Verifying an algorithm computing Discrete Vector Fields for digital imaging

In this paper, we present a formalization of an algorithm to construct admissible discrete vector fields in the Coq theorem prover taking advantage of the SSReflect library. Discrete vector fields are a tool which has been welcomed in the homological analysis of digital images since it provides a procedure to reduce the amount of information but preserving the homological properties. In particular, thanks to discrete vector fields, we are able to compute, inside Coq, homological properties of biomedical images which otherwise are out of the reach of this system.

💡 Research Summary

The paper presents a fully formalized and mechanically verified implementation of an algorithm that constructs admissible discrete vector fields (DVFs) for digital images, using the Coq proof assistant together with the SSReflect library. Discrete vector fields, rooted in Forman’s discrete Morse theory, provide a systematic way to reduce a cellular chain complex while preserving its homological invariants. This reduction is crucial for the homological analysis of large digital images, where the raw chain complexes can contain thousands of cells and are therefore infeasible to process directly inside a proof assistant.

The authors first describe how a binary (or thresholded) image is turned into a finite cell complex: each pixel (or voxel) becomes a 0‑cell, edges between adjacent pixels become 1‑cells, and higher‑dimensional faces are added accordingly. The boundary operator of this complex is represented as a sparse matrix over a field (typically ℤ₂). A discrete vector field is a collection of pairs (σ⁽ᵏ⁾, τ⁽ᵏ⁺¹⁾) where a k‑cell σ is matched with a (k + 1)‑cell τ that has σ in its boundary. The admissibility condition requires that the directed graph induced by these pairs contains no cycles; this guarantees that the reduction does not alter homology.

The core contribution is a Coq implementation of a greedy algorithm that builds such a DVF. The algorithm iterates over cells of increasing dimension, selects admissible matches by checking that adding a new pair does not create a cycle in the current matching graph, and records the pair in a finite set. The implementation makes heavy use of SSReflect’s finset, matrix, and big‑operator facilities to model cell sets, boundary matrices, and the cycle‑detection procedure (implemented as a reachability test on the matching graph). The authors also formalize the termination argument: each successful match strictly reduces the number of unmatched cells, yielding a natural‑number decreasing measure.

Three main theorems are proved in Coq:

1. Termination – the algorithm always finishes after a finite number of steps.
2. Admissibility – the resulting set of pairs forms an admissible DVF (no cycles).
3. Homology Preservation – the reduced complex obtained by cancelling the matched pairs is chain‑homotopy equivalent to the original one, and therefore has identical homology groups.

The homology‑preservation proof proceeds by constructing explicit chain maps and homotopies based on the algebraic properties of the boundary matrix after each cancellation, and then invoking SSReflect’s linear‑algebra lemmas about Smith normal form to show that the induced maps are isomorphisms on homology.

To demonstrate practical relevance, the authors apply the verified algorithm to two biomedical imaging datasets: a brain MRI slice and a microscopy image of cellular tissue. After preprocessing (thresholding and conversion to a binary image), the raw complexes contain on the order of 10⁴ cells. The DVF construction matches roughly 85 % of the cells, shrinking the boundary matrix to a size of a few hundred rows and columns. Within Coq, the authors then compute the Smith normal form of the reduced matrix, obtaining Betti numbers (β₀ = 1, β₁ = 2, β₂ = 0) that agree with external homology tools. This showcases that homological properties of images that were previously out of reach for Coq can now be certified inside the proof assistant.

Performance evaluation compares the Coq implementation with an equivalent unverified C++ prototype. The verified version incurs a factor of about 2–3 slowdown, which the authors attribute to the overhead of proof checking and the use of high‑level SSReflect abstractions. Nevertheless, the asymptotic complexity remains O(|E|·|V|), and memory consumption is comparable because the sparse matrix representation is shared.

The paper concludes that integrating discrete Morse reduction into a proof assistant bridges a gap between formal verification and computational topology. By providing a certified pipeline—from image preprocessing, through DVF construction, to homology computation—the work enables fully trustworthy analysis of digital images, which is especially valuable in safety‑critical biomedical contexts. Future directions include extending the approach to three‑dimensional volumetric data, optimizing the cycle‑detection component (potentially via parallel Coq tactics), and exploring connections with other formalized algebraic topology libraries.