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Scalar Function Topology Divergence: Comparing Topology of 3D Objects

Ilya Trofimov, Daria Voronkova, Eduard Tulchinskii, Evgeny Burnaev, Serguei Barannikov

TL;DR

The paper tackles the limitation of topology-based comparisons that ignore where topological features occur. It introduces Scalar Function Topology Divergence ($\mathrm{SFTD}$) and $\text{F-Cross-Barcode}_k(f,g)$, a localization-aware extension of persistent barcodes computed on a doubled graph or extended lattice, enabling multi-scale comparison of sublevel-set topology for scalar functions on graphs or Euclidean spaces. $\mathrm{SFTD}$ provides a differentiable loss (for parametric $f$ and $g$) and stability guarantees, and a visualization mechanism to localize topological differences via $\text{F-Cross-Barcodes}$. Empirically, $\mathrm{SFTD}$ improves 3D shape reconstruction and 2D segmentation over Betti-matching losses and standard barcodes, while offering faster computation and informative visual diagnostics. The approach broadens topology-aware learning in vision tasks, especially for graph-structured data and lattice domains, by capturing localization of features across multiple scales.

Abstract

We propose a new topological tool for computer vision - Scalar Function Topology Divergence (SFTD), which measures the dissimilarity of multi-scale topology between sublevel sets of two functions having a common domain. Functions can be defined on an undirected graph or Euclidean space of any dimensionality. Most of the existing methods for comparing topology are based on Wasserstein distance between persistence barcodes and they don't take into account the localization of topological features. The minimization of SFTD ensures that the corresponding topological features of scalar functions are located in the same places. The proposed tool provides useful visualizations depicting areas where functions have topological dissimilarities. We provide applications of the proposed method to 3D computer vision. In particular, experiments demonstrate that SFTD as an additional loss improves the reconstruction of cellular 3D shapes from 2D fluorescence microscopy images, and helps to identify topological errors in 3D segmentation. Additionally, we show that SFTD outperforms Betti matching loss in 2D segmentation problems.

Scalar Function Topology Divergence: Comparing Topology of 3D Objects

TL;DR

The paper tackles the limitation of topology-based comparisons that ignore where topological features occur. It introduces Scalar Function Topology Divergence () and , a localization-aware extension of persistent barcodes computed on a doubled graph or extended lattice, enabling multi-scale comparison of sublevel-set topology for scalar functions on graphs or Euclidean spaces. provides a differentiable loss (for parametric and ) and stability guarantees, and a visualization mechanism to localize topological differences via . Empirically, improves 3D shape reconstruction and 2D segmentation over Betti-matching losses and standard barcodes, while offering faster computation and informative visual diagnostics. The approach broadens topology-aware learning in vision tasks, especially for graph-structured data and lattice domains, by capturing localization of features across multiple scales.

Abstract

We propose a new topological tool for computer vision - Scalar Function Topology Divergence (SFTD), which measures the dissimilarity of multi-scale topology between sublevel sets of two functions having a common domain. Functions can be defined on an undirected graph or Euclidean space of any dimensionality. Most of the existing methods for comparing topology are based on Wasserstein distance between persistence barcodes and they don't take into account the localization of topological features. The minimization of SFTD ensures that the corresponding topological features of scalar functions are located in the same places. The proposed tool provides useful visualizations depicting areas where functions have topological dissimilarities. We provide applications of the proposed method to 3D computer vision. In particular, experiments demonstrate that SFTD as an additional loss improves the reconstruction of cellular 3D shapes from 2D fluorescence microscopy images, and helps to identify topological errors in 3D segmentation. Additionally, we show that SFTD outperforms Betti matching loss in 2D segmentation problems.
Paper Structure (28 sections, 6 theorems, 16 equations, 13 figures, 3 tables, 1 algorithm)

This paper contains 28 sections, 6 theorems, 16 equations, 13 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related work
  3. Preliminaries
  4. Scalar functions defined on a graph
  5. Method
  6. Scalar functions defined on a graph
  7. Scalar functions defined on a lattice
  8. Scalar Function Topology Divergence
  9. SFTD basic properties
  10. Visualization of F-Cross-Barcodes
  11. SFTD differentiation
  12. Experiments
  13. Synthetic data
  14. Functions with local minima in 2D.
  15. Lattices in 2D.
  16. ...and 13 more sections

Key Result

proposition thmcounterproposition

If $SFT\!D_k(f, g) = SFT\!D_k(g, f)=0$ for all $k \geq 0$, then the sublevel sets barcodes of functions $f$ and $g$ are the same in any degree. Moreover, in this case their topological features are located in the same places: the inclusions $f^{-1}((-\infty, \varepsilon])\subseteq (\min(f,g))^{-1}(

Figures (13)

  • Figure 1: 1D function and its sublevel sets.
  • Figure 2: A visualization of $\tilde{\mathcal{G}}$, $\tilde{\mathcal{L}}$.
  • Figure 3: Two scalar functions (a), (b) are indistinguishable by their persistence barcodes (d), (e). At the same time, $\text{SFTD}_0(f_1, f_2)=4.6$ see (f).
  • Figure 4: Two scalar functions (a), (b) are indistinguishable by their persistence barcodes (c), (d). At the same time, $\text{SFTD}_1(f_1, f_2)=1$, see (e).
  • Figure 5: Two scalar fields (a), (b) are indistinguishable by their persistence barcodes (d), (e). At the same time, $\text{SFTD}_1(f_1, f_2)=1$, see (f).
  • ...and 8 more figures

Theorems & Definitions (10)

  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • proposition thmcounterproposition
  • proof
  • proposition thmcounterproposition
  • proof
  • theorem thmcountertheorem
  • proof
  • proposition thmcounterproposition
  • proof