MSND简短笔记

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目录：

• TNRD区别于一般NRD
• MSND与TNRD的联系
• MSND用于高斯去噪及其推导
• MSND用于泊松去噪及其推导
• 实验

贡献/创新点

（1）Multi-scale procedure is useful, 尤其是heavy noise。
（2）描述了TNRD和MSND的广泛使用（通用poisson denoising）
（3）Extend the TNRD with a multi-scale strategy, 可以增大感受野。

TNRD相比于一般的nonlinear diffusion process的主要变化：

$$-----TRAINABLE-----$$

即：The TNRD framework is modeled by $\underbrace{\it{highly\; parameterized\;linear\;filter}}_{①}$ as well as $\underbrace{\it{highly\;parameterized\;influence\;functions}}_{②}$.
①②，they are learned from training data through a loss based approach.

$$\begin{cases} n_0={\rm{I}_0} \\[2ex] u_t=\underbrace{Prox_{{\frak{g}}^t}}_{reaction\, force} \left( u_{t+1}- \left( \underbrace{\sum\limits_{i=1}^{N_k}\left({{K_i^t}} \right)^{\top} \phi_i^t \left( K_i^t \, u_{t+1}\right)}_{diffusion \, force} +\underbrace{\psi^t \left( u_{t+1}, f\right)}_{reaction\, force}\right)\right)\\[2ex] t=1, \cdots, T \\[2ex] \end{cases} \tag{2.1}$$

其中，image

u∈RN
$u\in\Bbb{R}^N$;

Ki∈RN×N
$K_i\in\Bbb{R}^{N\times{N}}$, highly sparse matrix, a set of linear fitlers;



ϕ(Kiu)∈RN
$\phi\left(K_iu\right)\in\Bbb{R}^N$: a set of influence function / flux function;



Proxgt
$Prox_{\frak{g}^t}$: proximal mapping operation.





*TNRD结构

The reaction term(force) is designated to handle a certain smooth data term, while the term

g(u,f)
$\frak{g}\rm{\left(u,f\right)}$ is used to handle the problem with a non-smooth data term.

$${Prox}_{{\frak{g}}^t} \left( \tilde{u}\right)= \min_u{\frac{{\Vert u - \tilde{u}\Vert}_2^2}{2}} + {\frak{g}}^t \left( u, f\right)$$

$TNRD \quad \Longrightarrow \quad MSND$（增加了缩放尺度）

$$\sum_{l=0}^L{\sum_{i=1}^{N_k}{A_l^{\top} {\left( K_{l,i}^t \right)}^{\top}\phi_{l,i}^t\left( K_{l,i}^t \, A_l\,u_{t-1}\right)}}, \tag{3.1}$$

公式

(3.1)
$(3.1)$为公式

(2.1)
$(2.1)$diffusion force的更新部分；





*MSND结构

MSND的完整求解公式

$A_l$: Down-sampling the original image $u$ to different sizes.
$L$: The number of total scales.
The parameters to be trained in stage t $\varTheta^t=\lbrace{\{\{\theta_{l,i}^t, k_{l,i}^t\}_{l=1}^L\}_{i=1}^{N_k},\lambda^t\rbrace}$；其主要难点在于计算$\frac {\partial u_t } {\partial \varTheta^t}$和$\frac {\partial u_{t+1} } {\partial u_t}$