The Augmented Fast Marching Method for Level Set Reinitialization

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 $|\nabla\phi|^2=1$, where $\phi$ is the level set function, the method ensures that $\nabla(|\nabla\phi|)^2=0$ and $\nabla\nabla(|\nabla\phi|)^2=0$ 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.