空间方向与距离关系结合方法的研究
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摘要
目前,单一空间关系的研究已经有了非常丰富的积累,将多个空间关系结合起来进行表示和推理,有助于提高模型的表达能力及推理算法的准确性,是当前空间推理研究的热点,已有的结合方法主要集中在拓扑和方位、距离和大小、距离和方向以及位置和方向等方面。关于距离和方向关系结合的研究主要是基于点对象的,尚缺乏能够统一处理点、线段、区域等空间对象之间距离和方向关系的研究模型。
本文研究了适合于表示点、线段、区域等空间对象方向、距离结合关系模型的表示及推理。主要工作包括:(1)总结并分析了现有空间方向、距离关系模型的研究现状,详细介绍了空间对象方向关系模型中的方向关系统计模型及定性距离的表示方法;(2)结合文献[21]的方向关系模型和Clementini等提出的点对象距离关系模型,提出了一种同时适合于表示点、线段、区域等空间对象方向、距离结合关系的表达模型-SD模型;(3)在此基础上给出了基于SD模型的点对象方向、距离结合关系推理算法-CombinationDD算法;(4)基于分解综合的思想,进一步给出SD模型中区域之间方向距离结合关系推理算法-CombinationDR算法;(5)设计并实现了基于SD模型的演示系统。
本文提出的结合方向、距离关系的SD模型,能够对空间对象的方位关系进行更准确、详细地描述,既能定性地描述空间对象在相同距离关系下的方向差异,又能定量地表示出空间对象在同一方向关系下的差异。良好的可扩展性及统一的表达方法,使得我们提出的SD模型能够有效地提高空间信息表达的完备性和推理算法的准确度,更精确地对空间对象间的方向和距离关系进行表示及推理,对空间关系的结合方式及推理方法进行了有益的扩展与补充。本文的研究工作增强了对象之间空间关系的可区分性,提高了空间分析方法的准确性,对GIS对象间空间关系的分析和查询等有一定的理论意义与应用价值。
Spatial relation is the relation with spatial character between spatial objects. It usually consists of topological relations, order relations and metric relations. Theses relations are the bases of spatial data organizing, querying, analyzing and reasoning. Spatial relation is one of the most important theoretical problems in the fields of spatial reasoning, Geographical Information System, spatial databases and computer vision.
All spatial relations are not entirely independent of each other. They depict correlation characters of spatial objects in different aspects. There are very close connections among these spatial relations, and we need to consider multiple spatial relations in many practical applications. Now the researches on single spatial relation have already rich accumulation, it is a hot issue on current spatial reasoning researches that how to combine multiple spatial relations to represent spatial objects. Most of the existing methods focus on the combination of topology and orientation, distance and size, distance and direction, position and direction. However, there are less works on the combination of direction relations and distance relations. Most of the research about combination of direction and distance is based on the point reference model, very few relate to the region. We are in need of the direction relations and distance relations model which can deal with the point objects, line segments, regions and other spatial objects. In practical, it is a hot issue on current spatial reasoning researches that how to combine the combination of topology and orientation, distance and size, distance and direction, position and direction relations to represent complicated regions.
In this paper, we study and discussion about the presentation and reasoning of direction relations and distance relations between spatial objects. Firstly, we summarized and analyzed the study of spatial relation among objects in recent years. We describe the statistical model for directional relations between spatial objects and qualitative distance in detail. Base on the model proposed by reference 21, we improved the recently combination of direction and distance model. We made the reasoning model--SD model which can deal with spatial object like point, line segment and regions. In addition, we made the reasoning algorithm based on combination of direction relations and distance relations of point—CombinationDD algorithm. Based on the precise degree of research, we partition the line segment and regions into pieces based on the precise degree of research. So as to use the high efficiency point object research method. In the end, we use the statistic method to calculate the distance and direction as the result of spatial reasoning. The method can effectively improve the maturity of spatial information and precise of reasoning. Base the statistical model for directional relations, the new model can use the similitude expressions to describe points, line segment and regions. In the distance system, the reasoning method made for points objects can do better perform on line segment and regions.
The main work and results included in this paper are as follows:
(1) It is a short introduction on background and significance of this paper. We summarized and analyzed the state of arts on spatial relation among objects in recent years.
(2) We briefly introduce the theory background of spatial direction relations model and attributes. According to the need of research, comparing the characteristics of the above model, briefly describe the combination methods of direction and distance relations.
(3) Based on the statistical model for directional relations, we present a new model to describe the direction relations and distance relations between spatial objects. In order to illustrate the direction relations more detailed, effectively to improve the expression accuracy of the direction relations, made the direction and distance relations of point-to-point be easily extended to the regions. Combined with the statistical model of direction relations, starting from the space object, we improved the distance relations and direction relations research model that proposed by Clementini et al. by means of introducing the azimuth. We described in detail about the combination method of direction relations and distance relations when the spatial object is the point, line target, region, respectively. We also defined the basic concepts, symbols, definitions and the formula for calculating the correlation. Using the specific relationship between the azimuth and vector, we expanded the model based on the point of direction, distance relations to that based on the line segment and region, so as to form a unified the inference algorithm for the combination of direction and distance relations for the next step.
(4) We can get a better description of the spatial relations between objects by combination the statistical model of direction relations with distance relations. Depending on the situation, the direction and distance between point to point is divided into: the synthesis of the direction relationship with the same distance, the synthesis of the direction relationship with the different distances, the synthesis of the direction relationship with any angle direction. We proposed inference algorithm combining distance and direction of points based on an improved model including the basic idea of algorithm and ADL description language.
To analyze the combination method of the direction and distance between the space targets based on the idea of decomposition and synthesis. First of all, according to the level of detail, we decomposed the space target into smaller basic units. Then compute the distance and direction between these small units and that is, a set of distance and direction, or as a distribution. Finally, giving the statistical description of the distance and direction of this group, we could draw the distribution of entire space in the distance and direction, and make the reasoning and verification for further.
Doing research from space target itself, we can acquire the representation and inference more accurately to the distance and direction between space targets, so as to expand and add useful on the combination of spatial relations and reasoning methods.
(5) We designed and implemented the system based on the improvement of representation and reasoning model of combination direction relations and distance relations--SD model. This system is based on the MVC design ideas, and uses Java 5.0 and SWT technology. The modules of this system have low coupling between each other, and the interior of the modules has stronger cohesion. The whole system is of high efficiency.
The measurements of combination direction relations and distance relations model proposed in this paper can describe position relations of spatial objects more accurately and detailed. The combination method of direction relations and distance relations between spatial objects can not only distinguish different distance relations of spatial objects qualitatively, but also quantitative difference between spatial objects can be made under the same direction. This representation structure establishes good foundation for formal representation and reasoning of spatial objects. In a word, the study results of this paper have both theoretical and practical benefits, it can be applied to represent and analyze the spatial relations among objects in spatial reasoning, geographic information system.
本文研究了适合于表示点、线段、区域等空间对象方向、距离结合关系模型的表示及推理。主要工作包括:(1)总结并分析了现有空间方向、距离关系模型的研究现状,详细介绍了空间对象方向关系模型中的方向关系统计模型及定性距离的表示方法;(2)结合文献[21]的方向关系模型和Clementini等提出的点对象距离关系模型,提出了一种同时适合于表示点、线段、区域等空间对象方向、距离结合关系的表达模型-SD模型;(3)在此基础上给出了基于SD模型的点对象方向、距离结合关系推理算法-CombinationDD算法;(4)基于分解综合的思想,进一步给出SD模型中区域之间方向距离结合关系推理算法-CombinationDR算法;(5)设计并实现了基于SD模型的演示系统。
本文提出的结合方向、距离关系的SD模型,能够对空间对象的方位关系进行更准确、详细地描述,既能定性地描述空间对象在相同距离关系下的方向差异,又能定量地表示出空间对象在同一方向关系下的差异。良好的可扩展性及统一的表达方法,使得我们提出的SD模型能够有效地提高空间信息表达的完备性和推理算法的准确度,更精确地对空间对象间的方向和距离关系进行表示及推理,对空间关系的结合方式及推理方法进行了有益的扩展与补充。本文的研究工作增强了对象之间空间关系的可区分性,提高了空间分析方法的准确性,对GIS对象间空间关系的分析和查询等有一定的理论意义与应用价值。
Spatial relation is the relation with spatial character between spatial objects. It usually consists of topological relations, order relations and metric relations. Theses relations are the bases of spatial data organizing, querying, analyzing and reasoning. Spatial relation is one of the most important theoretical problems in the fields of spatial reasoning, Geographical Information System, spatial databases and computer vision.
All spatial relations are not entirely independent of each other. They depict correlation characters of spatial objects in different aspects. There are very close connections among these spatial relations, and we need to consider multiple spatial relations in many practical applications. Now the researches on single spatial relation have already rich accumulation, it is a hot issue on current spatial reasoning researches that how to combine multiple spatial relations to represent spatial objects. Most of the existing methods focus on the combination of topology and orientation, distance and size, distance and direction, position and direction. However, there are less works on the combination of direction relations and distance relations. Most of the research about combination of direction and distance is based on the point reference model, very few relate to the region. We are in need of the direction relations and distance relations model which can deal with the point objects, line segments, regions and other spatial objects. In practical, it is a hot issue on current spatial reasoning researches that how to combine the combination of topology and orientation, distance and size, distance and direction, position and direction relations to represent complicated regions.
In this paper, we study and discussion about the presentation and reasoning of direction relations and distance relations between spatial objects. Firstly, we summarized and analyzed the study of spatial relation among objects in recent years. We describe the statistical model for directional relations between spatial objects and qualitative distance in detail. Base on the model proposed by reference 21, we improved the recently combination of direction and distance model. We made the reasoning model--SD model which can deal with spatial object like point, line segment and regions. In addition, we made the reasoning algorithm based on combination of direction relations and distance relations of point—CombinationDD algorithm. Based on the precise degree of research, we partition the line segment and regions into pieces based on the precise degree of research. So as to use the high efficiency point object research method. In the end, we use the statistic method to calculate the distance and direction as the result of spatial reasoning. The method can effectively improve the maturity of spatial information and precise of reasoning. Base the statistical model for directional relations, the new model can use the similitude expressions to describe points, line segment and regions. In the distance system, the reasoning method made for points objects can do better perform on line segment and regions.
The main work and results included in this paper are as follows:
(1) It is a short introduction on background and significance of this paper. We summarized and analyzed the state of arts on spatial relation among objects in recent years.
(2) We briefly introduce the theory background of spatial direction relations model and attributes. According to the need of research, comparing the characteristics of the above model, briefly describe the combination methods of direction and distance relations.
(3) Based on the statistical model for directional relations, we present a new model to describe the direction relations and distance relations between spatial objects. In order to illustrate the direction relations more detailed, effectively to improve the expression accuracy of the direction relations, made the direction and distance relations of point-to-point be easily extended to the regions. Combined with the statistical model of direction relations, starting from the space object, we improved the distance relations and direction relations research model that proposed by Clementini et al. by means of introducing the azimuth. We described in detail about the combination method of direction relations and distance relations when the spatial object is the point, line target, region, respectively. We also defined the basic concepts, symbols, definitions and the formula for calculating the correlation. Using the specific relationship between the azimuth and vector, we expanded the model based on the point of direction, distance relations to that based on the line segment and region, so as to form a unified the inference algorithm for the combination of direction and distance relations for the next step.
(4) We can get a better description of the spatial relations between objects by combination the statistical model of direction relations with distance relations. Depending on the situation, the direction and distance between point to point is divided into: the synthesis of the direction relationship with the same distance, the synthesis of the direction relationship with the different distances, the synthesis of the direction relationship with any angle direction. We proposed inference algorithm combining distance and direction of points based on an improved model including the basic idea of algorithm and ADL description language.
To analyze the combination method of the direction and distance between the space targets based on the idea of decomposition and synthesis. First of all, according to the level of detail, we decomposed the space target into smaller basic units. Then compute the distance and direction between these small units and that is, a set of distance and direction, or as a distribution. Finally, giving the statistical description of the distance and direction of this group, we could draw the distribution of entire space in the distance and direction, and make the reasoning and verification for further.
Doing research from space target itself, we can acquire the representation and inference more accurately to the distance and direction between space targets, so as to expand and add useful on the combination of spatial relations and reasoning methods.
(5) We designed and implemented the system based on the improvement of representation and reasoning model of combination direction relations and distance relations--SD model. This system is based on the MVC design ideas, and uses Java 5.0 and SWT technology. The modules of this system have low coupling between each other, and the interior of the modules has stronger cohesion. The whole system is of high efficiency.
The measurements of combination direction relations and distance relations model proposed in this paper can describe position relations of spatial objects more accurately and detailed. The combination method of direction relations and distance relations between spatial objects can not only distinguish different distance relations of spatial objects qualitatively, but also quantitative difference between spatial objects can be made under the same direction. This representation structure establishes good foundation for formal representation and reasoning of spatial objects. In a word, the study results of this paper have both theoretical and practical benefits, it can be applied to represent and analyze the spatial relations among objects in spatial reasoning, geographic information system.
引文
[1]刘大有,胡鹤,王生生,谢琦.时空推理研究进展[J].软件学报, 2004,15(8): 1142-1149.
[2]谢琦,刘大有,虞强源,陈娟.定性方向关系模型研究进展[J].计算机科学, 2006, 33(11): 5-9.
[3] Frank AU. Qualitative spatial reasoning: Cardinal directions as an example[J]. International Journal of Geographical Information Systems, 1996, 10(3): 269-290.
[4] Hernandez Daniel, Clementini Eliseo, Felice Paolino Di. Qualitative Distances[A]. In: Lecture Notes in Computer Science 988. Pisa: Springer-Verlag, 1995: 45-57.
[5] Frank AU. Qualitative spatial reasoning about distances and directions in geographic space [J]. Jurnal of Visual Languages and Computing, 1992, 3(4): 343-371.
[6]曹菡,陈军.方向关系与距离关系的定性描述与推理[J].西安石油学院学报(自然科学版), 2001, 16(1): 68-72.
[7]陈军,赵仁亮. GIS空间关系的基本问题与研究进展[J].测绘学报, 1999, 28(2): 95-102.
[8] Zimmermann K. Enhancing qualitative spatial reasoning-combining orientation and distance[C]. Proceeding of COST'93, 1993: 69-76.
[9] L Latecki, R Rohrig. Orientation and Qualitative Angle for Spatial Reasoning[C]. Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI-93), 1993: 1544-1549.
[10] Hong JH, Egenhofer MJ, Frank AU. On the Robustness of Qualitative Distance and Direction Reasoning[C]. Annual Convention Exposition Technical Papers ACSM/ASPRS'95. Auto-Carto, 1995, 12(4): 301-310.
[11] Zimmermann K. Measuring without Measures: The Delta Calculus[C]. Spatial Information Theory: A theoretical basis for GIS, Proceeding of COSIT'95, 1995, 998: 59-68.
[12] Clementini E, PD Felice, Hernandez. Qualitative representation of positional information[J]. Artificial Intelligence, 1997, 95(2): 317-356.
[13] Ligozat G. Reasoning about cardinal directions[J]. Journal of Visual Languages and Computing, 1998, 9(1): 23-44.
[14] MT Escrig, F Toledo. Qualitative Spatial Reasoning: theory and practice: Application to Robot Navigation[M]. Hamburg: IOS Press, 1998.
[15] H Sturm, NY Suzuki, F Wolter, M Zakharyaschev. Semi-Qualitative Reasoning about Distances[C]. Logics in Artificial Intelligence. Proceeding of JELIA 2000, 2000: 37-56.
[16] Moratz R, Renz J, Wolter D. Qualitative spatial reasoning about line segments[C]. Proceedings of the 14th European Conference on Artifical Intelligence (ECAI 2000). Amsterdam: IOS Press, 2000: 234-238.
[17] Goyal RK, Egenhofer MJ. Cardinal Directions between Extended Spatial Objects[C]. IEEE Transactions on Data Knowledge and Data Engineering, 2000.
[18] Goyal R, Egenhofer MJ. Consistent queries over cardinal directions across different levels of detail[C]. Proceedings of the 11th International Workshop on Database and Expert Systems Applications, 2000: 876-880.
[19]王生生,刘大有,谢琦,王新颖.集成多方面信息的定性空间推理及应用[J].软件学报, 2003, (14): 1857-1862.
[20]邓敏,刘文宝,李俊杰,孙电.矢量GIS空间方向关系的演算模型[J].遥感学报, 2006, 10(6): 821-828.
[21] Deng M, Li ZL. A statistical model for directional relations between spatial objects[J]. GEOINFORMATICA, 2007, 12(2): 193-217.
[22] Nedas K, Egenhofer MJ, Wilmsen D. Metric Details of Topological Line-Line Relations[J]. International Journal of Geographical Information Science, 2007, 21(1): 21-48.
[23] Chen Xiaoyong, Doihara Takeshi, Nasu Mitsuru. Spatial Relations of Distances between Arbitrary Objects in 2D/3D Geographic Spaces based on the Hausdorff Metric[C]. In: The Proc. of towards three Dimensional, Temporal and Dynamic Spatial Data Modeling and Analysis. LIESMARS’95, 1995: 30-41.
[24] Gahegan Mark. Proximity Operators for Qualitative Spatial Reasoning[C]. In: Lecture Notes in Computer Science 988. Pisa: Springer-Verlag, 1995: 31-44.
[25] Frank AU. Qualitative spatial reasoning about cardinal directions[C]. Proceedings of Austrian Conference on Artificial Intelligence, 1991: 157-167.
[26] Hernandez D. Relative representation of spatial knowledge: The 2-D case[J]. Cognitiveand Linguistic Aspects of Geographic Space, 1991: 373-385.
[27] Hernandez D. Qualitative representation of spatial knowledge[M]. Vol. Lecture notes in Artificial Intelligence, Berlin: Springer, 1994.
[28] Freksa C. Using orientation information for qualitative spatial reasoning[C]. Proceedings of International Conference on Theories and Methods of Spatial-Temporal Reasoning in Geographic Space, 1992: 162-178.
[29] Zimmermann K. Enhancing qualitative spatial reasoning-combining orientation anddistance[C]. Proceedings of COSIT'93, 1993: 69-76.
[30] Zimmermann K, C Freksa. Qualitative spatial reasoning using orientation, distance, and path knowledge[J]. Applied Intelligence, 1996, 6(1): 49-58.
[31] Papadias D, T Sellis. Qualitative Representation of Spatial Knowledge in Two Dimensional Space[J]. VLDB Journal, 1994, 3(4): 479-516.
[32] Mukerjee A, Joe GA. qualitative model for space[C]. Proceedings 8th National Conference on Artificial Intelligence, 1990: 721-727.
[33]杜世宏,王桥,杨一鹏.一种定性细节方向关系的表达模型[J].中国图象图形学报, 2004, 9(12): 1496-1503.
[34]邓敏,张燕,李光强.空间方向关系描述模型及其GIS应用分析[J].计算机工程与应用, 2008, 44(7): 37-40.
[35]张恭庆,林源渠.泛函分析讲义[M].北京:北京大学出版社, 2004.
[36] Eliseo Clementini, Paolino Di Felice. Qualitative representation of positional information[J]. Artificial Intelligence, 1997, 95(2): 317-356.
[37] Longin Latecki, Ralf Rohrig. Orientation and qualitative angle for spatial reasoning[C]. In Proceedings of IJCAI-93, 1993: 1544-1549.
[2]谢琦,刘大有,虞强源,陈娟.定性方向关系模型研究进展[J].计算机科学, 2006, 33(11): 5-9.
[3] Frank AU. Qualitative spatial reasoning: Cardinal directions as an example[J]. International Journal of Geographical Information Systems, 1996, 10(3): 269-290.
[4] Hernandez Daniel, Clementini Eliseo, Felice Paolino Di. Qualitative Distances[A]. In: Lecture Notes in Computer Science 988. Pisa: Springer-Verlag, 1995: 45-57.
[5] Frank AU. Qualitative spatial reasoning about distances and directions in geographic space [J]. Jurnal of Visual Languages and Computing, 1992, 3(4): 343-371.
[6]曹菡,陈军.方向关系与距离关系的定性描述与推理[J].西安石油学院学报(自然科学版), 2001, 16(1): 68-72.
[7]陈军,赵仁亮. GIS空间关系的基本问题与研究进展[J].测绘学报, 1999, 28(2): 95-102.
[8] Zimmermann K. Enhancing qualitative spatial reasoning-combining orientation and distance[C]. Proceeding of COST'93, 1993: 69-76.
[9] L Latecki, R Rohrig. Orientation and Qualitative Angle for Spatial Reasoning[C]. Proceedings of the 13th International Joint Conference on Artificial Intelligence (IJCAI-93), 1993: 1544-1549.
[10] Hong JH, Egenhofer MJ, Frank AU. On the Robustness of Qualitative Distance and Direction Reasoning[C]. Annual Convention Exposition Technical Papers ACSM/ASPRS'95. Auto-Carto, 1995, 12(4): 301-310.
[11] Zimmermann K. Measuring without Measures: The Delta Calculus[C]. Spatial Information Theory: A theoretical basis for GIS, Proceeding of COSIT'95, 1995, 998: 59-68.
[12] Clementini E, PD Felice, Hernandez. Qualitative representation of positional information[J]. Artificial Intelligence, 1997, 95(2): 317-356.
[13] Ligozat G. Reasoning about cardinal directions[J]. Journal of Visual Languages and Computing, 1998, 9(1): 23-44.
[14] MT Escrig, F Toledo. Qualitative Spatial Reasoning: theory and practice: Application to Robot Navigation[M]. Hamburg: IOS Press, 1998.
[15] H Sturm, NY Suzuki, F Wolter, M Zakharyaschev. Semi-Qualitative Reasoning about Distances[C]. Logics in Artificial Intelligence. Proceeding of JELIA 2000, 2000: 37-56.
[16] Moratz R, Renz J, Wolter D. Qualitative spatial reasoning about line segments[C]. Proceedings of the 14th European Conference on Artifical Intelligence (ECAI 2000). Amsterdam: IOS Press, 2000: 234-238.
[17] Goyal RK, Egenhofer MJ. Cardinal Directions between Extended Spatial Objects[C]. IEEE Transactions on Data Knowledge and Data Engineering, 2000.
[18] Goyal R, Egenhofer MJ. Consistent queries over cardinal directions across different levels of detail[C]. Proceedings of the 11th International Workshop on Database and Expert Systems Applications, 2000: 876-880.
[19]王生生,刘大有,谢琦,王新颖.集成多方面信息的定性空间推理及应用[J].软件学报, 2003, (14): 1857-1862.
[20]邓敏,刘文宝,李俊杰,孙电.矢量GIS空间方向关系的演算模型[J].遥感学报, 2006, 10(6): 821-828.
[21] Deng M, Li ZL. A statistical model for directional relations between spatial objects[J]. GEOINFORMATICA, 2007, 12(2): 193-217.
[22] Nedas K, Egenhofer MJ, Wilmsen D. Metric Details of Topological Line-Line Relations[J]. International Journal of Geographical Information Science, 2007, 21(1): 21-48.
[23] Chen Xiaoyong, Doihara Takeshi, Nasu Mitsuru. Spatial Relations of Distances between Arbitrary Objects in 2D/3D Geographic Spaces based on the Hausdorff Metric[C]. In: The Proc. of towards three Dimensional, Temporal and Dynamic Spatial Data Modeling and Analysis. LIESMARS’95, 1995: 30-41.
[24] Gahegan Mark. Proximity Operators for Qualitative Spatial Reasoning[C]. In: Lecture Notes in Computer Science 988. Pisa: Springer-Verlag, 1995: 31-44.
[25] Frank AU. Qualitative spatial reasoning about cardinal directions[C]. Proceedings of Austrian Conference on Artificial Intelligence, 1991: 157-167.
[26] Hernandez D. Relative representation of spatial knowledge: The 2-D case[J]. Cognitiveand Linguistic Aspects of Geographic Space, 1991: 373-385.
[27] Hernandez D. Qualitative representation of spatial knowledge[M]. Vol. Lecture notes in Artificial Intelligence, Berlin: Springer, 1994.
[28] Freksa C. Using orientation information for qualitative spatial reasoning[C]. Proceedings of International Conference on Theories and Methods of Spatial-Temporal Reasoning in Geographic Space, 1992: 162-178.
[29] Zimmermann K. Enhancing qualitative spatial reasoning-combining orientation anddistance[C]. Proceedings of COSIT'93, 1993: 69-76.
[30] Zimmermann K, C Freksa. Qualitative spatial reasoning using orientation, distance, and path knowledge[J]. Applied Intelligence, 1996, 6(1): 49-58.
[31] Papadias D, T Sellis. Qualitative Representation of Spatial Knowledge in Two Dimensional Space[J]. VLDB Journal, 1994, 3(4): 479-516.
[32] Mukerjee A, Joe GA. qualitative model for space[C]. Proceedings 8th National Conference on Artificial Intelligence, 1990: 721-727.
[33]杜世宏,王桥,杨一鹏.一种定性细节方向关系的表达模型[J].中国图象图形学报, 2004, 9(12): 1496-1503.
[34]邓敏,张燕,李光强.空间方向关系描述模型及其GIS应用分析[J].计算机工程与应用, 2008, 44(7): 37-40.
[35]张恭庆,林源渠.泛函分析讲义[M].北京:北京大学出版社, 2004.
[36] Eliseo Clementini, Paolino Di Felice. Qualitative representation of positional information[J]. Artificial Intelligence, 1997, 95(2): 317-356.
[37] Longin Latecki, Ralf Rohrig. Orientation and qualitative angle for spatial reasoning[C]. In Proceedings of IJCAI-93, 1993: 1544-1549.