The existence of the front means that the level set function has positive and negative parts, then it has negative and positive values including zeroes. The level set function with this property is called a signed distance function. This property should be kept through the iterations in order not to lose the front. There are different solutions for this problem . We will discuss only the solution introduced by Osher et al. . It was proved that recomputing the level set function by solving Eq. 9.67 frequently enough will maintain the function as signed distance function:
where it contains the sign function sign. When the level set function is negative, the information flows one way and when it is positive, the information flows the other way. The net effect is to "straighten out" the level set function on either sides of the zero level set,
By solving this equation, the derivative of $ with respect to time will vanish resulting in Eq. 9.68. |V$ | = 1 denotes the measure for signed distance function.
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