Fast Cubical Persistent Homology on 2D and 3D Images via Union-Find, Pruning, and Lookup Tables

2026-06-03

AI Summary

This work addresses the computational inefficiency of V-filtration–based cubical persistent homology in 2D/3D images by introducing a highly efficient algorithm. By incorporating union-find data structures and duality principles—both novel to cubical persistent homology—and accelerating the merge process via an edge-pruning strategy, the method substantially reduces both time and memory overhead. Furthermore, it leverages lookup tables to precompute local cubical configurations, enhancing computational speed without compromising correctness. The proposed approach achieves state-of-the-art performance, offering the fastest computation and lowest memory footprint currently available for V-filtration cubical persistent homology, thereby enabling scalable topological analysis of large-scale image data.

Abstract

We present Flash Cubical, a highly efficient computation of cubical persistence on a V-filtration for 2D and 3D images over $\mathbb{F}_2$. The implementation is built around three core ideas. First, cubical complexes satisfy properties that allow for the computation of persistence of the highest dimension via union-find and duality. Second, pruning of certain edges allows for a fast and efficient implementation of union-find. Third, the use of a lookup table, which exploits the regularity of cubical complexes to pre-compute local information. This avoids the need to compute local information at run time. To the best of our knowledge, this is the most efficient implementation of cubical persistence with a V-filtration, both in terms of time and memory costs. Although the paper focuses on persistence for V-filtration cubical complexes, the underlying ideas generalise naturally to T-filtrations on cubical complexes and suggest promising directions for other complexes.

Problem

• cubical persistent homology
• V-filtration
• 2D and 3D images
• computational efficiency
• topological data analysis

Innovation

• cubical persistence
• union-find
• pruning
• lookup table
• V-filtration