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4.2.4 Reinitialization

As noted in the introduction to this chapter, there are two means by which the level set method can be kept stable for arbitrary speed functions. For nearly all applications of the level set method, one of these techniques must be used. One method involves using velocity extensions, and the other uses reinitialization. Both methods are frequently used, and there is disagreement as to which method is preferred. Recent advances in the level set method have resulted in either method producing good results. For balance, both methods are presented, with reinitialization treated here and velocity extensions to follow in Section 4.2.5.

Reinitialization was first introduced in [19], where it was observed that the only part of the level set function which is of interest is the portion immediately around the zero level set. While initially, the level set function can be constructed to be the signed distance function to the interface, most speed functions, F, will not preserve this property over time. This can lead to instability, and ultimately failure of the method. Reinitialization is, therefore, a process where the level set function is reconstructed to be the signed distance function.

˜

Let φ be the level set function, and let φ be the desired reconstructed level

˜

set function, then φ solves

This pair of equations is precisely the type of problem the fast marching method is designed to solve, with F ≡ 1 in Eq. 4.21. Furthermore, the function φ can be used to initialize the fast marching method, as described in Section 4.2.3. The

˜

solution φ of Eqs. 4.29 and 4.30 is now called reinitialized.

Early implementations of reinitialization suffered from accuracy, particularly in regions of high curvature. When the interface was reinitialized, there was significant error in the computed solution in Eq. 4.29. This was primarily due to the low-order accurate methods used for interpolating φ. More recent methods, such as the one presented in Section 4.2.3, significantly reduced this error, as illustrated in Fig. 4.8.

It has been observed recently [100] that for the specific application of reinitialization, it is not necessary to use the heap sort method. In fact, the same results can be achieved by simply taking a first-in-first-out strategy for the order of the grid points. In other words, instead of maintaining the binary tree and

the velocity field in the first place. This approach was introduced by Adalsteinsson and Sethian [3] as an alternative to the use of reinitialization. As noted in the introduction, velocity extensions also serve the purpose of constructing a velocity field for the entire domain of φ even when the speed, F, is defined only on the interface itself.

For velocity extensions, the objective is to force the velocity field, F, to be such that the signed distance function is preserved, i.e.

Differentiating Eq. 4.31 with respect to t, and using Eqs. 4.5 and 4.31, gives

Viewed geometrically, Eq. 4.32 makes sense because it requires the speed function normal to the interface to be constant along that normal. This effectively keeps the level sets of φ evenly spaced.

To solve Eq. 4.32, assume the function F is given on the zero level set of φ. The goal is to construct an extension velocity Fext, such that

The solution of Eq. 4.33 is done in a manner very similar to the fast marching method. The discretization of Eq. 4.33 is given by

min(D+xφi, j , 0)D+x Fiext, j + max(D−xφi, j , 0)D−x Fiext, j

+ min(D+yφi, j , 0)D+y Fiext, j + max(D−yφi, j , 0)D−y Fiext, j = 0. (4.34)

This is a linear equation in terms of the unknown Fiext, j and is easily solved. Note that Eq. 4.34 must be solved at the grid points xi, j in the order of increasing magnitude of φi, j similar to the fast marching method. This is easily accomplished using the same heap-sort strategy described in Section 4.2.3.

The initialization of Fiext, j on the grid points near the interface φ−1(0) is done using the bicubic interpolation method discussed in Section 4.2.3. Given a grid point xi, j , the point, y, on the interface φ−1(0) nearest to xi, j is computed using the bicubic interpolant. The value of Fiext, j must be the same as F(y), because the vector xi, j − y is orthogonal to the interface, and hence parallel to φ, so Fext must be constant along that vector. This populates the grid points adjacent to the interface, and the velocity extension algorithm can then proceed.

The algorithm for velocity extensions is therefore given by

1.Initialize the value of Fext at all the grid points adjacent to the interface

φ−1(0) using the bicubic interpolation algorithm discussed in Section 4.2.3. Place all those grid points in the accepted set A. Add all grid points adjacent to a point in the set A into the set T, and the remainder of the grid points are placed in the set D.

2.Take the grid point xi, j with the smallest value of |φi, j | from the set T and place it in set A. Solve Eq. 4.34 for Fiext, j at this xi, j . Take all neighbors of xi, j which are in D, and place them in T.

3.Repeat step 2 while T = .

Similar to the comments made in the previous section, the velocity extension method also can avoid the cost of the heap sort by taking the first-in-first-out strategy. Therefore, the computational cost for the velocity extension is the same as for reinitialization, O (N ).

4.2.6 Narrow Band Methods

There is another technique frequently used in level set methods that deserves attention. While it is not an essential part of the level set method it is useful in that it can significantly reduce the computational cost.

As noted earlier, switching from a parametric representation to the implicit representation used in the level set method also increased the computational cost. For example, if an evolving curve in the plane can be modeled with O (N ) points, then the corresponding level set representation would require O (N2) points, due to the higher dimension of the level set function. However, most of that increased computational cost is spent computing the evolution of φ in regions far from the φ = 0 interface of interest.

It was observed in [19] that it is not necessary to compute the evolution of φ everywhere, but only in the neighborhood of the φ = 0 interface. This observation effectively reduces the computation back to O (N ). This technique is called a narrow-band level set method, and was significantly refined in [2]. Basically, the evolution equation of φ is computed on a dynamically determined set of grid points where φ is small.

Not all applications will benefit from a narrow band implementation; it depends heavily on the cost of computing F, which can easily overwhelm the cost

of the rest of the level set method. However, if that is not the case, the reader should see [2] for a detailed description of the narrow band method.

4.2.7 Assembling the Basic Algorithm

To summarize this section, we conclude with an algorithmic description of the full level set method. The strategies of using reinitialization or velocity extensions are similar, so the algorithms will be presented together.

1.Initialize φ to be the signed distance function to the initial interface . If necessary, reinitialization can be used to enforce this condition.

2.Compute the speed function F. This is, obviously, application dependent.

3.Determine Fext from F (if using velocity extensions).

4.Advance φ in time using Eq. 4.5, with speed F (or Fext if using velocity extensions).

5.Reinitialize (if using reinitialization).

6.Return to step 2.

Note that reinitialization does not have to be done every time step, but can be done as needed, so step 5 can be skipped most of the time. The frequency of reinitialization is application dependent. Automated detectors for determining when reinitialization is required have been proposed, but the computational cost of these detectors is of the same order of magnitude as doing reinitialization in the first place. Consequently, it is preferable to simply choose an appropriate number of time steps between each reinitialization.

On the other hand, velocity extensions must be done every time step. The common belief that the velocity extension method is more accurate was certainly true when it was first introduced. However, improvements in the reinitialization process have closed the gap. Nonetheless, for the majority of applications, the speed function F is defined only on the front, so velocity extensions will be required every step anyway, just to be able to use Eq. 4.5. In that case, it makes more sense to use velocity extensions and never do reinitialization. For the remainder of the applications where stability is a concern, it is essentially a toss-up between the two.

represent the step edges of a stratified material [21]. Lower dimensional manifolds, such as one-dimensional curves in R3, can be captured by the intersection of two level surfaces [14]. Multiple distinct regions with interfaces that result in triple junctions can also be captured using multiple level set functions [17, 108]. Interfaces with boundaries, such as a finite-length crack in a plate, can also be represented using multiple level sets [107, 115, 117].

In other examples, multiple level surfaces of a single level set function are used. In [95], the level surfaces for each of the integers represent a different interface. In other applications, the entire continuous spectrum of level surfaces are used. For example, in [25], each level surface evolves to a surface of constant curvature, while in [20], the spectrum of evolving level surfaces is shifted in order to locate an unstable equilibrium surface.

The fast marching method has also been used in a variety of applications, resulting in dramatically increased speed in some computationally intensive calculations. For example, see the work on computing multiple travel-time arrivals in [46].

The range of applications for the level set and fast marching methods is now very wide, and still growing. Many times, variations of the method are required to make it fit the problem. In this section, some recent improvements and variations, which will be of general interest are presented.

4.3.1 Ordered Upwind Methods

In [101, 129], Sethian and Vladimirsky developed a novel extension of the fast marching method, making it applicable to a significantly wider class of problems.

Recall the fast marching method equation

It is important to recognize that this equation assumes that from any point x, the speed, F, is the same, independent of the direction the interface is traveling. In other words, the speed function is isotropic. Sethian and Vladimirsky have generalized the fast marching method so that the speed function can vary with direction, i.e. the speed function, F, may depend on φ. In this case, the speed function is called anisotropic. The generalized method is called the ordered upwind method, of which the fast marching method is a special case.