Sparse cubical complexes for efficient topology-preservation in image data

Alexander H. Berger; Marco Fontana; Daniel Rueckert; Johannes C. Paetzold; Laurin Lux; Ulrich Bauer

cs.CVSep 29, 2026

Sparse cubical complexes for efficient topology-preservation in image data

Authors: Alexander H. Berger, Marco Fontana, Daniel Rueckert, Johannes C. Paetzold, Laurin Lux, Ulrich Bauer

Organizations: Weill Cornell Medicine, New York, USA · Technical University of Munich, Munich, Germany · Department of Computing, Imperial College London, UK · Munich Center for Machine Learning (MCML), Munich, Germany · Cornell Tech, New York, USA · Munich Data Science Institute, Technical University of Munich, Munich, Germany

Abstract

Persistent homology (PH) is a frequently used tool for extracting and preserving topological information from image data, particularly in image segmentation, where preservation of topological structures is important. However, despite its general applicability across dimensionality, domains, and target structures, the runtime cost of PH-based methods often makes their practical use infeasible. In this work, we argue that this runtime cost is largely driven by processing information that is unimportant for downstream application (e.g. as optimization objective). We propose sparse cubical filtrations as an alternative foundation for PH computation, reducing subsequent computational costs by factors of up to 100 on real datasets. We show close agreement with the optimization signal of the dense counterpart and empirically evaluate our solution's effectiveness as an optimization objective in realistic training regimes where other PH-based objectives can practically not operate (i.e., 3D data with large patch sizes). We show how our solution improves topological accuracy by up to 80% across six diverse datasets while maintaining pixel- and region-based accuracy.

Figures & tables

Figure 1: Our approach, PH-based loss functions with sparse cubical complexes, significantly reduces topological errors in image segmentation at a fraction of the cost, making PH-based loss functions usable with large patch sizes and state-of-the-art training paradigms. (a) In realistic 3D training settings (here, on ATM’26), sparseBM is orders of magnitude faster than classical PH-based loss functions and its runtime is substantially less sensitive to patch size in the sparse regime considered here. (b) sparseBM significantly reduces topological error across domains with diverse topological targets and across 2D and 3D (lower is better).

Figure 2: Core idea of the sparse construction. A common retained cubical subcomplex

{\mathbb{S}}={\mathbb{C}}_{\tau}

is selected from the comparison filtration, while retained cells keep their original filtration values. The omitted region is represented implicitly rather than instantiated as the full cubical grid.

Figure 3: Exemplary comparison between a sparse and dense barcode. Corresponding dense intervals are indicated in grey. The intervals are: (a) kept with shifted death, (b) exact, (c) kept via label with shifted birth and death, (d) omitted, (e) kept exact via label

Figure 4: Barcode extraction time using dense and sparse cubical complexes on outputs of segmentation models.

dataset	loss	Dice $\uparrow$	BM err. $\downarrow$	clDice $\uparrow$	VOI $\downarrow$	NSD $\uparrow$	Train. time $\downarrow$
ATM’26 airway lumen $128^{3}$	Dice+CE	.9446 $\pm$ .0032	102 $\pm$ 12 ∗	.900 $\pm$ .002	.0064 $\pm$ .0003	.955 $\pm$ .004	1.00 $\times$
	clDice	.9421 $\pm$ .0048	84.8 $\pm$ 9.3 ∗	.906 $\pm$ .006	.0065 $\pm$ .0004	.953 $\pm$ .004	1.29 $\times$
	Skel. Recall	.9437 $\pm$ .0022	96.4 $\pm$ 6.5 ∗	.896 $\pm$ .003 ∗	.0065 $\pm$ .0001 ∗	.953 $\pm$ .001	1.03 $\times$
	warping †	.9413 $\pm$ .0038	94.6 $\pm$ 12.3 ∗	.896 $\pm$ .008	.0066 $\pm$ .0003	.951 $\pm$ .006	1.40 $\times$
	sparse BM (ours)	.9449 $\pm$ .0017	18.8 $\pm$ .5	.906 $\pm$ .006	.0061 $\pm$ .0001	.956 $\pm$ .004	1.15 $\times$
NISB-B cell interfaces $128^{2}{\times}64$	Dice+CE	.7847 $\pm$ .0001	367.3k $\pm$ 3.3k ∗	.766 $\pm$ .000	4.521 $\pm$ .000	.929 $\pm$ .000	1.00 $\times$

Table 1: Test-set results in 2D and 3D. Three seeds each, scored once on held-out data; mean

\pm

sd over seeds. Bold: best arm of the dataset. ∗ : sparse BM is significantly better than that arm on that metric (paired

t

-test over seeds,

p<0.05

). Train. time: wall clock of the whole training, as a multiple of Dice+CE. † : Runtime improved version of the authors’ released code.

Figure 5: SparseBM closely reproduces the optimization signal of dense Betti Matching. (a) shows the cosine similarity between their update steps, together with baseline comparisons to ComboLoss and to a different batch with the same loss. (b) shows the corresponding difference in persistence diagrams. Most affected features have very small persistence and lie in the background. Both plots are created from real samples during training on the ATM’26 dataset.

patch	loss	Dice $\uparrow$	BM err. $\downarrow$	clDice $\uparrow$	VOI $\downarrow$	NSD $\uparrow$	Train. time $\downarrow$
$128^{3}$	Dice+CE	.9446 $\pm$ .0032	102 $\pm$ 12 ∗	.900 $\pm$ .002	.0064 $\pm$ .0003	.955 $\pm$ .004	1.00 $\times$
$128^{3}$	sparse BM (ours)	.9449 $\pm$ .0017	18.8 $\pm$ .5	.906 $\pm$ .006	.0061 $\pm$ .0001	.956 $\pm$ .004	1.15 $\times$
$64^{3}$	Dice+CE	.9299 $\pm$ .0124	129 $\pm$ 20 ∗	.878 $\pm$ .014 ∗	.0075 $\pm$ .0011	.939 $\pm$ .013	1.00 $\times$
	DMT	.9407 $\pm$ .0013	66.5 $\pm$ 1.8 ∗	.906 $\pm$ .004	.0065 $\pm$ .0001	.957 $\pm$ .002	5.38 $\times$
	dense BM	.9358 $\pm$ .0062	25.1 $\pm$ 1.8 ∗	.904 $\pm$ .002 ∗	.0069 $\pm$ .0005	.949 $\pm$ .004	1.88 $\times$
	sparse BM (ours)	.9412 $\pm$ .0023	20.2 $\pm$ 2.1	.909 $\pm$ .000	.0063 $\pm$ .0001	.954 $\pm$ .001	1.19 $\times$

Table 2: Comparison to PH-based losses in a training setting with reduced patch size on the ATM’26 dataset. Same annotation as in Table 1 .

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 6: Ablation on the weight parameter on the NISB-B dataset (validation set, individual patches). Betti Matching error (left axis, blue) and Dice (right axis, orange) against the topology weight. The dashed line is the Dice+CE baseline, the dotted line the admissibility floor (reference

-\,0.010

) for model selection. The selected weight is the last point above the floor.

Figure 7: We observe that sparseBM’s optimal weight can be chosen based on the ratio between the gradient norm of sparseBM and ComboLoss.

$\tau$	kept (%)	runtime (ms) $\downarrow$	BM err. $\downarrow$	Dice $\uparrow$
0.5	1.26	48.6 $\pm$ 6.2	4.40	.9724
0.6	1.26	49.1 $\pm$ 7.0	4.06	.9734
0.7	1.26	49.1 $\pm$ 7.1	4.09	.9740
0.8 (default)	1.27	49.3 $\pm$ 6.7	3.96	.9732
0.9	1.28	49.1 $\pm$ 6.7	4.14	.9720
0.95	1.29	50.0 $\pm$ 7.4	3.92	.9733

Appendix

Table 3: Kept fraction, runtime, validation BM error and Dice with varying

\tau

on ATM’26 (validation). Bold: best

\tau

; the runtime differences between

\tau

values are within 2% (paired per micro-batch).

Figure 8: Fraction of included voxels as the sparsification threshold

\tau

increases. The y-axis is logarithmic.

dataset	modality	target	train / val / test	resolution	patch
ATM’26	chest CT	airway tree	193 / 50 / 53 volumes	native, 0.51–0.92 mm in-plane	$128^{3}$
NISB-B	synthetic EM	neuron boundaries	5 / 1 / 1 volumes a	$4.5\times 4.5\times 10$ nm (lifted)	$128^{2}\times 64$
BraTS-METS	brain MRI	metastases (tumour core)	733 / 181 / 162 cases	1 mm isotropic	$128^{3}$
MMWHS	cardiac CT	LV myocardium	12 / 4 / 4 volumes b	1.25 mm isotropic	$128^{3}$
FIVES	fundus photography	retinal vessels	478 / 120 / 200 images	native	$2048^{2}$
ACDC	cine MRI	LV myocardium	1506 / 396 / 1076 slices	native, 1.37–1.92 mm	$224^{2}$

Appendix

Table 4: Datasets. Train/val = fold 0 of the cross-validation split; test = held-out cases, scored once.

Figure 9: We interleave forward/backward passes with the loss calculation of two subsequent micro-batches to maximize GPU utilization and reduce runtime.

Figure 10: Runtime comparison for metric calculation between sparse and dense betti matching error. The runtime decrease translates to application on binary inputs.

loss	$\Delta$ Dice	$\Delta$ clDice	$\Delta$ TLD	$\Delta$ BD
Dice+CE	$-0.53$	$-1.51$	$+2.75$	$+4.32$
clDice	$-0.70$	$-1.13$	$+2.55$	$+4.15$
Skeleton Recall	$-0.71$	$-1.68$	$+2.20$	$+3.51$

Appendix

Table 5: Results on the use of sparse cubical complexes as a post-processing tool. Each cell describes the improvements after post-processing compared to the respective base model.

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Persistent Homology

Open paper AskPDF arXiv

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.

Abstract

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