图像去噪问题中的几类非光滑数值方法
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摘要
随着计算机技术的发展,图像处理问题在日常生活中扮演着越来越重要的角色.然而,由于图像在形成、传输、生成的过程中受外界因素的影响而导致质量的降低,因此有效地复原退化图像是图像处理中一个基本任务.一般地,图像复原方法可以归结为三类：基于小波的方法、基于概率统计的方法和基于偏微分方程的方法.其中,基于偏微分方程的图像复原模型由于具有自适应性比较强、贴近图像特性等优点,因而最近十几年来得到了快速地发展,已经扩展到几乎所有的图像处理领域.
通常情况下,由于图像复原问题是一个不适定的反问题,因此需要在正则化意义下建立一个适定的模型.为了使所建模型能更好地描述图像的特性,经常要求模型满足一定的数学性质,这样就增加了数值难度,所以寻求快速有效的数值算法是图像处理领域内一个重要的研究课题.由于图像去噪是图像复原问题中的一个重要环节,因此本论文主要研究基于偏微分方程的两个基本而又重要的图像去噪模型—Rudin-Osher-Fatemi(ROF)模型和Lysaker-Lundervold-Tai(LLT)模型及其快速的数值方法.我们的主要工作和创新成果如下：
由于ROF模型和LLT模型的对偶问题的最优性条件包含有非线性互补问题,因而我们可以利用Fisher-Burmeister NCP函数的一些数学性质将这个最优性条件转化为一个半光滑方程组.为了使算法达到全局收敛性,通过引入一个价值函数,我们提出用阻尼修正高斯牛顿法解这个半光滑方程组.另外,在算法中,我们引入一个修正参数使得搜索的步长增加,并且用预处理共轭梯度法来提高计算速度.同时,也给出了算法的全局收敛性及Q-超线性收敛速度的理论分析.
由于增广拉格朗日方法结合了拉格朗日方法和罚方法的优点,因此被广泛地应用到求解非光滑凸优化问题.对于LLT模型,我们首先将其转换为一个约束问题,然后基于增广拉格朗日方法得到该约束问题的最优性条件,并且指出这个最优性条件可以看作投影梯度法.因此,我们提出用投影梯度法解离散的LLT模型,并且指出经典的半隐式梯度下降法可以由投影梯度法得到.同时,我们还将投影梯度法推广到解纹理提取问题的混合模型(ROF模型和LLT模型).此外,我们将增广拉格朗日方法应用到非负约束图像去模糊问题,提出了一个积极集方法,并且证明了这个积极集方法可以归结到半光滑牛顿法.
最近,利用Bregman算法的思想,Goldstein和Osher提出了分裂Bregman算法解图像复原问题.在此基础上,我们把分裂Bregman方法推广到解各向异性LLT模型和LOT模型的第二步.虽然分裂Bregman方法具有一定的优势,但是由于该方法的每一次迭代都需要解一个偏微分方程,从而使得计算量大大增加.为了克服这种缺陷,利用投影算子和压缩算子的性质,我们提出一个新的快速有效的算法一投影算法.为了说明该算法的有效性,我们将本算法应用到解各向异性LLT模型,并且给出了此算法的收敛性的理论分析.特别地,我们指出基于分裂Bregman方法的投影算法可以归结到FBS算法的框架中.
With the development of computer technology, image processing problems play an increasingly important role in daily life. However, in the course of the image formation, transmission and generation, some external factors led to de-crease in the image quality, so it is one of important research topics in image processing to effectively restore degraded image. Normally, the image restoration methods can be summarized into three categories:wavelet-based methods, based on probability and statistics methods, and methods based on partial differential equations (PDEs). Among these approaches, the image restoration models based on the PDEs have some advantages such as relatively strong self-adaptability, close to the image characteristics, etc., so that this class of approaches has been rapid development over the past ten years and has been extended to almost all fields in image processing.
The image restoration problem is usually an ill-posed inverse problem so that a well-posed model is needed to build in the sense of the regularization. In order to build a model consistent with the characteristics of the image, the model is often required to satisfy certain mathematical properties so that the numerical difficulty is increased, therefore it is an important research topic in image processing to look for some rapid and effective numerical algorithms. Since the image denoising is an important part in the image restoration, in this dissertation, we focus on two basic and important denoising models—Rudin-Osher-Fatemi (ROF) model and Lysaker-Lundervold-Tai (LLT) model, and their rapid numerical methods. Our main work and the innovations are as follows:
Since the optimality conditions for the dual problem of the ROF model or the LLT model contain a nonlinear complementarity problem(NCP), we can apply some mathematics qualities of the Fisher-Burmeister NCP function to transform this optimality condition into a system of semismooth equations. In order to deduce global convergence of the proposed algorithm, by introducing a merit func-tion, we propose to use the damped modified Gauss-Newton method to solve the system of semi-smooth equations. In addition, in the algorithm, we introduce a modified parameter to increase the search step length and use the preconditioned conjugate gradient method to improve the calculation speed. At the same time, we also give theoretical analyses about the global convergence and the Q-superlinear convergence rate for the proposed algorithm.
Since the augmented Lagrangian approach combines the advantage of the Lagrangian method and penalty method, it is widely applied to solve the nons-mooth convex optimization problems. For the LLT model, we first convert it into a constrained problem and then attain the optimality conditions of the constrained problem based on the augmented Lagrangian approach. Moreover, we point out that the optimality conditions can be seen as a projected gradient method. So we propose to use the projected gradient method to solve the discretization LLT model and point out that the classic semi-implicit gradient descent method can be deduced from the projected gradient method. At the same time,, we extend the projected gradient method to solve a mixed model (mixing the ROF model and the LLT model) for texture extraction. In addition, we also apply the augmented Lagrange method to the image deblurring problem with nonnegative constraints and propose an active set strategy. Furthermore, we prove that the active set method can be fell into the framework of semismooth Newton methods.
Recently, by using the ideas of the Bregman iterative algorithm, Goldstein and Osher proposed a split Bregman method to solve the image restoration problems. On this basis, we extend the split Bregman method to solve the anisotropic LLT model and the second step of the LOT model. Although the split Bregman method has certain advantages, it has to suffer from solving a PDE at each iterative so that the computation costs are increased. In order to overcome this drawback, based on the properties of the projection operator and the shrink operator, we propose a new rapid and efficient algorithm—the projection algorithm. In order to illustrate the effectiveness of the projection algorithm, we apply this algorithm to solve the anisotropic LLT model. Furthermore, the theoretical analysis about the convergence of this algorithm is also given. In particular, we point out that the projection algorithm based on the split Bregman method can be fell into the framework of the FBS algorithm.
The dissertations is supported by the National Natural Science Foundation of China (No.60872129).
通常情况下,由于图像复原问题是一个不适定的反问题,因此需要在正则化意义下建立一个适定的模型.为了使所建模型能更好地描述图像的特性,经常要求模型满足一定的数学性质,这样就增加了数值难度,所以寻求快速有效的数值算法是图像处理领域内一个重要的研究课题.由于图像去噪是图像复原问题中的一个重要环节,因此本论文主要研究基于偏微分方程的两个基本而又重要的图像去噪模型—Rudin-Osher-Fatemi(ROF)模型和Lysaker-Lundervold-Tai(LLT)模型及其快速的数值方法.我们的主要工作和创新成果如下：
由于ROF模型和LLT模型的对偶问题的最优性条件包含有非线性互补问题,因而我们可以利用Fisher-Burmeister NCP函数的一些数学性质将这个最优性条件转化为一个半光滑方程组.为了使算法达到全局收敛性,通过引入一个价值函数,我们提出用阻尼修正高斯牛顿法解这个半光滑方程组.另外,在算法中,我们引入一个修正参数使得搜索的步长增加,并且用预处理共轭梯度法来提高计算速度.同时,也给出了算法的全局收敛性及Q-超线性收敛速度的理论分析.
由于增广拉格朗日方法结合了拉格朗日方法和罚方法的优点,因此被广泛地应用到求解非光滑凸优化问题.对于LLT模型,我们首先将其转换为一个约束问题,然后基于增广拉格朗日方法得到该约束问题的最优性条件,并且指出这个最优性条件可以看作投影梯度法.因此,我们提出用投影梯度法解离散的LLT模型,并且指出经典的半隐式梯度下降法可以由投影梯度法得到.同时,我们还将投影梯度法推广到解纹理提取问题的混合模型(ROF模型和LLT模型).此外,我们将增广拉格朗日方法应用到非负约束图像去模糊问题,提出了一个积极集方法,并且证明了这个积极集方法可以归结到半光滑牛顿法.
最近,利用Bregman算法的思想,Goldstein和Osher提出了分裂Bregman算法解图像复原问题.在此基础上,我们把分裂Bregman方法推广到解各向异性LLT模型和LOT模型的第二步.虽然分裂Bregman方法具有一定的优势,但是由于该方法的每一次迭代都需要解一个偏微分方程,从而使得计算量大大增加.为了克服这种缺陷,利用投影算子和压缩算子的性质,我们提出一个新的快速有效的算法一投影算法.为了说明该算法的有效性,我们将本算法应用到解各向异性LLT模型,并且给出了此算法的收敛性的理论分析.特别地,我们指出基于分裂Bregman方法的投影算法可以归结到FBS算法的框架中.
With the development of computer technology, image processing problems play an increasingly important role in daily life. However, in the course of the image formation, transmission and generation, some external factors led to de-crease in the image quality, so it is one of important research topics in image processing to effectively restore degraded image. Normally, the image restoration methods can be summarized into three categories:wavelet-based methods, based on probability and statistics methods, and methods based on partial differential equations (PDEs). Among these approaches, the image restoration models based on the PDEs have some advantages such as relatively strong self-adaptability, close to the image characteristics, etc., so that this class of approaches has been rapid development over the past ten years and has been extended to almost all fields in image processing.
The image restoration problem is usually an ill-posed inverse problem so that a well-posed model is needed to build in the sense of the regularization. In order to build a model consistent with the characteristics of the image, the model is often required to satisfy certain mathematical properties so that the numerical difficulty is increased, therefore it is an important research topic in image processing to look for some rapid and effective numerical algorithms. Since the image denoising is an important part in the image restoration, in this dissertation, we focus on two basic and important denoising models—Rudin-Osher-Fatemi (ROF) model and Lysaker-Lundervold-Tai (LLT) model, and their rapid numerical methods. Our main work and the innovations are as follows:
Since the optimality conditions for the dual problem of the ROF model or the LLT model contain a nonlinear complementarity problem(NCP), we can apply some mathematics qualities of the Fisher-Burmeister NCP function to transform this optimality condition into a system of semismooth equations. In order to deduce global convergence of the proposed algorithm, by introducing a merit func-tion, we propose to use the damped modified Gauss-Newton method to solve the system of semi-smooth equations. In addition, in the algorithm, we introduce a modified parameter to increase the search step length and use the preconditioned conjugate gradient method to improve the calculation speed. At the same time, we also give theoretical analyses about the global convergence and the Q-superlinear convergence rate for the proposed algorithm.
Since the augmented Lagrangian approach combines the advantage of the Lagrangian method and penalty method, it is widely applied to solve the nons-mooth convex optimization problems. For the LLT model, we first convert it into a constrained problem and then attain the optimality conditions of the constrained problem based on the augmented Lagrangian approach. Moreover, we point out that the optimality conditions can be seen as a projected gradient method. So we propose to use the projected gradient method to solve the discretization LLT model and point out that the classic semi-implicit gradient descent method can be deduced from the projected gradient method. At the same time,, we extend the projected gradient method to solve a mixed model (mixing the ROF model and the LLT model) for texture extraction. In addition, we also apply the augmented Lagrange method to the image deblurring problem with nonnegative constraints and propose an active set strategy. Furthermore, we prove that the active set method can be fell into the framework of semismooth Newton methods.
Recently, by using the ideas of the Bregman iterative algorithm, Goldstein and Osher proposed a split Bregman method to solve the image restoration problems. On this basis, we extend the split Bregman method to solve the anisotropic LLT model and the second step of the LOT model. Although the split Bregman method has certain advantages, it has to suffer from solving a PDE at each iterative so that the computation costs are increased. In order to overcome this drawback, based on the properties of the projection operator and the shrink operator, we propose a new rapid and efficient algorithm—the projection algorithm. In order to illustrate the effectiveness of the projection algorithm, we apply this algorithm to solve the anisotropic LLT model. Furthermore, the theoretical analysis about the convergence of this algorithm is also given. In particular, we point out that the projection algorithm based on the split Bregman method can be fell into the framework of the FBS algorithm.
The dissertations is supported by the National Natural Science Foundation of China (No.60872129).
引文
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