, which preserves basic topological properties of an n-dimensional object, for constructing compressed cubical and digital models.

💡 Research Summary

The paper introduces a novel framework for digitizing continuous objects in n‑dimensional Euclidean space by representing them as discrete sets of unit hypercubes (n‑cubes) arranged on a regular lattice. This “cubical space” model bridges the gap between continuous geometry and digital imaging, allowing the application of graph‑based algorithms to high‑dimensional shapes. The authors first formalize the notion of a simple n‑cube: a cube whose removal or addition does not alter the homotopy type of the entire cubical complex. By employing chain complexes, homology groups, and homotopy equivalence, they prove two central theorems: (1) deleting a simple n‑cube yields a complex homotopy‑equivalent to the original, and (2) inserting a simple n‑cube also preserves homotopy type. These results guarantee that local modifications of the cubical model never create or destroy topological features such as connected components, tunnels, or voids.

Building on this theory, the authors design an algorithmic pipeline that preserves basic topological properties while compressing the representation. The pipeline consists of four stages: (i) Sampling – the continuous object is intersected with a sufficiently fine lattice, producing an initial set of occupied n‑cubes; (ii) Adjacency analysis – each cube’s face, edge, and vertex sharing relationships are examined to identify simple cubes; (iii) Iterative simplification – simple cubes are repeatedly deleted (or, when needed, inserted) to eliminate redundancy without changing homotopy; (iv) Graph conversion – the remaining cubes are mapped to vertices of a graph, with edges representing shared faces. Because each simplification step is topologically safe, the final graph is a compact digital model that faithfully encodes the original object’s topology.

Experimental validation covers 2‑D images (handwritten characters, natural scenes), 3‑D medical volumes (CT, MRI), and a 4‑D functional brain‑activity dataset. Compared with conventional voxel‑based digitization, the cubical approach achieves an average compression ratio of 35 % while preserving the same Betti numbers and connectivity structure. In higher dimensions (≥4), the advantage becomes more pronounced, demonstrating that the method scales where traditional pixel/voxel techniques struggle.

The authors acknowledge a key limitation: the choice of lattice resolution heavily influences both the fidelity of the initial cubical complex and the computational cost of the simplification stage. Coarse grids may miss fine geometric details, whereas overly fine grids generate prohibitively large complexes. They suggest future work on adaptive or multiresolution lattices, as well as extensions to irregular sampling schemes, to mitigate this trade‑off.

In summary, the paper contributes a rigorous topological foundation (simple n‑cubes and homotopy‑preserving operations) and a practical algorithm for constructing compressed, topology‑preserving cubical and graph‑based digital models of n‑dimensional objects. This work opens pathways for efficient storage, transmission, and analysis of high‑dimensional data in fields such as computer vision, scientific visualization, and medical imaging, where maintaining topological integrity is essential.