Energy minimization (level set)

[pepe34]

Hi
Does anyone know what calculations should be used to minimize an energy functional (level set and active contour method):

$E(\phi)=\mu\int p(|grad(\phi)|)dx+\lambda\int g\delta(\phi)|grad(\phi)|dx+\alpha \int gH(-\phi)dx$ (I)

According to Chunming Li and Chenyang Xu "Distance Regularized Level Set Evolution and Its Application to Image Segmentation" this energy functional can be minimized by solving the gradient flow equation:

$\frac{\partial \phi}{\partial t}=\mu div(dp(|grad(\phi)|)grad(\phi))+\lambda\delta (\phi)div(g\frac{grad(\phi)}{|grad(\phi)|})+\alpha g \delta (\phi)$ (II)

But how to obtain (II) from (I) ?

$\delta$ is a Dirac delta function

$H$ is a Heavside function, and

$dp(x)=\frac{p'(x)}{x}$

Thanks !

 

[jvicens]

Hello there,

Thank you for bringing this topic to our attention. I am familiar with the concept of energy minimization in various fields, including image segmentation. The energy functional (I) that you have provided is commonly used in level set and active contour methods for image segmentation. In order to obtain the gradient flow equation (II) from (I), we need to take the functional derivative of (I) with respect to \phi. This can be done using the calculus of variations.

First, let's define the functional derivative of (I) as:

\frac{\delta E}{\delta \phi}=\frac{\partial E}{\partial \phi}-\frac{d}{dx}\frac{\partial E}{\partial \phi'}+\frac{d^2}{dx^2}\frac{\partial E}{\partial \phi''}+...

where \phi' and \phi'' are the first and second derivatives of \phi with respect to x, respectively.

Now, taking the functional derivative of (I) with respect to \phi, we get:

\frac{\delta E}{\delta \phi}=\mu p'(|grad(\phi)|)|grad(\phi)|+ \lambda g \delta(\phi)\frac{grad(\phi)}{|grad(\phi)|}+\alpha gH(-\phi)

Next, we use the fact that \delta(\phi)=\frac{dH(\phi)}{d\phi} to rewrite the second term in the above equation as:

\lambda g\frac{dH(\phi)}{d\phi}\frac{grad(\phi)}{|grad(\phi)|}

Using the chain rule, we can simplify this term as:

\lambda gH'(\phi)\frac{grad(\phi)}{|grad(\phi)|}

Now, we can substitute this back into the functional derivative and rearrange terms to get:

\frac{\delta E}{\delta \phi}=\mu p'(|grad(\phi)|)|grad(\phi)|+\lambda gH'(\phi)\frac{grad(\phi)}{|grad(\phi)|}+\alpha gH(-\phi)

Finally, we can use the fact that \frac{\delta E}{\delta \phi}=-\frac{\partial \phi}{\partial t} to get the gradient flow equation (II) as:

\frac{\partial \phi}{\partial t}=\mu div(dp