I'm working through a topic concerning diffusion regularisation. The following energy function is given.
$$ E(\mathbf u) = (u - f)^2 + \lambda \mathbf{ |\nabla u |}^2 $$
I understand that $(u-f)^2$ is a term that we need, because we want our resulting image to be mostly the same, while the regularistion term $\lambda \mathbf{ |\nabla u |}^2$ should smooth out the noise. I'm having problems understanding the derivative for the regularisation term though. It is given by:
$$ \frac{\delta \frac{1}{2} \mathbf{ |\nabla u |}^2 }{\delta \mathbf{u}} = -div(\mathbf{\nabla u}) = -\Delta u$$
Again, I understand that the Laplace operator is given by $div(\mathbf{\nabla u})$, but I don't see how the derivation ends up being the divergence. As far as I know the Laplace operator is the sum of all unmixed second partial derivatives, while $\nabla u$ should be a vector of the derivatives with respect to $\delta x$ and $\delta y$. How do you take the derivative with respect to $\delta u$?
Thanx for your help