Morse理论在图像边缘检测中的应用研究
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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。
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.
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。
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.
引文
[1]J.Milnor, Morse Theory, Princeton University Press,1963, pp.43-115.
[2]J.Zeng, S.J Li, An Ekeland's variational principle for set-valued mappings with applications, Journal of Computational and Applied Mathematics, Vol.230(2),2009, pp.477-484.
[3]L.V Alfors, Sario L, Riemann surfaces, Princeton:Prince on University Press19 60,pp.99-113.
[4]David J.Kriegman, Jean Ponce, Computing exact aspect graphs of curved objects:solids of revolution, International Journal of Computer Vision,5:2,1990,pp.119-135.
[5]V.I.Arnold,V.V.Goryunov,O.Lyashko,SingularityTheroy Ⅱ,Vol.39.Springer-Verlag, 1993,pp.44-74.
[6]Maunder, C.R.F., Algebraic Topology, Cambridge University Press,1980,pp.55-1 12.
[7]张筑生,微分拓扑新讲,北京：北京大学出版社,2003,pp.32-77.
[8]JAMES J.CLARK, Singularity theory and phantom edges in scale space, Pattern analysis and machine intelligence, VOL.10(5), September 1988,pp.720-728.
[9]李忠,复分析导引,北京：北京大学出版社,2005,pp.59-88.
[10]张恭庆,郭懋正,泛函分析讲义(下册),北京：北京大学出版社,2004,pp.25-56.
[11]陈维桓,微分几何,北京：北京大学出版社,2006,pp.55-98.
[12]尤承业,基础拓扑学讲义,北京：北京大学出版社,2005,pp.56-64.
[13]宋飞,Morse理论在单复变函数论中的简单应用,数学及其应用新进展-2010年全国数学与信息科学研究生学术研讨会论文集,2010.
[14]丘维声,高等代数(第二版)上册,北京：高等教育出版社,2002.3,pp.77-145.
[15]冈萨雷斯,数字图像处理(Matlab中文版),北京：电子工业出版社,2005.9,pp.11-97.
[16]L.D.Griffin,M.Lillholm, Scale-Space 2003,Springer-Verlag Berlin Heidelberg 2003,2003,pp.33-43.
[17]Tomoyuki Nieda, Alexander Pasko, Detection of Critical Points for Shape Meta morphosis Animation, Proceedings of the 10th International Multimedia Modeling Con ference,2004,pp.57-63.
[18]Yoshihisa Shinagawa,L. Kunii, A Reeb graph-based representation for non-sequential construction of topologically complex shapes, Computer & Graphics ,Vol.22(2～3),1998,pp.255-268.
[19]F. Lazarus, A. Verroust, Level set diagrams of polyhedral objects, Proc. Siggraph , August 2001,pp.130-140.
[20]John C Hart, C code for intersection and surface normal, Texturing and Model,Vol.1(3),2003,pp.627-629.
[21]Zou Jiancheng, Finite determination and universal unfolding of bifurcation problems, Acta Mathematical Scientist,数学物理学报英文版,1998, Vol.18 (4), pp.399-403.
[22]James Murray, John C Hart, On the local and general influence on the body, of increased and diminished atmospheric pressure, The Lancet, Vol.23 (604),1985,pp.909-917.
[23]伍鸿熙,陈志华,紧黎曼曲面引论,北京：科学出版社,1983,pp.22-115.
[24]余建明,邹建成,奇点理论浅引,数学进展,Vol.27,NO.4,1998.8,pp.56-60
[25]Tosiyasu L.Kunii, Chapter 20 Cognitive technology and differential topology: the importance of shape features, Advances in Psychology, Vol.113(2),1996,pp.337-345.
[26]Marek. Kossowski, The Boy-Gauss-bonnet Theorems for c(infinite)-Singular Surfaces with Limiting Tangent Bundle, Annals of Global Analysis and Geometry ,2002,pp.19-29.
[27]Nainn-Ping Ke, Singularity theory and stability robustness analysis, UKACC International Conference on Control'98,Vol.1(3),1998,pp.75-79.
[28]A. Gullstrand,"Zur kenntnis der kreispunkte," Acta Mathematical,Vol. 29,1999,pp.55-104.
[29]Thurston W P, The Geometry and Topology of Three Manifolds, Princeton Lectu re Notes,1979,pp.55-99.
[30]L. Kunii, Modeling contact of two complex objects,with an application to characterizing dental articulation, Computer& Graphics,Vol.19(1),1995,pp.21-28.
[31]姜伯驹,同调论,北京：北京大学出版社,2007,pp.113-186.
[32]M.D Carmo, Differential Geometry of Curves and Surface, Prentice-Hall:New Jersey,1976, pp.35-55.
[2]J.Zeng, S.J Li, An Ekeland's variational principle for set-valued mappings with applications, Journal of Computational and Applied Mathematics, Vol.230(2),2009, pp.477-484.
[3]L.V Alfors, Sario L, Riemann surfaces, Princeton:Prince on University Press19 60,pp.99-113.
[4]David J.Kriegman, Jean Ponce, Computing exact aspect graphs of curved objects:solids of revolution, International Journal of Computer Vision,5:2,1990,pp.119-135.
[5]V.I.Arnold,V.V.Goryunov,O.Lyashko,SingularityTheroy Ⅱ,Vol.39.Springer-Verlag, 1993,pp.44-74.
[6]Maunder, C.R.F., Algebraic Topology, Cambridge University Press,1980,pp.55-1 12.
[7]张筑生,微分拓扑新讲,北京：北京大学出版社,2003,pp.32-77.
[8]JAMES J.CLARK, Singularity theory and phantom edges in scale space, Pattern analysis and machine intelligence, VOL.10(5), September 1988,pp.720-728.
[9]李忠,复分析导引,北京：北京大学出版社,2005,pp.59-88.
[10]张恭庆,郭懋正,泛函分析讲义(下册),北京：北京大学出版社,2004,pp.25-56.
[11]陈维桓,微分几何,北京：北京大学出版社,2006,pp.55-98.
[12]尤承业,基础拓扑学讲义,北京：北京大学出版社,2005,pp.56-64.
[13]宋飞,Morse理论在单复变函数论中的简单应用,数学及其应用新进展-2010年全国数学与信息科学研究生学术研讨会论文集,2010.
[14]丘维声,高等代数(第二版)上册,北京：高等教育出版社,2002.3,pp.77-145.
[15]冈萨雷斯,数字图像处理(Matlab中文版),北京：电子工业出版社,2005.9,pp.11-97.
[16]L.D.Griffin,M.Lillholm, Scale-Space 2003,Springer-Verlag Berlin Heidelberg 2003,2003,pp.33-43.
[17]Tomoyuki Nieda, Alexander Pasko, Detection of Critical Points for Shape Meta morphosis Animation, Proceedings of the 10th International Multimedia Modeling Con ference,2004,pp.57-63.
[18]Yoshihisa Shinagawa,L. Kunii, A Reeb graph-based representation for non-sequential construction of topologically complex shapes, Computer & Graphics ,Vol.22(2～3),1998,pp.255-268.
[19]F. Lazarus, A. Verroust, Level set diagrams of polyhedral objects, Proc. Siggraph , August 2001,pp.130-140.
[20]John C Hart, C code for intersection and surface normal, Texturing and Model,Vol.1(3),2003,pp.627-629.
[21]Zou Jiancheng, Finite determination and universal unfolding of bifurcation problems, Acta Mathematical Scientist,数学物理学报英文版,1998, Vol.18 (4), pp.399-403.
[22]James Murray, John C Hart, On the local and general influence on the body, of increased and diminished atmospheric pressure, The Lancet, Vol.23 (604),1985,pp.909-917.
[23]伍鸿熙,陈志华,紧黎曼曲面引论,北京：科学出版社,1983,pp.22-115.
[24]余建明,邹建成,奇点理论浅引,数学进展,Vol.27,NO.4,1998.8,pp.56-60
[25]Tosiyasu L.Kunii, Chapter 20 Cognitive technology and differential topology: the importance of shape features, Advances in Psychology, Vol.113(2),1996,pp.337-345.
[26]Marek. Kossowski, The Boy-Gauss-bonnet Theorems for c(infinite)-Singular Surfaces with Limiting Tangent Bundle, Annals of Global Analysis and Geometry ,2002,pp.19-29.
[27]Nainn-Ping Ke, Singularity theory and stability robustness analysis, UKACC International Conference on Control'98,Vol.1(3),1998,pp.75-79.
[28]A. Gullstrand,"Zur kenntnis der kreispunkte," Acta Mathematical,Vol. 29,1999,pp.55-104.
[29]Thurston W P, The Geometry and Topology of Three Manifolds, Princeton Lectu re Notes,1979,pp.55-99.
[30]L. Kunii, Modeling contact of two complex objects,with an application to characterizing dental articulation, Computer& Graphics,Vol.19(1),1995,pp.21-28.
[31]姜伯驹,同调论,北京：北京大学出版社,2007,pp.113-186.
[32]M.D Carmo, Differential Geometry of Curves and Surface, Prentice-Hall:New Jersey,1976, pp.35-55.