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G-LoG Bi-filtration for Medical Image Classification

Qingsong Wang, Jiaxing He, Bingzhe Hou, Tieru Wu, Yang Cao, Cailing Yao

TL;DR

The G-LoG (Gaussian-Laplacian of Gaussian) bi-filtration is defined to generate the features more suitable for multi-parameter persistence module, and it is proved the interleaving distance on the persistence modules obtained from the authors' bi-filtrations on the bounded functions is stable with respect to the maximum norm of the bounded functions.

Abstract

Building practical filtrations on objects to detect topological and geometric features is an important task in the field of Topological Data Analysis (TDA). In this paper, leveraging the ability of the Laplacian of Gaussian operator to enhance the boundaries of medical images, we define the G-LoG (Gaussian-Laplacian of Gaussian) bi-filtration to generate the features more suitable for multi-parameter persistence module. By modeling volumetric images as bounded functions, then we prove the interleaving distance on the persistence modules obtained from our bi-filtrations on the bounded functions is stable with respect to the maximum norm of the bounded functions. Finally, we conduct experiments on the MedMNIST dataset, comparing our bi-filtration against single-parameter filtration and the established deep learning baselines, including Google AutoML Vision, ResNet, AutoKeras and auto-sklearn. Experiments results demonstrate that our bi-filtration significantly outperforms single-parameter filtration. Notably, a simple Multi-Layer Perceptron (MLP) trained on the topological features generated by our bi-filtration achieves performance comparable to complex deep learning models trained on the original dataset.

G-LoG Bi-filtration for Medical Image Classification

TL;DR

The G-LoG (Gaussian-Laplacian of Gaussian) bi-filtration is defined to generate the features more suitable for multi-parameter persistence module, and it is proved the interleaving distance on the persistence modules obtained from the authors' bi-filtrations on the bounded functions is stable with respect to the maximum norm of the bounded functions.

Abstract

Building practical filtrations on objects to detect topological and geometric features is an important task in the field of Topological Data Analysis (TDA). In this paper, leveraging the ability of the Laplacian of Gaussian operator to enhance the boundaries of medical images, we define the G-LoG (Gaussian-Laplacian of Gaussian) bi-filtration to generate the features more suitable for multi-parameter persistence module. By modeling volumetric images as bounded functions, then we prove the interleaving distance on the persistence modules obtained from our bi-filtrations on the bounded functions is stable with respect to the maximum norm of the bounded functions. Finally, we conduct experiments on the MedMNIST dataset, comparing our bi-filtration against single-parameter filtration and the established deep learning baselines, including Google AutoML Vision, ResNet, AutoKeras and auto-sklearn. Experiments results demonstrate that our bi-filtration significantly outperforms single-parameter filtration. Notably, a simple Multi-Layer Perceptron (MLP) trained on the topological features generated by our bi-filtration achieves performance comparable to complex deep learning models trained on the original dataset.
Paper Structure (9 sections, 2 theorems, 16 equations, 3 figures, 3 tables)

This paper contains 9 sections, 2 theorems, 16 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Overview
  3. Motivation
  4. Contributions
  5. Preliminaries
  6. G-LoG Bifiltration and its stability
  7. Experiments
  8. Results
  9. Conclusions and future work

Key Result

Theorem 2.5

$d_{I}$ is stable.

Figures (3)

  • Figure 1: Bi-parameter persistence modules $H_0$ and $H_1$ generated by G-LoG bi-filtration from a medical tissue image.
  • Figure 2: Bi-parameter persistence modules $H_0$, $H_1$ and $H_2$ generated by G-LoG bi-filtration from a 3D medical organ volume.
  • Figure 4: Classification pipeline using G-LoG bi-filtration

Theorems & Definitions (9)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Micheal-2015
  • Definition 3.1
  • Theorem 3.2
  • proof
  • Remark 3.3