Simplex Free Adaptive Tree Fast Sweeping and Evolution Methods for Solving Level Set Equations in Arbitrary Dimension

Abstract

We introduce simplex free adaptive tree numerical methods for solving static and time dependent Hamilton-Jacobi equations arising in level set problems in arbitrary dimension. The data structure upon which our method is built is a generalized n-dimensional binary tree, but it does not require the complicated splitting of cubes into simplices (aka generalized n-dimensional triangles or hypertetrahedrons) that current tree based methods require. It has enough simplicity that minor variants of standard numerical Hamiltonians developed for uniform grids can be applied, yielding consistent, monotone, convergent schemes. Combined with the fast sweeping strategy, the resulting tree based methods are highly efficient and accurate. Thus, without changing more than a few lines of code when changing dimension, we have obtained results for calculations in up to n = 7 dimensions.

Delaunay Deformable Models: Topology-Adaptive Meshes Based on the Restricted Delaunay Triangulation

“…Much effort has been done to alleviate this drawback, leading to the narrow band methodology [1] and to the PDE-based fast local level set method [36]. More recently, octree decompositions have been proposed [10,19,27,28] to circumvent the typically fixed uniform sampling of the level set method, in order to reach high resolution (typically an effective resolution of 512 3 ) while keeping the computational and memory cost sustainable. However, these methods somewhat lose the simplicity of the original level set method, as an efficient implementation of such tree-based methods turns out to be a tricky task.…”

Legendre-transform-based fast sweeping methods for static Hamilton–Jacobi equations on triangulated meshes

“…If Hðx; pÞ ¼ jpjHðx; p jpj Þ ¼ jpjVðxÞ, then the eikonal equation for isotropic wave propagation results. Fast sweeping methods are a family of efficient methods for solving static Hamilton-Jacobi equations [26,23,9,10,5,18,19,27,12], and some essential ideas of these methods may trace back to [20,3]. In [26] the fast sweeping method was systematically analyzed for eikonal equations.…”

“…In [26] the fast sweeping method was systematically analyzed for eikonal equations. Since then the fast sweeping methods have undergone intensive development for general static Hamilton-Jacobi equations in [26,23,9,10,5,18,19,27,12] and have found many different applications; see [11,8], for examples. On the other hand, the fast marching method and its relatives consist of another family of numerical methods for solving static Hamilton-Jacobi equations [24,21,7,22].…”

“…On a triangulated mesh, however, such natural orderings no longer exist. In [18,5,19] some novel ordering strategies based on reference points and l p -distances have been proposed, which are demonstrated to be effective.…”