Blind Image Deblurring Using Dark Channel Prior

使用暗通道先验的盲图像去模糊(Blind Image Deblurring Using Dark Channel Prior)

摘要 Abstract

Our work is inspired by the interesting observation that the dark channel of blurred images is less sparse.

dark channels (smallest values in a local neighborhood) of blurred images are less dark

when a dark pixel is averaged with neighboring high-intensity pixels during the blur process, its intensity increases.

This new term favors clean images over blurred images in the restoration process.

Optimizing the new L0-regularized dark channel term is
challenging

The $L_0$ norm is highly non-convex and the optimization involves a non-linear minimum operation.

dark channel prior

Dark Channel

  • Dark channel [He et al., CVPR 2009]




N(x) denote a patch centered at pixel x




Dark Channel-Compute the minimum intensity in a patch of an image

当图像$I$是彩色$RGB$图

$$ D(I)(x)=\left. min \right|_{y \in N(x)} (\left. min \right|_{c \in r,g,b}I^c(y))$$

$x$ $y$ 表示像素位置

$N(x)$ 图像块以$x$为中心

$I^c$图像$I$的$c$颜色通道

针对与彩色图来说的暗通道就是指,这一个图像块中亮度最低的那个像素,且是每个像素三个通道最低的那个

当图像$I$是灰色图

$$ D(I)(x)=\left. min \right|_{y \in N(x)} I(y)$$

针对与灰色图来说的暗通道就是指,这一个图像块中亮度最低的那个像素

Convolution(Blur) Operation

$$ B(x)=\sum_{z \in\Omega_k}I(x+[\frac{s}{2}]-z)k(z)$$

$\Omega_k$ 模糊核的领域 the domain of blur kernel

$s$ 模糊核的尺寸 size of blur kernel

$k$ 模糊核

$[]$ 取整操作

$$k(z) \ge 0$$

$$\sum_{z \in\Omega_k}k(z)=1$$

Proposition 1:





$$ B(x) \ge \left. min \right|_{y \in N(x)} I(y)$$

Property 1:

$$ D(B)(x) \ge D(I)(x)$$

$D(B)$ denote the dark channel of the clear images

$D(I)$ denote the dark channel of the blurred images

Property 2:

$Ω$ denote the domain of an image $I$

If there exist some pixels $x ∈ Ω$ such that I(x) = 0, we have:

$$ ||D(B)||_{0} > ||D(I)||_{0}$$

Blurred images have less sparse dark channels than clear images

总结:模糊的图像含强度更低的、更稀疏的暗通道

$$ \left. min \right| _{I,k} ||I \otimes k-B||^{2}_{2}+\gamma||k||^{2}_{2}+\mu||\nabla I||_{0}+\lambda ||D(I)||_{0}$$

L0 norm and non-linear min operator

Optimization

  • Update latent image I:

$$ \left. min \right| _{I} ||I \otimes k-B||^{2}_{2}+\mu||\nabla I||_{0}+\lambda ||D(I)||_{0}$$

  • Half-quadratic splitting 半二次方分裂

$$ \left. min \right| _{I,u,g} ||I \otimes k-B||^{2}_{2}+\alpha||\nabla I-g||^{2}_{2}+\beta||D(I)-u||^{2}_{2}+\mu||g||_{0}+\lambda ||u||_{0}$$

  • Alternative minimization 交替最小化

$$ \left. min \right| _{I} ||I \otimes k-B||^{2}_{2}+\beta||D(I)-u||^{2}_{2}+\mu||g||_{0}$$

$$ \left. min \right| _{u,g} \alpha||\nabla I-g||^{2}_{2}+\beta||D(I)-u||^{2}_{2}+\mu||g||_{0}+\lambda ||u||_{0}$$

  • u, g sub-problem 划分为子问题

$$\begin{cases}
\left. min \right| _{u} \beta||D(I)-u||^{2}_{2}+\lambda ||u||_{0}\\
&&&&\vdots\\
\left. min \right| _{g} \alpha||\nabla I-g||^{2}_{2}+\mu||g||_{0}
\end{cases}$$

$u$和$g$独立

$$u=\begin{cases}
D(I), & |D(I)|^{2} \ge \frac{λ}{β}\
0, & otherwise
\end{cases}$$

$$g=\begin{cases}
\nabla I, & |\nabla I|^{2} \ge \frac{μ}{α}\
0, & otherwise
\end{cases}$$

  • I sub-problem 划分为子问题

Our observation

$D(I)=MI$

$$ y=\left. min \right|_{z \in N(x)} I(z)$$

$$M(x,z)=\begin{cases}
1, & z=y\
0, & otherwise
\end{cases}$$

参考论文

[1]Blind Image Deblurring Using Dark Channel Prior
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