图像去噪与图像分割中的数学方法

发布时间：2018-10-29 22:20

【摘要】：偏微分方程和变分法已经成为图像处理领域中的两个十分重要的工具.对于偏微分方程来说,我们可以直接构造偏微分方程来解决图像恢复和图像分割中的问题.而对于变分法,图像恢复和图像分割问题往往转变为一个能量泛函极小化问题,这种方法可以整合一些有用的信息,比如,先验形状和边界正则化.在本文中,我们一方面提出高效的数学模型,并且注重快速算法在图像处理中的应用;另一方面给出一些模型的理论分析,从而这些模型能够在理论上得到支撑.本文的主要研究内容分为以下两个部分:第一部分:图像去噪中的数学方法1,基于分数阶保真项的偏微分方程我们提出一个新的用于图像去噪的偏微分方程,该模型结合了分数阶保真项和全局灰度值保真项.这两种保真项的结合能够衡量图像间灰度值变化的相似性,并且可以消除阶梯效应,增强图像的纹理信息.实验表明,基于分数阶保真项的模型在去除图像噪声上要比基于梯度保真项的模型好.2,基于时滞正则化的偏微分方程我们提出一个基于时滞正则化的非线性偏微分方程,该方程能够在去除图像噪声的同时保存图像的纹理信息.由于时滞正则化纳入滤波过程中,我们可以获取迭代过程中每一步的图像.时滞正则化方法能够较好地替代仅从图像本身构造扩散系数的高斯滤波平滑方法,并且能够防止过度平滑.最后,我们利用Galerkin方法给出方程解的存在性与唯一性的证明.第二部分:图像分割中的数学方法1,基于图像恢复和Mumford-Shah模型的图像分割我们提出一个基于图像恢复和Mumford-Shah模型的图像分割方法,该方法可以有效地提取模糊图像或者噪声图像的感兴趣区域.由于交替迭代算法便于计算,而且具备了收敛性质,因此,我们选取交替迭代算法用于求解我们的模型.在理论上,我们给出了极小化能量泛函解的存在性,唯一性的证明.2,基于图像曲面的平均曲率正则化的Mumford-Shah模型和阈值的图像分割我们首先提出一个改进的Mumford-Shah模型,该模型的正则项由图像曲面的平均曲率的L1范数所取代.这种改进的Mumford-Shah模型不仅能够去除噪声,而且能够保存物体的几何形状,特别是,物体的拐角处.利用Augmented Lagrangian算法求解该模型,我们得到一个平滑图像u.最后,图像分割则是通过选取合适的阈值实现的.实验表明,我们的方法能够更好地提取物体的边界,特别是,物体拐角处的边界.
[Abstract]:Partial differential equations and variational methods have become two very important tools in the field of image processing. For partial differential equations, we can construct partial differential equations directly to solve the problems of image restoration and image segmentation. For variational methods, image restoration and image segmentation problems are often transformed into an energy functional minimization problem, which can integrate some useful information, such as prior shape and boundary regularization. In this paper, on the one hand, we propose efficient mathematical models, and focus on the application of fast algorithms in image processing; on the other hand, we give some theoretical analysis of the models, so that these models can be supported theoretically. The main research contents of this paper are as follows: the first part: mathematical method in image denoising. Based on partial differential equation of fractional fidelity term, we propose a new partial differential equation for image denoising. The model combines fractional fidelity terms and global grayscale fidelity terms. The combination of these two fidelity terms can measure the similarity of grayscale changes between images, eliminate the ladder effect, and enhance the texture information of images. Experimental results show that the model based on fractional fidelity term is better than the model based on gradient fidelity term. 2. Based on the partial differential equation of time-delay regularization, we propose a nonlinear partial differential equation based on time-delay regularization. This equation can save image texture information while removing image noise. Because delay regularization is incorporated into the filtering process, we can obtain images of each step in the iterative process. The time-delay regularization method can replace the Gao Si filtering smoothing method which only constructs diffusion coefficients from the image itself and can prevent excessive smoothing. Finally, we use the Galerkin method to prove the existence and uniqueness of the solution of the equation. The second part: the mathematical method of image segmentation. 1. We propose an image segmentation method based on image restoration and Mumford-Shah model, which is based on image restoration and image segmentation based on Mumford-Shah model. This method can effectively extract regions of interest from blurred or noisy images. Because the alternating iteration algorithm is easy to calculate and has the convergence property, we choose the alternating iteration algorithm to solve our model. In theory, we give the existence and uniqueness of minimized energy functional solutions. 2. Based on the Mumford-Shah model of mean curvature regularization of image surface and the threshold of image segmentation, we first propose an improved Mumford-Shah model. The canonical term of the model is replaced by the L1 norm of the mean curvature of the image surface. The improved Mumford-Shah model can not only remove noise, but also preserve the geometric shape of the object, especially at the corner of the object. Using Augmented Lagrangian algorithm to solve the model, we get a smooth image u. Finally, the image segmentation is realized by selecting the appropriate threshold. Experiments show that our method can better extract the boundary of the object, especially at the corner.
【学位授予单位】：浙江大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：TP391.41

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