Sparse cubical complexes for efficient topology-preservation in image data

2026-09-29

AI Summary

This study addresses the prohibitive computational overhead of persistent homology (PH) in image segmentation, which hinders its scalability to large-scale 3D data. To overcome this limitation, we introduce sparse cubical complexes into PH computation for the first time and propose a topology-aware deep learning framework based on sparse cubical filtration. By eliminating redundant information, this approach accelerates topological feature extraction and optimization. The proposed method achieves an effective balance between computational efficiency and topological precision, yielding a hundredfold speedup and an 80% improvement in topological accuracy. Furthermore, it enables end-to-end training on large-scale 3D data, establishing a new paradigm for the scalable application of topological methods in 3D vision.

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.

Problem

• Persistent Homology
• Image Segmentation
• Topological Preservation
• Computational Cost
• Cubical Complexes

Innovation

• Sparse Cubical Filtrations
• Persistent Homology
• Topology Preservation
• Image Segmentation
• Computational Efficiency