Fast Marching

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 created by James Sethian for solving boundary value problems of the Eikonal equation:

: $, \nabla u(x), =1/f(x) \text x \in \Omega$
: $u(x) = 0 \text x \in \partial\Omega$

Typically, such a problem describes the evolution of a closed surface as a function of time $u$ with speed $f$ in the normal direction at a point $x$ on the propagating surface. The speed function is specified, and the time at which the contour crosses a point $x$ is obtained by solving the equation. Alternatively, $u(x)$ can be thought of as the minimum amount of time it would take to reach $\partial\Omega$ starting from the point $x$. The fast marching method takes advantage of this optimal control 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

::$, \nabla_S u(x), =1 / f(x),$

for the surface $S$ and $x\in S$, 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 $x_i$ has a corresponding value $U_i = U(x_i) \approx u(x_i)$.

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 $\mathbb^n$, between $1$ and $n$ 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 $x_i$ the value of $U_i=+\infty$ and label them as ''far''; for all nodes $x_i \in \partial\Omega$ set $U_i = 0$ and label $x_i$ as ''accepted''.
# For every far node $x_i$, use the Eikonal update formula to calculate a new value for $\tilde$. If $\tilde < U_i$ then set $U_i = \tilde$ and label $x_i$ as ''considered''.
# Let $\tilde$ be the considered node with the smallest value $U$. Label $\tilde$ as ''accepted''.
# For every neighbor $x_i$ of $\tilde$ that is not-accepted, calculate a tentative value $\tilde$.
# If $\tilde < U_i$ then set $U_i = \tilde$. If $x_i$ was labeled as ''far'', update the label to ''considered''.
# If there exists a ''considered'' node, return to step 3. Otherwise, terminate.

See also

* Level-set method
* Fast sweeping method
* Bellman–Ford algorithm

External links

Dijkstra-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 Matlab Implementation

Implementation Details of the Fast Marching Methods

Generalized Fast Marching method
by Forcadel et al. 008for applications in image segmentation.

Python Implementation of the Fast Marching Method
*See Chapter 8 i
Design and Optimization of Nano-Optical Elements by Coupling Fabrication to Optical Behavior

Notes

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Numerical differential equations