利用等距同构建立多尺度空间实体相似性度量模型
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摘要
从度量几何学的观点,建立多尺度空间实体等距同构模型。将多尺度空间实体分别看作刚性图形和弹性图形,利用等距同构不变性的概念,构建多尺度空间实体几何相似性图形距离和拓扑相似性图形距离。通过统一几何和拓扑相似性图形距离,构成一个二元集值距离,作为多尺度空间实体相似性的评价指标。通过对不同复杂性的面状和线状空间实体多尺度表达图形的几何和拓扑相似性度量实验表明,该方法能同时顾及多尺度空间实体几何和拓扑结构的改变,且符合空间实体的多尺度抽象规律。
From the perspective of metric geometry, this paper proposes an isomorphic model for multiscale spatial object representation. Firstly, the multi-scale spatial objects are considered as rigid shapes and non-rigid shapes respectively. Then, the shape distances based on geometry similarity and topology similarity are constructed by the concept of isometry invariance. Finally, the two kinds of similarity shape distances are combined to form a two-element set and are used as an evaluated criterion of multi-scale spatial object similarity. Experiments on different complexity levels of spatial line and polygon objects show that, the two similarity metrics can encode both geometrical and topological changes, and be in accord with the abstract principles of multi-scale spatial object representation.
From the perspective of metric geometry, this paper proposes an isomorphic model for multiscale spatial object representation. Firstly, the multi-scale spatial objects are considered as rigid shapes and non-rigid shapes respectively. Then, the shape distances based on geometry similarity and topology similarity are constructed by the concept of isometry invariance. Finally, the two kinds of similarity shape distances are combined to form a two-element set and are used as an evaluated criterion of multi-scale spatial object similarity. Experiments on different complexity levels of spatial line and polygon objects show that, the two similarity metrics can encode both geometrical and topological changes, and be in accord with the abstract principles of multi-scale spatial object representation.
引文
[1] Ding Hong. A Study on Spatial Similarity Theory and Calculation Model[D]. Wuhan:Wuhan University,2004(丁虹.空间相似性理论与计算模型的研究[D].武汉：武汉大学，2004)
[2] An Xiaoya,Sun Qun,Xiao Qiang,et al. A Shape Multilevel Description Method and Application in Measuring Geometry Similarity of Multi-scale Spatial Data[J]. Acta Geodaetica et Cartographica Sinica,2011,40(4):495-501(安晓亚，孙群，肖强，等.一种形状多级描述方法及在多尺度空间数据几何相似性度量中的应用[J].测绘学报，2011,40(4):495-501)
[3] Bronstein A M,Bronstein M M,Kimmel R. Topology-Invariant Similarity of Nonrigid Shapes[J].International Journal of Computer Vision,2009,81(3):281-301
[4] Xia Min,Liu Hongshen. The Shape Recognition System Based on Wavelet Descriptors and Neural Networks[J]. Computer Technology and Development,2007,17(3):106-108(夏敏，刘宏申.基于小波描述子和神经网络的形状识别[J].计算机技术与发展，2007,17(3):106-108)
[5] Osada R,Funkhouser T,Chazelle B,et al. Shape Distributions[J]. ACM Transactions on Graphics(TOG),2002,21(4):807-832
[6] Elad A,Kimmel R. On Bending Invariant Signatures for Surfaces[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence,2003,25(10):1 285-1 295
[7] Reuter M,Wolter F E,Peinecke N. Laplace-Beltrami Spectra as“Shape-DNA”of Surfaces and Solids[J]. Computer-Aided Design,2006,38(4):342-366
[8] Abbas I,Grussenmeyer P,Hottier P. Contr?le de la Planimétrie D′une Base de Données Vectorielle:Une Nouvel1e Méthode Basée sur la Distance de Hausdorff:la Méthode du Contr?le Linéaire[J].Bulletin SFPT,1995,137(1):6-11
[9] Wang Qiao,Wu Hehai. The Research on Fractal Method of Automatic Generalization of Map Polygons[J]. Journal of Wuhan Technical University of Surveying and Mapping,1996,21(1):59-63(王桥，毋河海.地图图斑群自动综合的分形方法研究[J].武汉测绘科技大学学报，1996,21(1):59-63)
[10] Ai Tinghua,Shuai Yun,Li Jingzhong. A Spatial Query Based on Shape Similarity Cognition[J]. Acta Geodaetica et Cartographica Sinica, 2009,38(4):300-310(艾廷华，帅赟，李精忠.基于形状相似性识别的空间查询[J].测绘学报，2009,38(4):300-310)
[11] Chen Zhanlong,Wu Liang,Xie Zhong,et al. Similarity Measurement of Multi-holed Regions Using Constraint Satisfaction Problem[J]. Geomatics and Information Science of Wuhan University,2018,43(5):745-751(陈占龙，吴亮，谢忠，等.利用约束满足问题进行多洞面实体相似性度量[J].武汉大学学报·信息科学版，2018,43(5):745-751)
[12] Ai Tinghua,Li Jingzhong. The Lifespan Model of GIS Data Representation over Scale Space[J]. Geomatics and Information Science of Wuhan University,2010,35(7):757-762,781(艾廷华，李精忠.尺度空间中GIS数据表达的生命期模型[J].武汉大学学报·信息科学版，2010,35(7):757-762,781)
[13] Wang Jinggeng. Topology, Nicknamed “Rubber Geometry”—One of the Extracurricular Lectures for Secondary School Students[J]. Mathematical Bulletin,1986(1):33-36(王敬庚.拓扑，外号叫“橡皮几何学”——对中学生的课外讲座之一[J].数学通报，1986(1):33-36)
[14] Liu Mingxue. On the Isometric Isomorphism of Probabilistic Metric Spaces[J]. Applied Mathematics and Mechanics,2002,23(5):548-550(刘明学.关于概率度量空间的等距同构[J].应用数学和力学，2002,23(5):548-550)
[15] Gromov M. Metric Structures For Riemannian and Non-riemannian Spaces[J]. Progr Math,1999(3):353-353
[16] Borg I, Groenen P. Modern Multidimensional Scaling:Theory and Applications[J]. Journal of Educational Measurement,2003,40(3):277-280
[17] Linial N,London E,Rabinovich Y. The Geometry of Graphs and Some of Its Algorithmic Applications[J]. Combinatorica,1995,15(2):215-245
[2] An Xiaoya,Sun Qun,Xiao Qiang,et al. A Shape Multilevel Description Method and Application in Measuring Geometry Similarity of Multi-scale Spatial Data[J]. Acta Geodaetica et Cartographica Sinica,2011,40(4):495-501(安晓亚，孙群，肖强，等.一种形状多级描述方法及在多尺度空间数据几何相似性度量中的应用[J].测绘学报，2011,40(4):495-501)
[3] Bronstein A M,Bronstein M M,Kimmel R. Topology-Invariant Similarity of Nonrigid Shapes[J].International Journal of Computer Vision,2009,81(3):281-301
[4] Xia Min,Liu Hongshen. The Shape Recognition System Based on Wavelet Descriptors and Neural Networks[J]. Computer Technology and Development,2007,17(3):106-108(夏敏，刘宏申.基于小波描述子和神经网络的形状识别[J].计算机技术与发展，2007,17(3):106-108)
[5] Osada R,Funkhouser T,Chazelle B,et al. Shape Distributions[J]. ACM Transactions on Graphics(TOG),2002,21(4):807-832
[6] Elad A,Kimmel R. On Bending Invariant Signatures for Surfaces[J]. IEEE Transactions on Pattern Analysis and Machine Intelligence,2003,25(10):1 285-1 295
[7] Reuter M,Wolter F E,Peinecke N. Laplace-Beltrami Spectra as“Shape-DNA”of Surfaces and Solids[J]. Computer-Aided Design,2006,38(4):342-366
[8] Abbas I,Grussenmeyer P,Hottier P. Contr?le de la Planimétrie D′une Base de Données Vectorielle:Une Nouvel1e Méthode Basée sur la Distance de Hausdorff:la Méthode du Contr?le Linéaire[J].Bulletin SFPT,1995,137(1):6-11
[9] Wang Qiao,Wu Hehai. The Research on Fractal Method of Automatic Generalization of Map Polygons[J]. Journal of Wuhan Technical University of Surveying and Mapping,1996,21(1):59-63(王桥，毋河海.地图图斑群自动综合的分形方法研究[J].武汉测绘科技大学学报，1996,21(1):59-63)
[10] Ai Tinghua,Shuai Yun,Li Jingzhong. A Spatial Query Based on Shape Similarity Cognition[J]. Acta Geodaetica et Cartographica Sinica, 2009,38(4):300-310(艾廷华，帅赟，李精忠.基于形状相似性识别的空间查询[J].测绘学报，2009,38(4):300-310)
[11] Chen Zhanlong,Wu Liang,Xie Zhong,et al. Similarity Measurement of Multi-holed Regions Using Constraint Satisfaction Problem[J]. Geomatics and Information Science of Wuhan University,2018,43(5):745-751(陈占龙，吴亮，谢忠，等.利用约束满足问题进行多洞面实体相似性度量[J].武汉大学学报·信息科学版，2018,43(5):745-751)
[12] Ai Tinghua,Li Jingzhong. The Lifespan Model of GIS Data Representation over Scale Space[J]. Geomatics and Information Science of Wuhan University,2010,35(7):757-762,781(艾廷华，李精忠.尺度空间中GIS数据表达的生命期模型[J].武汉大学学报·信息科学版，2010,35(7):757-762,781)
[13] Wang Jinggeng. Topology, Nicknamed “Rubber Geometry”—One of the Extracurricular Lectures for Secondary School Students[J]. Mathematical Bulletin,1986(1):33-36(王敬庚.拓扑，外号叫“橡皮几何学”——对中学生的课外讲座之一[J].数学通报，1986(1):33-36)
[14] Liu Mingxue. On the Isometric Isomorphism of Probabilistic Metric Spaces[J]. Applied Mathematics and Mechanics,2002,23(5):548-550(刘明学.关于概率度量空间的等距同构[J].应用数学和力学，2002,23(5):548-550)
[15] Gromov M. Metric Structures For Riemannian and Non-riemannian Spaces[J]. Progr Math,1999(3):353-353
[16] Borg I, Groenen P. Modern Multidimensional Scaling:Theory and Applications[J]. Journal of Educational Measurement,2003,40(3):277-280
[17] Linial N,London E,Rabinovich Y. The Geometry of Graphs and Some of Its Algorithmic Applications[J]. Combinatorica,1995,15(2):215-245