The fast marching methodJ.A. Sethian. A Fast Marching Level Set Method for Monotonically Advancing Fronts, Proc. Natl. Acad. Sci., 93, 4, pp.1591--1595, 1996 /ref> is a
numerical method
In numerical analysis, a numerical method is a mathematical tool designed to solve numerical problems. The implementation of a numerical method with an appropriate convergence check in a programming language is called a numerical algorithm.
Mathem ...
boundary value problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...
s of the Eikonal equation:
:
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Typically, such a problem describes the evolution of a closed surface as a function of time with speed in the normal direction at a point on the propagating surface. The speed function is specified, and the time at which the contour crosses a point is obtained by solving the equation. Alternatively, can be thought of as the minimum amount of time it would take to reach starting from the point . The fast marching method takes advantage of this
optimal control
Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and ...
interpretation of the problem in order to build a solution outwards starting from the "known information", i.e. the boundary values.
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
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for the surface and , were introduced by Ron Kimmel and James Sethian.
Image:Fast_marching_maze.png, Maze as speed function shortest path
Image:Fast_marching_multi_stencil_2nd_order.png, Distance map multi-stencils with random source points
Algorithm
First, assume that the domain has been discretized into a mesh. We will refer to meshpoints as nodes. Each node has a corresponding value .
The algorithm works just like Dijkstra's algorithm but differs in how the nodes' values are calculated. In Dijkstra's algorithm, a node's value is calculated using a single one of the neighboring nodes. However, in solving the PDE in , between and of the neighboring nodes are used.
Nodes are labeled as ''far'' (not yet visited), ''considered'' (visited and value tentatively assigned), and ''accepted'' (visited and value permanently assigned).
# Assign every node the value of and label them as ''far''; for all nodes set and label as ''accepted''.
# For every far node , use the Eikonal update formula to calculate a new value for . If then set and label as ''considered''.
# Let be the considered node with the smallest value . Label as ''accepted''.
# For every neighbor of that is not-accepted, calculate a tentative value .
# If then set . If was labeled as ''far'', update the label to ''considered''.
# If there exists a ''considered'' node, return to step 3. Otherwise, terminate.