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

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

where $\phi_0$ is the function to be re-initialized.

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

• if $\phi_0$ is not smooth or $\phi_0$ 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 $\phi$ is to a signed distance function in $\Omega \subset \mathbb{R}^2$. This item $\mathcal{P}$ is added to energy function as internal energy. Together with the external energy for function $\phi$, the total energy function is %

To solve the evolution equation, we need to minimize the functional $\mathcal{E}$, and the evolution function has a relationship with first variation of $\mathcal{E}$, which is

Hence, we need to consider of the derivative at the right hand of the equation $(6)$.

According to the original paper, the first variation of the functional $\mathcal{E}$ 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 $(5)$ should correspond to the three items in equation $(7)$.

## Deal with $\mathcal{P}$

According to equation $(3)$, we have

Let $F_{\mathcal{P}} = \dfrac{1}{2}\big(\vert\nabla\phi\vert^2 - 2\vert\nabla\phi\vert + 1\big)$, then we introduce a small variable $\epsilon$ and an arbitrary function $h$ which satisfies $h\vert_{\partial\Omega}=0$, we have

Now calculate partial derivation

and consider of the original internal energy function $\mathcal{P}$, we have

According to Green Equation

and the pre-set condition $h\vert_{\partial \Omega}=0$, the equation $(9)$ becomes

When $\mathcal{P}$ reaches minimal, the equation $(11)$ reaches zero, together with $(6)$, we have

## Deal with $\mathcal{L}_g$

According to original paper, the function $\mathcal{L}_g$ is defined as

Just like the manner of previous process, we can define a function $F_{\mathcal{L}} = g\delta(\phi)\vert\nabla\phi\vert$, and we introduce a small variable $\epsilon$ and an arbitrary function $h$ which satisfies $h\vert_{\partial\Omega}=0$, we have

Then we can get

which can be simplified as

and consider of the original energy function $\mathcal{L}$, similar to $(9, 10, 11)$, we have

together with $(6)$, similar to the process of $(12)$, we have

## Deal with $\mathcal{A}_g$

According to original paper, the function $\mathcal{A}_g$ is defined as

We still define $\epsilon$ and arbitrary fuinction $h$, then

then we can deduce that

Obviously the solution is

Now we can get the form of the final equation: