Level sets was invented by Osher and Sethian  to handle the topology changes of curves. A simple representation is that a surface intersects with the zero plane to give the curve. When this surfaces changes the curve changes. The surface can be described by the following equation:
fax, t) > 0 if x e tt, fax, t) < 0if x / tt, and fa(x, t) = 0 if x e V, (9.49)
where fa represents the surface function, tt denotes the set of points where the function is positive, and V represents the set of points at which the function is zero. In Fig. 9.21, an example of a surface and its intersection with the zero plane is shown. This intersection is called the front. The surface changes with time, resulting in different fronts. So the level set function is positive at some points, negative at other points, and zero at the front r. The time as extra dimension is added to the problem to track the changes of the front. The topology changes of the curve are handled naturally by this presentation as we see from Fig. 9.22. The first row represents the surface and the zero plane at different time samples and the second row represents the resulting curves. The front is initially two ellipses, then the two ellipses merge to make a closed curve and it changes and so on. This representation allows the front to merge and break.
Figure 9.21: Change of the level set function with time resulting in different
Z=Q(x,y,t2) * Add extra dimension to the problem
Figure 9.21: Change of the level set function with time resulting in different curves.
To get an equation describing the change of the curve or the front with time, we will start with the asssumption that the level set function is zero at the front as follows:
and then compute its derivative which is also zero, d_l + a*a_x + way = 0, (9.51)
d t d x d t d y d t Converting the terms to the dot product form of the gradient vector and the x and y derivatives vector, we get a* + (#,#). (dx djA = 0 (9.52)
d t \dx dy) \dt d t J Multiplying and dividing by |V* | and takeing the other part to be F, we get the following equation:
Where F, the speed function, is given by
The selection of the speed function is very important to keep the change of the front smooth and also it is application dependent. Equation 9.55 represents speed function containing the mean curvature k. The positive sign means that the front is shrinking and the negative sign means that the front is expanding and e is selected to be a small value for smoothness. The curvature term allows the front to merge and break and also handles sharp corners,
Where k is given by
. _ *xx*y - 2*x*y*xy + *yy*l k = (*x + *2y?/2 ■ (9.56)
In 3D, the front will be an evolving surface rather than an evolving curve.
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