发布时间: 2018-09-16 摘要点击次数: 全文下载次数: DOI: 10.11834/jig.180011 2018 | Volume 23 | Number 9 图像处理和编码

 收稿日期: 2018-01-08; 修回日期: 2018-03-26 基金项目: 国家自然科学基金项目（51774281）；江苏省六大人才高峰资助项目（2015-ZBZZ-009）；徐州市重点研发项目（KC16GZ013） 第一作者简介: 程德强, 1979年生, 男, 教授, 主要研究方向为图像处理与机器视觉。E-mail:chengdq@cumt.edu.cn;邵丽蓉, 女, 硕士研究生, 主要研究方向为图像质量评价。E-mail:18356285372@163.com;陈亮亮, 男, 硕士研究生, 主要研究方向为图像识别、图像超分辨率重建。E-mail:15062197925@163.com. 中图法分类号: TP391 文献标识码: A 文章编号: 1006-8961(2018)09-1285-08

# 关键词

Super resolution reconstruction algorithm based on kernel sparse representation and atomic correlation
Cheng Deqiang, Liu Weilong, Shao Lirong, Chen Liangliang
School of Information and Control Engineering, China University of Mining and Technology, Xuzhou 221008, China
Supported by: National Natural Science Foundation of China(51774281)

# Abstract

Objective To overcome the low efficiency of dictionary atom screening and the poor effect of image reconstruction results in some super-resolution methods based on sparse representation, which are mostly unconsidered in atom screening, this paper proposes a super-resolution reconstruction algorithm. This algorithm is based on a combination of kernel method and dictionary atomic correlation, which fully uses the correlation between the dictionary and image, and selects the atoms, which significantly contributes to the reconstruction results and improves the efficiency and effect of the reconstruction. Method First, a set of low-and high-resolution samples are obtained by pre-processing applied on the high-resolution images. Low-and high-resolution dictionaries are learned by using a dictionary learning algorithm. Second, the dictionary atom is uncorrelated to improve the ability of the dictionary atom to express. Third, by using the low-resolution dictionary, the kernel method and dictionary atom screening method are used for sparse representation, to set thresholds to screen for highly correlated atoms, eliminate low-correlation atoms, and then use the normal atoms for normalized processing. The resulting high-and low-resolution dictionary atoms are incoherent, thereby eliminating the similarity between dictionary atoms, enhancing the expressive power of dictionary atoms, and helping to select the next dictionary atoms. In the process of solving the representation coefficient, selecting the appropriate atoms from the low-resolution dictionary to the support set, which is the largest part of the computation, is necessary. When updating the support set, the dictionary of low-resolution images is trained from other images, which leads to the large contribution of some atoms to the samples. The atoms with low correlation often do not contribute during the iteration process, but each iteration has considerable computation costs. At the same time, for the image blocks that need to be restored, a number of highly correlated atomic pairs have a major contribution to reconstruction. To reduce the computational complexity and improve the reconstruction effect, this paper improves the traditional method by using the correlation of the residual and atom to conduct efficient dictionary selection. Finally, the sparse representation problem is solved to obtain sparse coefficients, and the super-resolution image is recovered by using these coefficients. High-resolution image blocks are obtained by using high-resolution dictionary and coefficient representation coefficients, and then, high-resolution images are synthesized by utilizing image blocks. Result The performance of algorithm reconstruction is measured by PSNR, structural similarity, and time and compared with Yang, MSDSC, and SDCKR algorithms. In the experiment, the following test chart is analyzed in detail, and the ImageNet standard image database is trained to obtain additional detailed experimental results. The experimental results show that, compared with the contrast method, the image reconstruction time is increased by 22.2%, the image structure similarity is increased by 9.06%, and the PSNR is increased by 2.30 dB. The original method based on dictionary learning for dictionary selection has a certain blindness. The atom and reconstruction image correlation degree is low, and the reconstruction effect is poor. This method can reduce the dictionary sparse representation of time consumption and improve the accuracy of sparse representation. In the super-resolution reconstruction of the classical image reconstruction algorithm, the effect is not ideal and the reconstruction time is too long. The main reason is that the dictionary selection efficiency is low, aiming at the abovementioned problem. For the improvement of dictionary learning algorithm in solving the sparse coefficient method in the process of nuclear innovation and the introduction of machine learning and new atom selection method, this paper presents a test with the commissioning of a large number of practical engineering images. The experimental results show that this method can improve the reconstruction effect and reduce the time required for reconstruction. Conclusion Compared with the same algorithm of dictionary learning, the reconstruction time of this algorithm is also less. Experiments have proven that in this method, the reconstruction time of image sparse representation process is significantly reduced. The reconstruction effect is also improved, with good reconstruction efficiency and effectiveness under the condition of few training samples, which is suitable for practical use.

# Key words

sparse representation; super-resolution reconstruction; kernel method; atomic correlation; unrelated processing

# 1.1 图像退化降质模型

 ${\mathit{\boldsymbol{Y}}_k} = {\mathit{\boldsymbol{C}}_k}{\mathit{\boldsymbol{B}}_k}{\mathit{\boldsymbol{H}}_k}{\mathit{\boldsymbol{K}}_k} + {\mathit{\boldsymbol{N}}_k}$ (1)

# 1.2 图像稀疏表示模型

 $\begin{array}{*{20}{c}} {\min {{\left\| \mathit{\boldsymbol{\alpha }} \right\|}_0},}&{{\rm{s}}.\;{\rm{t}}.\;\mathit{\boldsymbol{X}} = \mathit{\boldsymbol{Da}}} \end{array}$ (2)

# 2.1 稀疏字典对训练

 $\begin{array}{*{20}{c}} {\left\{ {\mathit{\boldsymbol{D}},\mathit{\boldsymbol{A}}} \right\} = \mathop {\arg \min }\limits_{\mathit{\boldsymbol{D}},\mathit{\boldsymbol{A}}} \left\| {\mathit{\boldsymbol{X}} - \mathit{\boldsymbol{DA}}} \right\|_{\rm{F}}^2}\\ {{\rm{s}}.\;{\rm{t}}.\;\;\forall i = 1,2, \cdots ,k,{{\left\| {{\mathit{\boldsymbol{\alpha }}^i}} \right\|}_0} \le T} \end{array}$ (4)

 $\left\{ {{\mathit{\boldsymbol{D}}_{\rm{H}}},\mathit{\boldsymbol{A}}} \right\} = \mathop {\arg \min }\limits_{{\mathit{\boldsymbol{D}}_{\rm{H}}},\mathit{\boldsymbol{A}}} \left\| {\mathit{\boldsymbol{X}} - {\mathit{\boldsymbol{D}}_{\rm{H}}}\mathit{\boldsymbol{A}}} \right\|_{\rm{F}}^2$ (5)

 $\left\{ {{\mathit{\boldsymbol{D}}_{\rm{L}}},\mathit{\boldsymbol{A}}} \right\} = \mathop {\arg \min }\limits_{{\mathit{\boldsymbol{D}}_{\rm{L}}},\mathit{\boldsymbol{A}}} \left\| {\mathit{\boldsymbol{X}} - {\mathit{\boldsymbol{D}}_{\rm{L}}}\mathit{\boldsymbol{A}}} \right\|_{\rm{F}}^2$ (6)

 $\frac{1}{M}\left\{ {{\mathit{\boldsymbol{D}}_{\rm{H}}},\mathit{\boldsymbol{A}}} \right\} + \frac{1}{N}\left\{ {{\mathit{\boldsymbol{D}}_{\rm{L}}},\mathit{\boldsymbol{A}}} \right\} = \min \left\| {\mathit{\boldsymbol{Z}} - \mathit{\boldsymbol{DA}}} \right\|_{\rm{F}}^2$ (7)

 $\left\{ \begin{array}{l} \mathit{\boldsymbol{Z}} = {\sqrt M ^{ - 1}}\mathit{\boldsymbol{X/}}{\sqrt N ^{ - 1}}\mathit{\boldsymbol{Y}}\\ \mathit{\boldsymbol{D}} = {\sqrt M ^{ - 1}}{\mathit{\boldsymbol{D}}_{\rm{H}}}\mathit{\boldsymbol{/}}{\sqrt N ^{ - 1}}{\mathit{\boldsymbol{D}}_{\rm{L}}} \end{array} \right.$ (8)

$M$$N分别为样本中图像块向量形式的维度。可以用\mathit{\boldsymbol{Z}}作为K-SVD[16]算法的输入，获得字典\mathit{\boldsymbol{D}}，再用式(8)求得高、低分辨率字典。 为了提高字典的原子间的非相干性，本文引入基于梯度的方法[17]，在字典训练的过程中对字典原子处理，设{\mathit{\boldsymbol{D}}_i}是第i次迭代的字典，即  {\mathit{\boldsymbol{D}}_i} = \mathop {\arg \min }\limits_{{\mathit{\boldsymbol{D}}_i}} \left\| {\mathit{\boldsymbol{D}}_i^{\rm{T}}{\mathit{\boldsymbol{D}}_i} - \mathit{\boldsymbol{I}}} \right\|_{\rm{F}}^2 (9) 式中, \mathit{\boldsymbol{I}}是一个单位矩阵，梯度\mathit{\boldsymbol{E}}{\bf{:}} = \left\| {\mathit{\boldsymbol{D}}_i^{\rm{T}}{\mathit{\boldsymbol{D}}_i}-\mathit{\boldsymbol{I}}} \right\|_{\rm{F}}^2，插入到\mathit{\boldsymbol{D}} \leftarrow \mathit{\boldsymbol{D}}-\eta \nabla \mathit{\boldsymbol{E}}  \mathit{\boldsymbol{D}}_i^{{\rm{new}}} = \mathit{\boldsymbol{D}}_i^{{\rm{old}}} - \eta \mathit{\boldsymbol{D}}_i^{{\rm{old}}}\left( {\mathit{\boldsymbol{D}}_i^{{\rm{Told}}}\mathit{\boldsymbol{D}}_i^{{\rm{old}}} - \mathit{\boldsymbol{I}}} \right) (10) 由此得到的高、低分辨率字典原子具有不相干性，消除了字典原子间的相似性，增强了字典原子的表达能力，有利于接下来字典原子的选取。 # 2.2 稀疏表示系数求解 在求解表示系数的过程中，需从低分辨率字典中选取合适的原子加入到支撑集，这是计算量最大的部分。在更新支撑集的时候，由于低分辨率图像的字典是从其他图像训练得到的，容易使得某些原子对样本的表示贡献度非常低。当相关度为零时，该原子甚至会导致字典无法更新，相关性极低的原子在迭代过程中往往无贡献但每次迭代耗费计算量。同时对于需要恢复的图像块，往往有若干相关性极高的原子对重建有主要的贡献。为了减小计算复杂度，同时改善重建效果，本文改进了传统方法，利用残差与原子的相关性进行高效的字典选取[18] 本文引入机器学习中的核方法，将所需处理的数据通过非线性映射\mathit{\Phi }映射到高维(甚至无穷维)空间\mathfrak{R}中，\mathfrak{R}是一个希尔伯特空间，则非线性映射\mathit{\Phi }对应的核函数表示为  \begin{array}{*{20}{c}} {\left( {\mathit{\Phi }\left( \mathit{\boldsymbol{x}} \right),\mathit{\Phi }\left( {\mathit{\boldsymbol{x'}}} \right)} \right) = }\\ {\mathit{\Phi }{{\left( \mathit{\boldsymbol{x}} \right)}^{\rm{T}}}\mathit{\Phi }\left( {\mathit{\boldsymbol{x'}}} \right) = k\left( {\mathit{\boldsymbol{x}},\mathit{\boldsymbol{x'}}} \right)} \end{array} (11) 式中，k为核函数，其求解不需知道具体的映射\mathit{\Phi }，并且与数据的维度无关，只需选取合适的核函数k，可以减小向量内积的计算。 同时引入相似度公式，设，\mathit{\boldsymbol{X}} = \{ {\mathit{\boldsymbol{X}}_1}, {\mathit{\boldsymbol{X}}_2}, \cdots, {\mathit{\boldsymbol{X}}_n}\}$$\mathit{\boldsymbol{Y}} = \{ {\mathit{\boldsymbol{Y}}_1}, {\mathit{\boldsymbol{Y}}_2}, \cdots, {\mathit{\boldsymbol{Y}}_n}\}$，其相似度为

 $\rho = \mathit{\boldsymbol{XY/}}\sqrt {{\mathit{\boldsymbol{X}}^2}} \sqrt {{\mathit{\boldsymbol{Y}}^2}}$ (12)

 $\begin{array}{*{20}{c}} {{\rho _i} = \mathit{\Phi }\left( \mathit{\boldsymbol{r}} \right)\mathit{\Phi }\left( {{\mathit{\boldsymbol{d}}_{{\rm{Li}}}}} \right)/\sqrt {{\mathit{\Phi }^2}\left( \mathit{\boldsymbol{r}} \right)} \sqrt {{\mathit{\Phi }^2}\left( {{\mathit{\boldsymbol{d}}_{{\rm{Li}}}}} \right)} = }\\ {k\left( {\mathit{\boldsymbol{r}},{\mathit{\boldsymbol{d}}_{{\rm{Li}}}}} \right)/\sqrt {k\left( {\mathit{\boldsymbol{r}},\mathit{\boldsymbol{r}}} \right)} \sqrt {k\left( {{\mathit{\boldsymbol{d}}_{{\rm{Li}}}},{\mathit{\boldsymbol{d}}_{{\rm{Li}}}}} \right)} } \end{array}$ (15)

1) 利用式(14)求出残差$\mathit{\boldsymbol{r}}$，用式(15)求解相关系数${\rho _i}$

2) if ${\rho _i}$>${\rho _a}$

then do正则化处理相应原子加入支撑集

else if ${\rho _a}$${\rho _i}$${\rho _b}$

then do $\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}$$\mathit{\boldsymbol{ \boldsymbol{\varLambda} }}$$\mathop {{\rm{arg}}\;{\rm{max}}}\limits_i \;{\rho _i}$

else do除去对应原子

end if

3) 再利用式(13)更新稀疏表示系数${\mathit{\boldsymbol{\alpha }}_k}$

4) 重复以上步骤，直至收敛。

# 2.3 重建

 $\mathit{\boldsymbol{X}} = {\mathit{\boldsymbol{Y}}_{\rm{L}}} + {\left( {\sum\limits_k {\mathit{\boldsymbol{R}}_k^{\rm{T}}{\mathit{\boldsymbol{R}}_k}} } \right)^{ - 1}}\left( {\sum\limits_k {\mathit{\boldsymbol{R}}_k^{\rm{T}}{\mathit{\boldsymbol{R}}_k}} } \right)$ (16)

# 3 实验分析

Table 1 Comparison of test images reconstruction on PSNR, SSIM and time

 图像 方法 PSNR/dB SSIM 时间/s Lena Yang 33.78 0.901 0 289.0 MSDSC 34.71 0.954 6 202.5 SDCKR 32.32 0.865 1 238.2 本文 36.65 0.989 3 129.2 Bike Yang 34.45 0.867 0 198.8 MSDSC 35.51 0.932 3 280.1 SDCKR 29.12 0.875 1 170.2 本文 36.12 0.977 4 231.1 Girl Yang 34.77 0.816 5 265.9 MSDSC 35.54 0.860 4 370.1 SDCKR 34.81 0.894 9 347.9 本文 36.81 0.954 9 267.9 Grope Yang 26.87 0.855 4 399.3 MSDSC 27.14 0.897 1 339.5 SDCKR 29.89 0.822 8 430.0 本文 28.89 0.912 8 340.0

Table 2 Comparison of Set 5 images reconstruction on average PSNR, SSIM and time

 方法 PSNR/dB SSIM 时间/s Yang 25.22 0.765 5 321.6 MSDSC 26.56 0.867 0 265.2 SDCKR 25.67 0.849 2 310.5 本文 28.12 0.902 2 232.6

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