Algorithm for solving boundary value problems of the Eikonal equation
The fast marching method[1] is a numerical method created by James Sethian for solving boundary value problems of the Eikonal equation:


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 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

for the surface
and
, were introduced by Ron Kimmel and James Sethian.
Algorithm
First, assume that the domain has been discretized into a mesh. We will refer to mesh points 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.
See also
External links
Notes
- ^ J.A. Sethian. A Fast Marching Level Set Method for Monotonically Advancing Fronts, Proc. Natl. Acad. Sci., 93, 4, pp.1591--1595, 1996. [1]