Accurate Tracking of Monotonically Advancing Fronts

A wide range of computer vision applications such as distance field computation, shape from shading, and shape representation require an accurate solution of a particular Hamilton-Jacobi (HJ) equation, known as the Eikonal equation. Although the fast marching method (FMM) is the most stable and consistent method among existing techniques for solving such equation, it suffers from large numerical error along diagonal directions as well as its computational complexity is not optimal. In this paper, we propose an improved version of the FMMthat is both highly accurate and computationally efficient for Cartesian domains. The new method is called the multi-stencils fast marching (MSFM), which computes the solution at each grid point by solving the Eikonal equation along several stencils and then picks the solution that satisfies the fast marching causality relationship. The stencils are centered at each grid point x and cover its entire nearest neighbors. In 2D space, 2 stencils cover the 8-neighbors of x, while in 3D space, 6 stencils cover its 26-neighbors. For those stencils that are not aligned with the natural coordinate system, the Eikonal equation is derived using directional derivatives and then solved using a higher order finite difference scheme.