基于变指数分数阶全变差和整数阶全变差的图像恢复算法

doi:10.6040/j.issn.1671-9352.0.2019.358

《山东大学学报(理学版)》 ›› 2019, Vol. 54 ›› Issue (11): 115-126.doi: 10.6040/j.issn.1671-9352.0.2019.358

• • 上一篇

基于变指数分数阶全变差和整数阶全变差的图像恢复算法

王迎美¹,王桢东²,李功胜¹

1.山东理工大学数学与统计学院, 山东淄博 255049;2.山东大学数学学院, 山东济南 250100

发布日期:2019-11-06
作者简介:王迎美(1987— ),女,博士,讲师,硕士生导师,研究方向为医学图像处理. E-mail:yingmeiwang@sdut.edu.cn
基金资助:
山东省自然科学基金资助项目(ZR2018BA014);淄博市校城融合发展计划资助项目(2017ZBXC117)

An image restoration algorithm based on variable exponential fractional order total variation and integer order total variation

WANG Ying-mei¹, WANG Zhen-dong², LI Gong-sheng¹

1. School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, Shandong, China;
2. School of Mathematics, Shandong University, Jinan 250100, Shandong, China

Published:2019-11-06

摘要/Abstract

摘要： 结合变指数全变差(totalvariation, TV)和整数阶TV,提出一种变分图像恢复算法。该变分问题的能量泛函主要分为三个部分:变指数p(x)的分数阶TV正则化项、整数阶TV正则化项和数据保真项。该模型中的指数p(x)是与图像的梯度信息有关的函数。在理论上,由于分数阶导数和整数阶导数的结合,使得所提方法不仅能有效地去除图像噪音,保护图像的边界高频信息,还能更好地保留图像的纹理细节等中低频信息,同时可以极大地消除图像处理中产生的阶梯效应和散斑效应。在模型的求解上,利用变分法可以简单地将极小化泛函的优化问题转化为梯度下降流方程。最后,通过模拟数据和真实数据对本文所提方法进行了验证。试验结果表明,该方法可以去除噪声的同时,有效保持边界和纹理细节,并且对噪声是鲁棒的,具有一定的实际应用价值。

关键词: 分数阶全变差, 整数阶全变差, 变指数, 变分方法, 梯度下降流

Abstract: This paper proposes a new variational image restoration algorithm based on variable exponential total variation(TV)and integer order TV. The energy functional of the variational problem is mainly composed of three parts: a fractional order TV regularization term of variable exponent p(x), an integer order TV regularization term and a data fidelity term. The exponential p(x) in this model is a function which related to the gradient information of the image. As the combination of the fractional order derivative and integral derivative, the proposed method can effectively remove the noise of the image, protect the image boundary, and also better retain the image texture details. At the same time, this method can greatly eliminate the staircase effect and the speckle effect. To solve the model, using the variational method, the optimization problem can be simply transformed into a gradient descent flow. Finally, to validate the effectiveness of our proposed method, we give the experiments with simulated data and real data. The experimental results show that this method can effectively remove noise, keep boundary and texture details and is robust to noise. And it has certain practical application value.

Key words: fractional order total variation, integer order total variation, variable exponent, variational method, gradient descent flow

中图分类号:

王迎美,王桢东,李功胜. 基于变指数分数阶全变差和整数阶全变差的图像恢复算法[J]. 《山东大学学报(理学版)》, 2019, 54(11): 115-126.

WANG Ying-mei, WANG Zhen-dong, LI Gong-sheng. An image restoration algorithm based on variable exponential fractional order total variation and integer order total variation[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(11): 115-126.

参考文献

[1] JUNG M, BRESSON X, CHAN T F, et al. Nonlocal Mumford-Shah regularizers for color image restoration[J]. IEEE Transactions on Image Processing, 2011, 20(6):1583-1598.
[2] TIKHONOV A N, ARSENIN V Y. Solutions of ill-posed problems[M]. New York: Wiley, 1977.
[3] RUDIN L I, OSHER S, FATEMI E. Nonlinear total variation based noise removal algorithms[J]. Physica D, 1992, 60(1/2/3/4):259-268.
[4] YOU Yuli, XU Wenyuan, TANNENBAUM A, et al. Behavioral analysis of anisotropic diffusion in image processing[J]. IEEE Transactions on Image Processing, 1996, 5(11):1539-1553.
[5] STRONG D M, CHAN T F. Edge-preserving and scale-dependent properties of total variation regularization[J]. Inverse Problems, 2003, 19(6):165-187.
[6] NIKOLOVA M. Weakly constrained minimization: application to the estimation of images and signals-involving constant regions[J]. Journal of Mathematical Imaging and Vision, 2004, 21(2):155-175.
[7] BLOMGREN P, CHAN T F, MULET P, et al. Total variation image restoration: numerical methods and extensions[C] // International Conference on Image Processing. New York: IEEE, 1997: 384-387.
[8] BLOMGREN P, CHAN T F. Color TV: total variation methods for restoration of vector-valued images[J]. IEEE Transactions on Image Processing, 1998, 7(3):304-309.
[9] CHEN Yunmei, LEVINE S, RAO M. Variable exponent, linear growth functionals in image restoration[J]. SIAM Journal on Applied Mathematics, 2006, 66(4):1383-1406.
[10] LI Fang, LI Zhibin, PI Ling. Variable exponent functionals in image restoration[J]. Applied Mathematics and Computation, 2010, 216(3):870-882.
[11] NINNESS B. Estimation of 1/f noise[J]. IEEE Transactions on Information Theory, 1998, 44(1):32-46.
[12] UNSER M. Splines, a perfect fit for signal and image processing[J]. IEEE Signal Processing Magazine, 1999, 16(6):22-38.
[13] UNSER M, BLU T. Fractional splines and wavelets[J]. SIAM Review, 2000, 42(1):43-67.
[14] CUESTA E, FINAT J. Image processing by means of a linear integro-differential equation[C] // International Conference Visualization, Imaging and Image Processing. Benalmadena, Spain: IASTED, 2003: 438-442.
[15] DUITS R, FELSBERG M, FLORACK L. α scale spaces on a bounded domain[C] // International Conference on Scale Space Methods in Computer Vision. Berlin: Springer, 2003: 494-510.
[16] DIDAS S, BURGETH B, IMIYA A, et al. Regularity and scale-space properties of fractional high order linear filtering[C] // International Conference on Scale Space Methods in Computer Vision. Berlin: Springer, 2005: 13-25.
[17] BAI Jian, FENG Xiangchu. Fractional-order anisotropic diffusion for image denoising[J]. IEEE Transactions on Image Processing, 2007, 16(10):2492-2502.

多维度评价

Viewed

Full text

Abstract

Cited

Shared

Discussed

基于变指数分数阶全变差和整数阶全变差的图像恢复算法

An image restoration algorithm based on variable exponential fractional order total variation and integer order total variation

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 8

多维度评价

本文评价

推荐阅读 0

[1]	方小珍,孙爱文,王敏,束立生. 广义多线性算子在变指数空间上的有界性[J]. 《山东大学学报(理学版)》, 2019, 54(4): 6-16.
[2]	陈丽珍,冯晓晶,李刚. 一类Schrödinger-Poisson系统非平凡解的存在性[J]. 《山东大学学报(理学版)》, 2019, 54(10): 74-78.
[3]	张申贵. 四阶变指数椭圆方程Navier边值问题的多解性[J]. 山东大学学报（理学版）, 2018, 53(2): 32-37.
[4]	赵欢,周疆. 变指数Herz型Hardy空间上的多线性Calderón-Zygmund算子交换子[J]. 山东大学学报（理学版）, 2018, 53(10): 42-50.
[5]	马小洁,赵凯. 分数次Hardy算子交换子在变指数空间的加权有界性[J]. 山东大学学报（理学版）, 2017, 52(11): 106-110.
[6]	王杰,瞿萌,束立生. Littlewood-Paley算子及其交换子在变指数Herz空间上的有界性[J]. 山东大学学报（理学版）, 2016, 51(4): 9-18.
[7]	马洁,潘振宽*,魏伟波,国凯. 含Euler弹性项图像修复变分模型的快速Split Bregman算法[J]. J4, 2013, 48(05): 70-77.
[8]	潘振宽,魏伟波,张海涛 . 基于梯度和拉普拉斯算子的图像扩散变分模型[J]. J4, 2008, 43(11): 11-16 .