Title: Accelerating Cubical Persistent Homology for 2D and 3D Images Using Union-Find, Pruning, and Lookup Tables

Abstract:

This paper introduces Flash Cubical, a novel and highly optimized approach to calculating cubical persistence on V-filtrations for two- and three-dimensional images over the field $\mathbb{F}_2$. The architecture of this implementation relies on three fundamental strategies. First, it leverages specific properties of cubical complexes, utilizing duality and union-find algorithms to determine persistence for the highest dimension. Second, the method employs edge pruning to significantly enhance the speed and efficiency of the union-find operations. Third, it incorporates a lookup table that capitalizes on the inherent regularity of cubical complexes to pre-calculate local information, thereby eliminating the need for time-consuming computations during runtime. To the best of our knowledge, this represents the most efficient solution for cubical persistence with a V-filtration regarding both memory usage and processing time. While the study primarily addresses V-filtration cubical complexes, the core principles proposed here can be naturally extended to T-filtrations on cubical complexes and offer promising avenues for application to other types of complexes.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC