﻿ 基于分数阶偏微分方程的图像放大模型
 文章快速检索 高级检索
 浙江大学学报(理学版)  2016, Vol. 43 Issue (5): 550-553  DOI:10.3785/j.issn.1008-9497.2016.05.010 0

### 引用本文 [复制中英文]

[复制中文]
GAO Ran , GU Cong , LI Shenghong . 2016. Image zooming model based on fractional-order partial differential equation[J]. Journal of Zhejiang University(Science Edition) , 43(5): 550-553. DOI: 10.3785/j.issn.1008-9497.2016.05.010.
[复制英文]

### 文章历史

1. 中原工学院 理学院, 河南 郑州 450007
2. 浙江大学 数学系, 浙江 杭州 310027

Image zooming model based on fractional-order partial differential equation
GAO Ran1 , GU Cong1 , LI Shenghong2
Abstract: A new image zooming model based on the fractional-order partial differential equation is proposed, which adopts the idea of total variation. Simulation results show that the new model is capable of preserving the characteristics of image edge, and it can retain more texture details than the integer order partial differential equation model. The model is therefore effective and practical for image zooming.
Key words: fractional-order    partial differential equation(PDE)    variation    image zooming

 \begin{align} & E\left( u \right)={{\int }_{\Omega }}{{(u_{xx}^{2}+u_{xy}^{2}+u_{yx}^{2}+u_{yy}^{2})}^{12}}dxdy+ \\ & \frac{\lambda }{2}{{\int }_{\Omega }}{{(u-{{u}_{0}})}^{2}}\cdot {{\chi }_{{{\Omega }_{1}}}}(u-{{u}_{0}})dxdy, \\ \end{align} (1)

1 相关理论 1.1 分数阶导数

 ${{D}^{p}}f\left( x \right)=\underset{h\to 0}{\mathop{\lim }}\,\frac{\sum\limits_{k\ge 0}^{n}{{}}{{\left( -1 \right)}^{k}}\left( \begin{matrix} p \\ k \\ \end{matrix} \right)f\left( x-kh \right)}{{{h}^{p}}},$ (2)

 $\left( \begin{matrix} p \\ k \\ \end{matrix} \right)=\frac{\Gamma \left( p+1 \right)}{\Gamma \left( k+1 \right)\Gamma \left( p-k+1 \right)}.$

 ${{D}^{p}}f\left( x \right)\approx \sum\limits_{k=0}^{K-1}{{}}{{\left( -1 \right)}^{k}}\left( \begin{matrix} p \\ k \\ \end{matrix} \right)f\left( x-k \right).$ (3)
1.2 分数阶导数的差分格式

 $\frac{{{\partial }^{p}}u}{\partial {{x}^{p}}}=\underset{h\to 0}{\mathop{\lim }}\,\frac{\sum\limits_{k\ge 0}^{n}{{}}{{\left( -1 \right)}^{k}}\left( \begin{matrix} p \\ k \\ \end{matrix} \right)u\left( x-kh,y \right)}{{{h}^{p}}},$ (4)
 $\frac{{{\partial }^{p}}u}{\partial {{y}^{p}}}=\underset{h\to 0}{\mathop{\lim }}\,\frac{\sum\limits_{k\ge 0}^{n}{{}}{{\left( -1 \right)}^{k}}\left( \begin{matrix} p \\ k \\ \end{matrix} \right)u\left( x,y-kh \right)}{{{h}^{p}}},$ (5)
 \begin{align} & \frac{{{\partial }^{p}}u\left( x,y \right)}{\partial {{x}^{p}}}\approx u\left( x,y \right)+\left( -p \right)u\left( x-1,y \right)+ \\ & \frac{-p\left( -p+1 \right)}{2}u\left( x-2,y \right)+\cdots + \\ & \frac{\Gamma \left( -p+1 \right)}{n!\Gamma \left( -p+n+1 \right)}u\left( x-n,y \right), \\ \end{align} (6)
 \begin{align} & \frac{{{\partial }^{p}}u\left( x,y \right)}{\partial {{y}^{p}}}\approx u\left( x,y \right)+\left( -p \right)u\left( x,y-1 \right)+ \\ & \frac{-p\left( -p+1 \right)}{2}u\left( x,y-2 \right)+\cdots + \\ & \frac{\Gamma \left( -p+1 \right)}{n!\Gamma \left( -p+n+1 \right)}u\left( x,y-n \right). \\ \end{align} (7)
2 基于分数阶的图像放大算法 2.1 模型的建立

 \begin{align} & E\left( u \right)-{{\int }_{\Omega }}\left| {{D}^{p}}u \right|dxdy+\frac{\lambda }{2}{{\int }_{\Omega }}{{\left( u-{{u}_{0}} \right)}^{2}}\times \\ & {{\chi }_{{{\Omega }_{1}}}}\left( u-{{u}_{0}} \right)dxdy, \\ \end{align} (8)

 $-{{D}^{p}}\left( \frac{{{D}^{p}}u}{|{{D}^{p}}u|} \right)+\lambda (u-{{u}_{0}}){{\chi }_{{{\Omega }_{1}}}}(u-{{u}_{0}})=0,$ (9)

 \left\{ \begin{align} & \frac{\partial u}{\partial t}={{D}^{p}}\left( \frac{{{D}^{p}}u}{|{{D}^{p}}u|} \right)\lambda (u-{{u}_{0}}){{\chi }_{{{\Omega }_{1}}}}(u-{{u}_{0}}), \\ & u\left( 0,x,y \right)={{u}_{0}}\left( x,y \right) \\ & {{\left. \frac{\partial u}{\partial n} \right|}_{\partial \Omega }}=0, \\ \end{align} \right. (10)

2.2 模型的算法

3 数值离散及实验结果分析

 \begin{align} & {{u}^{n+1}}={{u}^{n}}+\Delta t\left[ D_{x}^{p}\left( \frac{D_{x}^{p}{{u}^{n}}}{|D_{x}^{p}{{u}^{n}}|} \right)+D_{y}^{p}\frac{D_{y}^{p}{{u}^{n}}}{|{{D}^{p}}_{y}{{u}^{n}}|} \right]+ \\ & \Delta t\lambda ({{u}^{n}}-{{u}_{0}}){{\chi }_{{{\Omega }_{1}}}}({{u}^{n}}-{{u}_{0}}), \\ \end{align} (11)

 \begin{align} & D_{x}^{p}u_{ij}^{n}=u_{ij}^{n}+\left( -p \right)u_{i-1,j}^{n}+\frac{-p\left( -p+1 \right)}{2}u_{i-2,j}^{n}, \\ & D_{x}^{p}u_{ij}^{n}=u_{ij}^{n}+\left( -p \right)u_{i,j-1}^{n}+\frac{-p\left( -p+1 \right)}{2}u_{i,j-2}^{n}, \\ & |{{D}^{p}}(u_{ij}^{n})|=\sqrt{{{(D_{x}^{p}(u_{ij}^{n}))}^{2}}+{{({{D}^{p}}_{y}(u_{ij}^{n}))}^{2}}+\varepsilon }. \\ \end{align}

 图 1 图像放大效果 Fig. 1 Results of image zooming

 图 2 图像放大前后灰度值曲线比较 Fig. 2 Comparison of different zooming methods

4 结 论

 [1] 付树军, 阮秋琦, 王文洽. 偏微分方程(PDEs)模型在图像处理中的若干应用[J]. 计算机工程与应用, 2005 (2) : 33–35. FU Shujun, YUAN Qiuqi, WANG Wenqia. Applications of PDEs model in image analysis and processing: A survey[J]. Computer Engineering and Applications, 2005 (2) : 33–35. (0) [2] 张亶, 陈刚. 基于偏微分方程的图像处理[M]. 北京: 高等教育出版社, 2004 . ZHANG Dan, CHEN Gang. Image Processing Based on PDEs[M]. Beijing: Higher Education Press, 2004 . (0) [3] CHEN Y M, VEMURI B C, WANG L. Image denoising and segmentation via nonlinear diffusion[J]. Computers and Mathematics with Applications, 2000, 39 (5/6) : 131–149. (0) [4] WEICKERT J. Coherence-enhancing diffusion filtering[J]. International Journal of Computer Vision, 1999, 31 (2/3) : 111–127. DOI:10.1023/A:1008009714131 (0) [5] 阮秋琦, 忤冀颖. 数字图像处理中的偏微分方程方法[J]. 信号处理, 2012, 28 (3) : 301–314. YUAN Qiuqi, WU Jiying. Partial differential equation method on digital image processing[J]. Signal Processing, 2012, 28 (3) : 301–314. (0) [6] KIM H, CHA Y, KIM S. Curvature interpolation method for image zooming[J]. IEEE Transactions on Image Processing, 2011, 20 (7) : 1895–1903. DOI:10.1109/TIP.2011.2107523 (0) [7] 朱宁, 吴静, 王忠谦. 图像放大的偏微分方程方法[J]. 计算机辅助设计与图形学学报, 2005, 17 (19) : 1941–1945. ZHU Ning, WU Jing, WANG Zhongqian. Image zooming based on partial differential equations[J]. Journal of Computer-Aided Design & Computer Graphics, 2005, 17 (19) : 1941–1945. (0) [8] CRANDALL M G, ISHII H, LIONS P L. User's guide to viscosity solutions of second order partial linear differential equations[J]. Bull Amer Math Soc, 1992, 27 : 1–67. DOI:10.1090/S0273-0979-1992-00266-5 (0) [9] RUDIN L, OSHER S. Feature-oriented image enhancement using shock filters[J]. SIAM J Numerical Analysis, 1990, 27 : 919–940. DOI:10.1137/0727053 (0) [10] GAO Ran, SONG Jinping, TAI Xuecheng. Image zooming algorithm based on partial differential equations technique[J]. International Journal of Numerical Analysis and Modeling, 2009 (6) : 5–18. (0) [11] 蒲亦非, 王卫星. 数字图像的分数阶微分掩模及其数值运算规则[J]. 自动化学报, 2007, 33 (11) : 1128–1135. PU Yifei, WANG Weixing. Fractional differential masks of digital image and their numerical implementation algorithms[J]. Acta Automatica Sinica, 2007, 33 (11) : 1128–1135. (0) [12] 杨柱中, 周激流, 黄梅, 等. 基于分数阶微分的边缘检测[J]. 四川大学学报:工程科学版, 2008, 40 (1) : 152–156. YANG Zhuzhong, ZHOU Jiliu, HUANG Mei, et al. Edge detection based on fractional differential[J]. Journal of Sichuan University:Engineering Science Edition, 2008, 40 (1) : 152–156. (0) [13] 艾必刚, 罗以宁, 蒋涛, 等. 分数阶微分梯度算子在图像增强中的应用[J]. 四川大学学报:自然科学版, 2009, 46 (2) : 343–347. AI Bigang, LUO Yining, JIANG Tao, et al. Applications of fractional-order differential gradient Operator in Image Enhancement[J]. Journal of Sichuan University: Natural Science Edition, 2009, 46 (2) : 343–347. (0) [14] PODLUBNY I. Fractional Differential Equations[M]. New York: Academic Press, 1998 . (0)