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Computing geodesic paths encoding a curvature prior for curvilinear structure tracking.

Da Chen1, Jean-Marie Mirebeau2, Minglei Shu1

  • 1Shandong Artificial Intelligence Institute, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250014, China.

Proceedings of the National Academy of Sciences of the United States of America
|August 7, 2023
PubMed
Summary
This summary is machine-generated.

We developed an efficient method to compute optimal curves using an enhanced elastica energy model with a curvature prior. This approach guarantees global optimality by solving a Hamilton-Jacobi-Bellman equation for accurate centerline tracking.

Keywords:
curvature priorcurvilinear structure trackingfast-marching methodsecond-order geodesic pathvariant of the Euler–Mumford elastica model

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Area of Science:

  • Computational geometry
  • Image analysis
  • Differential geometry

Background:

  • Minimizing curve energy is crucial for shape analysis and image segmentation.
  • Traditional elastica models may not capture complex geometric features effectively.
  • Incorporating data-driven priors can enhance curve evolution models.

Purpose of the Study:

  • To introduce an efficient method for computing curves that minimize a modified Euler-Mumford elastica energy.
  • To enhance the bending energy with a user-defined, data-driven curvature prior.
  • To accurately track curvilinear structures in image data.

Main Methods:

  • Numerical computation of the viscosity solution to a static Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE).
  • Derivation of the explicit Hamiltonian for the modified elastica model.
  • Discretization of the HJB PDE using an adaptive finite difference scheme.
  • Solving the HJB PDE via a generalized fast-marching method.
  • Practical estimation of curvature prior values from image data.

Main Results:

  • An efficient and globally optimal curve computation method was developed.
  • The proposed method effectively incorporates curvature priors for enhanced energy minimization.
  • Accurate tracking of curvilinear structure centerlines was demonstrated.
  • Numerical experiments validated the method's performance on synthetic and real image data.

Conclusions:

  • The enhanced elastica model with curvature priors offers advantages for complex geometric structures.
  • The proposed numerical method efficiently computes globally optimal curves.
  • This approach shows promise for advanced image analysis and computational geometry applications.