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A Fast and Scalable Computational Topology Framework for the Euler Characteristic

Daniel J. Laky, Victor M. Zavala

TL;DR

This paper tackles the bottleneck of computing the Euler Characteristic ($\chi$) for high-resolution 2D and 3D fields in contexts like microscopy, molecular dynamics, and hyperspectral imaging. It introduces a parallel vertex-contribution framework that generalizes to 3D cubical complexes, enabling fast EC computation and the derivation of additional descriptors such as perimeter, area, and volume from vertex contributions. The authors demonstrate 2–3 order-of-magnitude speedups over GUDHI and competitive performance with CHUNKYEuler, along with low-memory variants that support streaming data and out-of-core analysis. The approach provides practical real-time or high-throughput topology analysis, with open-source code to facilitate adoption and extension to broader descriptors and higher dimensions ($4$D and beyond).

Abstract

The Euler characteristic (EC) is a powerful topological descriptor that can be used to quantify the shape of data objects that are represented as fields/manifolds. Fast methods for computing the EC are required to enable processing of high-throughput data and real-time implementations. This represents a challenge when processing high-resolution 2D field data (e.g., images) and 3D field data (e.g., video, hyperspectral images, and space-time data obtained from fluid dynamics and molecular simulations). In this work, we present parallel algorithms (and software implementations) to enable fast computations of the EC for 2D and 3D fields using vertex contributions. We test the proposed algorithms using synthetic data objects and data objects arising in real applications such as microscopy, 3D molecular dynamics simulations, and hyperspectral images. Results show that the proposed implementation can compute the EC a couple of orders of magnitude faster than ${\tt GUDHI}$ (an off-the-shelf and state-of-the art tool) and at speeds comparable to ${\tt CHUNKYEuler}$ (a tool tailored to scalable computation of the EC). The vertex contributions approach is flexible in that it compute the EC as well as other topological descriptors such as perimeter, area, and volume (${\tt CHUNKYEuler}$ can only compute the EC). Scalability with respect to memory use is also addressed by providing low-memory versions of the algorithms; this enables processing of data objects beyond the size of dynamic memory. All data and software needed for reproducing the results are shared as open-source code.

A Fast and Scalable Computational Topology Framework for the Euler Characteristic

TL;DR

This paper tackles the bottleneck of computing the Euler Characteristic () for high-resolution 2D and 3D fields in contexts like microscopy, molecular dynamics, and hyperspectral imaging. It introduces a parallel vertex-contribution framework that generalizes to 3D cubical complexes, enabling fast EC computation and the derivation of additional descriptors such as perimeter, area, and volume from vertex contributions. The authors demonstrate 2–3 order-of-magnitude speedups over GUDHI and competitive performance with CHUNKYEuler, along with low-memory variants that support streaming data and out-of-core analysis. The approach provides practical real-time or high-throughput topology analysis, with open-source code to facilitate adoption and extension to broader descriptors and higher dimensions (D and beyond).

Abstract

The Euler characteristic (EC) is a powerful topological descriptor that can be used to quantify the shape of data objects that are represented as fields/manifolds. Fast methods for computing the EC are required to enable processing of high-throughput data and real-time implementations. This represents a challenge when processing high-resolution 2D field data (e.g., images) and 3D field data (e.g., video, hyperspectral images, and space-time data obtained from fluid dynamics and molecular simulations). In this work, we present parallel algorithms (and software implementations) to enable fast computations of the EC for 2D and 3D fields using vertex contributions. We test the proposed algorithms using synthetic data objects and data objects arising in real applications such as microscopy, 3D molecular dynamics simulations, and hyperspectral images. Results show that the proposed implementation can compute the EC a couple of orders of magnitude faster than (an off-the-shelf and state-of-the art tool) and at speeds comparable to (a tool tailored to scalable computation of the EC). The vertex contributions approach is flexible in that it compute the EC as well as other topological descriptors such as perimeter, area, and volume ( can only compute the EC). Scalability with respect to memory use is also addressed by providing low-memory versions of the algorithms; this enables processing of data objects beyond the size of dynamic memory. All data and software needed for reproducing the results are shared as open-source code.
Paper Structure (15 sections, 4 equations, 19 figures, 8 tables, 4 algorithms)

This paper contains 15 sections, 4 equations, 19 figures, 8 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Methodology
  3. Parallel Implementation
  4. 3D Field Processing
  5. Low-Memory Processing
  6. Computational Results
  7. Analysis of Synthetic Random Fields
  8. Memory analysis
  9. Microscopy Case Study
  10. Molecular Dynamics Data
  11. Hyperspectral Image Data
  12. Discussion
  13. Conclusions
  14. Acknowledgements
  15. 3D Algorithms

Figures (19)

  • Figure 1: Cubical simplexes relevant to 2D and 3D field processing.
  • Figure 2: Liquid crystal micrograph (image) represented as a cubical simplicial complex; this is done by assigning pixel values to faces which are connected by lower dimensional cubical simplexes, edges, and vertices.
  • Figure 3: Example 2D field undergoing the process of filtration over the full range of pixel intensity values.
  • Figure 4: EC curve for a 2D field. Note that, in this example, we see the emergence of a hole in the simplicial complex at a filtration level of $c = 6$.
  • Figure 5: Types of connectivity; on the left is vertex-adjacency (or 8-C) and, on the right, is edge-adjacency or 4-C.
  • ...and 14 more figures