Symmetric fast marching schemes for better numerical isotropy.

Area of Science:

• Numerical analysis
• Computational mathematics
• Partial differential equations

Background:

• Fast marching methods (FMM) solve the Eikonal equation for cost estimation.
• Current FMM use discontinuous models, introducing directional bias (anisotropy) in discrete computations.
• This anisotropy deviates from the isotropic nature of the continuous partial differential equation (PDE).

Purpose of the Study:

• To develop and evaluate methods for removing directional bias in FMM.
• To enhance the isotropy of discrete FMM implementations.
• To demonstrate the significance of bias removal in FMM applications.

Main Methods:

• Proposed two novel schemes to address directional bias in FMM.
• Scheme 1: Continuous interpolation of traveling cost to avoid front direction bias.
• Scheme 2: Upsampling traveling cost on a higher resolution grid to mitigate bias.

Main Results:

• Both proposed schemes successfully remove directional bias in cost computation.
• The discrete implementation of FMM becomes more isotropic with the new methods.
• Demonstrated the significance of bias removal in various FMM applications.

Conclusions:

• The proposed methods effectively eliminate anisotropy in discrete FMM.
• These advancements ensure discrete computations align with the underlying isotropic continuous PDE.
• Improved accuracy and reliability of FMM in computational applications.