| 基于矩阵秩最小化和变量变换的图像恢复方法 | |
| Alternative Title | Rank Minimization and Variable Transformation Based Methods for Image Restoration |
| 严辉银 | |
| Subtype | 博士 |
| Thesis Advisor | 黄玉梅 |
| 2018-09-30 | |
| Degree Grantor | 兰州大学 |
| Place of Conferral | 兰州 |
| Degree Name | 博士 |
| Degree Discipline | 计算数学 |
| Keyword | 图像恢复 矩阵秩最小化问题 核范数 加权核范数 加性高斯噪声 乘性噪声 分块匹配 正则化方法 非局部方法 交替迭代方法 Box-Cox变换 最大似然估计 块匹配三维变换方法(BM3D) |
| Abstract | 在数字图像的形成与传输中, 由于成像系统自身特性或外界环境中不可避免因素的影响, 图像应用中得到的图像常常是受到噪声和模糊污染的失真图像. 图像恢复就是要利用失真图像信息, 构造有效方法获得尽可能接近真实图像的复原图像. 图像恢复广泛的存在于很多数字图像应用中, 是数字图像处理的基本问题, 因此对图像恢复问题的研究具有重要的理论和实际意义. 本论文考虑了受到加性高斯噪声污染的图像恢复问题以及受到乘性噪声污染的图像去噪问题,构建了基于矩阵秩最小化和变量变换的图像恢复方法. 图像恢复问题近几十年来得到了广泛的研究, 并构造了很多有效的方法, 比如滤波器方法, 正则化方法, 稀疏表示方法, 非局部方法等. 通过求解矩阵秩最小化问题进行图像复原, 是近年来在图像恢复中采用的一个重要方法. 矩阵秩最小化问题是一个带约束的优化问题, 要在一定约束条件下求解得到一个具有低秩性质的矩阵. 由于对这个问题求解的困难性, 常用核范数最小化问题的解来近似矩阵秩最小化问题的解. 矩阵的核范数是矩阵奇异值之和, 是矩阵秩的近似. 在本论文中, 我们用矩阵秩作为正则项, 构造了矩阵秩最小化问题的新方法, 证明了所提模型可以通过对观察矩阵的奇异值进行硬阈值运算求解. 然后基于图像的自相似性和分块匹配算法, 利用所构造的矩阵秩最小化方法进行了图像中加性高斯噪声的去除, 并在对数域中对图像的Gamma乘性噪声进行了去除. 数值实验结果表明, 所构造的矩阵秩最小化方法在有效去除噪声的同时, 很好的保存了图像的边界和细节信息.对同时受到模糊和加性高斯噪声污染的图像, 构造了基于加权核范数最小化的恢复模型. 加权核范数是矩阵奇异值的加权和, 加权核范数最小化比核范数最小化方法能更有效的去除图像中的噪声. 通过先对图像进行块匹配, 得到图像块匹配矩阵, 利用这些图像块匹配矩阵构造了能量函数, 其中正则项是图像块匹配矩阵的加权核范数之和. 然后设计了交替迭代法求解所提模型, 分析并证明了迭代法的收敛性. 所提模型在迭代法的求解运算中, 避免了一般基于核范数最小化图像恢复中出现的方程组的病态性, 提高了运算性能. 实验结果表明所构造的方法能很好的去除图像中的噪声和模糊.考虑了利用变量变换去除非高斯型噪声的方法. 对非高斯型噪声先进行变量变换, 将其转化为高斯噪声或者近似高斯噪声, 然后利用已有的去除加性高斯噪声的有效方法去除噪声, 是对非高斯噪声进行去除的常用方法. Box-Cox变换是在数理统计中常用来将非高斯分布的数据转化为具有高斯分布特性数据的重要变换. Box-Cox变换中的变换参数是决定变换后的数据能否较好服从高斯分布的关键. 本文中, 针对乘性噪声问题, 构造了基于Box-Cox变换的去除方法. 先设计了选取Box-Cox变换中最优变换参数的最大似然估计方法. 然后利用Box-Cox变换, 将Gamma乘性噪声去除问题转化为高斯分布的加性噪声去除问题, 应用块匹配三维变换方法(BM3D, Block Matching 3D Filtering)去除了变换后的加性噪声. BM3D方法是一种高效的加性高斯噪声去除方法. 利用无偏的Box-Cox逆变换, 从去除了加性噪声的变换图像中, 得到去除了乘性噪声的复原图像. 理论分析和数值实验结果表明, 基于变量变换的方法能有效去除图像中的非高斯噪声, 图像中的Gamma乘性噪声通过Box-Cox变换可以得到去除。 |
| Other Abstract | During the formation and transformation process, images are inevitably degraded by noise and blur caused by intrinsic properties of the electronic imaging system or unavoidable effects from external environments. By utilizing the information of contaminated images, the aim of image restoration is to construct efficient methods to obtain recovered images which are as close as possible to the original ones. Image restoration problem arises in many digital imaging applications, it is a basic problem in image processing. Therefore, the study of image restoration is of greatly theoretical and practical significance. In this paper, the restoration for images contaminated by both additive Gaussian noise and blur, as well as multiplicative noise removal problems are considered. Image restoration methods based on matrix rank minimization and variable transformation are constructed. In recent decades, many efficient methods for image restoration have been proposed. These methods include filtering method, regularization method, sparse representation and non-local methods etc. Restoration methods based on rank minimization are important ones which have been extensively developed in recent years. Rank minimization problem is a constrained optimization problem which aims at getting a low-rank matrix under certain constraints. Because of the difficulty in solving the rank minimization problem, its solution is always approximated by the solution of the nuclear norm minimization problem. A matrix's nuclear norm is the sum of all singular values of the matrix and it is used as an approximation of the matrix's rank. In this paper, we propose a new method for rank minimization problem by utilizing matrix's rank as the regularization term in the energy function.We also prove that the proposed model can be solved by hard-thresholding operation on the observed matrix's singular values. By utilizing image self-similarity and image block matching scheme, we apply the proposed rank minimization method to remove white Gaussian additive noise in images. Gamma multiplicative noise is also removed in logarithm domain. Numerical results illustrate that the proposed rank minimization method can remove noises in images efficiently.Meanwhile, edges and details in images are also preserved.An new image restoration model is also proposed based on weighted nuclear norm minimization for restoring images contaminated by both blur and Gaussian additive noise. Weighted nuclear norm is the weighted sum of all singular values of a matrix which is more efficient than nuclear norm minimization for image noise removal. For each image block in the observed image, its block matching matrix can be formed by its similar blocks obtained through matching process in the image. The block matching matrices are used to establish the energy function of the proposed model. The regularization term is the sum of the block matching matrices' weighted nuclear norm. We further design an alternating iterative algorithm to solve the proposed model. Convergence of the algorithm is also analyzed and proved. Ill-conditioned problem does not appear in the proposed algorithm for the proposed image restoration model while such problem often arises in most of the nuclear norm based minimization problems. The computational efficiency is greatly improved. Numerical results show that the image recovered quality by the proposed method is good.Non-Gaussian noise removal problem by variable transformation is also considered. By transforming non-Gaussian noises into noises with Gaussian or near Gaussian distribution, the existing efficient Gaussian additive noise removal methods can be applied to remove these transformed noises. The inverse transformation are then used to get the final recovered images. Box-Cox transformation is an important data transformation which is often used in mathematical statistics to transform non-Gaussian distribution data into Gaussian distribution data. The selection of the parameter in Box-Cox transformation is very important since it determines if the transformed data follows Gaussian distribution. In this paper, we devise a method for multiplicative noise removal based on Box-Cox transformation. Firstly, we design a maximum likelihood method to determinethe optimal transform parameter in Box-Cox transformation for multiplicative denoising problem. Then we convert the Gamma multiplicative noise removal problem into a Gaussian additive noise removal problem by using Box-Cox transformation and block matching three dimensional filtering method (BM3D) is applied to remove the transformed noise. BM3D is an effective Gaussian additive noise removal method.The final multiplicative denoised image is obtained from the additive denoised image by using an unbiased inverse Box-Cox transformation. Both theoretical analysis and experimental results demonstrate that the variable transformation based methods can remove non-Gaussian noises in images efficiently. Box-Cox transformation based multiplicative noise removal method is also very efficient. |
| Pages | 103 |
| URL | 查看原文 |
| Language | 中文 |
| Document Type | 学位论文 |
| Identifier | https://ir.lzu.edu.cn/handle/262010/342216 |
| Collection | 数学与统计学院 |
| Affiliation | 数学与统计学院 |
| First Author Affilication | School of Mathematics and Statistics |
| Recommended Citation GB/T 7714 | 严辉银. 基于矩阵秩最小化和变量变换的图像恢复方法[D]. 兰州. 兰州大学,2018. |
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