# Scalar Function Topology Divergence: Comparing Topology of 3D Objects

arXiv: https://arxiv.org/abs/2407.08364 · HTML: https://mlsanity.com/paper/2407.08364/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 47 references that are in the map.
- First posted on arXiv 11 Jul 2024.

## Builds on (most cited first, 25 of 25)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Swin UNETR: Swin Transformers for Semantic Segmentation of Brain Tumors in MRI Images](https://mlsanity.com/paper/2201.01266/) (2022)
- [The RSNA-ASNR-MICCAI BraTS 2021 Benchmark on Brain Tumor Segmentation and Radiogenomic Classification](https://mlsanity.com/paper/2107.02314/) (2021)
- [Self-Supervised Pre-Training of Swin Transformers for 3D Medical Image Analysis](https://mlsanity.com/paper/2111.14791/) (2021)
- [Topology-Preserving Deep Image Segmentation](https://mlsanity.com/paper/1906.05404/) (2019)
- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [Topological Autoencoders](https://mlsanity.com/paper/1906.00722/) (2019)
- [PersLay: A Neural Network Layer for Persistence Diagrams and New Graph Topological Signatures](https://mlsanity.com/paper/1904.09378/) (2019)
- [MarrNet: 3D Shape Reconstruction via 2.5D Sketches](https://mlsanity.com/paper/1711.03129/) (2017)
- [Topology-Aware Segmentation Using Discrete Morse Theory](https://mlsanity.com/paper/2103.09992/) (2021)
- [Graph Filtration Learning](https://mlsanity.com/paper/1905.10996/) (2019)
- [Localization in the Crowd with Topological Constraints](https://mlsanity.com/paper/2012.12482/) (2020)
- [Z-GCNETs: Time Zigzags at Graph Convolutional Networks for Time Series Forecasting](https://mlsanity.com/paper/2105.04100/) (2021)
- [Representation Topology Divergence: A Method for Comparing Neural Network Representations](https://mlsanity.com/paper/2201.00058/) (2021)
- [Topologically faithful image segmentation via induced matching of persistence barcodes](https://mlsanity.com/paper/2211.15272/) (2022)
- [Link Prediction with Persistent Homology: An Interactive View](https://mlsanity.com/paper/2102.10255/) (2021)
- [Topology-Aware Uncertainty for Image Segmentation](https://mlsanity.com/paper/2306.05671/) (2023)
- [Manifold Topology Divergence: a Framework for Comparing Data Manifolds](https://mlsanity.com/paper/2106.04024/) (2021)
- [Learning Topology-Preserving Data Representations](https://mlsanity.com/paper/2302.00136/) (2023)
- [Learning Topological Interactions for Multi-Class Medical Image Segmentation](https://mlsanity.com/paper/2207.09654/) (2022)
- [Capturing Shape Information with Multi-Scale Topological Loss Terms for 3D Reconstruction](https://mlsanity.com/paper/2203.01703/) (2022)
- [ToDD: Topological Compound Fingerprinting in Computer-Aided Drug Discovery](https://mlsanity.com/paper/2211.03808/) (2022)
- [giotto-ph: A Python Library for High-Performance Computation of Persistent Homology of Vietoris-Rips Filtrations](https://mlsanity.com/paper/2107.05412/) (2021)
- [Improving Self-supervised Molecular Representation Learning using Persistent Homology](https://mlsanity.com/paper/2311.17327/) (2023)
- [Barcodes as Summary of Loss Function Topology](https://mlsanity.com/paper/1912.00043/) (2019)

## Abstract (arXiv)

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.