Minimization problem as PDE In the article "An Image Interpolation Scheme for Repetitive Structures" Luong, Ledda and Philips propose the following approach to denoising digital image.
They consider that regularized total variation minimization problem
$$\hat I(x)=\arg\min_{I(x)}[f(\nabla I(x))+\lambda\cdot g(H*I(x)-I_0(x))] \tag{3}$$
can be transformed to the partial differential equation:
$$\frac{\partial I(x, t)}{\partial t}=f_{I}' (\nabla I(x, t))+\lambda \cdot g_{I}'(H*I(x, t)-I(x, 0))) \tag{4}$$
I can't find foundation of such transformation and I can't agree with the equivalence of these two problems.
Moreover the researchers believe appropriate to take $f(\cdot)=||\cdot||_{L^2}$ (or maybe $||\cdot||_{L^1}$) and $g(\cdot)=||\cdot||_{L^1}$. And I can't understand how they're going to find corresponding derivatives in such case.
Could you help me understand these considerations?
 A: First, this is an engineering paper, not a math paper. One shouldn't expect rigorous math from engineers. 
Second, I don't think they take $f(\cdot)=\|\cdot\|_{L^2}$. They use "the total
variation (TV): $\rho_R(I(x, t)) = |\nabla I(x, t)|$". Which is basically the $L^1$ norm of the gradient. 
Third, what they really mean by 

The minimization problem of equation 3 could be transformed to the following
  partial diﬀerential equation (pde) which produces iteratively diffused images
  $I(x, t)$ starting from the initialisation image $I(x,0)$

is that they are going to run gradient descent in search for minimizer.  In their words, "the pde of equation 4 is iteratively applied to update the blurred and
noisy image in the restoration process." The right hand side of (4) should have a minus sign, otherwise it looks like they are going for maximum instead of minimum. 
As for how they are going to take the derivative of non-differentiable functions - in the actual computation it's a finite difference of some sort. 
