[Paper Review] The Augmented Fast Marching Method for Level Set Reinitialization

David Salač|arXiv (Cornell University)|Nov 29, 2011

Advanced Numerical Methods in Computational Mathematics11 references3 citations

TL;DR

This paper introduces the Augmented Fast Marching Method (AFMM), a novel level set reinitialization technique that computes the signed distance function and its first- and second-order derivatives with high accuracy. By enforcing not only |∇ϕ|² = 1 but also ∇(|∇ϕ|²) = 0 and ∇∇(|∇ϕ|²) = 0, the method ensures smooth curvature fields even on coarse grids, achieving second-order accuracy for the level set and gradient fields and first-order for curvature.

ABSTRACT

Including derivative information in the modelling of moving interfaces has been proposed as one method to increase the accuracy of numerical schemes with minimal additional cost. Here a new level set reinitialization technique using the fast marching method is presented. This augmented fast marching method will calculate the signed distance function and up to the second-order derivatives of the signed distance function for arbitrary interfaces. In addition to enforcing the condition $| ablaϕ|^2=1$, where $ϕ$ is the level set function, the method ensures that $ abla(| ablaϕ|)^2=0$ and $ abla abla(| ablaϕ|)^2=0$ are also satisfied. Results indicate that for both two- and three-dimensional interfaces the resulting level set and curvature field are smooth even for coarse grids. Convergence results show that using first-order upwind derivatives and the augmented fast marching method result in a second-order accurate level set and gradient field and a first-order accurate curvature field.

Motivation & Objective

• Address the need for accurate and smooth level set reinitialization in moving interface problems, particularly where curvature and its derivatives are critical.
• Overcome limitations of standard fast marching methods that fail to preserve higher-order derivative information during reinitialization.
• Ensure the level set function and its derivatives remain smooth and consistent with the Eikonal equation and geometric constraints.
• Enable second-order accuracy in the level set and gradient fields while maintaining first-order accuracy for curvature, even on coarse grids.
• Support numerical simulations of complex interfaces, such as vesicles, where bending forces depend on high-order derivatives of the level set function.

Proposed method

• Extend the fast marching method (FMM) to include derivative information by solving the Eikonal equation |∇ϕ| = 1 while enforcing additional constraints on the gradient and Hessian of |∇ϕ|².
• Introduce augmented state variables that track the first- and second-order derivatives of the signed distance function during the fast marching propagation.
• Use upwind finite differences to approximate spatial derivatives of ϕ, ∇ϕ, and ∇∇ϕ, ensuring causality and stability in the solution process.
• Maintain three node sets—accepted, trial, and distant—throughout the algorithm to ensure correct propagation order and convergence.
• Enforce geometric consistency by requiring ∇(|∇ϕ|²) = 0 and ∇∇(|∇ϕ|²) = 0 at each grid point, which enforces smoothness of the gradient magnitude.
• Apply the method in both two- and three-dimensional domains, using consistent discretization schemes for the Eikonal and derivative constraints.

Experimental results

Research questions

• RQ1 Can the fast marching method be extended to compute not only the signed distance function but also its first- and second-order derivatives with high accuracy?
• RQ2 Does enforcing ∇(|∇ϕ|²) = 0 and ∇∇(|∇ϕ|²) = 0 lead to smoother curvature fields and improved numerical stability in level set simulations?
• RQ3 What is the convergence rate of the augmented method for the level set function, its gradient, and curvature on structured Cartesian grids?
• RQ4 Can the method achieve second-order accuracy for the level set and gradient fields while maintaining first-order accuracy for curvature on coarse meshes?
• RQ5 How does the method perform in simulating complex interfaces such as vesicles, where curvature and its derivatives govern physical forces?

Key findings

• The augmented fast marching method successfully computes the signed distance function and up to second-order derivatives of the level set function with high accuracy.
• The method ensures that ∇(|∇ϕ|²) = 0 and ∇∇(|∇ϕ|²) = 0 are satisfied, resulting in smooth curvature fields even on coarse grids.
• Convergence studies show second-order accuracy for the level set and gradient fields when using first-order upwind derivatives.
• The curvature field is found to be first-order accurate, consistent with theoretical expectations for such schemes.
• Numerical results in both two and three dimensions confirm that the method produces smooth, stable, and geometrically consistent level set representations.
• The method enables accurate simulation of interfaces governed by curvature-dependent forces, such as vesicle membranes, by preserving high-order derivative information.