Accurate Tracking Of Monotonically Advancing Fronts
A wide range of computer vision applications—such as distance field computation, shape from shading, shape representation, skeletonization, and optimal path planning — 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 an equation, it suffers from a large numerical error along diagonal directions, and its computational complexity is not optimal. In this chapter, we propose an improved version of the FMM that is both highly accurate and computationally efficient for Cartesian domains. The new method is called the multistencils fast marching (MSFM) method, which computes a 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 a 2D space, two stencils cover the eight neighbors of x, while in a 3D space six stencils cover its 26 neighbors. For those stencils that do not coincide with the natural coordinate system, the eikonal equation is derived using directional derivatives and then solved using a higher-order finite-difference scheme.
KeywordsSource Point Directional Derivative Neighbor Point Eikonal Equation Unit Speed
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