## The Level Set Representation

At the heart of the level set method is the implicit representation of the interface. If the interface is given by r, T can then be represented by a function 0, called the level set function, defined by the signed distance function

Here dT (x) is the distance from the point x to the interface T, and the sign is determined so that it is negative on the inside and positive on the outside. At any time, the interface can be recovered by locating the set

For example, a circle interface and the corresponding level set function representation are shown in Fig. 4.1.

For most applications, this representation works well, but there are interfaces which cannot use it. For example, interfaces with triple junctions or any interface which does not have a clearly defined inside and outside cannot easily

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Figure 4.1: Example of a level set representation of a circle.

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Figure 4.1: Example of a level set representation of a circle.

be represented using a level set function. However, the level set method, with some modifications, can even be applied to these cases as well. These variations will be discussed in Section 4.3.

Once the level set function, 0, is constructed, the evolution equation for the interface must be rewritten in terms of 0. Given the interface r, let F(x) be the speed of the interface in the direction of the normal (see Fig. 4.2). Let x(t) be a point on the interface which evolves with the interface, then 0(x(t), t) = 0 for all t. Differentiating with respect to t gives

Now, the evolution of x(t) can be described by dx

where n is the unit normal to the interface. Use the fact that the unit normal can also be computed to be n = V0/||V0||, and substituting this with Eq. 4.4 into

Figure 4.2: Illustration of the relationship between 0(x, t), x, and F.

Eq. 4.3 gives the level set evolution equation, d0

This is the key evolution equation that was introduced in [85]. Through this equation, the motion of the interface r(t) is captured through Eq. 4.5 so that at any time t,

One key observation about Eq. 4.5 is that we have implicitly assumed that the function F is known over the entire domain of 0. Very often, this is not the case, and F is only defined on the interface itself. However, this problem can be solved by using velocity extensions, which will be discussed in Section 4.2.5.