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 应用科技  2020, Vol. 47 Issue (4): 14-19, 25  DOI: 10.11991/yykj.202002004 0

### 引用本文

XIANG Jianhong, WEI Junhao. An improved image denoising method based on ADMM[J]. Applied Science and Technology, 2020, 47(4): 14-19, 25. DOI: 10.11991/yykj.202002004.

### 文章历史

An improved image denoising method based on ADMM
XIANG Jianhong, WEI Junhao
College of Information and Communication Engineering, Harbin Engineering University, Harbin 150001, China
Abstract: In order to solve the problem that there may be effects of factors such as noise and excessive transmission time overhead in the transmission of the image, which will result in poor image restoration effect, a denoising and reconstruction algorithm is proposed based on entropy function, which is based on the principle of maximum entropy in mathematics. A new fast denoising algorithm is proposed by using the idea of divide and conquer of alternating direction method of multipliers. The experimental simulation complying with normalized mean square error and peak signal to noise ratio evaluation criteria verifies the superiority of the proposed algorithm. The experimental results show that the proposed method has a good effect and has a certain use in denoising.

1 一种基于熵函数的重构算法

 $\mathop {\min }\limits_x F\left( x \right) = \sum\limits_{i = 1}^n {{{\left| {{f_i}\left( x \right)} \right|}^p}} {\rm{, }} {0 \leqslant p \leqslant 1}$ (1)

 $F\left( x \right) = \displaystyle\sum\limits_{i = 1}^n {|{f_i}\left( x \right){|^p}} = \displaystyle\sum\limits_{i = 1}^n {{{\left[ {\max \left\{ {{f_i}\left( x \right) - {f_i}\left( x \right)} \right\}} \right]}^p}}$ (2)

 $\mathop {\lim }\limits_{q \to \infty } \left[ {\dfrac{1}{q}\ln \left( {{{\rm{e}}^{qt}} + {{\rm{e}}^{ - qt}}} \right)} \right] = \mathop {\lim }\limits_{q \to \infty } \dfrac{{t{{\rm{e}}^{qt}} - t{{\rm{e}}^{ - qt}}}}{{{{\rm{e}}^{qt}} + {{\rm{e}}^{ - qt}}}} = \max \left( {t, - t} \right) = |t|$ (3)

 $\mathop {\min }\limits_x F\left( {x,q} \right) = {\displaystyle\sum\limits_{i = 1}^n {\left[ {\dfrac{1}{q}\ln \left( {{{\rm{e}}^{q{f_i}\left( x \right)}} + {{\rm{e}}^{ - q{f_i}\left( x \right)}}} \right)} \right]} ^p}$ (4)

 $\begin{array}{c} \dfrac{1}{q}\ln \left( {{{\rm{e}}^{q{f_i}\left( x \right)}} + {{\rm{e}}^{ - q{f_i}\left( x \right)}}} \right) = \dfrac{1}{q}\ln \left[ {{{\rm{e}}^{q|{f_i}\left( x \right)|}}\left( {1 + {{\rm{e}}^{ - 2q|{f_i}\left( x \right)|}}} \right)} \right] =\\ {\rm{ }} |{f_i}\left( x \right)| + \dfrac{1}{q}\ln \left( {1 + {{\rm{e}}^{ - 2q|{f_i}\left( x \right)|}}} \right) \\ \end{array}$

 $\begin{array}{c} {\nabla _x}F\left( {x,q} \right) =\\ {\rm{ }} {\displaystyle\sum\limits_{i = 1}^n {p\left[ {\dfrac{1}{q}\ln \left( {{{\rm{e}}^{q{f_i}\left( x \right)}} + {{\rm{e}}^{ - q{f_i}\left( x \right)}}} \right)} \right]} ^{p - 1}} \dfrac{{{{\rm{e}}^{q{f_i}\left( x \right)}} - {{\rm{e}}^{ - q{f_i}\left( x \right)}}}}{{{{\rm{e}}^{q{f_i}\left( x \right)}} + {{\rm{e}}^{ - q{f_i}\left( x \right)}}}}{\nabla _x}{f_i}\left( x \right) =\\ {\rm{ }} {\displaystyle\sum\limits_{i = 1}^n {p\left[ {|{f_i}\left( x \right)| + \frac{1}{q}\ln \left( {1 + {{\rm{e}}^{ - 2q|{f_i}\left( x \right)|}}} \right)} \right]} ^{p - 1}} \dfrac{{1 - {{\rm{e}}^{ - 2q{f_i}\left( x \right)}}}}{{1 + {{\rm{e}}^{ - 2q{f_i}\left( x \right)}}}}{\nabla _x}{f_i}\left( x \right) \\ \end{array}$

 ${L_p}\left( {{{x}},\lambda ,q} \right) = \frac{1}{2}\left\| {{{\varPhi x}} - {{y}}} \right\|_2^2 + \lambda F\left( {{{x}},q} \right)$

1) 循环： $t = 1,2, \cdots ,{\mathit{\Upsilon }}$

a) 通过求解式(5)来更新 ${{{x}}^{\left( t \right)}}$ ，得到 ${{{x}}^{\left( {t + 1} \right)}}$

 $\begin{array}{c} p{\left[ {|{{{x}}^{\left( {t + 1} \right)}}| + \dfrac{1}{q}\ln \left( {1 + {{\rm{e}}^{ - 2q|{{{x}}^{\left( {t + 1} \right)}}|}}} \right)} \right]^{p - 1}} \dfrac{{1 - {{\rm{e}}^{ - 2q{f_i}\left( x \right)}}{{{x}}^{\left( {t + 1} \right)}}}}{{1 + {{\rm{e}}^{ - 2q{{{x}}^{\left( {t + 1} \right)}}}}}}+ \\ {\rm{ }} \dfrac{1}{\lambda }{{{\varPhi }}^{\rm{T}}}\left( {{{\varPhi }}{{{x}}^{\left( t \right)}} - {{y}}} \right) = 0 \\ \end{array}$ (5)

b) 更新 $\;\beta = {\beta _0}{10^{ - \mu \left( {t + 1} \right)}}$ ，其中 $\;\mu = \dfrac{{\log \left( {{{{\beta _0}} / {{\beta _{\mathit{\Upsilon }}}}}} \right)}}{{{\mathit{\Upsilon }} + 1}}$

c) 判断是否满足迭代终止条件： $\left| {{{{x}}^{\left( {t + 1} \right)}} - {{{x}}^{\left( t \right)}}} \right| < \xi$ 或者 $\;\beta = {\beta _{\mathit{\Upsilon }} }$ ，其中 $\xi$ 是一个很小的正常数， ${\mathit{\Upsilon }}$ 是使迭代满足终止条件的最小值。若满足，结束循环；若不满足，返回1)；

2) 得到稀疏目标信号的解： ${{x}} = {{{x}}^{\left( {t + 1} \right)}}$

1) 外部循环： $t = 1,2, \cdots ,{\mathit{\Upsilon }}$

a) 内部循环： $k = 0,1, \cdots ,{\rm{iter}}$ ，iter为迭代次数；

 ${{{x}}^{\left( {k + 1} \right)}} = \arg \mathop {\min }\limits_{{x}} \frac{1}{2}\left\| {{{y}} - {{\varPhi }}{{{x}}^{\left( k \right)}}} \right\|_2^2{\rm{ + }}\frac{\lambda }{{\rm{2}}}\left\| {{{{x}}^{\left( k \right)}} - \left( {{{{v}}^{\left( k \right)}} - {{{u}}^{\left( k \right)}}} \right)} \right\|_2^2$
 ${{{v}}^{\left( {k + 1} \right)}} = {\rm{BM3D}}\left( {1,\frac{{{y}}}{{\max \left( {{y}} \right)}},\sigma } \right)$

 ${{{u}}^{\left( {k + 1} \right)}} = {{{u}}^{\left( k \right)}} - \left( {{{{x}}^{\left( {k + 1} \right)}} - {{{v}}^{\left( {k + 1} \right)}}} \right)$

b) 将a)得到的结果赋值给 ${{{x}}^{\left( t \right)}}$ ，然后通过求解式(6)来更新 ${{{x}}^{\left( t \right)}}$ ，得到 ${{{x}}^{\left( {t + 1} \right)}}$

 $\begin{array}{c} p{\left[ {|{{{x}}^{\left( {t + 1} \right)}}| + \dfrac{1}{q}\ln \left( {1 + {{\rm{e}}^{ - 2q|{{{x}}^{\left( {t + 1} \right)}}|}}} \right)} \right]^{p - 1}} \dfrac{{1 - {{\rm{e}}^{ - 2q{f_i}\left( x \right)}}{{{x}}^{\left( {t + 1} \right)}}}}{{1 + {{\rm{e}}^{ - 2q{{{x}}^{\left( {t + 1} \right)}}}}}}+ \\ {\rm{ }} \dfrac{1}{\lambda }{{{\varPhi }}^{\rm{T}}}\left( {{{\varPhi }}{{{x}}^{\left( t \right)}} - {{y}}} \right) = 0 \\ \end{array}$ (6)

c) 更新 $\beta = {\beta _0}{10^{ - \mu \left( {t + 1} \right)}}$ ，式中 $\mu = \dfrac{{\log \left( {{{{\beta _0}} / {{\beta _{\mathit{\Upsilon }} }}}} \right)}}{{{\mathit{\Upsilon }} + 1}}$

d) 判断是否满足迭代终止条件： $\left| {{{{x}}^{\left( {t + 1} \right)}} - {{{x}}^{\left( t \right)}}} \right| < \xi$ 或者 $\;\beta = {\beta _{\mathit{\Upsilon }} }$ ，其中 $\xi$ 是一个很小的正常数。若满足，结束循环；若不满足，返回1)；

2) 得到稀疏目标信号的解： ${{x}} = {{{x}}^{\left( {t + 1} \right)}}$

2.1 参数的选择

 Download: 图 5 范数 $p$ 对算法的影响

 Download: 图 6 参数 $\lambda$ 对算法的影响
2.2 实验仿真分析

3 结论

1)首先根据熵函数提出了一种新的重构算法，即基于熵函数的重构算法——MEA-RA算法。

2)然后进行实验对比，对比的算法也都是与此相关的一些算法，经过仿真结果可以知道MEA-RA 算法具有很好的性能，而且时间复杂度相对也比较低。