I want to solve the image dehazing problem using ADMM.
I want to use the proximal algorithm to optimize each element.
I refer to this treatise: Efficient image dehazing with boundary constraint and contextual regularization.
From the above paper, we solve the following optimal solution problem in order to erase the haze added to the image.
$$\min\frac{1}{2}\Vert{x-g}\Vert_2^2+\Vert w\circ (D x)\Vert_1 \tag 1$$
Where, $x$ is reconstructed haze map, $g$ is detected haze map, $D$ is the differential operator ($3☓3 $ kirsch and Laplacian operator for preserving image edges and corners) , $\circ$ is the element-wise multiplication operator, and $w$ is the weighting function which is related to the squared difference between the two neighboring pixels.
Equation (1) can be stably solved using the intermediate variable $r$ as follows:
$$r=(w\circ (Dx^{(k)}) )\tag2$$
Then, the augmented Lagrangian of equations (1):
$$L(x^{(k)},r,y)=\frac{1}{2}\Vert{x^{(k)}-g}\Vert_2^2+\Vert{r}\Vert_1-y(r-w\circ (Dx^{(k)}) )+\frac{ρ}{2}\Vert{r-w\circ (Dx^{(k)}) }\Vert_2^2 \tag3$$
Where y is the Lagrange multiplier of the constraint $r$ in the row and column directions.
To solve with ADMM, equation (3) is applied to the sub-problems, (the other two variables are omitted.) This question is about how to solve this equation.
[x-update]
$$x^{(k+1)}=\mbox{argmin}\frac{1}{2}\Vert{x^{(k)}-g}\Vert_2^2-y(r-w\circ (Dx^{(k)}) )+\frac{ρ}{2}\Vert{r-w\circ (Dx^{(k)}) }\Vert_2^2 \tag4$$
[What I want you to tell me]
①To calculate the gradient of Eq. (4), when Eq. (4) is differentiated and set to 0,I don't know how to handle $w$ and $D$ operator. Will they be transposed???
② Should I solve this equation directly using MATLB etc.?
[I edited it.]
Vectors => small letters.
Matrices => capital letters.
$D$ is matrix form convolution operator.(if $g$ is $n×n$,then $D$ is $n^{2}× n^{2}$.)