几类全变差型图像恢复模型研究
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摘要
图像处理技术是对获取的退化图像进行有效的分析和处理,是帮助人类更好地认知世界的重要途径,它已经被广泛应用于航空航天、生物医学工程、目标识别、地理测绘等重要领域.然而,由于获取方式的局限性经常导致获取的退化图像具有复杂的结构特征,这就给建立有效的数学模型和数值算法获取优质图像带来了极大的挑战.
在众多的图像处理技术中,全变差型的图像处理模型虽然具有一定的局限性,但是由于具有良好的数学性质和高效的数值算法,因此在众多的图像处理领域内得到了广泛的关注.本学位论文拟从几类全变差型图像恢复模型出发,针对图像去噪、图像去模糊、图像修补、图像去乘性噪声等问题,提出新的模型和数值算法.主要创新点如下：
(1)针对含有两个非光滑项的凸优化问题,我们通过变量代换将该问题转化为一个约束优化问题,提出交替方向乘子法,并从理论上分析了算法的收敛性和有效性.然而,交替方向乘子法在数值计算时需要求解多个方程,这大大降低了该算法的有效性.基于投影算子和压缩阈值算子的联系,我们提出了临近点算法来求解该凸优化问题,并给出了算法收敛所需要的条件.事实上,临近点算法是考虑初始凸优化问题的对偶问题,再引入简洁的投影算子,这样可以将对偶变量直接投影到一个凸单位球上,因此该算法高效稳定.
(2)基于全变差函数空间的定义,我们给出自适应全变差函数空间的定义,并分析了该空间的数学性质.由于一阶全变差模型ROF模型在保持图像边缘的同时,会在图像的渐变区域产生模糊现象,而二阶全变差模型LLT模型在保持图像渐变区域的光滑性的同时,会引起图像边缘区域模糊,因此我们基于自适应全变差函数空间提出了凸结合这两个模型的混合模型,并将其应用于图像去噪、图像去模糊、图像修补等领域.另外,我们建议用交替方向乘子法和临近点方法分别求解该混合模型,同时对这两种算法的有效性进行了比较.特别地,我们通过变量替换将图像去模糊、图像修补、图像去乘性噪声等问题归结到去加性噪声问题的框架中,从而扩展了数值求解的应用范围.事实上,在所提出的混合模型中,我们引入边缘检测函数作为凸结合参数,这样就可以使得新模型在图像边缘区域ROF模型占优,而在图像的渐变区域LLT模型占优,因此该模型表现为局部性,从而具有自适应性和鲁棒性.
(3)针对传统的LOT模型需要求解两个偏微分方程(PDE)的数值缺陷,基于分裂Bregman方法具有高效、稳定、需要内存小的数值优势,我们建议用该算法求解LOT模型的第二步,从而改进了数值实验结果.事实上,LOT模型的第一步是求解非凸优化问题的法向场问题,因此除了求解对应的PDE之外,很难找到有效的数值算法.为此,基于ROF模型的最优性条件中的单位法向量和其对偶问题中对偶变量的等价关系,我们提出用对偶变量来代替LOT模型第一步中的单位法向场,得到了一个基于对偶变量的两步模型,并通过数值实验验证了所提模型的有效性.
Image processing technology is the effective analysis and processing for de-graded images. It can help mankind to have a better understanding of the world and has been widely applied to many important fields, such as aerospace engineer-ing, biomedical, object identification and geographical mapping. However, since the acquirement strategies of image have some limitations, there usually exist com-plex structure features in the degraded images. This brings enormous challenges for constructing effective mathematical models and numerical algorithms in order to get superior image from the degraded image.
Among the numerous image processing technology, although the models based on the total variation have some limitations, they have given rise to the exten-sive attention of researchers in many fields of image processing. That is because this class of models has good mathematical properties and efficient numerical al-gorithms. In this dissertation, we start with several types of restoration models based on the total variation and propose the new restoration models and numerical algorithms for image denoising, image deblurring, image inpainting, multiplicative noise removal and so on. Our main innovations are as follows:
(1) For the convex optimization problem with two nonsmooth terms, we transform it into a unconstrained problem by using variable substitution and propose the alternating direction method of multipliers. What's more, we theoretically analyze the convergence and validity of this algorithm. However, when we nu-merically implement this algorithm, several equations need to be solved. This greatly reduces the efficiency of the proposed algorithm. Based on the relation of the projection operator and the shrinkage threshold operator, we provide a proximal point method to solve this convex optimization problem, and give the required conditions for the convergence of this method. In fact, the proximal point method means that the concise projection operator is introduced into the dual problem of the initial convex optimization problem. Hence, the dual variable can be projected into a convex unit ball. This makes the proximal point method effective and stable.
(2) Based on the definition of the total variation function space, we give the weighted total variation function space and discuss the mathematical prop-erties of this space. The ROF model is known to keep edges well. But it has the undesirable staircase effect in the smooth regions. The LLT model can avoid this undesirable effect, but cause the edges blurring. In view of these facts, based on the weighted total variation function space, we suggest a hybrid model by using the convex combination of these two models. It has been used to image denoising, image deblurring, image inpainting and so on. Moreover, we solve this model by the alternating direction method of multipliers and the proximal point method respectively and compare the effectiveness of these two methods. Especially, with the help of variable substitution, we extend the solvable application range by converting the problem of image deblurring, image inpainting and multiplicative noise removal into the framework of the additive noise removal. In fact, we draw the edge indicator function into the hybrid model which can be seen as the convex combination parameter. Thus, the ROF model plays an important role at the edges and the LLT model does at the smooth region. That is to say, the hybrid model displays localization. Hence, it has adaptability and robustness.
(3) The drawback of the conventional LOT model is that it needs to solve two partial differential equations (PDEs). Since the split Bregman method has the advantages of efficiency, stablility and small memory footprint, we extend this method to solve the second step of the LOT model. Therefore, the numerical results have been improved. Actually, it is difficult to look for other effective algorithms in order to solve the first step of the LOT model expect solving its corresponding PDE. That is because it is to seek the smoothed flow field of a nonconvex optimization problem. As a result of the equivalence relation between the unit normal vector in the optimality condition of the ROF model and the dual variable in its dual problem, we substitute the dual variable for the smoothed flow field in the first step of the LOT model. Consequently, we obtain a two step model based on the dual variable. In addition, we provide some examples to illustrate the efficiency of the proposed model.
在众多的图像处理技术中,全变差型的图像处理模型虽然具有一定的局限性,但是由于具有良好的数学性质和高效的数值算法,因此在众多的图像处理领域内得到了广泛的关注.本学位论文拟从几类全变差型图像恢复模型出发,针对图像去噪、图像去模糊、图像修补、图像去乘性噪声等问题,提出新的模型和数值算法.主要创新点如下：
(1)针对含有两个非光滑项的凸优化问题,我们通过变量代换将该问题转化为一个约束优化问题,提出交替方向乘子法,并从理论上分析了算法的收敛性和有效性.然而,交替方向乘子法在数值计算时需要求解多个方程,这大大降低了该算法的有效性.基于投影算子和压缩阈值算子的联系,我们提出了临近点算法来求解该凸优化问题,并给出了算法收敛所需要的条件.事实上,临近点算法是考虑初始凸优化问题的对偶问题,再引入简洁的投影算子,这样可以将对偶变量直接投影到一个凸单位球上,因此该算法高效稳定.
(2)基于全变差函数空间的定义,我们给出自适应全变差函数空间的定义,并分析了该空间的数学性质.由于一阶全变差模型ROF模型在保持图像边缘的同时,会在图像的渐变区域产生模糊现象,而二阶全变差模型LLT模型在保持图像渐变区域的光滑性的同时,会引起图像边缘区域模糊,因此我们基于自适应全变差函数空间提出了凸结合这两个模型的混合模型,并将其应用于图像去噪、图像去模糊、图像修补等领域.另外,我们建议用交替方向乘子法和临近点方法分别求解该混合模型,同时对这两种算法的有效性进行了比较.特别地,我们通过变量替换将图像去模糊、图像修补、图像去乘性噪声等问题归结到去加性噪声问题的框架中,从而扩展了数值求解的应用范围.事实上,在所提出的混合模型中,我们引入边缘检测函数作为凸结合参数,这样就可以使得新模型在图像边缘区域ROF模型占优,而在图像的渐变区域LLT模型占优,因此该模型表现为局部性,从而具有自适应性和鲁棒性.
(3)针对传统的LOT模型需要求解两个偏微分方程(PDE)的数值缺陷,基于分裂Bregman方法具有高效、稳定、需要内存小的数值优势,我们建议用该算法求解LOT模型的第二步,从而改进了数值实验结果.事实上,LOT模型的第一步是求解非凸优化问题的法向场问题,因此除了求解对应的PDE之外,很难找到有效的数值算法.为此,基于ROF模型的最优性条件中的单位法向量和其对偶问题中对偶变量的等价关系,我们提出用对偶变量来代替LOT模型第一步中的单位法向场,得到了一个基于对偶变量的两步模型,并通过数值实验验证了所提模型的有效性.
Image processing technology is the effective analysis and processing for de-graded images. It can help mankind to have a better understanding of the world and has been widely applied to many important fields, such as aerospace engineer-ing, biomedical, object identification and geographical mapping. However, since the acquirement strategies of image have some limitations, there usually exist com-plex structure features in the degraded images. This brings enormous challenges for constructing effective mathematical models and numerical algorithms in order to get superior image from the degraded image.
Among the numerous image processing technology, although the models based on the total variation have some limitations, they have given rise to the exten-sive attention of researchers in many fields of image processing. That is because this class of models has good mathematical properties and efficient numerical al-gorithms. In this dissertation, we start with several types of restoration models based on the total variation and propose the new restoration models and numerical algorithms for image denoising, image deblurring, image inpainting, multiplicative noise removal and so on. Our main innovations are as follows:
(1) For the convex optimization problem with two nonsmooth terms, we transform it into a unconstrained problem by using variable substitution and propose the alternating direction method of multipliers. What's more, we theoretically analyze the convergence and validity of this algorithm. However, when we nu-merically implement this algorithm, several equations need to be solved. This greatly reduces the efficiency of the proposed algorithm. Based on the relation of the projection operator and the shrinkage threshold operator, we provide a proximal point method to solve this convex optimization problem, and give the required conditions for the convergence of this method. In fact, the proximal point method means that the concise projection operator is introduced into the dual problem of the initial convex optimization problem. Hence, the dual variable can be projected into a convex unit ball. This makes the proximal point method effective and stable.
(2) Based on the definition of the total variation function space, we give the weighted total variation function space and discuss the mathematical prop-erties of this space. The ROF model is known to keep edges well. But it has the undesirable staircase effect in the smooth regions. The LLT model can avoid this undesirable effect, but cause the edges blurring. In view of these facts, based on the weighted total variation function space, we suggest a hybrid model by using the convex combination of these two models. It has been used to image denoising, image deblurring, image inpainting and so on. Moreover, we solve this model by the alternating direction method of multipliers and the proximal point method respectively and compare the effectiveness of these two methods. Especially, with the help of variable substitution, we extend the solvable application range by converting the problem of image deblurring, image inpainting and multiplicative noise removal into the framework of the additive noise removal. In fact, we draw the edge indicator function into the hybrid model which can be seen as the convex combination parameter. Thus, the ROF model plays an important role at the edges and the LLT model does at the smooth region. That is to say, the hybrid model displays localization. Hence, it has adaptability and robustness.
(3) The drawback of the conventional LOT model is that it needs to solve two partial differential equations (PDEs). Since the split Bregman method has the advantages of efficiency, stablility and small memory footprint, we extend this method to solve the second step of the LOT model. Therefore, the numerical results have been improved. Actually, it is difficult to look for other effective algorithms in order to solve the first step of the LOT model expect solving its corresponding PDE. That is because it is to seek the smoothed flow field of a nonconvex optimization problem. As a result of the equivalence relation between the unit normal vector in the optimality condition of the ROF model and the dual variable in its dual problem, we substitute the dual variable for the smoothed flow field in the first step of the LOT model. Consequently, we obtain a two step model based on the dual variable. In addition, we provide some examples to illustrate the efficiency of the proposed model.
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