The fast marching method solves boundary value problems that describe how a closed curve evolves as a function of time t with a speed f(x) in the normal direction at a point x on the curve. The speed is given but the time it takes for the contour to pass the point x is calculated by the equation. Different points may have different speeds. The region grows outward from the seed of the area. The more the points considered, the better the contour. The fast marching method makes use of a stationary surface that gives the arrival information of the moving boundary. From the initial curve, build a little bit of the surface upwards, with each iteration always starting from the initial position. It's called fast because it uses a fast heap sort algorithm that can tell the proper data point to update and hence to build the surface. For each of the N points at any given height h, the next point is determined and always in the order from the priority queue. The surface grows with height h in a monotonically increasing order because previously evaluated grid points are not revisited. This is because of non-negative speed and because the heap guarantees that the grid point with the minimum distance is selected first out of the data points in the sweep of N points at any level. Hence the entire operation of determining the next contour takes NlogN time.
This is a numerical method because it tries to view the curve as a set of discrete markers. The fast marching method has applications in image processing such as for region growing.
(Courtesy: Sethian @ Berkeley)
In the fast march method, let us say there are N points on the zero-level of the surface we track in a N*N*N unit distance grid. The curve is pushed outwards in unit-distances. Then, each step involves the following:
1) The point with the smallest distance is removed from the priority queue and its value is frozen so that we don't have to revisit it again and our direction is strictly outward.
2) Points are added to the priority queue to maintain unit-thickness
3) The distance of the neighbors of the removed point are recomputed. The finite difference is a multivariable quadratic equation.
Each point in the N*N*N is removed once from the priority queue.
(courtesy: Sean Mauch and David Breen from Caltech)
There are several data structures used by such level set methods. For example, there are :
1) narrow band : This data structure restricts contour propagation calculations to a thin band of data points adjacent to the boundary instead of the three dimensional surface. The data points in this narrow band are rebuild again for each step of the propagation as mentioned but not detailed above.
2) sparse field : This sparse field level set method uses a set of linked lists instead of the priority queue in the narrow band. The linked lists are populated with the same active data points around the boundary as above.
3) sparse block : This divides the volume into small cubic blocks of a set of active data points each. Then another coarse grid stores the pointers only to those blocks that meet the zero-level narrow band
4) Octree : The octree level set method uses a tree of nested cubes and the leaf nodes of this tree contain the distance value.  5) The Run length encoding compresses the region away from the narrow band with the sign but the compression is not applied to the narrow band. An additional acceleration lookup table based on the number of runs per cross section is also used.
(courtesy: Wikipedia)