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Topology-Guaranteed Image Segmentation: Enforcing Connectivity, Genus, and Width Constraints

Wenxiao Li, Xue-Cheng Tai, Jun Liu

TL;DR

Topological fidelity in segmentation is addressed by a width-aware topological energy that preserves Betti numbers $\beta_0$ and $\beta_1$ while embedding width via smooth morphological gradients; the method integrates into variational and deep learning frameworks; experiments show improved connectivity, genus preservation, and width attributes with practical impact for medical and road-network imaging.

Abstract

Existing research highlights the crucial role of topological priors in image segmentation, particularly in preserving essential structures such as connectivity and genus. Accurately capturing these topological features often requires incorporating width-related information, including the thickness and length inherent to the image structures. However, traditional mathematical definitions of topological structures lack this dimensional width information, limiting methods like persistent homology from fully addressing practical segmentation needs. To overcome this limitation, we propose a novel mathematical framework that explicitly integrates width information into the characterization of topological structures. This method leverages persistent homology, complemented by smoothing concepts from partial differential equations (PDEs), to modify local extrema of upper-level sets. This approach enables the resulting topological structures to inherently capture width properties. We incorporate this enhanced topological description into variational image segmentation models. Using some proper loss functions, we are also able to design neural networks that can segment images with the required topological and width properties. Through variational constraints on the relevant topological energies, our approach successfully preserves essential topological invariants such as connectivity and genus counts, simultaneously ensuring that segmented structures retain critical width attributes, including line thickness and length. Numerical experiments demonstrate the effectiveness of our method, showcasing its capability to maintain topological fidelity while explicitly embedding width characteristics into segmented image structures.

Topology-Guaranteed Image Segmentation: Enforcing Connectivity, Genus, and Width Constraints

TL;DR

Topological fidelity in segmentation is addressed by a width-aware topological energy that preserves Betti numbers and while embedding width via smooth morphological gradients; the method integrates into variational and deep learning frameworks; experiments show improved connectivity, genus preservation, and width attributes with practical impact for medical and road-network imaging.

Abstract

Existing research highlights the crucial role of topological priors in image segmentation, particularly in preserving essential structures such as connectivity and genus. Accurately capturing these topological features often requires incorporating width-related information, including the thickness and length inherent to the image structures. However, traditional mathematical definitions of topological structures lack this dimensional width information, limiting methods like persistent homology from fully addressing practical segmentation needs. To overcome this limitation, we propose a novel mathematical framework that explicitly integrates width information into the characterization of topological structures. This method leverages persistent homology, complemented by smoothing concepts from partial differential equations (PDEs), to modify local extrema of upper-level sets. This approach enables the resulting topological structures to inherently capture width properties. We incorporate this enhanced topological description into variational image segmentation models. Using some proper loss functions, we are also able to design neural networks that can segment images with the required topological and width properties. Through variational constraints on the relevant topological energies, our approach successfully preserves essential topological invariants such as connectivity and genus counts, simultaneously ensuring that segmented structures retain critical width attributes, including line thickness and length. Numerical experiments demonstrate the effectiveness of our method, showcasing its capability to maintain topological fidelity while explicitly embedding width characteristics into segmented image structures.
Paper Structure (28 sections, 4 theorems, 66 equations, 12 figures, 3 tables, 1 algorithm)

This paper contains 28 sections, 4 theorems, 66 equations, 12 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The related works
  3. Variational segmentation model
  4. Mathematical morphology for image processing
  5. Persistent homology
  6. The proposed theoretical and numerical approaches
  7. Smooth form of morphological gradient
  8. Width-aware topological (WT) energy
  9. Incorporating width-aware topological preservation to variational models and deep neural networks for image segmentation
  10. Topology energy for varitional segmentation model
  11. Topology energy for deep neural network segmentation models
  12. Numerical experiments
  13. Test for width-aware topological (WT) energy
  14. Test on traditional segmentation model
  15. Test on deep neural network segmentation models
  16. ...and 13 more sections

Key Result

Theorem 3.3

For function $u_{\ell}: \Omega\rightarrow [0,1]$, $\mathbb{X}^{(t)}_{\bm b, \beta_k},\mathbb{X}^{(t)}_{\bm d, \beta_k}$ is the set of birth and death critical points of $u_{\ell}$, we have where $k_M(y,x)$ and $k_m(y,x)$ is defined at def:smooth_ker.

Figures (12)

  • Figure 1: Persistent homology energy to constrain one connected component. From left to right: image of CHASEDB1 chase, ground-truth, results of UNetunet, results of UNet with PH energyph. Compared to the result of UNet, PH links the unconnected regions by a single-pixel-width line.
  • Figure 1: Iterative process to minimize topological energy. The energy without width information \ref{['eq:phenergy']} has the gradient for backpropagation only at the critical point, but our width-aware topological energy \ref{['eq:wtenergy']} consider the neighborhood of the critical point.
  • Figure 1: Performance of topological energy. The first row constrains one connected component and the other rows are one connect component and two holes.
  • Figure 1: A schematic diagram of persistent homology (the foreground is represented in black and the background is white). As the threshold $t$ continuously decreases, the topological structures in the image change continuously (corresponding to the black regions). Three new connected components are born at $t = 243$. When $t$ reaches 241, The two lower regions in the figure are merged, causing one of the components to die. This gives rise to a birth-death pair: $(b_3^0, d_3^0) = (243, 241)$. Then, the upper and lower regions merge into a single area at $t=65$, resulting in $(b_2^0, d_2^0) = (243, 65)$. Similarly, two distinct loops (1-dimensional features) are born at $t = 198$ and $t = 26$, respectively, and both are filled (die) at $t = 1$. Their corresponding birth-death pairs are $(b_1^1, d_1^1) = (198, 1)$ and $(b_2^1, d_2^1) = (26, 1)$.
  • Figure 2: Performance on MINST dataset. The first row is the original image, second row is PH results and the thrid row is our proposed WT method.
  • ...and 7 more figures

Theorems & Definitions (10)

  • Definition 2.1: Grayscale erosion and dilation morphgradient
  • Definition 2.2: Internal gradient and external gradient morphgradient
  • Definition 3.1: Smooth morphological kernel
  • Definition 3.2: Smooth external and internal gradient
  • Theorem 3.3: Varitional of $\mathcal{Q}_{\varepsilon}(u_{\ell},u^{(t)}_{\ell},\beta_k)$
  • Remark 3.4
  • Theorem 3.5: Duality of $L_1$
  • Theorem 3.6
  • Theorem 3.7
  • Remark 3.8