图象空间中的拓扑结构及其性质
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摘要
数学形态学(Mathematical Morphology)是一种用于图象处理和模式识别领域的新方法。在形态学中，所考虑的图象通常被视为n维欧氏空间R~n中的集合。由于在视觉上我们无法区别一个集合和它在欧氏拓扑下的闭包，所以可以进一步假定物体的图象是R~n中的闭集。然而，实际的图象总是包含各种各样“噪音”，换句话说，实际中得到的集合总是有误差的。因此图象变换对这些误差的不敏感性对应用来说就显得极为重要。与此密切相关的一个概念就是连续性。然而，连续性是一个拓扑性质，所以为了了解形态变换的连续性，就必须在图象空间中建立一个严格的拓扑结构。形态学中，一种被称之为HM拓扑的拓扑结构已经建立，然而它并非完善。论文首先在HM拓扑的基础上，讨论了一个更自然的符合实际需要的拓扑结构。由于数学形态学变换都是由几何方式定义的，因而这些变换的性能必将与几何密切相关。更确切的说，对于解决形态学问题，几何占有十分重要的地位，它不仅是工具，也常常是目的。因此论文还研究了形态学算子的代数性质以及与之相关的拓扑和几何性质：形态学骨架的性质；凸集以及凸包的形态学运算的几何性质；凸集的Minkowski函数及其性质；Steiner公式及其推广。
The theory of Mathematical Morphology is developed for image management and pattern analysis. In actual management , the image under consideration is always considered as a set in Rn. Because we can not discern a set from its convex closure , we can consider the image as a closed set . However , the image for management usually comprise some "noisy data" , in another word , we always get some sets with error in practice. Therefore, the insensitive to those errors is important in image transformation and a relevant concept is continuity. In order to understand the continuity, we must set up an accurate geometrical and topological structure to process those sets .In the present paper , we shall discuss another more crucial topological structure on the base of HM topology which has been set up in Mathematical Morphology . Because the transformations in Mathematical Morphology are defined by means of geometry , the properties of these transformations are bound to relevant to geometry. Exactly , the geometry plays an important role in solving Mathematical Morphology problems. It is not only the tool , but the goal. In the paper , algebraical properties of morphology operation together with topological and geometrical properties are discussed : the properties of skeletons ; the operation properties of convex set and convex closure ; Minkowski function for convex set and its properties ; Steiner formulae and its developing.
The theory of Mathematical Morphology is developed for image management and pattern analysis. In actual management , the image under consideration is always considered as a set in Rn. Because we can not discern a set from its convex closure , we can consider the image as a closed set . However , the image for management usually comprise some "noisy data" , in another word , we always get some sets with error in practice. Therefore, the insensitive to those errors is important in image transformation and a relevant concept is continuity. In order to understand the continuity, we must set up an accurate geometrical and topological structure to process those sets .In the present paper , we shall discuss another more crucial topological structure on the base of HM topology which has been set up in Mathematical Morphology . Because the transformations in Mathematical Morphology are defined by means of geometry , the properties of these transformations are bound to relevant to geometry. Exactly , the geometry plays an important role in solving Mathematical Morphology problems. It is not only the tool , but the goal. In the paper , algebraical properties of morphology operation together with topological and geometrical properties are discussed : the properties of skeletons ; the operation properties of convex set and convex closure ; Minkowski function for convex set and its properties ; Steiner formulae and its developing.
引文
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[3] Gao zhi min and Y. Yasui. The continuity on pattern recognitions, Math. Japonica, Vol.53, No. 2(2001), 311-316.
[4] J. Serra. Image analysis and mathematical morphology, Academic Press, London, 1982.
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[7] 唐常青等，“数学形态学方法及其应用”，科学出版社，1990．
[8] 崔屹，“图象处理与分析专著：数学形态学方法及应用”，科学出版社，2000．
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[14] Ronse, C., et al., The algebraic basis of mathematical morphology. CVGIP, Vol. 51, No. 4, 1990.
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[19] Song, J. S. and Delp, E.J., The analysis of morphological filters with multiple structuring elements, CVGIP, Vol. 35, NO. 3, 1986.
[20] Skolnick, M.M., Applications of morphological transformations to the analysis of two-dimensional celectrophoretic gels of biological materials, CVGIP, Vol. 35, No. 3. 1986.
[21] Werman, M. and Peley, S., Multiresolution texture signature using minmax operations, Proc. 7ICPR, 1984.
[22] Yamado, H. and Kaswand, T., Transparent object extraction from regular textured backgrounds by using binary parallel operations, GVGI, Bol. 40, No. 1, 1987.
[23] Yokoi, S., et al. m On generalized distance transformation of digilized pictures, IEEE Trans. on PAMI, Vol. 3, No. 4, 1981.
[24] Zhuang, X. H. and Haralick, R.M., Morphological structuring element decomposition, CVGIP, Vol. 35, No. 3, 1986.
[25] 丁晓青，图象形态谱分析方法与物体形状表示，电子学报，Vol．17，No．6，1989．
[26] 吴敏金，图形形态变换与图象形态分析，电子学报，Vol，18，No．3，1990．
[27] 吴敏金等，汉字S-E坐标提取及其骨架重构，华东师大学报，1986．
[28] 万嘉若等，广义结构运算，自动化学报，Vol．12，No．3，1986．
[29] 任德麟，积分几何引论，上海科技出版社，1988．
[30] 苏步青等，微分几何，人民教育出版社，1979．
[2] Gao zhi min and Y. Yasui. Topological structure for the space consisting of images, Math. Japonica. Vol. 53, No. 2(2001), 305-310.
[3] Gao zhi min and Y. Yasui. The continuity on pattern recognitions, Math. Japonica, Vol.53, No. 2(2001), 311-316.
[4] J. Serra. Image analysis and mathematical morphology, Academic Press, London, 1982.
[5] 吴敏金，“图象形态学”，上海科学技术文献出版社，1991．
[6] 龚炜，石青云等，“数字空间中的数学形态学-理论及应用”，科学出版社，1990．
[7] 唐常青等，“数学形态学方法及其应用”，科学出版社，1990．
[8] 崔屹，“图象处理与分析专著：数学形态学方法及应用”，科学出版社，2000．
[9] Arcelli, C, al, From local maxima to connected skeletons, IEEE Trans. on PAMI, 1981.
[10] Dill, A, R. et, al., Multiple resolution skeletons, IEEE Trans. on PAMI, 1987.
[11] Maragos, P., et al., Morphological skeleton representation and coding of binary imager, IEEE Trans. on ASSP, Vol. 34,1986.
[12] Maragos, P., et al., A representation theory for morphological image and signal processing, IEEE Trans. on PAMI, Vol. 11, No. 6, 1989.
[13] Piper, L. j., et al., Erosion and dilation of binary images using interval coding, Pattern Recognition, Vol. 9, No. 3, 1989.
[14] Ronse, C., et al., The algebraic basis of mathematical morphology. CVGIP, Vol. 51, No. 4, 1990.
[15] Santalo, K.a., Integral Geometry and Geometric Probability, Addision Wesley, Reading, Mass, 1976.
[16] J.Serra and B, Lay, Square to hexagenal lattice conversion, Signal Processing. Bol. 9, No. 1, 1985.
[17] Shapiro, L. G., et al., Shape recognition with mathematical morphology, Proc. 8ICPR, 1986.
[18] Shin, F. Y. and Mitchell, Q.R., Automated fast recognition and location of arbitrarily shaped objects by image morphology, Proc. CVPRC, 1988.
[19] Song, J. S. and Delp, E.J., The analysis of morphological filters with multiple structuring elements, CVGIP, Vol. 35, NO. 3, 1986.
[20] Skolnick, M.M., Applications of morphological transformations to the analysis of two-dimensional celectrophoretic gels of biological materials, CVGIP, Vol. 35, No. 3. 1986.
[21] Werman, M. and Peley, S., Multiresolution texture signature using minmax operations, Proc. 7ICPR, 1984.
[22] Yamado, H. and Kaswand, T., Transparent object extraction from regular textured backgrounds by using binary parallel operations, GVGI, Bol. 40, No. 1, 1987.
[23] Yokoi, S., et al. m On generalized distance transformation of digilized pictures, IEEE Trans. on PAMI, Vol. 3, No. 4, 1981.
[24] Zhuang, X. H. and Haralick, R.M., Morphological structuring element decomposition, CVGIP, Vol. 35, No. 3, 1986.
[25] 丁晓青，图象形态谱分析方法与物体形状表示，电子学报，Vol．17，No．6，1989．
[26] 吴敏金，图形形态变换与图象形态分析，电子学报，Vol，18，No．3，1990．
[27] 吴敏金等，汉字S-E坐标提取及其骨架重构，华东师大学报，1986．
[28] 万嘉若等，广义结构运算，自动化学报，Vol．12，No．3，1986．
[29] 任德麟，积分几何引论，上海科技出版社，1988．
[30] 苏步青等，微分几何，人民教育出版社，1979．