Tracking the Front

Now, the solution is to find the front iteratively at different time steps. We get the front by intersecting the surface with the zero plane. We need to track this front by getting the length of the front or getting the area enclosed. This information is very important in the segmentation problem as we will see in the next sections. Simply the enclosed area contains all the points at which the level set function is greater than or equal to zero and the points of the front are the points at which the level set function is zero. Applying the heaviside step and delta functions is very useful in getting the area and the front respectively. For numerical implementation, it is desirable to replace the heaviside and the delta functions by some counterparts. Approximations of these two functions are used to handle smoothness problem as follows:

Figure 9.23: (a) The plot of the heaviside and delta functions for a specific value of a, (b) the narrow band points, (c) the level set function, (d) applying the heaviside step function, and (e) applying the delta function.

Figure 9.23: (a) The plot of the heaviside and delta functions for a specific value of a, (b) the narrow band points, (c) the level set function, (d) applying the heaviside step function, and (e) applying the delta function.

In Fig. 9.23(a), the two functions are plotted for a = 0.5. The value of a is always taken to be 1.5Ax to make the band equal to 3Ax where Ax is the mesh size, which is always 1. The enclosed area (A) and the length of the interface or front (L) are calculated as follows:

where D is the domain. A proof of Eq. 9.66 to be the length of the front is found in [53].

In Fig. 9.23(b), the red line represents the front and the yellow area represents the points around the front where this area is called the narrow band. In (c), (d), and (e) an example of a level set function and application of the heaviside step and delta functions are shown.

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