Morse理论在图像边缘检测中的应用研究

【作者】 宋飞；

【导师】 邹建成；

【作者基本信息】 北方工业大学 ， 应用数学， 2011， 硕士

【摘要】 Morse理论是微分拓扑中非常有用的工具,它也是近些年来人们研究的热点之一。与此同时Morse理论也帮助人们解决了一系列数学其他分支中比较困难和艰深的问题。利用Morse理论提供的大范围分析手段和对流形整体分解的思想,人们可以对一般微分流形的几何和拓扑性质有更深层次的了解。近些年来,Morse理论在偏微分方程理论和Lie群理论中的广泛应用引起了更多人的注意。但是令人遗憾的是,Morse理论与另一门很有用处的学科：单复变函数论并没有很好的结合。本文试图利用构造一维复流形上典型的Morse函数并且利用Morse理论对微分流形整体结构的影响给黎曼曲面上的一个重要定理：单值化定理一个新的证明。单值化定理对单复变函数论的帮助是非常巨大的,通过万有覆盖的相关性质,我们可以对微分流形的非常好的基本模型Riemann曲面和在数学上具有非常良好结构的Teichmuller空间理论进行深入分析。从中可以看出Morse理论对黎曼曲面复结构的影响。另外,本文还对Morse引理进行了推广。我们给出了一个比较容易验证的充分条件,将原本仅仅针对一个Morse函数成立的局部规范化形式推广至多个函数同时成立的情形,并给出了相关的证明。这样一来,Morse引理就在此条件下被较好的推广了。本文的基本工作包括如下几个方面：第一,系统介绍了Morse理论。第二,利用Morse理论给单值化定理一个相对简化的证明。第三,给定了一个充分条件,并在此条件下推广了Morse引理。第四,利用Morse引理给出了一条数字图像中物体边缘线右等价分类的描述,从而使其得到规范化。第五,用一个实例简单说明了上述理论。更多还原

【Abstract】 Morse theory is a useful tool in differential topology. Recently, Morse theory becomes a hot spot of mathematical research. Meanwhile, Morse theory solved a series of problems that is difficult in other mathematical subjects. Morse theory provides lots of analytical methods and the overall decomposition approach, with which people can gain deeper understanding of a differential manifold with its geometrical and topological properties. Recently, Morse theory’s applications in the theory of partial differential equations and Lie groups gained more people’s attention.But unfortunately, Morse theory and another very useful subject in mathematical: single complex analysis theory had not significant connection. This paper attempts to use the methods that construct a Morse function on one-dimensional complex manifold and by noticing the influence in the whole structure of the manifold caused by Morse theory, We reproved a very important theorem on Riemann surface: Uniformization theorem,which is very important in the field of single-variable complex analysis. With the help of universal cover, we may be able to give a good model for differential structure of Riemann surface and Teichmuller space which is of elegant structure. It shows the obvious influence on Riemann surface’s complex structure caused by Morse function.In addition, this paper also put forward a generalized Morse lemma and proved it. We promoted the Morse lemma which originally just effective in the situation of single-variable to the situation that is effective in several variables. We put forward a sufficient condition which is relatively easy to validate, and we gave its proof. Then Morse lemma has been promoted dramatically.The basic work of this paper includes the following aspects:(1)We introduced Morse theory by detail.(2)We reproved Unifromization theorem by Morse theory.(3)We popularized Morse lemma and put forward a sufficient condition for its promotion.(4) We put forward a right-equivalent descriptions on the edge of an object in the image by using Morse lemma so that we can make it normalized. (5)We demonstrated our algorithm by an example.更多还原