Computing geodesic paths encoding a curvature prior for curvilinear structure tracking

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.