﻿ 自适应分数阶算子的图像增强
 光电工程  2019, Vol. 46 Issue (9): 180517      DOI: 10.12086/oee.2019.180517

1. 石家庄铁道大学电气与电子工程学院，河北 石家庄 050043;
2. 中国人民解放军32142部队，河北 石家庄 050000

Image enhancement of adaptive fractional operator
Li Shuai1, Wang Weiming1, Liu Xianhong2, Yan Deli1
1. School of Electrical and Electronics Engineering, Shijiazhuang Tiedao University, Shijiazhuang, Hebei 050043, China;
2. Unit 32142 of PLA, Shijiazhuang, Hebei 050000, China
Abstract: In order to highlight the texture details of the image while preserving the smooth region and saving the time to determine the fractional differential order, an improved adaptive fractional differential operator is proposed. Firstly, the classical Tiansi template is decomposed into four different directions, which are respectively convolved with the pixels to be processed to achieve the effect of enhancing the texture details of the image. Secondly, the current situation of the optimal differential order is determined by the experiment for the Tiansi operator. The local feature information of the image constructs a fractional order model with an adaptive ability, which can obtain more detailed information than the original image. The experimental results of multiple sets of different scene images show that the constructed adaptive fractional differential operators effectively enhance the texture details of the image. The subjective visual effects and objective evaluation indexes of the adaptive fractional differential operators are better than the original images. The average gradient, information entropy and contrast in the objective evaluation index are increased by 190.3%, 8.1%, and 18.3%, respectively. The average gradient and contrast are 45.0% and 9.6% higher than that of the Tiansi operator.
Keywords: image enhancement    fractional differential    Tiansi operator    adaptive fractional

1 引言

2 分数阶微分理论

 $_a^GD{}_t^vf(t) = \mathop {\lim }\limits_{h \to 0} \frac{1}{{{h^v}}}\sum\limits_{m = 0}^{\left[ {\frac{{t - a}}{h}} \right]} {{{( - 1)}^m}\frac{{\Gamma (v + 1) \cdot f(t - mh)}}{{m!\Gamma (v - m + 1)}}} ,$ (1)

 $\frac{{{{\rm{d}}^v}(t)}}{{{\rm{d}}{t^v}}} \approx f(t) + ( - v)f(t - 1) + \frac{{( - v)( - v + 1)}}{2}f(t - 2)\\ + \cdots + \frac{{\Gamma ( - v + 1)}}{{m!\Gamma ( - v + m + 1)}}f(t - m)。$ (2)

 $\left\{ \begin{gathered} {a_0} = 1 \\ {a_1} = - v \\ {a_2} = \frac{{( - v)( - v + 1)}}{2} \\ {a_3} = \frac{{( - v)( - v + 1)( - v + 2)}}{6} \\ \vdots \\ {a_n} = \frac{{\Gamma ( - v + 1)}}{{m!\Gamma ( - v + m + 1)}} \\ \end{gathered} \right.。$ (3)

 图 1 5×5的Tiansi模板 Fig. 1 5×5 Tiansi template

 $T = 8{a_0} + 8{a_1} + 8{a_2} = 8 - 12v + 4{v^2}。$ (4)

3 自适应分数阶微分算子的构造 3.1 改进的Tiansi算子

 图 2 按坐标轴分解为四个方向模板 Fig. 2 Decomposition into four orientation templates by coordinate axes

 $s = \frac{{{M_1} + {M_2} + {M_3} + {M_4}}}{{8 - 12v + 4{v^2}}} = \frac{{\frac{{{M_1} + {M_2} + {M_3} + {M_4}}}{4}}}{{\frac{{8 - 12v + 4{v^2}}}{4}}}。$ (5)

 $\frac{{{M_1} + {M_2} + {M_3} + {M_4}}}{4} = \max ({M_1}, {M_2}, {M_3}, {M_4}),$

 $s = \frac{{\max ({M_{\rm{1}}}, {M_{\rm{2}}}, {M_{\rm{3}}}, {M_{\rm{4}}})}}{{2 - 3v + {v^2}}},$ (6)

 图 3 Tiansi算子与改进算子处理后的对比图像 Fig. 3 Contrast images processed by Tiansi operator and improved operator

3.2 结合局部特性构造自适应分数阶阶次的微分算子

 $f\left( {G, L, C} \right) = \frac{G}{{G + L + C}}G + \frac{L}{{G + L + C}}L + \frac{C}{{G + L + C}}C,$ (7)

f=0时，待处理的像素点一定位于平滑区域，此时阶次最小，取v=0.01；

f=1时，待处理的像素点一定位于边缘区域，此时阶次最大，取v=0.99：

 $\left\{ \begin{gathered} 0.01 = a \times {0^2} + b \\ 0.99 = a \times {1^2} + b \\ \end{gathered} \right.,$

 $v = 0.98 \times {f^2} + 0.01,$ (8)

 图 4 自适应分数阶微分算子的具体处理步骤 Fig. 4 Concrete processing steps of adaptive fractional differential operator

4 实验与分析

4.1 对Lena图像进行质量评价 4.1.1 主观评价

 图 5 经过各种算法处理后的Lena图像 Fig. 5 Lena images processed by various algorithms

4.1.2 客观评价

 原图像 Laplacian Tiansi算子 文献[7]算子 改进算子 自适应算子 信息熵 5.4601 6.8303 7.6604 7.1410 7.5440 7.5471 平均梯度 6.0157 18.7322 12.2129 21.7388 20.1014 14.0116 图像对比度 43.8420 50.0691 47.1296 42.5253 51.9007 49.3702 SSIM \ 0.5260 0.7922 0.5569 0.5454 0.6703

4.2 对两种不同场景图像进行客观评价

4.2.1 对雾霾天气图像进行客观评价

 图 6 经过各种算法处理后的雾霾图像1 Fig. 6 Haze image 1 processed by various algorithms

 原图像 Laplacian Tiansi算子 文献[7]算子 改进算子 自适应算子 信息熵 7.3193 7.6280 7.6257 7.3430 7.6484 7.5170 平均梯度 3.6461 10.4439 7.7882 13.7948 12.6791 12.1317 图像对比度 41.6897 47.3064 45.0094 41.0306 50.0650 50.4051 SSIM \ 0.7546 0.8716 0.6219 0.6125 0.7571

 图 7 经过各种算法处理后的雾霾图像2 Fig. 7 Haze image 2 processed by various algorithms

 原图像 Laplacian Tiansi算子 文献[7]算子 改进算子 自适应算子 信息熵 6.4407 6.5971 6.5237 6.4629 6.6755 6.6928 平均梯度 1.4535 4.7734 3.2798 6.9128 5.9013 5.4956 图像对比度 19.2248 19.5839 19.5757 16.0659 19.9807 19.4519 SSIM \ 0.8326 0.9244 0.7104 0.6749 0.8229
4.2.2 对夜间图像进行客观评价

 图 8 经过各种算法处理后的夜间图像1 Fig. 8 Night image 1 processed by various algorithms

 原图像 Laplacian Tiansi算子 文献[7]算子 改进算子 自适应算子 信息熵 7.0607 7.1267 7.0884 6.8329 7.1588 7.1336 平均梯度 1.6423 3.8517 2.9670 5.6480 4.8916 4.7185 图像对比度 37.0751 38.0947 37.7044 29.7156 38.8709 38.8115 SSIM \ 0.8822 0.9460 0.7510 0.7256 0.8498

 图 9 经过各种算法处理后的夜间图像2 Fig. 9 Night image 2 processed by various algorithms

 原图像 Laplacian Tiansi算子 文献[7]算子 改进算子 自适应算子 信息熵 5.4228 5.5114 5.4795 5.2203 5.5392 5.4268 平均梯度 1.7520 3.5697 3.0279 4.5573 4.2556 4.1458 图像对比度 9.8958 11.4134 10.8980 11.1497 12.3368 13.2254 SSIM \ 0.9321 0.9654 0.8405 0.8282 0.9177

5 结论

 [1] Svoboda T, Kybic J, Hlavac V. Image Processing, Analysis & and Machine Vision: A MATLAB Companion[M]. CL-Engineering, 2007: 712-717. [2] Zhou S B, Wang L P, Yin X H. Applications of fractional partial differential equations in image processing[J]. Journal of Computer Applications, 2017, 37(2): 546-552. 周尚波, 王李平, 尹学辉. 分数阶偏微分方程在图像处理中的应用[J]. 计算机应用, 2017, 37(2): 546-552 [Crossref] [3] Gao R, Gu C, Li X. Image zooming model based on fractional-order partial differential equation[J]. Journal of Discrete Mathematical Sciences and Cryptography, 2017, 20(1): 55-63. [Crossref] [4] Huang G, Chen Q L, Xu L, et al. Realization of adaptive image enhancement with variable fractional order differentials[J]. Journal of Shenyang University of Technology, 2012, 34(4): 446-454. 黄果, 陈庆利, 许黎, 等. 可变阶次分数阶微分实现图像自适应增强[J]. 沈阳工业大学学报, 2012, 34(4): 446-454 [Crossref] [5] Pu Y F. Application of fractional differential approach to digital image processing[J]. Journal of Sichuan University (Engineering Science Edition), 2007, 39(3): 124-132. 蒲亦非. 将分数阶微分演算引入数字图像处理[J]. 四川大学学报(工程科学版), 2007, 39(3): 124-132 [Crossref] [6] Pu Y F, Siarry P, Chatterjee A, et al. A fractional-order variational framework for retinex: fractional-order partial differential equation-based formulation for multi-scale nonlocal contrast enhancement with texture preserving[J]. IEEE Transactions on Image Processing, 2018, 27(3): 1214-1229. [Crossref] [7] Zhang S Y, Xie Y Y, Zhang X, et al. Enhancement of fuzzy traffic video images based on fractional differential[J]. Optics and Precision Engineering, 2014, 22(3): 779-786. 张绍阳, 解源源, 张鑫, 等. 基于分数阶微分的模糊交通视频图像增强[J]. 光学精密工程, 2014, 22(3): 779-786 [Crossref] [8] Wang C L, Lan L B, Zhou S B. Adaptive fractional differential and its application to image texture enhancement[J]. Journal of Chongqing University, 2011, 34(2): 32-37. 汪成亮, 兰利彬, 周尚波. 自适应分数阶微分在图像纹理增强中的应用[J]. 重庆大学学报, 2011, 34(2): 32-37 [Crossref] [9] Yang Z Z, Zhou J L, Yu X Y, et al. Image enhancement based on fractional differentials[J]. Journal of Computer-Aided Design & Computer Graphics, 2008, 20(3): 343-348. 杨柱中, 周激流, 晏祥玉, 等. 基于分数阶微分的图像增强[J]. 计算机辅助设计与图形学学报, 2008, 20(3): 343-348 [Crossref] [10] Huang G, Xu L, Pu Y F. Summary of research on image processing using fractional calculus[J]. Journal of Computer Applications, 2012, 29(2): 414-420, 426. 黄果, 许黎, 蒲亦非. 分数阶微积分在图像处理中的研究综述[J]. 计算机应用研究, 2012, 29(2): 414-420, 426 [Crossref] [11] Ma Q T, Dong F F, Kong D X. A fractional differential fidelity-based PDE model for image denoising[J]. Machine Vision and Applications, 2017, 28(5-6): 635-647. [Crossref] [12] Pu Y F, Wang W X, Zhou J L, et al. Fractional differential approach to detecting textural features of digital image and its fractional differential filter implementation[J]. Science in China Series F: Information Sciences, 2008, 51(9): 1319-1339. [Crossref] [13] Yu P, Hao C C. Foggy image enhancement by combined fractional differential and multi-scale retinex[J]. Advances in Laser and Optoelectronics, 2018, 55(1): 272-277. 余萍, 郝成成. 基于分数阶微分和多尺度Retinex联合的雾霭图像增强算法[J]. 激光与光电子学进展, 2018, 55(1): 272-277 [Crossref] [14] Pu Y F. Fractional-order euler-Lagrange equation for fractional-order variational method: A necessary condition for fractional-order fixed boundary optimization problems in signal processing and image processing[J]. IEEE Access, 2016, 4: 10110-10135. [Crossref] [15] Wu R F, Xuan S B, Jing Q. Fractional differential image enhancement algorithm based on local feature[J]. Computer Engineering and Applications, 2014, 50(3): 160-164. 吴瑞芳, 宣士斌, 荆奇. 基于局部特征的分数阶微分图像增强方法[J]. 计算机工程与应用, 2014, 50(3): 160-164 [Crossref] [16] Wang Z, Bovik A C, Sheikh H R, et al. Image quality assessment: from error visibility to structural similarity[J]. IEEE Transactions on Image Processing, 2004, 13(4): 600-612. [Crossref]