I would like to solve the following Image Deconvolution equation by ADMM. $$\mathbf { \min\frac{1}{2}\Vert{Cx-b}\Vert_2^2+\Vert w\circ (D x)\Vert_1 \tag 1}$$
Where, $x$ is a vector of unknown pixel values, $b$ is measurements,and $C$ is the point spread kernel, $D$ is the differential operator , $\circ$ is the element-wise multiplication operator, and $w$ is the weighting function which is related to the squared difference between the two neighboring pixels.
From the Stanford - EE 367 / CS 448I: Computational Imaging and Display Notes: Image Deconvolution (lecture 6), in ADMM notation,, the TV-regularized deconvolution problem is, $$\mathbf { \min\frac{1}{2}\Vert{Cx-b}\Vert_2^2+λ\Vert z\Vert_1 \tag 2}$$ $$\mathbf { subject\; to\;Dx-z=0 }$$
where, $D$ is represents the finite differences approximation of the horizontal and vertical image gradients.
[For the x-update equation(2)]
The proximal operator $\mathbf { prox_{f,ρ}}$ is following a quadratic program
$$\mathbf {prox_{f,p}(v)=argmin\frac{1}{2}\Vert{Cx-b}\Vert_2^2+\frac{ρ}{2}\Vert Dx-v\Vert_2^2 ,\qquad v=z-u \tag 3}$$
I write the objective function$(3)$ out as
$$\mathbf {\frac{1}{2}(Cx-b)^T(Cx-b)+\frac{ρ}{2}(Dx-v)^T(Dx-v) }$$
$$\mathbf {= \frac{1}{2}(x^TC^TCx-2x^TC^Tb+b^Tb)+ \frac{ρ}{2}(x^TD^TDx-2x^TD^Tv+v^Tv) \tag 4}$$
The gradient of Eq. 4,and , equated to zero, results in the normal equations, $$\mathbf {x= (C^TC+ρD^TD)^{-1} (C^Tb+ρD^Tv) \tag 5}$$
[What I want you to tell me]
How can I express Eq.(6) as Eq. (5)? I tried to expand equation (1) as follows, is it correct?
$$\mathbf {prox_{f,p}(v)=argmin\frac{1}{2}\Vert{Cx-b}\Vert_2^2+\frac{ρ}{2}\Vert w\circ Dx-v\Vert_2^2 ,\qquad v=z-u \tag 6}$$
$$\mathbf {\frac{1}{2}(Cx-b)^T(Cx-b)+\frac{ρ}{2}(w\circ Dx-v)^T(w\circ Dx-v) }$$ $$\mathbf {= \frac{1}{2}(x^TC^TCx-2x^TC^Tb+b^Tb)+\\ \frac{ρ}{2}(x^T(D^T\circ w^T)(w\circ D)x-2x^T(D^T\circ w^T)v+v^Tv) \tag 7}$$
The gradient of Eq. 7,and , equated to zero, results in the normal equations, $$\mathbf {x= (C^TC+ρx^T(D^T\circ w^T)(w\circ D))^{-1} (C^Tb+ρ(D^T\circ w^T)v) \tag 8}$$
Eq.8 is True? Thanks for your answer.
How