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?
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
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.