I have this optimization problem:

Where $X$ is the output image and $Y$ is the input image.

Let's say the input image is $y$ and the output image is $x$ (transform image to vector) then the problem can be rewritten:

Where $D_h$ is the horizontal Derivative Operator, $D_v$ is the vertical Derivative Operator and $1$ is vector of ones.

Then the solution is given by:

$\hat{x} = { \left( I + \lambda {D}_{h}^{T} {D}_{h} + \lambda {D}_{v}^{T} {D}_{v} \right) }^{-1} \left( \lambda {D}_{h}^{T} {D}_{h} y + \lambda {D}_{v}^{T} {D}_{v} y + 255 \cdot \boldsymbol{1} \right)$

My question is given the input y how to apply $Dh$ and $Dv$ to this specific equation.

Thanks for your reply.

I have this optimization problem:

$$ \arg \min_{ X \left( i, j \right) } \sum_{i, j} \left\| X \left( i, j \right) - 255 \right\|_{2}^{2} + \lambda \sum_{i, j} \left\| \nabla X \left( i, j \right) - \nabla Y \left( i, j \right) \right\|_{2}^{2} $$

Where $ X $ is the output image and $ Y $ is the input image.

Let's say the input image is $ y $ and the output image is $ x $ (Transform image to vector) then the problem can be rewritten:

$$ \hat{x} = \arg \min_{x} \frac{1}{2} {\left\| x - 255 \cdot \boldsymbol{1} \right\|}_{2}^{2} + \frac{\lambda}{2} {\left\| {D}_{h} \left( x - y \right) \right\|}_{2}^{2} + \frac{\lambda}{2} {\left\| {D}_{v} \left( x - y \right) \right\|}_{2}^{2} $$

Where $D_h$ is the horizontal Derivative Operator, $D_v$ is the vertical Derivative Operator and $1$ is vector of ones.

Then the solution is given by:

$$ \hat{x} = { \left( I + \lambda {D}_{h}^{T} {D}_{h} + \lambda {D}_{v}^{T} {D}_{v} \right) }^{-1} \left( \lambda {D}_{h}^{T} {D}_{h} y + \lambda {D}_{v}^{T} {D}_{v} y + 255 \cdot \boldsymbol{1} \right) $$

My question is given the input $ y $ how to apply $ {D}_{h} $ and $ {D}_{v} $ to this specific equation.

Thanks for your reply.