﻿ 基于非凸加权<i>L<sub>p</sub></i>范数稀疏误差约束的图像去噪算法
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 智能系统学报  2019, Vol. 14 Issue (3): 500-507  DOI: 10.11992/tis.201804057 0

引用本文

XU Jiucheng, WANG Nan, WANG Yuyao, et al. Non-convex weighted-Lp-norm sparse-error constraint for image denoising [J]. CAAI Transactions on Intelligent Systems, 2019, 14(3): 500-507. DOI: 10.11992/tis.201804057.

文章历史

1. 河南师范大学 计算机与信息工程学院，河南 新乡 453007;
2. 河南省高校计算智能与数据挖掘工程技术研究中心，河南 新乡 453007

Non-convex weighted-Lp-norm sparse-error constraint for image denoising
XU Jiucheng 1,2, WANG Nan 1,2, WANG Yuyao 1,2, XU Zhanwei 1,2
1. College of Computer and Information Engineering, He’nan Normal University, Xinxiang 453007, China;
2. Engineering Technology Research Center for Computing Intelligence and Data Mining in Colleges and University of He’nan Province, Xinxiang 453007, China
Abstract: Due to noise during image denoising, it is difficult to learn accurate prior knowledge. Therefore, obtaining a desirable sparse coefficient proves to be difficult. To solve this problem, this paper proposes an image denoising method based on the non-convex weighted-lp-norm sparse-error constraint. This algorithm decomposes the coefficient-solving process into two sub-problems. First, the algorithm solves the sparse coefficient in the lp norm by the generalized soft threshold value algorithm and then uses the surrogate algorithm to solve the sparse coefficient in the sparse-error constraint. Finally, the algorithm obtains a robust sparse coefficient according to its average value. The experimental results show that the proposed algorithm features a high peak signal-to-noise ratio and high efficiency in terms of the running time. Simultaneously, a desirable visual perception is obtained.
Key words: image denoising    sparse representation    sparse coefficient    prior knowledge    l1 norm    non-convex weighted lp norm    sparse error constraint    peak signal-to-noise ratio

1 基本概念

 ${N_x} = \arg \mathop {\min }\limits_{{N_i}} \sum\limits_{i = 1}^n {(\frac{1}{2}\parallel {X_i} - {D_i}{N_i}\parallel _{\rm{F}}^2} + \lambda \parallel {N_i}{\parallel _{1}})$ (1)

 ${M_y} = \arg \mathop {\min }\limits_{{M_i}} \sum\limits_{i = 1}^n {(\frac{1}{2}\parallel {Y_i} - {{\hat X}_i}\parallel _{\rm{F}}^2} + \lambda \parallel {M_i}{\parallel _{{\rm{ }}1}})$ (2)

1.1 非凸加权lp范数最小化

 ${{{M}}_y} = \arg \mathop {\min }\limits_{{{{M}}_i}} \sum\limits_{i = 1}^n {(\frac{1}{2}\parallel {Y_i} - {{\hat X}_i}\parallel _{\rm{F}}^2} + \parallel {{{W}}_i} \cdot {{{M}}_i}{\parallel _p})$

1.2 稀疏误差

 $R = M - N$

 ${M_y} = \arg \mathop {\min }\limits_{{M_i}} \sum\limits_{i = 1}^n {(\frac{1}{2}\parallel {Y_i} - \hat X\parallel _{\rm{F}}^2} + \gamma \parallel {M_i} - {N_i}{\parallel _p})$ (5)

 ${M_y} = \arg \mathop {\min }\limits_{{M_i}} \sum\limits_{i = 1}^n {(\frac{1}{2}\parallel {Y_i} - \hat X\parallel _{\rm{F}}^2} + \gamma \parallel {M_i} - {N_i}{\parallel _{{\rm{ }}1}})$
2 基于非凸加权lp范数的稀疏误差约束算法

 $\begin{array}{c} {M_y} = \arg \mathop {\min }\limits_{{M_i}} \displaystyle\sum\limits_{i = 1}^n {(\frac{1}{2}\parallel {Y_i} - \hat X\parallel _F^2} + \\ \parallel {W_i} \cdot {M_i}{\parallel _p} + \gamma \parallel {M_i} - {N_i}{\parallel _{{\rm{ }}1}}) = \\ \arg \mathop {\min }\limits_{{{{u}}_i}} (\parallel {{{y}}_i} - {{{D}}_i}{{{u}}_i}\parallel _2^2 + \parallel {{{w}}_i} \cdot {{{u}}_i}{\parallel _p} + \\ \gamma \parallel {{{u}}_i} - {{{v}}_i}{\parallel _{{\rm{ }}1}}) \end{array}$ (7)

 $\begin{array}{c} {M_y} = \arg \mathop {\min }\limits_{{{{u}}_i}} \{ \parallel {{{y}}_i} - {{{D}}_i}{{{u}}_i}\parallel _2^2 + \displaystyle\frac{1}{2}(\parallel {{{w}}_i} \cdot {{{u}}_{pi}}{\parallel _p} + \\ \gamma \parallel {{{u}}_{ri}} - {{{v}}_i}{\parallel _{{\rm{ }}1}})\} \end{array}$

 ${{{u}}_{pi}} = S_p^{{\rm{GST}}}({\delta _i};{{{w}}_i})$

 ${{{u}}_{ri}} = {S_\gamma }({{D}}_i^{ - 1}{\hat {{x}}_i} - {{{v}}_i}) + {{{v}}_i}$ (10)

1) 利用公式 ${{{Y}}^{t + 1}} = {\hat {{X}}^t} + \lambda ({{Y}} - {\hat {{X}}^t})$ 对输入的噪声图像Y进行归一化；

2) 判断Y t+1YB的结构相似性与Y tYB的结构相似性的差值是否小于阈值τ

3) 若 ${\rm{SSIM}}({{{Y}}^{t + 1}},{{{Y}}_B}) - {\rm{SSIM}}({{{Y}}^t},{{{Y}}_B}) < \tau$ ，在图像Y t+1中进行相似块选取；否则，在图像YB中进行相似块选取；

4) 通过kNN方法对每一个图像块yiYBi进行筛选，构建相似组 ${{{Y}}_i}^{t + 1}$ ${{{Y}}_{Bi}}^{t + 1}$

5) 对相似组Yi进行PCA操作，构建字典 ${{{D}}_i}^{t + 1}$

6) 由YBi=DiNiYi=Diδi分别求解第t+1次稀疏系数 ${{{N}}_i}^{t + 1}$ ${{{\delta}} _i}^{t + 1}$

7) 根据 ${{{w}}_i} = c \times 2\sqrt 2 {{{\sigma}} ^2}/{{{\sigma}} _i}$ 求解第t+1次权重 ${W_i}^{t + 1}$

8) 利用广义软阈值算法求解lp范数中的稀疏系数upi

9) 利用代理算法来求解误差约束中的稀疏系数uri

10) 通过求解upiuri的均值来计算第t+1次稀疏系数 ${{{M}}_i}^{t + 1}$

11) 根据 ${{{X}}_i}^{t + 1} = {{{D}}_i}^{t + 1}{{{M}}_i}^{t + 1}$ 求得估计图像块 ${{{X}}_i}^{t + 1}$

12) 根据加权平均合并图像块 ${{{X}}_i}^{t + 1}$ 形成去噪图像 ${\hat {{X}}^{t + 1}}$

13) 结束。

3 实验分析

 Download: 图 1 去噪实验的所有测试图像 Fig. 1 All test images for the denoising experiments

 Download: 图 2 House测试图像的去噪结果 Fig. 2 Denoising results of House test images
4 结束语

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