Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Learning Topological Representations for Deep Image Understanding

Xiaoling Hu

TL;DR

This dissertation leverage the mathematical tools from topological data analysis, i.e., persistent homology and discrete Morse theory, to develop principled methods for better segmentation and uncertainty estimation, which will become powerful tools for scalable annotation.

Abstract

In many scenarios, especially biomedical applications, the correct delineation of complex fine-scaled structures such as neurons, tissues, and vessels is critical for downstream analysis. Despite the strong predictive power of deep learning methods, they do not provide a satisfactory representation of these structures, thus creating significant barriers in scalable annotation and downstream analysis. In this dissertation, we tackle such challenges by proposing novel representations of these topological structures in a deep learning framework. We leverage the mathematical tools from topological data analysis, i.e., persistent homology and discrete Morse theory, to develop principled methods for better segmentation and uncertainty estimation, which will become powerful tools for scalable annotation.

Learning Topological Representations for Deep Image Understanding

TL;DR

This dissertation leverage the mathematical tools from topological data analysis, i.e., persistent homology and discrete Morse theory, to develop principled methods for better segmentation and uncertainty estimation, which will become powerful tools for scalable annotation.

Abstract

In many scenarios, especially biomedical applications, the correct delineation of complex fine-scaled structures such as neurons, tissues, and vessels is critical for downstream analysis. Despite the strong predictive power of deep learning methods, they do not provide a satisfactory representation of these structures, thus creating significant barriers in scalable annotation and downstream analysis. In this dissertation, we tackle such challenges by proposing novel representations of these topological structures in a deep learning framework. We leverage the mathematical tools from topological data analysis, i.e., persistent homology and discrete Morse theory, to develop principled methods for better segmentation and uncertainty estimation, which will become powerful tools for scalable annotation.
Paper Structure (95 sections, 2 theorems, 32 equations, 53 figures, 22 tables, 1 algorithm)

This paper contains 95 sections, 2 theorems, 32 equations, 53 figures, 22 tables, 1 algorithm.

Table of Contents

  1. Introduction and Overview
  2. Topological Loss for Image Segmentation
  3. Beyond Pixel-Wise Representation
  4. Outline
  5. Related Work
  6. Topological Data Analysis and Persistent Homology
  7. Attention Mechanism
  8. Deep Image Segmentation
  9. Topology-Aware Deep Image Segmentation
  10. Trojan Detection
  11. Segmentation Uncertainty
  12. Topology-Preserving Deep Image Segmentation via TopoLoss
  13. Introduction
  14. Method
  15. Topology and Persistent Homology
  16. ...and 80 more sections

Key Result

Theorem 1

When the loss function $L_{topo}(f,g)$ is zero, the segmentation by thresholding $f$ at 0.5 has the same Betti number as $g$.

Figures (53)

  • Figure 1: Illustration of the importance of topological correctness in a neuron image segmentation task. The goal of this task is to segment membranes that partition the image into regions corresponding to neurons. (a) an input neuron image. (b) ground truth segmentation of the membranes (dark blue) and the result neuron regions. (c) result of a baseline method without topological guarantee fakhry2016deep. Small pixel-wise errors lead to broken membranes, resulting in the merging of many neurons into one. (d) Our method produces the correct topology and the correct partitioning of neurons hu2019topology.
  • Figure 2: Illustration of the critical points identified by different methods. (a): Original image. (b): GT mask. (c): Predicted likelihood map. (d): Segmentation (Thresholded mask from likelihood map). (e): Critical points identified by hu2019topology. (f): Critical points identified by our homotopy warping. Please zoom-in for better viewing.
  • Figure 3: Illustration of the importance of topological correctness in a neuron image segmentation task. The goal of this task is to segment membranes that partition the image into regions corresponding to neurons. (a) an input neuron image. (b) ground truth segmentation of the membranes (dark blue) and the result neuron regions. (c) result of a baseline method without topological guarantee fakhry2016deep. Small pixel-wise errors lead to broken membranes, resulting in the merging of many neurons into one. (d) Our method produces the correct topology and the correct partitioning of neurons.
  • Figure 4: An overview of our method.
  • Figure 5: Illustration of topology and topology of a likelihood. For visualization purposes, the higher the function values are, the darker the area is. (a) an example segmentation $X$ with two connected components and one handle. (b) The ground truth with one connected component and two handles. It can also be viewed as a binary valued function $g$. (c) a likelihood map $f$ whose segmentation (bounded by the red curve) is $X$. The landscape views near the broken bridge and handle are drawn. Critical points are highlighted in the segmentation. (d) another likelihood map $f'$ with the same segmentation as $f$. But the landscape views reveal that $f'$ is worse than $f$ due to deeper gaps.
  • ...and 48 more figures

Theorems & Definitions (2)

  • Theorem 1: Topological Correctness
  • Lemma 1