Fast marching method

The fast marching method is a numerical method for solving boundary value problems of the Eikonal equation:

$F(x)|\nabla T(x)|=1.$

Typically, such a problem describes the evolution of a closed curve as a function of time $T$ with speed $F(x)$ in the normal direction at a point $x$ on the curve. The speed function is specified, and the time at which the contour crosses a point $x$ is obtained by solving the equation.

The algorithm is similar to Dijkstra's algorithm and uses the fact that information only flows outward from the seeding area.

This problem is a special case of level set methods. More general algorithms exist but are normally slower.

Extensions to non-flat (triangulated) domains solving:

$F(x)|\nabla_S T(x)|=1, \,\, \mbox{for the surface} \,\, S, \, \mbox{and} \,\, x\in S.$

was introduced by Ron Kimmel and James Sethian.

• Fast marching maze.png

  Maze as speed function shortest path

• Fast marching multi stencil 2nd order.png

  Distance map multi-stencils with random source points

See also

• Level set method

External links

• Djikstra-like Methods for the Eikonal Equation J.N. Tsitsiklis, 1995
• The Fast Marching Method and its Applications by James A. Sethian
• Multi-Stencils Fast Marching Methods
• Multi-Stencils Fast Marching Matlab Implementation
• Implementation Details of the Fast Marching Methods
• Generalized Fast Marching method by Forcadel et al. [2008] for applications in image segmentation.
• See Chapter 8 in Design and Optimization of Nano-Optical Elements by Coupling Fabrication to Optical Behavior