【作者】 王伟；

【导师】 沈纯理；

【作者基本信息】 华东师范大学 ， 基础数学， 2010， 博士

【摘要】 本文的研究基于几何变分理论和偏微分方程理论,主要讨论了图像处理中的几个常见问题,如图像去噪,图像分割,图像去模糊,图像增强等。基本的思路是根据不同的背景提出相应的能量泛函,并通过求解Euler-Lagrange方程极小化能量泛函。对于求解Euler-Lagrange方程的方法,本文也给出了一些快速算法,使得数值计算更加稳定。具体地,本文的研究结果包括下面几个方面：1.基于对Nonlocal—BV函数的分析,以ROF模型为例,讨论了包含Nonlocal-TV正则项的泛函极小解的存在唯一性及其相应的热扩散方程解的存在唯一性。应用方面,我们将量子集BV函数推广到Nonlocal的情形,从而提高了图像分割的效果。2.讨论了基于frame的图像去模糊问题,通过分析未知函数在空间域和频率域上正则项的不同选取,提出了三种不同的盲去模糊模型进行对比研究,并引进快速算法,详细介绍了模型的数值实现过程。3.在经典的图像去噪模型-ROF模型和图像分割模型-测地活动环路模型的基础上,引进张量投票技术,分别对平衡参数和停止函数作了修改,使得它们在保留纹理的去噪和对比度较低的图像分割方面更加有效。4.基于Retinex理论,我们提出了一种分解光照函数和反射函数的新模型,使用快速算法求解达到了很好的图像增强效果,并讨论了模型解的存在唯一性,算法的收敛性等理论。更多还原

【Abstract】 This paper is based on variational method and partial differential equations. We focus on some common problems in image processing, such as denoising, segmentation, deblurring and image enhancement. The mean idea is to deal with an energy minimization problem by looking for the solution of the related Euler-Lagrange equation. We also give some fast algorithms which are stable and efficient to solve the equations. The mean results are as follows:1. We study the Nonlocal-BV function space and give some properties. Then the nonlocal-ROF model is discussed about the following topics:existence and uniqueness of the solution in nonlocal-BV space, existence and uniqueness of the solution to the related heat equation. We give the definition of the nonlocal quantum BV function, and show the applications in image segmentation numerically.2. We discuss the frame-based image delurring, including different choice of the regu-larization term from space domain and frequency domain. In order to compare the choice, we give three different models, then we explain them in detail including the numerical implement.3. We introduce some applications of Tensor Voting method by combining it with clas-sical models. we modify Geodesic Active Contour model by adding influence of Tensor Voting in the stopping function during the detecting procedure, and we im-prove ROF model with the same idea. Numerical results show the effectiveness.4. Based on the Retinex theory, we give a new model to decompose the image into illu-mination function and reflection function. We also give some existence and conver-gence theory about the new model. Some fast algorithms help us get good numerical results.更多还原