Fast Marching Method

In the family of boundary value problem solving methods, Fast Marching method is another numerical method. It solves the Eikonal equation

Introduction

Fast marching method was introduced by Sethian. It was originally designed for forward only propagation of the boundary with positive values for speed, but has later been revised  for including both positive and negative values for speed and this is now called  Generalized Fast Marching Method.

Motivation

This method allows us to convert the problem of  continuously forward or backward moving front into a stationary formulation where the arrival surface gives the front at zero –level. Furthermore, using a grid with unit-distance and  numerical techniques, this method is very fast, efficient and highly stable.

Terminology

Arrival surface: The arrival surface results from the travel time function that gives the arrival of the front at a given point

Narrow band: This is the set of active data points being considered for the calculation of the next level of the front. The travel time of each of the data points is computed based on the gradient between the previous point and the current as well as the gradient between current and the next point.

Initial Data

The speed at a given data point is known. Each data point can have  a different speed

Algorithm

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.

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.

we did not discuss the partial differential equation (PDE) or how to solve it numerically. Specifically, we did not discuss the distance function (time cost) based on which the next data point is selected. Here we mention it with an example.

We start from a data point whose travel time is frozen i.e. we don't change it again and proceed outwards to the neighboring points from it. We put these neighboring points in a priority queue where the points are sorted based on the minimum travel time. Then the top point in this priority queue is removed which gives the minimum travel time. This point is then frozen. The neighboring points are updated with the Euclidean norm of the gradients. The gradients are along the x,y,z axes and are computed as derivatives (inverse of speed) defined as the difference in the travel time between the new point and the old point on the same axes over a unit distance. We take the maximum of this gradient. The gradients are also taken between the next point and the new point in the normal direction. We take the minimum of this gradient. The Eucledian norm is nothing but a sum of squares just like we measure a hypotenuse. So we take the squares of the maximum and minimum respectively for each of the three axes. This gives us a quadratic  in travel time t at the new point and speed v that has a well-defined solution as one of either the travel time at the previous point + (unit-distance / speed) or the next point + (unit-distance/speed). In the fast marching method, we simplify this computation to take forward only direction and at zero-level plane so only x and y co-ordinates.

We repeat this for all other neighboring points around the one we removed from the priority queue.

It might so happen that one or more of the neighboring points are in the active set  i.e. in the priority queue. If we take a new point for which we want to compute the travel time as the center, then there are four neighbours around it and one of them is the smallest of all the trial values. Then there are four cases we need to consider.  1) where none of the neighbors are in the active set, 2) one of the neighbors is in the active set 3) two of the neighbors are in the active set and 4) all three of these neighbors are in the active set.

In the case where none of the neighbors are in the active set, the travel time of the new point is the sum of that of A and the inverse speed function f (f is based on the unit-distance).

In the case where one of the neighboring points b is already in the active set, then we know that its travel time is more than the one we just removed and also that the travel time of the new point will be lesser than the sum of the travel time for b and the inverse speed function f.

If two of the neighbors are in the active set and one of them is frozen, then this point is in the forward direction and will be the contributor to the inverse speed function since the frozen value will not be changed and this degenerates to the case 1)

If three of the neighbors are in the active set, the smallest values in each co-ordinate direction is taken and this degenerates to the cases discussed above.

Thus we obtain the travel times of the neighboring points as well and add them to the priority queue

The neighboring points are typically referred to as the narrow band and we locate the grid point with the minimum travel time.

 

Computational Complexity

The entire operation of determining the next contour takes NlogN time.

 

Error Analysis

There are two limitations of this approach.

This method does not apply to continuous boundaries

There is a potential large error in the first step of the numerical calculation.

Courtesy : Al Khalifa, S. Fomel : Fast Marching Implementation, Sethian : Fast Marching Method.