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

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.

• The paper introduces Flash Cubical, a C++ implementation that combines dual union-find, local pruning, and lookup tables to compute cubical persistence efficiently over mathbb{F}_2.
• The method outperforms CubicalRipser and GUDHI on tested V-filtration data, delivering roughly 314 speedups and 28 memory reductions, especially for 3D volumes.
• The approach remains specialized to regular 2D and 3D cubical images, with 3D H_1 matrix reduction, incomplete general duality proofs, and open opportunities for GPU, higher-dimensional, and alpha-complex extensions.

Overview

This paper presents Flash Cubical, a C++ implementation of cubical persistent homology over $\mathbb{F}_2$ for 2D and 3D images under the V-filtration (the lower-star filtration assigning pixel values to vertices). The implementation rests on three ideas: (1) computing top-dimensional persistence via union-find on a dual graph, justified by a cohomology duality argument; (2) pruning zero-persistence and already-classified edges before union-find; and (3) precomputing local lower-star information in lookup tables that exploit the regularity of cubical grids. Benchmarks against CubicalRipser and GUDHI show Flash Cubical is the fastest and most memory-efficient option for V-filtrations across all tested cases, with the largest gains on 3D volumes.

Background: filtrations and tie-breaking

The paper distinguishes the V-filtration, where $f_V(\sigma) = \max_{v \in \sigma} f_V(v)$, from the T-filtration, where pixel values attach to top-dimensional cells and $f_T(\sigma)$ is the minimum over incident top cells. Since persistence requires a fully refined filtration with exactly one cell per step, two tie-break levels are introduced: a first-order tie-break for equal pixel values, and a second-order tie-break ordering cells within a vertex's lower star $L(v)$. A key definitional choice is that pruning decisions refer to zero-persistence in the semi-refined filtration (first-order tie-break only); pairs arising from first-order ties are not removed by local simplification because they are harder to detect locally. The final algorithm still reports pairs against the original V-filtration values, omitting any pair whose birth and death coincide there.

Dual union-find for top-dimensional persistence

The central theoretical observation is that when every $(d-1)$-dimensional facet has one or two $d$-dimensional co-facets — true of all cubical complexes built from images — top-dimensional persistence reduces to union-find on a dual graph. Top cells become dual vertices; facets become dual edges; facets with a single co-facet connect to an added "infinite" vertex. Processing dual edges in reverse filtration order places the computation in the cohomology setting, where union-find yields the same persistence pairs as matrix reduction but at near-linear cost.

The appendix sketches a proof over $\mathbb{Z}/p\mathbb{Z}$: coboundary vectors of facets have at most two nonzero entries ($V_r - V_s$ or $V_r$), and an injective linear embedding into one extra dimension converts single-cofacet vectors into the same two-sparse form. Linear dependence of such vectors is then exactly connectivity in the dual graph, so the span test becomes a find operation, and pairing follows by taking the youngest dual vertex of a negative edge. The authors note this argument is grounded in prior work on dual cubical complexes (Garin et al., 2020), but claim it plausibly extends beyond cubical complexes — e.g., to alpha filtrations — while providing only elements of a proof rather than a complete general theorem.

In 2D this means both $H_0$ and $f_V(\sigma) = \max_{v \in \sigma} f_V(v)$0 are computed entirely by union-find; in 3D, $f_V(\sigma) = \max_{v \in \sigma} f_V(v)$1 and $f_V(\sigma) = \max_{v \in \sigma} f_V(v)$2 use union-find, leaving only a reduced $f_V(\sigma) = \max_{v \in \sigma} f_V(v)$3 problem for matrix reduction.

Pruning and lookup tables

Before union-find runs, two edge classes are eliminated: direct zero-persistence negative edges (paired locally within a lower star) and positive edges, which would otherwise trigger fruitless find operations. In 2D an additional "cross-pruning" step removes all $f_V(\sigma) = \max_{v \in \sigma} f_V(v)$4-negative edges from the $f_V(\sigma) = \max_{v \in \sigma} f_V(v)$5 dual computation, since both dimensions operate on the same edge set; $f_V(\sigma) = \max_{v \in \sigma} f_V(v)$6 is computed first because it produces the most pairs.

The lookup table exploits grid regularity: a vertex's entire lower star and second-order tie-break order are determined by which of its neighbours (8 in 2D, 26 in 3D) precede it in the semi-refined filtration. Although a naive count gives $f_V(\sigma) = \max_{v \in \sigma} f_V(v)$7 configurations in 3D, collapsing configurations where absent adjacent vertices make diagonal presence irrelevant reduces this to 15,935 precomputed entries covering classification (locally positive/negative, zero-persistence, direct) and tie-break order. This is the one component that does not generalize beyond cubical complexes; the authors also note that for simplicial complexes, apparent pairs (Kaji et al., 2020) offer a cheaper analogue of the pruning criterion.

Algorithms

The 2D pipeline is: lookup table query → local pruning → $f_V(\sigma) = \max_{v \in \sigma} f_V(v)$8 union-find → cross-pruning → $f_V(\sigma) = \max_{v \in \sigma} f_V(v)$9 dual union-find. The 3D pipeline is: lookup → pruned $f_T(\sigma)$0 union-find → pruned $f_T(\sigma)$1 dual union-find → pruned $f_T(\sigma)$2 matrix reduction, where edges paired in $f_T(\sigma)$3 and squares paired in $f_T(\sigma)$4 are removed from the row and column sets respectively before reduction.

Experimental results

Benchmarks (single-threaded, Intel i7-12650H, 16 GB RAM) compare Flash Cubical against CubicalRipser 0.0.33 and GUDHI 3.12.0 on synthetic uniform noise, Lena, DIV2K, and the Fuel, Bonsai, and Aneurism volumes. Representative results:

Case	Flash time (s) / MB	GUDHI	CRipser
Synthetic 2D large ($f_T(\sigma)$5)	0.11 / 23	1.95 / 266	0.34 / 208
Synthetic 3D large ($f_T(\sigma)$6)	2.21 / 239	13.79 / 958	8.29 / 783
Bonsai ($f_T(\sigma)$7)	0.53 / 194	7.28 / 1384	2.94 / 440
Aneurism ($f_T(\sigma)$8)	0.28 / 189	6.16 / 1519	2.62 / 437

Flash Cubical is fastest and leanest in every tested case, with speedups of roughly 3–14× over CubicalRipser and memory reductions of 2–8×; gains are largest on 3D real data. Two reference comparisons temper these claims. First, in reduced modes that sacrifice a dimension (Flash skipping $f_T(\sigma)$9, CubicalRipser skipping $L(v)$0), Flash drops from 2.21 s to 0.7 s on synthetic $L(v)$1 data, but gains on Bonsai and Aneurism are modest — indicating much of the runtime on real volumes lies outside $L(v)$2. Second, GUDHI's dedicated T-filtration module computes 2D persistence roughly twice as fast as Flash Cubical at comparable memory (e.g., 0.06 s vs 0.11 s on synthetic $L(v)$3). The authors state plainly that for 2D images where either filtration is acceptable, GUDHI's T-filtration module is the more efficient choice; Flash Cubical's advantage is specific to V-filtrations and to 3D.

Limitations and open questions

Several caveats bear directly on the results. The duality-based union-find argument is supported only by "elements of a proof"; a full proof, particularly for settings beyond cubical complexes, is not given. Zero-persistence detection operates in the semi-refined filtration, so pairs created by first-order tie-breaks are not pruned locally. The lookup-table approach requires strong regularity and does not transfer to point-cloud-derived complexes. The 3D $L(v)$4 computation remains standard matrix reduction on a reduced problem, and the authors identify it as needing a better algorithm. Whether the combination of duality, pruning, and apparent-pair detection yields practical gains for alpha filtrations is explicitly left open, as are higher-dimensional cubical extensions, GPU parallelization of the lookup and union-find phases, and multithreaded computation of the top and bottom dimensions.

Conclusion

Flash Cubical demonstrates that combining cohomological duality, aggressive local pruning, and precomputed lower-star tables yields substantial time and memory improvements for V-filtration cubical persistence, particularly on 3D volumes, where it outperforms existing tools by large margins. Its scope is deliberately narrow — 2D/3D image data over $L(v)$5 under the V-filtration — and the paper is candid that GUDHI's T-filtration module remains preferable for 2D users indifferent to filtration type. The main open problems are a complete proof of the duality-to-union-find reduction in general, an efficient 3D $L(v)$6 method, and extension of the framework to T-filtrations and alpha complexes.