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Re-initialization-free Level Set Method via Molecular Beam Epitaxy Equation Regularization for Image Segmentation

Fanghui Song, Jiebao Sun, Shengzhu Shi, Zhichang Guo, Dazhi Zhang

TL;DR

A high-order level set variational segmentation method integrated with molecular beam epitaxy (MBE) equation regularization that can avoid the re-initialization in the evolution process and regulate the smoothness of the segmented curve and keep the segmentation results independent of the initial curve selection is proposed.

Abstract

Variational level set method has become a powerful tool in image segmentation due to its ability to handle complex topological changes and maintain continuity and smoothness in the process of evolution. However its evolution process can be unstable, which results in over flatted or over sharpened contours and segmentation failure. To improve the accuracy and stability of evolution, we propose a high-order level set variational segmentation method integrated with molecular beam epitaxy (MBE) equation regularization. This method uses the crystal growth in the MBE process to limit the evolution of the level set function, and thus can avoid the re-initialization in the evolution process and regulate the smoothness of the segmented curve. It also works for noisy images with intensity inhomogeneity, which is a challenge in image segmentation. To solve the variational model, we derive the gradient flow and design scalar auxiliary variable (SAV) scheme coupled with fast Fourier transform (FFT), which can significantly improve the computational efficiency compared with the traditional semi-implicit and semi-explicit scheme. Numerical experiments show that the proposed method can generate smooth segmentation curves, retain fine segmentation targets and obtain robust segmentation results of small objects. Compared to existing level set methods, this model is state-of-the-art in both accuracy and efficiency.

Re-initialization-free Level Set Method via Molecular Beam Epitaxy Equation Regularization for Image Segmentation

TL;DR

A high-order level set variational segmentation method integrated with molecular beam epitaxy (MBE) equation regularization that can avoid the re-initialization in the evolution process and regulate the smoothness of the segmented curve and keep the segmentation results independent of the initial curve selection is proposed.

Abstract

Variational level set method has become a powerful tool in image segmentation due to its ability to handle complex topological changes and maintain continuity and smoothness in the process of evolution. However its evolution process can be unstable, which results in over flatted or over sharpened contours and segmentation failure. To improve the accuracy and stability of evolution, we propose a high-order level set variational segmentation method integrated with molecular beam epitaxy (MBE) equation regularization. This method uses the crystal growth in the MBE process to limit the evolution of the level set function, and thus can avoid the re-initialization in the evolution process and regulate the smoothness of the segmented curve. It also works for noisy images with intensity inhomogeneity, which is a challenge in image segmentation. To solve the variational model, we derive the gradient flow and design scalar auxiliary variable (SAV) scheme coupled with fast Fourier transform (FFT), which can significantly improve the computational efficiency compared with the traditional semi-implicit and semi-explicit scheme. Numerical experiments show that the proposed method can generate smooth segmentation curves, retain fine segmentation targets and obtain robust segmentation results of small objects. Compared to existing level set methods, this model is state-of-the-art in both accuracy and efficiency.
Paper Structure (19 sections, 1 theorem, 46 equations, 24 figures, 5 tables, 1 algorithm)

This paper contains 19 sections, 1 theorem, 46 equations, 24 figures, 5 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Motivation
  3. Level set segmentation method and re-initialization
  4. Distance regularization term for re-initialization-free
  5. Molecular Beam Epitaxy regularization term
  6. Main method
  7. The MBE regularization term
  8. Two applications of MBE regularization method
  9. The MBE-GAC model
  10. The MBE-RSF model
  11. Numerical algorithms
  12. Scalar auxiliary variable (SAV) scheme
  13. The discretization of space
  14. Experimental results
  15. Independence of initial curve
  16. ...and 4 more sections

Key Result

Theorem 1

The SAV scheme eq:sav is first-order unconditionally energy stable in the sense that: where $\tilde{\mathcal{E}}(\eta, r)=\dfrac{1}{2}\left(\eta, \mathcal{L} \eta\right)+r^{2}$ is the modified energy.

Figures (24)

  • Figure 1: The plots of the diffusion coefficient of the DR1, DR2 of $|\nabla\phi|$.
  • Figure 2: The $|\nabla \phi|$ of segmentation result of different regularization term: (a)DR1 (b)MBE
  • Figure 3: Segmentation results of different time steps: (a) $\tau=0.01$; (b) $\tau=0.1$; (c) $\tau=0.5$; (d) $\tau=1$.
  • Figure 4: Corresponding modified energy $\tilde{\mathcal{E}}$ of different time steps: (a) $\tau=0.01$; (b) $\tau=0.1$; (c)$\tau=0.5$; (d) $\tau=1$.
  • Figure 5: Segmentation results of the DR1-GAC model and MBE-GAC model with same initial level set functions. Upper row: binary function as the initial level set function; lower row: signed distance function as the initial level set function.
  • ...and 19 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof