About formula derivation x_hat and x_bar

[chensming]

Hi, bro, a nice work! But I am not very understanding of a problem. In the section "2.2 RANGE-NULL SPACE DECOMPOSITION", I have seen this equation

And in "3.1 DENOISING DIFFUSION NULL-SPACE MODEL", in this context, I don't understand the difference between $\hat{x}$ and $\bar{x}$ and here the $y=Ax$?

[wyhuai]

Hi, thanks for your attention!

Yes, $y=Ax$, $\hat{x}$ is our final solution, and $\bar{x}$ is the only unknown variable that needs to be solved.

[chensming]

Oh, thanks. And I would like to know if you have tried some non-linear recovery tasks such as rain removal, fog removal and light enhancement. Do you have some suggestions for that?

[wyhuai]

Tasks like rain removal are difficult since the degradation A is unknown and is hard to predict. I guess you may refer to recent supervised methods for such degradation-agnostic problems.

[chensming]

Tasks like rain removal are difficult since the degradation A is unknown and is hard to predict. I guess you may refer to recent supervised methods for such degradation-agnostic problems.

Oh. I have another problem, here I don't know how to get this equation:

The context of it is as followings:

[wyhuai]

For this equation, the left side is our variance, the right side is the originally defined variance. Note that this equation is a simplified one, inaccurate. You may also refer to the appendix of Unlimited-Size Diffusion Restoration for accurate calculation of the variance calculation.

[chensming]

For this equation, the left side is our variance, the right side is the originally defined variance. Note that this equation is a simplified one, inaccurate. You may also refer to the appendix of Unlimited-Size Diffusion Restoration for accurate calculation of the variance calculation.

Oh, thanks! You are so kind! Hope to see you on campus someday. ^_^