Alessio Borgi

Alessio Borgi

PhD in Graph Neural Networks and Generative AI 👨🏻‍💻🤖

Cubical Complexes: Topology on Grids and Images

2 minute read

Published:

TL;DR: A cubical complex builds topology from elementary intervals [k, k+1] and their products — hypercubes. This is the native structure for image data: pixels are 0-cubes, edges between adjacent pixels are 1-cubes, and pixel squares are 2-cubes. Cubical persistence is implemented directly in CubicalRipser and GUDHI and runs faster on image data than any simplicial approach.

Elementary Cubes and Cubical Complexes

An elementary interval is either a point \([k,k] = \{k\}\) (degenerate) or a unit interval \([k, k+1]\) (non-degenerate) for \(k \in \mathbb{Z}\). An elementary cube \(Q\) in \(\mathbb{R}^d\) is a product of elementary intervals:

$$Q = I_1 \times I_2 \times \cdots \times I_d$$

The dimension of \(Q\) is the number of non-degenerate intervals in the product. A cubical complex \(K\) is a finite collection of elementary cubes closed under the operation of taking faces (sub-cubes obtained by fixing one or more coordinates).

For a 2D image:

  • 0-cubes: individual pixels (vertices)
  • 1-cubes: shared edges between adjacent pixels (horizontal and vertical)
  • 2-cubes: 2×2 pixel squares (faces)

Cubical Boundary Maps

The boundary map for cubical complexes mirrors the simplicial case. For a 1-cube \([k, k+1] \times \{j\}\):

\[\partial([k,k+1] \times \{j\}) = [\{k+1\} \times \{j\}] - [\{k\} \times \{j\}]\]

For a 2-cube \([k,k+1] \times [j, j+1]\):

\[\partial = [k+1,k+1]\times[j,j+1] - [k,k]\times[j,j+1] + [k,k+1]\times[j+1,j+1] - [k,k+1]\times[j,j]\]

The property \(\partial \circ \partial = 0\) holds, giving a chain complex and hence well-defined homology groups \(H_n(K)\) with the same topological interpretation as in the simplicial case.

Sub-Level Set Filtrations on Images

For an image \(f: \{1,\ldots,m\} \times \{1,\ldots,n\} \to \mathbb{R}\) (grayscale), define the sub-level set filtration:

\[K^\alpha = \{Q \in K : \max_{p \in Q} f(p) \leq \alpha\}\]

As \(\alpha\) increases, cubes enter one by one in order of their maximum pixel value. The persistent homology of this filtration:

  • \(H_0\) bars: connected components of bright regions (or dark, depending on convention).
  • \(H_1\) bars: loops/rings in the image (e.g., circular blobs, voids).

This is directly useful for feature detection in medical images, materials science, and astronomical data.

Key Insight: Cubical persistence avoids the $$O(n^2)$$ edge creation cost of Vietoris-Rips on image data. A 512×512 image has 262,144 pixels; building a Vietoris-Rips complex would create ~34 billion edges. Cubical complexes only include edges between adjacent pixels — O(n) edges — making computation feasible. CubicalRipser handles millions of voxels.

References

  • T. Kaczynski, K. Mischaikow, M. Mrozek, Computational Homology, Springer, 2004. The standard reference for cubical homology.
  • S. Suzuki, Y. Miyata, “CubicalRipser: Software for Computing Persistent (Co)homology of Image Data,” arXiv:2005.12692, 2020.

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