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

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


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

$x$ $y$ 表示像素位置

$N(x)$ 图像块以$x$为中心




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


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


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

L0 norm and non-linear min operator


  • 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

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


D(I), & |D(I)|^{2} \ge \frac{λ}{β}\
0, & otherwise

\nabla I, & |\nabla I|^{2} \ge \frac{μ}{α}\
0, & otherwise

  • I sub-problem 划分为子问题

Our observation


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

1, & z=y\
0, & otherwise


[1]Blind Image Deblurring Using Dark Channel Prior
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