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偏微分方程在图像处理中的应用
The Application of Partial Differential Equation in Image Processing
【作者】 闵志方;
【导师】 徐长发;
【作者基本信息】 华中科技大学 , 计算数学, 2006, 硕士
【摘要】 本文主要研究的是偏微分方程在数字图像处理方面的应用。文章首先详细阐述了尺度空间的概念和性质,并以热传导方程所确定高斯尺度空间为例给出了连续尺度空间与偏微分方程的联系。接着文章由简单的线性扩散方程开始,介绍了非线性同质扩散和非线性异质扩散,不仅给出了相应的具体扩散方程模型,而且对它们在图像处理中的不同效果作了比较。然后,文章分析了梯度能量最小化与异质扩散方程的必然联系,从能量分析法切入,构造变分问题,从而得到原始模型。紧接着,文章分析了非线性滤波器与异质扩散方程的必然联系,从讨论各种扩散方程的格式开始,逐个分析其滤波原理,发现其相应扩散方程在频域滤波方面的不足,为弥补这些不足,从构造滤波器着手,以PDE的时域迭代方式实现。最后,根据对偏微分方程各项与图像处理的相关规律出发,以去噪、提升边缘、保留原图信息为目标改进了模型。对于图像处理的难点——边缘的去噪去模糊,在文章中得到较好的解决。结果表明:经过该方法处理后的图像在提升信噪比方面十分显著。本文从数学角度建立的扩散模型不仅产生了文中的新算法,而且为人们在该领域的后续研究提供了参考。本文是在塑性成型模拟及模具技术国家重点实验室开放基金的资助下进行的。文中的算法经过matlab编程实现,证实了新算法有效性。
【Abstract】 This article mainly studies the application of the digital image process based on PDE. Firstly, the article expatiates on the concept and property of scale space, and give the relationship between continuous scale space and the PDE model, for example, the heat exchange equation ascertain the gauss scale space. Following with the simple linear diffusion equation, it introduces the nonlinear isotropic diffusion and the nonlinear anisotropic diffusion. It not only presents corresponding idiographic diffusion equation models, but also compares the different effects of them in image processing.Afterward, the article analyses the inevitable relationship between the minimum energy of grads and anisotropic diffusion equations. It begins with the method of energy analysis, then constructs variational problem, gets the original model at last.Then, the article analyses the inevitable relationship between the nonlinear filters and anisotropic diffusion equations. It starts with the discussion of format of each kinds of diffusion equations, then analyses their filtering theory one by one, then discovers the deficiency of each diffusion equations in frequency domain. To remedy the deficiency, it goes on with the construction of filter, makes the filter come true with the iterative method in time domain.Finally, according to the rule between each items of PDE and image processing, it improves the original model for the goal of noise-removing, edge enhancement, remaining the original image’s information. The difficulty of image processing---edge’s noise-removing and blur-removing, will be answered preferably by this article. As the result indicate, the image processed by this method is quite notable in the aspect of SNR enhancement. This article analyses the function of each PDE’s items in image processing from the mathematical view, by which we constructs the diffusion model which not only generates the new arithmetic in the article, but also paves the way for people’s further research in this field.This dissertation is funded by the opening foundation of the state key laboratory of plastic forming simulation and die and mould technology. The arithmetic of the paper is carried out by programming in matlab, and is verified very effective.
- 【网络出版投稿人】 华中科技大学 【网络出版年期】2008年 03期
- 【分类号】TP391.41
- 【被引频次】1
- 【下载频次】718