# 顶刊TPAMI2022｜复旦大学研究团队提出基于贝叶斯理论的图像超分辨率网络BayeSR

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## 2.本文方法

### 2.1 图像退化建模

$\mathbf{y}=\mathbf{A}(\mathbf{x}+\mathbf{z})+\mathbf{n}$

### 2.2 后验分布的变分推理

$\breve{q}(\boldsymbol{\psi}) \in \underset{q(\boldsymbol{\psi})}{\arg \min } \mathrm{KL}(q(\boldsymbol{\psi}) \| p(\boldsymbol{\psi} \mid \mathbf{y}))$

\begin{aligned} \breve{q}(\mathbf{m}) & =\mathcal{N}\left(\mathbf{m} \mid \breve{\boldsymbol{\mu}}_{m}, \operatorname{diag}\left(\breve{\boldsymbol{\sigma}}_{m}^{2}\right)\right) \\ \breve{q}(\boldsymbol{\rho}) & =\prod_{i=1}^{d_{y}} \mathcal{G}\left(\rho_{i} \mid \breve{\beta}_{\rho i}, \breve{\boldsymbol{\alpha}}_{\rho i}\right) \\ \breve{q}(\mathbf{x}) & =\mathcal{N}\left(\mathbf{x} \mid \breve{\boldsymbol{\mu}}_{x}, \operatorname{diag}\left(\breve{\boldsymbol{\sigma}}_{x}^{2}\right)\right) \\ \breve{q}(\boldsymbol{v}) & =\prod_{i=1}^{d_{u}} \mathcal{G}\left(v_{i} \mid \breve{\beta}_{v i}, \breve{\boldsymbol{\alpha}}_{v i}\right) \\ \breve{q}(\mathbf{z}) & =\mathcal{N}\left(\mathbf{z} \mid \breve{\boldsymbol{\mu}}_{z}, \operatorname{diag}\left(\breve{\boldsymbol{\sigma}}_{z}^{2}\right)\right) \\ \breve{q}(\boldsymbol{\omega}) & =\prod_{i=1}^{d_{u}} \mathcal{G}\left(\omega_{i} \mid \breve{\beta}_{\omega i}, \breve{\boldsymbol{\alpha}}_{\omega i}\right) \end{aligned}

\begin{aligned} \operatorname{KL}(\breve{q}(\boldsymbol{\psi}) \| p(\boldsymbol{\psi} \mid \mathbf{y})) & =\mathbb{E}[\log \breve{q}(\boldsymbol{\psi})]-\mathbb{E}[\log p(\boldsymbol{\psi} \mid \mathbf{y})] \\ & =\mathbb{E}[\log \breve{q}(\boldsymbol{\psi})]-\mathbb{E}[\log p(\boldsymbol{\psi}, \mathbf{y})]+\log p(\mathbf{y}) \end{aligned}

\begin{aligned} & \min _{\breve{q}(\boldsymbol{\psi})} \mathbb{E}[\log \breve{q}(\boldsymbol{\psi})]-\mathbb{E}[\log p(\boldsymbol{\psi}, \mathbf{y})] \\ = & \min _{\breve{q}(\boldsymbol{\psi})} \mathrm{KL}(\breve{q}(\boldsymbol{\psi}) \| p(\boldsymbol{\psi}))-\mathbb{E}[\log p(\mathbf{y} \mid \boldsymbol{\psi})] \end{aligned}

$\min _{\breve{q}(\boldsymbol{\psi})} \operatorname{KL}(\breve{q}(\boldsymbol{\psi}) \| p(\boldsymbol{\psi}))-\mathbb{E}_{\breve{q}(\boldsymbol{\rho})}[\log p(\mathbf{y} \mid \boldsymbol{\psi})]$

## 参考

[1] S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin, “An iterative regularization method for total variation-based image restoration,” Multiscale Model. Simul., vol. 4, no. 2, pp. 460–489, 2005.