Scalar Function Topology Divergence: Comparing Topology of 3D Objects

TL;DR

The paper introduces Scalar Function Topology Divergence (SFTD), a novel topological measure for comparing 3D objects that accounts for feature localization, improving shape reconstruction and segmentation accuracy.

Contribution

It presents SFTD, a new topology comparison tool that considers feature localization, with applications in 3D shape reconstruction and segmentation.

Findings

SFTD improves 3D shape reconstruction from 2D images.

SFTD outperforms Betti matching loss in 2D segmentation.

SFTD helps identify topological errors in 3D segmentation.

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.