非凸函数在图像复原中的应用
发布时间：2019-03-20 19:24
【摘要】：图像信息是人类认识世界的重要信息来源,然而由于在图像成像条件和图像传输过程中存在各种不利因素致使图像质量下降,从而影响图像的使用及其后续处理.如何从退化图像复原出清晰的、内容丰富的图像是人们所普遍关注的问题,这正是图像复原要解决的问题.图像复原是图像处理领域中重要的研究内容之一.通常情况下,由于图像复原问题是一个不适定的反问题,这就需要利用先验信息将不适定问题正则化处理转化为适定模型.同时,自然图像统计学显示图像边缘分布既不全是Gaussian分布也不全是Laplacian分布,而是类似于Hyper-Laplacian分布,即先验信息是非凸的.本论文正是基于各种非凸势函数,包括Lipschitz非凸函数与non-Lipschitz非凸函数,建立相应的非凸非光滑优化模型,采用交替最小化算法求解,分析其收敛性.本论文的主要工作与取得的创新性成果主要有:针对Lipschitz非凸正则函数与加性噪声,建立L2+Lipschitz正则函数的非凸能量函数.先采用非凸累进算法处理,相应的非凸累进能量函数随着变系数增大而由凸能量函数趋近于原目标非凸能量函数.对每一个固定的系数,代理能量函数分别采用四种交替最小化算法求解,在求解过程中,为了保证海森阵正定性,仅考虑能量函数海森阵中的正定部分.同时,将四种交替最小化算法归结为一种模式处理,且选择其中一个算法,利用Kurdyka-Lojasiewicz不等式分析该算法的收敛性,且成功分析了代理能量函数随着变系数改变而趋于原目标非凸能量函数时的收敛性;针对non-Lipschitz拟范数?p(0p1)正则函数与乘性噪声,建立L1+TVp的非凸能量函数,采用变量分离与邻近点交替最小化算法处理.而对含TVp项的子问题(去噪模型),先采用Huber函数处理?p范数,再对相应的欧拉方程采用原始对偶牛顿法求解对偶向量,进而得到广义正定海森阵,接下来用信赖域法求该子问题的最优解,且分析了处理该子问题的算法具有超线性收敛性.之后,利用KurdykaLojasiewicz不等式对整个算法,进行了收敛性分析;给出了三种近年来出现的非凸函数模型应用于图像复原,如箱式约束非凸最小化模型,p-压缩算子(0p1)与修正的非凸最小化模型,且针对不同噪声分别给出了相应的算法与相关收敛性分析.在数值试验中,分别验证了各种算法的有效性.特别的,在第三章中由数值试验分析了变系数取值区间,在第四章中分析了拟范数?p作为正则函数处理乘性噪声时,发现当p=12时,图像复原效果最优,以及其他一些非凸模型的特殊效果,如箱式约束非凸模型在特定的图像区域能改善复原效果,修正非凸模型能加速处理速度.
[Abstract]:Image information is an important source of information for human beings to understand the world. However, due to various unfavorable factors in image imaging conditions and image transmission, the image quality is degraded, thus affecting the use of images and their subsequent processing. How to recover clear and rich images from degraded images is a common concern, which is exactly the problem to be solved in image restoration. Image restoration is one of the important research contents in the field of image processing. Usually, since the image restoration problem is an ill-posed inverse problem, it is necessary to transform the regularization of the ill-posed problem into a well-posed model by using prior information. At the same time, natural image statistics show that image edge distribution is not all Gaussian distribution or Laplacian distribution, but similar to Hyper-Laplacian distribution, that is, prior information is non-convex. Based on various non-convex potential functions, including Lipschitz nonconvex function and non-Lipschitz nonconvex function, the corresponding non-convex and non-smooth optimization model is established and solved by alternating minimization algorithm, and its convergence is analyzed. The main work and innovative results of this thesis are as follows: for Lipschitz nonconvex regularization function and additive noise, the non-convex energy function of L2-Lipschitz regular function is established. First, the non-convex cumulant algorithm is used to deal with it. The corresponding nonconvex cumulant energy function approaches to the non-convex energy function of the original target with the increase of the variable coefficient. For each fixed coefficient, the proxy energy function is solved by four alternating minimization algorithms. In order to ensure the positive definiteness of the Haysen matrix, only the positive definite part of the energy function is considered. At the same time, the four alternative minimization algorithms are reduced to a kind of pattern processing, and one of them is chosen to analyze the convergence of the algorithm by using Kurdyka-Lojasiewicz inequality. The convergence of the proxy energy function tends to the non-convex energy function of the original target with the change of the variable coefficient, and the convergence of the proxy energy function is analyzed successfully. For the non-Lipschitz quasi-norm p (0p1) regularization function and multiplicative noise, the non-convex energy function of L1-TVp is established, and the algorithm of variable separation and alternating minimization of adjacent points is used to deal with it. For the sub-problem (de-noising model) with TVp term, the Huber function is first used to deal with the p-norm, then the primal dual Newton method is used to solve the dual vector for the corresponding Euler equation, and then the generalized positive definite Hessen matrix is obtained. Next, the trust region method is used to find the optimal solution of the sub-problem, and the superlinear convergence of the algorithm for dealing with the sub-problem is analyzed. Then, the convergence of the whole algorithm is analyzed by using KurdykaLojasiewicz inequality. Three nonconvex function models, such as box constrained nonconvex minimization model, p-contractive operator (0p1) and modified nonconvex minimization model, are presented in this paper. The corresponding algorithm and related convergence analysis are given for different noises. In numerical experiments, the validity of each algorithm is verified. In the third chapter, the range of variable coefficients is analyzed by numerical experiments. In the fourth chapter, when the quasi-norm? p is used as a regular function to deal with multiplicative noise, it is found that the image restoration effect is the best when p is 12. And some other special effects of non-convex model, such as box constrained non-convex model can improve the restoration effect in a specific image area, and the modified non-convex model can accelerate the processing speed.
【学位授予单位】：湖南大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：TP391.41;O174.13
[Abstract]:Image information is an important source of information for human beings to understand the world. However, due to various unfavorable factors in image imaging conditions and image transmission, the image quality is degraded, thus affecting the use of images and their subsequent processing. How to recover clear and rich images from degraded images is a common concern, which is exactly the problem to be solved in image restoration. Image restoration is one of the important research contents in the field of image processing. Usually, since the image restoration problem is an ill-posed inverse problem, it is necessary to transform the regularization of the ill-posed problem into a well-posed model by using prior information. At the same time, natural image statistics show that image edge distribution is not all Gaussian distribution or Laplacian distribution, but similar to Hyper-Laplacian distribution, that is, prior information is non-convex. Based on various non-convex potential functions, including Lipschitz nonconvex function and non-Lipschitz nonconvex function, the corresponding non-convex and non-smooth optimization model is established and solved by alternating minimization algorithm, and its convergence is analyzed. The main work and innovative results of this thesis are as follows: for Lipschitz nonconvex regularization function and additive noise, the non-convex energy function of L2-Lipschitz regular function is established. First, the non-convex cumulant algorithm is used to deal with it. The corresponding nonconvex cumulant energy function approaches to the non-convex energy function of the original target with the increase of the variable coefficient. For each fixed coefficient, the proxy energy function is solved by four alternating minimization algorithms. In order to ensure the positive definiteness of the Haysen matrix, only the positive definite part of the energy function is considered. At the same time, the four alternative minimization algorithms are reduced to a kind of pattern processing, and one of them is chosen to analyze the convergence of the algorithm by using Kurdyka-Lojasiewicz inequality. The convergence of the proxy energy function tends to the non-convex energy function of the original target with the change of the variable coefficient, and the convergence of the proxy energy function is analyzed successfully. For the non-Lipschitz quasi-norm p (0p1) regularization function and multiplicative noise, the non-convex energy function of L1-TVp is established, and the algorithm of variable separation and alternating minimization of adjacent points is used to deal with it. For the sub-problem (de-noising model) with TVp term, the Huber function is first used to deal with the p-norm, then the primal dual Newton method is used to solve the dual vector for the corresponding Euler equation, and then the generalized positive definite Hessen matrix is obtained. Next, the trust region method is used to find the optimal solution of the sub-problem, and the superlinear convergence of the algorithm for dealing with the sub-problem is analyzed. Then, the convergence of the whole algorithm is analyzed by using KurdykaLojasiewicz inequality. Three nonconvex function models, such as box constrained nonconvex minimization model, p-contractive operator (0p1) and modified nonconvex minimization model, are presented in this paper. The corresponding algorithm and related convergence analysis are given for different noises. In numerical experiments, the validity of each algorithm is verified. In the third chapter, the range of variable coefficients is analyzed by numerical experiments. In the fourth chapter, when the quasi-norm? p is used as a regular function to deal with multiplicative noise, it is found that the image restoration effect is the best when p is 12. And some other special effects of non-convex model, such as box constrained non-convex model can improve the restoration effect in a specific image area, and the modified non-convex model can accelerate the processing speed.
【学位授予单位】：湖南大学
【学位级别】：博士
【学位授予年份】：2015
【分类号】：TP391.41;O174.13
【相似文献】
相关期刊论文 前10条
1 宋寅卯;蔡开穗;刘磊;;图像复原研究方法及进展[J];科技创新与生产力;2011年10期
2 李斐;;相位差图像复原技术研究[J];物理学报;2012年23期
3 钱霖;一个数字图像复原公式的推导[J];苏州大学学报(自然科学版);2004年02期
4 王艳;郭永彩;高潮;;基于空间域共轭梯度法的盲目图像复原[J];光学技术;2007年01期
5 曹雷;陈洪斌;邱琪;张建林;任戈;徐智勇;张彬;;盲图像复原研究现状[J];中国光学;2014年01期
6 陈华;陈婷;谢敏;;基于小波包分析二次阈值去噪图像复原方法[J];光学技术;2008年02期
7 陈波;程承旗;濮国梁;耿则勋;杨阳;;面向图像复原的广义岭估计模式法波前重构[J];红外与激光工程;2012年02期
8 朱昱,王春涛,刘政凯;匀速直线运动降质图像复原中的误差改善[J];环境遥感;1992年04期
9 苏宏伟;陈华;;单画幅欠采样图像复原中的插值作用[J];广西科学院学报;2006年04期
10 洪汉玉;;红外气动退化图像复原校正的复合算法研究[J];应用光学;2008年06期
相关会议论文 前10条
1 赵剡;张春晓;许东;;基于弱表层模型的降晰函数辨识与图像复原[A];2007年光电探测与制导技术的发展与应用研讨会论文集[C];2007年
2 黎明;杨杰;王辉;蔡念;;一种彩色图像复原新方法:基于滑动窗口的支持向量回归算法[A];2006年全国光电技术学术交流会会议文集（D 光电信息处理技术专题）[C];2006年
3 侯榆青;张欢;杨旭朗;陈q，
本文编号：2444551
本文链接：https://www.wllwen.com/kejilunwen/yysx/2444551.html
