【作者】 高飞；

【导师】 刘婧；

【作者基本信息】 大连海事大学 ， 应用数学， 2011， 硕士

【摘要】 用偏微分方程进行图像处理始于20世纪90年代,几十年来,以P-M模型为基础,经过众多专家学者的不懈努力,Catte模型、Alvarez-Lions-Morel模型、全变差(TV)模型等很多成熟与高效的算法被提出并广泛应用在图像复原、图像分割、图像重构、图像识别、图像分析等方面。图像去噪作为图像处理中的一个经典的研究领域,其关键在于去除噪声的同时,保持重要的图像特征,如图像边缘,尽可能减少噪声对后续的图像处理的影响。本文对基于偏微分方程的图像去噪方法作了较为深入的研究,给出了两个新的图像去噪模型,在实验中新模型取得了比较理想的去噪效果。本文的组织结构如下：第1章概述了课题的背景、意义、发展与现状,PDE与泛函变分的关系。第2章简述了与数字图像处理相关的基本知识,主要包括数字图像的分类,噪声模型,图像质量的主、客观评价标准及两种传统的滤波方法。第3章介绍了几种经典的偏微分方程图像去噪模型。第4章作为本文的主要工作提出了两个新的图像去噪模型,在各向同性扩散模型和全变差(TV)模型的基础上,通过一个加权函数的作用,得出了一个既有效去除噪声,又很好的保持图像的纹理特征和边缘信息的新模型一,它继承了两者的优点,弥补了彼此的不足。针对二阶方程和四阶方程的特点,得到了新模型二,在平坦区域发挥了二阶方程的作用加速了平滑,在边缘区域保护边缘,在图像的渐变区域发挥了四阶方程的作用,保护了细节。实验结果表明与其他模型相比,新模型的去噪效果有很大进步。但在实际应用中,噪声的情况要比实验中更为复杂,因此,要达到比较理想的去噪效果还需要利用多种去噪方法。第5章为总结与展望。更多还原

【Abstract】 Image processing based on partial differential equations began in the 1990s, and for decades, many mature and highly efficient algorithms based on the P-M model have been proposed through the tireless efforts of many experts and scholars, for example, the Catte model, Alvarez-Lions-Morel model, Total Variation (TV) model and so on. These models are widely used in image restoration, image segmentation, image reconstruction, image analysis and so on. As a classic area of research, the key of image denoising is to remove the noise, and retain the important features, such as image edges, of the image at the same time, and to minimize the impact of subsequent image processing from noise as far as possible. This paper made a more depth study on image denoising based on partial differential equations, and two new image denoising models are given, which have achieved the ideal denoising in the simulations. This paper is organized as follows:The chapter one describes the subject background, significance and the current research situation, the relations of PDE and functional variation. In chapter two, the basic knowledge of digital image processing are briefly summarized, including digital image classification, noise models, the subjective and objective denoising evaluation criterions, and two conventional methods of filtering noise. In chapter three, several classical image denoising methods based on partial differential equations are introduced. The main work is the chapter four. Two new image denoising models are proposed, we get the first model with weighted function, based on both the isotropic diffusion model and the Total Variation (TV) model, and this new model inherits the advantages of both and makes up their disadvantages, such as removing the noise effectively and retaining the image texture and edge information. According to the characteristics of the second-order equation and the fourth-order equation, we proposed the second new model. In the flat areas of the image processing, this model plays the role of the second-order equation, accelerating the smooth and protecting the edge in the regional area. In the gradient region of the image processing, it plays the role of the fourth-order equation, protecting the details. Compared with the other models, simulation results show that the denoising results of the new models have a lot of progress. However, the situation of the noise is more complex in practice than in the simulations, therefore, many denoising methods should be used when we want to get a more ideal performance. In chapter five summary and outlook are described.更多还原