Level Set Evolution Equation


In traditional level set methods, evolution equation can be written as

During the evolution steps, the signed distance function is re-initialized to make a more accurate result. The standard re-initialization method is to solve level set equation

where is the function to be re-initialized.

However, this re-initialization method does not work well:

  • if is not smooth or is much steeper on one side of the interface than the other
  • if the level set function is far away from a signed distance function

From the practical view, the re-initialization process can be quite complicated, expensive and have subtle side effects. The paper propose a integral item

as a metric to characterize how close a function is to a signed distance function in . This item is added to energy function as internal energy. Together with the external energy for function , the total energy function is

To solve the evolution equation, we need to minimize the functional , and the evolution function has a relationship with first variation of , which is

Hence, we need to consider of the derivative at the right hand of the equation .

According to the original paper, the first variation of the functional can be represented as the following equation, which can be seen as the result we are going to derive

It is clear that the three items in equation should correspond to the three items in equation .

Deal with

According to equation , we have

Let , then we introduce a small variable and an arbitrary function which satisfies , we have

Now calculate partial derivation

and consider of the original internal energy function , we have

According to Green Equation

and the pre-set condition , the equation becomes

When reaches minimal, the equation reaches zero, together with , we have

Deal with

According to original paper, the function is defined as

Just like the manner of previous process, we can define a function , and we introduce a small variable and an arbitrary function which satisfies , we have

Then we can get

which can be simplified as

and consider of the original energy function , similar to , we have

together with , similar to the process of , we have

Deal with

According to original paper, the function is defined as

We still define and arbitrary fuinction , then

then we can deduce that

Obviously the solution is

Now we can get the form of the final equation: