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Efficient computation of topological integral transforms

Vadim Lebovici, Steve Oudot, Hugo Passe

TL;DR

This work tackles the lack of exact, weighted-topology transforms for grayscale data by introducing eucalc, a fast, parallelizable C++ library (with Python binding) that computes the Euler characteristic transform, Radon transform, and hybrid transforms on weighted axis-aligned cubical complexes. Leveraging piecewise linear Morse theory and Euler calculus, the authors reduce computation to a finite set of critical points and precompute across the $2^n$ sign patterns, enabling exact transform representations with per-direction evaluation near linear in the number of vertices. Empirical results on large grayscale datasets show significant speedups over discretized baselines and demonstrate robustness to grayscale inputs, while parallelization further accelerates preprocessing and evaluation. The approach yields practical, exact transforms with potential to enable exact inverse operations and broader extensions to general polytopal complexes in higher dimensions. The work thus provides a scalable, exact toolkit for topological shape analysis of images, with clear implications for computer vision and biomedical imaging applications.

Abstract

Topological integral transforms have found many applications in shape analysis, from prediction of clinical outcomes in brain cancer to analysis of barley seeds. Using Euler characteristic as a measure, these objects record rich geometric information on weighted polytopal complexes. While some implementations exist, they only enable discretized representations of the transforms, and they do not handle weighted complexes (such as for instance images). Moreover, recent hybrid transforms lack an implementation. In this paper, we introduce Eucalc, a novel implementation of three topological integral transforms -- the Euler characteristic transform, the Radon transform, and hybrid transforms -- for weighted cubical complexes. Leveraging piecewise linear Morse theory and Euler calculus, the algorithms significantly reduce computational complexity by focusing on critical points. Our software provides exact representations of transforms, handles both binary and grayscale images, and supports multi-core processing. It is publicly available as a C++ library with a Python wrapper. We present mathematical foundations, implementation details, and experimental evaluations, demonstrating Eucalc's efficiency.

Efficient computation of topological integral transforms

TL;DR

This work tackles the lack of exact, weighted-topology transforms for grayscale data by introducing eucalc, a fast, parallelizable C++ library (with Python binding) that computes the Euler characteristic transform, Radon transform, and hybrid transforms on weighted axis-aligned cubical complexes. Leveraging piecewise linear Morse theory and Euler calculus, the authors reduce computation to a finite set of critical points and precompute across the sign patterns, enabling exact transform representations with per-direction evaluation near linear in the number of vertices. Empirical results on large grayscale datasets show significant speedups over discretized baselines and demonstrate robustness to grayscale inputs, while parallelization further accelerates preprocessing and evaluation. The approach yields practical, exact transforms with potential to enable exact inverse operations and broader extensions to general polytopal complexes in higher dimensions. The work thus provides a scalable, exact toolkit for topological shape analysis of images, with clear implications for computer vision and biomedical imaging applications.

Abstract

Topological integral transforms have found many applications in shape analysis, from prediction of clinical outcomes in brain cancer to analysis of barley seeds. Using Euler characteristic as a measure, these objects record rich geometric information on weighted polytopal complexes. While some implementations exist, they only enable discretized representations of the transforms, and they do not handle weighted complexes (such as for instance images). Moreover, recent hybrid transforms lack an implementation. In this paper, we introduce Eucalc, a novel implementation of three topological integral transforms -- the Euler characteristic transform, the Radon transform, and hybrid transforms -- for weighted cubical complexes. Leveraging piecewise linear Morse theory and Euler calculus, the algorithms significantly reduce computational complexity by focusing on critical points. Our software provides exact representations of transforms, handles both binary and grayscale images, and supports multi-core processing. It is publicly available as a C++ library with a Python wrapper. We present mathematical foundations, implementation details, and experimental evaluations, demonstrating Eucalc's efficiency.
Paper Structure (23 sections, 6 theorems, 15 equations, 7 figures, 2 tables)

This paper contains 23 sections, 6 theorems, 15 equations, 7 figures, 2 tables.

Table of Contents

  1. Contributions
  2. Preliminaries
  3. Weighted polytopal complexes
  4. Integral transforms
  5. Naive algorithm
  6. Euler critical points
  7. Implementation and asymptotic analysis
  8. Implementation of cubical complexes
  9. Preprocessing of critical points and values
  10. Exact representation and evaluation
  11. Experimental evaluation
  12. Data set example
  13. Asymptotic analysis
  14. Parallelization
  15. Conclusion
  16. ...and 8 more sections

Key Result

Lemma 1

Let $\bkernel:\R\to$ be a primitive of $\kernel$. As a consequence,

Figures (7)

  • Figure 1: Step-by-step computation times for varying size $m$ of $m\times m$ images.
  • Figure 2: Computation times for varying number of critical points excluding (\ref{['fig:crit-pts-eval-dem']}) and including (\ref{['fig:crit-pts-total-dem']}) initialization and preprocessing steps.
  • Figure 3: Computation times (s) for varying number of directions of our and 's implementation of the ECT on a single image of the fashion-MNIST dataset.
  • Figure 4: Vectorization time (s) with respect to dimension for a $1000\times 1000$ image with $10^6$ critical points.
  • Figure 5: Number of critical points in two random settings. Red and blue symbols represent respectively classical and ordinary critical points. Dots and crosses respectively represent these numbers for the uniform setting and for the Gaussian random field setting.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Remark 4
  • Lemma 5
  • Lemma 6
  • Lemma 7