空间关系模糊描述及组合推理的理论和方法研究
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摘要
现实世界是无限复杂和巨大的系统，其中的现象和过程之间存在着各种复杂的关系，是一种连续而自然的现实模型；而GIS是一个有限的、离散的系统，其中的数字模型是被人们经过认知、抽象和描述后的离散模型。这种有限与无限、连续与离散、原始与抽象、自然与认知之间的差异，导致了GIS中的数据、分析模型、处理方法和可视化等带有不确定性。一方面，由于受数据、认知和分析处理中不确定性的影响，空间关系也具有不确定性；另一方面，现有的空间关系描述方法所能够描述的概念与人们经常使用的空间语言之间总存在着一些差异。由于现有的空间关系描述方法是基于二值逻辑的确定性方法，基本上不能描述这种不确定性和差异性，因而这些方法计算得到的对象间的空间关系和现实世界中对应现象间的关系不一致，从而使得基于空间关系的空间数据查询和处理所得到的结果与现实相差较大或不是用户所期望的，限制了空间关系在GIS中的使用。基于这样的认识，本文提出了空间关系的模糊描述方法和一种细节方向关系的描述和推理模型，模糊方法能够描述空间关系的不确定性，而细节方向关系能够以自然语言的方式处理空间关系，从而减小了描述方法所能描述的概念与自然语言的差异性，并从理论、方法、技术和原型实验等几个方面进行了研究和论述，主要内容包括：
(1)全面分析了影响空间关系不确定性的因素，指出空间关系不确定性受空间数据不确定性、认知不确定性和分析处理不确定性等3个方面的影响。空间关系描述必须全面考虑这3个因素，而不是分开处理，即这3个不确定性因素应该在一个统一的框架下分析和处理。
(2)提出了模糊形态运算。把基于经典集合的数学形态学基本算子扩展到基于模糊集的模糊形态算子，使之能够处理模糊数据。模糊形态算子能够同时处理模糊和非模糊数据，能够提高处理的精度。
(3)分析了空间数据位置不确定性和属性不确定性对拓扑关系的影响，提出了对象拓扑空间模糊划分隶属函数及拓扑关系模糊描述的隶属函数，形成了模糊九交模型。模糊九交模型能把模糊对象和精确对象的拓扑关系集中在一个框架中统一描述，能够同时描述位置不确定性和属性确定性对拓扑关系的影响；研究了模糊拓扑关系的矢量和栅格算法，该矢量算法能够保证有较高的精度和较高的速度。
(4)把粗糙集方法引入方向关系处理，提出了方向关系粗糙推理方法。模糊对象和精确对象可以用下粗和上粗两个集合统一描述，用它们的差集体现了数据的不确定性；同样，方向关系也可以用下粗和上粗两个集合来近似表示，它们的差集体现了方向关系的不确定性；方向关系推理的结果也可以用下粗和上粗两个集合来近似表示，它们的差集表示了推理结果的不确定性。
(5)分析了空间数据位置不确定性、属性不确定性及认知不确定性对方向关系的影响，提出了方向关系的模糊隶属函数，形成了方向关系模糊描述矩阵，并研究了模糊方向关系模糊描述的性质。模糊描述方法能够把模糊对象和精确对象集中在一个框架内，通过隶属度来统一描述数据的模糊性、认知的模糊性和分析处理的模糊性。
(6)提出了一种新型方向关系描述模型一一细节方向关系模型。细节方向
关系包括内部、边界和环部等3个方向关系，把它们和外部方向关系、拓扑关系
组合使用，不仅可以描述现有方法不能描述的空间关系概念，而且可以提高空间
关系的分辨率。另外，由于细节方向关系还能够描述参照对象的形状和岛屿及其
位置，因而可以描述参照对象形状比较复杂时的方向关系，还可以描述一些与凹
部有关的空间关系概念，如“线对象穿过面对象的东部凹区”等。
(7)基于细节方向关系，研究了自然语言空间关系的形式化定义。从自然
语言空间关系所涉及到的基本拓扑和方向术语出发，结合细节方向关系的优势，
首先研究了基本方向术语的空间范围划分;然后把拓扑关系、外部方向关系和细
节方向关系组合，给出了自然语一言空间关系的定性定义;最后从外部方向关系的
细化及拓扑关系和细节方向关系的度量两方面入手，对自然语言空间关系的定义
作了进一步的改进和限制，从而形成了最终的形式化定义。
(8)基于细节方向关系，组合外部、内部、边界和环部方向关系，提出一
种基于1个参照系的方向关系推理方法，弥补了现有的基于2个参照系的方向关
系推理方法的不足。把方向关系推理分为四种类型，利用组合表、定理、证明和
实例相结合的方式，提出了一个完整的基于细节方向关系的方向关系推理方法。
(9)提出了分辨两个方向关系之间关系的九交模型，分别研究了根据单种
方向关系和组合多种方向关系推理拓扑关系的理论和方法。九交模型能够区分基
于1个参照系的两个内部方向关系间的50种不同关系，共有5大类;能够区分
两个边界方向关系间的61种不同的关系，共有巧大类，其中每个大类对应着一
条推理规则。该方法包括11张组合表、13张图例、27条规则，全面论述了根据
单种和多种方向关系推理拓扑关系的方法和结果。
(10)提出了细节方向关系的不确定性描述方法。分别利用粗糙集和模糊集
方法对细节方向关系的不确定性作了处理，它们均能一个框架中描?
Real world, in which there are lots of complex relations between phenomena and processes, is an infinitive complex and huge system, and also a continuous and natural real model. However, GIS, in which spatial data are managed, analyzed and processed, is a finite and discrete system, and the digital model stored in GIS is discrete through cognition, abstract and presentation steps. The differences between the infinite and the finite, the continuous and the discrete and the natural and the cognition result in the uncertainty of spatial data, analysis models, process methods and visualization. On the one hand, impacted by the uncertainty of data, cognition and methods, spatial relations are also uncertain. On the other hand, there always exist differences between the concepts described by existing methods of describing spatial relations and spatial language often used by human. Because existing methods of describing spatial relations are certain and can not describe that uncertainty and differences, the relatio
ns between two spatial objects computed by existing methods are inconsistent with the ones among entities in real world, which makes the results got from query and process based on spatial relations are clearly different with real relations or not the expected ones. Based on above points, the fuzzy methods for describing spatial relations and a model for describing and reasoning direction relations are proposed, and the works covering the issues of theoretic consideration, technology development, and application prototypes are presented. The fuzzy methods can describe the uncertainty of spatial relations, while the detailed direction relations can process the spatial relations in natural language, therefore can decrease the differences between the concepts expressed by existing methods and spatial language .The main contents of this dissertation include following points.
(1) The three sources of the uncertainty of spatial relations, including the uncertainty of spatial data, the fuzziness of cognition and the uncertainty of process of spatial relations, are proposed. In addition, these three sources must be processed in a united method, not in different methods.
(2) The fuzzy morphological operators processing fuzzy data are proposed. The basic operators of crisp mathematical morphology based on classic sets are extended to fuzzy operators based on fuzzy set. Therefore, the fuzzy operators can process fuzzy data.
(3) First, the impacts of uncertainty of position and attribute upon topological relations are analyzed. Second, the fuzzy membership functions of dividing topological space and describing topological relations are proposed. Third, a fuzzy 9-intersection model is formed based on the fuzzy membership functions. The fuzzy 9-intersection model can describe the topological relations between fuzzy objects and crisp objects in a united framework, and deal with the impact of uncertainty of
position and attribute on topological relations. Finally, the vector and raster algorithms of fuzzy 9-intersection model are researched. The vector algorithms have better precision and computation speed than raster ones.
(4) Introducing the methods of rough set into the description of direction relations, the rough methods of reasoning direction relations are proposed. Both fuzzy objects and crisp objects can be expressed by two rough sets, down and up rough sets, and the difference set between up and down rough set shows the uncertainty of spatial data. In the same way, the direction relations also can be approximated by a down and an up rough set, the difference between these two rough sets express the uncertainty of direction relations. The reasoning of direction relations is described by two rough sets, and the difference set represents the uncertainty of reasoning direction relations.
(5) After analyzing the impacts of uncertainty of position, attribute and cognition upon direction relations, first, the fuzzy membership functions of direction relations and there similarities are presented; seco
(1)全面分析了影响空间关系不确定性的因素，指出空间关系不确定性受空间数据不确定性、认知不确定性和分析处理不确定性等3个方面的影响。空间关系描述必须全面考虑这3个因素，而不是分开处理，即这3个不确定性因素应该在一个统一的框架下分析和处理。
(2)提出了模糊形态运算。把基于经典集合的数学形态学基本算子扩展到基于模糊集的模糊形态算子，使之能够处理模糊数据。模糊形态算子能够同时处理模糊和非模糊数据，能够提高处理的精度。
(3)分析了空间数据位置不确定性和属性不确定性对拓扑关系的影响，提出了对象拓扑空间模糊划分隶属函数及拓扑关系模糊描述的隶属函数，形成了模糊九交模型。模糊九交模型能把模糊对象和精确对象的拓扑关系集中在一个框架中统一描述，能够同时描述位置不确定性和属性确定性对拓扑关系的影响；研究了模糊拓扑关系的矢量和栅格算法，该矢量算法能够保证有较高的精度和较高的速度。
(4)把粗糙集方法引入方向关系处理，提出了方向关系粗糙推理方法。模糊对象和精确对象可以用下粗和上粗两个集合统一描述，用它们的差集体现了数据的不确定性；同样，方向关系也可以用下粗和上粗两个集合来近似表示，它们的差集体现了方向关系的不确定性；方向关系推理的结果也可以用下粗和上粗两个集合来近似表示，它们的差集表示了推理结果的不确定性。
(5)分析了空间数据位置不确定性、属性不确定性及认知不确定性对方向关系的影响，提出了方向关系的模糊隶属函数，形成了方向关系模糊描述矩阵，并研究了模糊方向关系模糊描述的性质。模糊描述方法能够把模糊对象和精确对象集中在一个框架内，通过隶属度来统一描述数据的模糊性、认知的模糊性和分析处理的模糊性。
(6)提出了一种新型方向关系描述模型一一细节方向关系模型。细节方向
关系包括内部、边界和环部等3个方向关系，把它们和外部方向关系、拓扑关系
组合使用，不仅可以描述现有方法不能描述的空间关系概念，而且可以提高空间
关系的分辨率。另外，由于细节方向关系还能够描述参照对象的形状和岛屿及其
位置，因而可以描述参照对象形状比较复杂时的方向关系，还可以描述一些与凹
部有关的空间关系概念，如“线对象穿过面对象的东部凹区”等。
(7)基于细节方向关系，研究了自然语言空间关系的形式化定义。从自然
语言空间关系所涉及到的基本拓扑和方向术语出发，结合细节方向关系的优势，
首先研究了基本方向术语的空间范围划分;然后把拓扑关系、外部方向关系和细
节方向关系组合，给出了自然语一言空间关系的定性定义;最后从外部方向关系的
细化及拓扑关系和细节方向关系的度量两方面入手，对自然语言空间关系的定义
作了进一步的改进和限制，从而形成了最终的形式化定义。
(8)基于细节方向关系，组合外部、内部、边界和环部方向关系，提出一
种基于1个参照系的方向关系推理方法，弥补了现有的基于2个参照系的方向关
系推理方法的不足。把方向关系推理分为四种类型，利用组合表、定理、证明和
实例相结合的方式，提出了一个完整的基于细节方向关系的方向关系推理方法。
(9)提出了分辨两个方向关系之间关系的九交模型，分别研究了根据单种
方向关系和组合多种方向关系推理拓扑关系的理论和方法。九交模型能够区分基
于1个参照系的两个内部方向关系间的50种不同关系，共有5大类;能够区分
两个边界方向关系间的61种不同的关系，共有巧大类，其中每个大类对应着一
条推理规则。该方法包括11张组合表、13张图例、27条规则，全面论述了根据
单种和多种方向关系推理拓扑关系的方法和结果。
(10)提出了细节方向关系的不确定性描述方法。分别利用粗糙集和模糊集
方法对细节方向关系的不确定性作了处理，它们均能一个框架中描?
Real world, in which there are lots of complex relations between phenomena and processes, is an infinitive complex and huge system, and also a continuous and natural real model. However, GIS, in which spatial data are managed, analyzed and processed, is a finite and discrete system, and the digital model stored in GIS is discrete through cognition, abstract and presentation steps. The differences between the infinite and the finite, the continuous and the discrete and the natural and the cognition result in the uncertainty of spatial data, analysis models, process methods and visualization. On the one hand, impacted by the uncertainty of data, cognition and methods, spatial relations are also uncertain. On the other hand, there always exist differences between the concepts described by existing methods of describing spatial relations and spatial language often used by human. Because existing methods of describing spatial relations are certain and can not describe that uncertainty and differences, the relatio
ns between two spatial objects computed by existing methods are inconsistent with the ones among entities in real world, which makes the results got from query and process based on spatial relations are clearly different with real relations or not the expected ones. Based on above points, the fuzzy methods for describing spatial relations and a model for describing and reasoning direction relations are proposed, and the works covering the issues of theoretic consideration, technology development, and application prototypes are presented. The fuzzy methods can describe the uncertainty of spatial relations, while the detailed direction relations can process the spatial relations in natural language, therefore can decrease the differences between the concepts expressed by existing methods and spatial language .The main contents of this dissertation include following points.
(1) The three sources of the uncertainty of spatial relations, including the uncertainty of spatial data, the fuzziness of cognition and the uncertainty of process of spatial relations, are proposed. In addition, these three sources must be processed in a united method, not in different methods.
(2) The fuzzy morphological operators processing fuzzy data are proposed. The basic operators of crisp mathematical morphology based on classic sets are extended to fuzzy operators based on fuzzy set. Therefore, the fuzzy operators can process fuzzy data.
(3) First, the impacts of uncertainty of position and attribute upon topological relations are analyzed. Second, the fuzzy membership functions of dividing topological space and describing topological relations are proposed. Third, a fuzzy 9-intersection model is formed based on the fuzzy membership functions. The fuzzy 9-intersection model can describe the topological relations between fuzzy objects and crisp objects in a united framework, and deal with the impact of uncertainty of
position and attribute on topological relations. Finally, the vector and raster algorithms of fuzzy 9-intersection model are researched. The vector algorithms have better precision and computation speed than raster ones.
(4) Introducing the methods of rough set into the description of direction relations, the rough methods of reasoning direction relations are proposed. Both fuzzy objects and crisp objects can be expressed by two rough sets, down and up rough sets, and the difference set between up and down rough set shows the uncertainty of spatial data. In the same way, the direction relations also can be approximated by a down and an up rough set, the difference between these two rough sets express the uncertainty of direction relations. The reasoning of direction relations is described by two rough sets, and the difference set represents the uncertainty of reasoning direction relations.
(5) After analyzing the impacts of uncertainty of position, attribute and cognition upon direction relations, first, the fuzzy membership functions of direction relations and there similarities are presented; seco
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