计算数学
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计算数学  2017, Vol. 39 Issue (4): 393-406    DOI:
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基于分数阶微积分正则化的图像处理
陈云1,2, 郭宝裕1,2, 马祥园1,2
1. 中山大学数学学院;
2. 中山大学广东省计算科学重点实验室, 广州 510275
IMAGE PROCESSING BASED ON REGULARIZATION WITH FRACTIONAL CALCULUS
Chen Yun1,2, Guo Baoyu1,2, Ma Xiangyuan1,2
1. School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China;
2. Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University, Guangzhou 510275, China
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摘要 全变分正则化方法已被广泛地应用于图像处理,利用此方法可以较好地去除噪声,并保持图像的边缘特征,但得到的优化解会产生"阶梯"效应.为了克服这一缺点,本文通过分数阶微积分正则化方法,建立了一个新的图像处理模型.为了克服此模型中非光滑项对求解带来的困难,本文研究了基于不动点方程的迫近梯度算法.最后,本文利用提出的模型与算法进行了图像去噪、图像去模糊与图像超分辨率实验,实验结果表明分数阶微积分正则化方法能较好的保留图像纹理等细节信息.
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关键词分数阶   正则化方法   图像处理   迫近梯度算法     
Abstract: Total variation (TV) regularization has been widely used for ill-posed image processing problems to suppress noise and preserve edge features. However, traditional TV regularization tends to yield the "staircasing artifacts", causing piecewise constant approximation of the true image. To overcome this difficulty, we considered a non-smooth convex optimization model based on regularization method with fractional calculus for several typical image processing problems. A proximal gradient algorithm was developed to overcome the difficulty of non-smoothness for the proposed convex optimization model, and solve the optimization model. Numerical experiments demonstrated that the developed method is effective to preserve texture features of images for image denoising, image deblurring and image super-resolution restoration.
Key wordsfractional calculus   regularization   image processing   proximal gradient algorithm   
收稿日期: 2017-01-06;
引用本文:   
. 基于分数阶微积分正则化的图像处理[J]. 计算数学, 2017, 39(4): 393-406.
. IMAGE PROCESSING BASED ON REGULARIZATION WITH FRACTIONAL CALCULUS[J]. Mathematica Numerica Sinica, 2017, 39(4): 393-406.
 
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