基于几何代数的多维统一GIS数据模型研究
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摘要
支持复杂地理对象及连续地理现象的一体化表达、建模与模拟是GIS与地学分析研究的热点。GIS处理对象由二维向三维乃至高维扩展成为GIS发展的必然趋势。受制于地理对象与现象自身结构的复杂性、空间参照系统的多样性、维度扩展复杂性和不同维度运算不统一等因素,现有空间数据模型在多维对象的自适应表达、实体对象空间索引、多维统一的空间分析方法与算法构建框架以及多维统一的GIS系统实现与地学应用等方面仍显不足。基于严密的数学理论,对数据模型的底层理论基础及实现技术进行创新,突破现有GIS数据模型的上述局限性,建立多维统一表达和计算框架是现阶段GIS空间数据模型创新的可能途径。
本论文引入以维度运算为基础的几何代数理论,借鉴其在相关学科的成功经验,构建基于多维统一框架下可支撑地理全景分析的新型空间数据模型；研究多维地理对象的自适应表达与一体化建模；探索相应的数据组织、存储与检索机制以及对应的多维空间分析统一计算模型,在此基础上构建相应的原型系统并进行应用示范。论文研究有助于推动GIS数据模型的理论和技术创新,为GIS应用提供维度可扩展与可制定的基础；为地理现象发展演化过程的表达、建模与模拟提供全新的技术支撑；也为拓展多维统一的空间分析方法体系提供新的思路。论文主要研究内容与主要成果如下：
(1)利用共形几何代数(CGA)空间中Gras smann分级结构与几何对象维度分级结构的一致性,基于内积、外积和几何积实现了内蕴不同维度层次构建及度量关系的几何形体自适应表达,给出了常用几何对象在CGA中的内、外积表达,探讨了几何对象内外积表达间的对偶关系以及不同几何对象间基于几何意义的相互转化；建立了不同维度地理对象与对应的几何代数基本要素(Blades)间的映射关系；利用多重向量实现了不同维度几何对象的统一表达与存储,在代数空间中实现了对不同维度、不同类型地理对象的统一表达与运算；以此为基础,构建了基于共形几何代数的多维GIS空间数据模型的整体架构,探讨了数据的存储结构和编辑、更新机制；对多个基准几何对象及两个不同规模的三维场景进行建模的结果显示,本文所提的数据模型具有结构清晰、几何意义明确,占用空间小且可有效支撑数学运算等优势。
(2)利用几何代数中球在表达形式上的简洁性、几何意义明确性、参数更新的动态性以及几何关系运算的便捷性,构建了边界约束的非相交离散球树(BRNO-ST)多维统一空间索引；探讨了三维空间中多维实体对象包含边界约束的非相交离散球实体填充与剖分算法,实现剖分粒度与表达精度的有效平衡；构建了包含球体积修正的批量Neural Gas层次聚类算法(Vol-BGNG-GA),实现对填充球快速、稳健以及相对均匀的分割,并以此为基础构建了索引树；探讨了基于BRNO-ST实体对象表面及其内部任意位置及区域的检索策略,并结合相关几何代数算子,实现了有限时间约束条件下多维实体对象最近邻距离的近似层次检索动态实体对象相交检测算法。
(3)对几何代数相关算子进行梳理与扩展,构建了适用于GIS空间分析与地学分析的算子库与算法库；探讨了多维空间对象间几何度量、空间位置、空间拓扑关系的计算策略；利用几何代数对多维几何对象及其基本度量和空间关系的统一表达,构建了多维空间对象几何与拓扑关系的批量计算方法,实现了多维GIS几何和拓扑分析功能；针对多维对象的运动表达与分析问题,构建了基于Versor的空间变换与运动的统一表达；探讨了奇数阶Versor的指数表达及其几何意义,并构建了其运动过程线性插值方法；构建了面向多维GIS空间关系分析的统一计算框架,给出了适用于多维空间关系统一计算的算子库与算法库及实现的流程框架；基于三维小区数据的实例分析显示该框架在几何和拓扑关系运算上具有简明、高效等特点,具备支撑大规模多维GIS分析的潜力。
(4)基于本论文所提出的数据模型、索引以及空间关系计算框架,构建了多维统一GIS空间分析原型系统；探讨了原型系统的整体架构、层次体系与主要功能模块；结合地学数据特征及几何代数运算需求,构建了几何代数核心计算引擎,并实现了其与常见GIS空间数据类型间的数据接口。基于插件机制实现多维空间分析模型构建与集成框架；对地理空间与几何代数空间的相互转换、数据I/0与数据管理以及运动场景模拟等主要功能模块进行了系统实现与功能展现；最后基于南极洲海-地-冰系统耦合演化过程进行综合性应用示范,实现了数据组织、存储、检索、算法构建、地学分析的有效整合。
本论文研究显示,几何代数可用于进行多维地理对象的统一表达与分析,并可进行多维空间的不依赖于坐标的几何计算；本文所构建数据模型在结构上具有多维统一性与一致性,在表达上具有简明性与几何意义明确性,且可有效支撑坐标无关的多维统一几何计算。进而可针对特定的分析问题和分析需求,构建合适的数学空间,并基于基本算子进行算法抽象与构建,将可形成理论架构—数据模型—数据分析—数据表达有机融合的基本框架与应用平台,并促进以多维统一为特征的GIS分析技术的发展。
Supporting the intergrated expression, modeling and simulation of complex geometric phonemena is one of the hottest topics of GIS and geographical researches. It is clearly the trend that the GIS research dimensions will be extended from two dimensions to three dimensions and even to higher. Because the geographical phonemena itself are complex, existing data models should have various kinds of coordinations system and heterogenous computational framework, which will occur difficulties when the dimesions are extended. Commonly used data models are still limited in the area of adaptive expression of multidimensional objects, the spatial index of solid objects, the unified multidimensional spatial analysis methods and algorithms, the system implementation and application of multidimensional GIS systems. Make innovations on the theoretical foundations and implementation new technologies, based on new mathematical theories, will provide a potensial way to break through the above limitations of existed GIS spatial data models and construct unified multidimensional expression and computational framework.
The Clifford Algebra is introduced to constructed new unified multidimensional spatial data model, referenced on its successfully application at mathematical and physical areas. The adaptive expression and intergrated modeling of multidimensional objects and according data managementare first researched. The storing and indexing mechanism, associated multidimensional unified computational model and protype software system based on the above results are also proposed. The above researches will benefit and innovate in the GIS spatial data model theory and technology, which provided new technogical supporting for the expression, modeling and simulation of geographical phonemena. The research also provdes new potentially ways to extend the unfied multidimensional spatial analysis methods. The main contents and results of this paper are summarized as follow:
(1) The Grassmann structure is consisted with the dimensional structure of geometrical expression under conformal geometric algebra (CGA). Based on this, the adaptive geometric expression model by inner and outer product, which intergrates the dimensional construction level and metric relations, is proposed. The dual relations between the inner and outer product expression and the reciprocal transformation of different geometric objects based on their geometric meannings are discussed. Both inner and outer product expression of common geometric objects, the mapping relations between geographical objects of different dimensions and their associated basic elements of geometric algbra (Blades) and the unified expression and storage of multidimensional objects based on multivector data structures are given. Based on these, we constructed the main architecture of our multidimesion spatial data model based on geometric algebra. The storage structure, the editing and updating mechanism of multidimensional data are discussed. The application on several standard geometric objects and two 3D geographical sences of different scales suggest that our data model have simple structure and clear geometric meaning, which reduce the space occupation as well as support for mathematical analysis.
(2) The sphere expression under geometric algebra has clear geometric meaning, dynamical updateable parameters and simple geometric relations calculation. Taking advantage of these, the Boundry restrictednon-overlap sphere tree spatial index (BRNO-ST) were proposed. The boundary restruction and the sphere filling algorithm based on non-overlaping spheres, which achieved the blance between granularity of subdivision and expression precision, are proposed. The volume adjusted adaptive batch Neural Gas herichical cluster algorithm (Vol-BGNG-GA), that can quickly, roubustly and relatively uniformly classify the filling sphere, and the level structure of the index trees are then constructed. The query mechanism of the any loction and ranges on and in the solid objects based on the BRNO-ST index are provided. With the help of geometric algebra operators, the hierarcal approximatical nearest linkage distance and dynamical overlapping query under limited time restrictions are also implemented.
(3) Existing geometric algebra operators are filtered and extended to construct the operators and operations that suited for GIS spatial analysis and geographical analysis. The computational mechanism of geometric metric, spatial relations and spatial topology are also discussed. Take advantage of the unified expression of multidimensional geometric objects and metric & spatial relations, the batch multidimensional geometric and topological relations algorithms are proposed. The basic GIS geometric and topological analysis methods are also constructed. We also constructed the unified transformation and expression based on the versor equations for the motion expression and analysis. The motion interpolation algorithm are constructed based on the exponentional expression and their geometric meaning of the even grade versor. The unified multidimensional GIS spatial relation computation framework is proposed based on the unified operator and algorithm libraries and their implementation framework. The case studies on the three dimensional housing estate suggest the framework are simple and efficient in geometric and topology relation computations, which has the potential to support large scale GIS analysis.
(4) The unified multidimensional GIS spatial analysis propo type system is constructed based on the data model, spatial index and spatial relations computation framework proposed above. The main architecture, layer structures and main function moduals are discussed. The core geometric algebra computation engine and the according data interface with other common GIS spatial data files are implementated to fulfill the characteristics of geographical data and the needs of geometric algebra computation. A pluging system is also constructed for implementing the constructionand intergration the multidimensional spatial analysis framework. Main functional moduals, such as the space transformation from geographicalspace in geometric algebra space, data I/O and data management, dynamical sence simulation etc. are impelemented and demonstrated. Finally, a comprehensive application based on the intergrated sea-land-ice coupled dynamical interaction and evolutionary process are proposed, which suggest that our approach can well intergrated the processes from data organization, storage, search and query, algorithem construction and the geographical analysis.
The studies proposed in this paper suggest that geometric algebra can express geometric objects in different dimensions effectively, which also can support simply and effectively computation in multidimesioal space in a coordination free geometric computation. The data model constructed in this paper is simple and of clear geometric meaning, which can also support for unified multidimensional coordinate-free geometric computation. Based on specific problems and requires, we can constructed suited computational space and using basic operators to abstract ad construct the algorithms. Our work can form a basic framework and application platform that will intergrated the underline theories, data models, data analysis and expressions, which will imporove the new GIS analysis technology that are multidimensional unified.
本论文引入以维度运算为基础的几何代数理论,借鉴其在相关学科的成功经验,构建基于多维统一框架下可支撑地理全景分析的新型空间数据模型；研究多维地理对象的自适应表达与一体化建模；探索相应的数据组织、存储与检索机制以及对应的多维空间分析统一计算模型,在此基础上构建相应的原型系统并进行应用示范。论文研究有助于推动GIS数据模型的理论和技术创新,为GIS应用提供维度可扩展与可制定的基础；为地理现象发展演化过程的表达、建模与模拟提供全新的技术支撑；也为拓展多维统一的空间分析方法体系提供新的思路。论文主要研究内容与主要成果如下：
(1)利用共形几何代数(CGA)空间中Gras smann分级结构与几何对象维度分级结构的一致性,基于内积、外积和几何积实现了内蕴不同维度层次构建及度量关系的几何形体自适应表达,给出了常用几何对象在CGA中的内、外积表达,探讨了几何对象内外积表达间的对偶关系以及不同几何对象间基于几何意义的相互转化；建立了不同维度地理对象与对应的几何代数基本要素(Blades)间的映射关系；利用多重向量实现了不同维度几何对象的统一表达与存储,在代数空间中实现了对不同维度、不同类型地理对象的统一表达与运算；以此为基础,构建了基于共形几何代数的多维GIS空间数据模型的整体架构,探讨了数据的存储结构和编辑、更新机制；对多个基准几何对象及两个不同规模的三维场景进行建模的结果显示,本文所提的数据模型具有结构清晰、几何意义明确,占用空间小且可有效支撑数学运算等优势。
(2)利用几何代数中球在表达形式上的简洁性、几何意义明确性、参数更新的动态性以及几何关系运算的便捷性,构建了边界约束的非相交离散球树(BRNO-ST)多维统一空间索引；探讨了三维空间中多维实体对象包含边界约束的非相交离散球实体填充与剖分算法,实现剖分粒度与表达精度的有效平衡；构建了包含球体积修正的批量Neural Gas层次聚类算法(Vol-BGNG-GA),实现对填充球快速、稳健以及相对均匀的分割,并以此为基础构建了索引树；探讨了基于BRNO-ST实体对象表面及其内部任意位置及区域的检索策略,并结合相关几何代数算子,实现了有限时间约束条件下多维实体对象最近邻距离的近似层次检索动态实体对象相交检测算法。
(3)对几何代数相关算子进行梳理与扩展,构建了适用于GIS空间分析与地学分析的算子库与算法库；探讨了多维空间对象间几何度量、空间位置、空间拓扑关系的计算策略；利用几何代数对多维几何对象及其基本度量和空间关系的统一表达,构建了多维空间对象几何与拓扑关系的批量计算方法,实现了多维GIS几何和拓扑分析功能；针对多维对象的运动表达与分析问题,构建了基于Versor的空间变换与运动的统一表达；探讨了奇数阶Versor的指数表达及其几何意义,并构建了其运动过程线性插值方法；构建了面向多维GIS空间关系分析的统一计算框架,给出了适用于多维空间关系统一计算的算子库与算法库及实现的流程框架；基于三维小区数据的实例分析显示该框架在几何和拓扑关系运算上具有简明、高效等特点,具备支撑大规模多维GIS分析的潜力。
(4)基于本论文所提出的数据模型、索引以及空间关系计算框架,构建了多维统一GIS空间分析原型系统；探讨了原型系统的整体架构、层次体系与主要功能模块；结合地学数据特征及几何代数运算需求,构建了几何代数核心计算引擎,并实现了其与常见GIS空间数据类型间的数据接口。基于插件机制实现多维空间分析模型构建与集成框架；对地理空间与几何代数空间的相互转换、数据I/0与数据管理以及运动场景模拟等主要功能模块进行了系统实现与功能展现；最后基于南极洲海-地-冰系统耦合演化过程进行综合性应用示范,实现了数据组织、存储、检索、算法构建、地学分析的有效整合。
本论文研究显示,几何代数可用于进行多维地理对象的统一表达与分析,并可进行多维空间的不依赖于坐标的几何计算；本文所构建数据模型在结构上具有多维统一性与一致性,在表达上具有简明性与几何意义明确性,且可有效支撑坐标无关的多维统一几何计算。进而可针对特定的分析问题和分析需求,构建合适的数学空间,并基于基本算子进行算法抽象与构建,将可形成理论架构—数据模型—数据分析—数据表达有机融合的基本框架与应用平台,并促进以多维统一为特征的GIS分析技术的发展。
Supporting the intergrated expression, modeling and simulation of complex geometric phonemena is one of the hottest topics of GIS and geographical researches. It is clearly the trend that the GIS research dimensions will be extended from two dimensions to three dimensions and even to higher. Because the geographical phonemena itself are complex, existing data models should have various kinds of coordinations system and heterogenous computational framework, which will occur difficulties when the dimesions are extended. Commonly used data models are still limited in the area of adaptive expression of multidimensional objects, the spatial index of solid objects, the unified multidimensional spatial analysis methods and algorithms, the system implementation and application of multidimensional GIS systems. Make innovations on the theoretical foundations and implementation new technologies, based on new mathematical theories, will provide a potensial way to break through the above limitations of existed GIS spatial data models and construct unified multidimensional expression and computational framework.
The Clifford Algebra is introduced to constructed new unified multidimensional spatial data model, referenced on its successfully application at mathematical and physical areas. The adaptive expression and intergrated modeling of multidimensional objects and according data managementare first researched. The storing and indexing mechanism, associated multidimensional unified computational model and protype software system based on the above results are also proposed. The above researches will benefit and innovate in the GIS spatial data model theory and technology, which provided new technogical supporting for the expression, modeling and simulation of geographical phonemena. The research also provdes new potentially ways to extend the unfied multidimensional spatial analysis methods. The main contents and results of this paper are summarized as follow:
(1) The Grassmann structure is consisted with the dimensional structure of geometrical expression under conformal geometric algebra (CGA). Based on this, the adaptive geometric expression model by inner and outer product, which intergrates the dimensional construction level and metric relations, is proposed. The dual relations between the inner and outer product expression and the reciprocal transformation of different geometric objects based on their geometric meannings are discussed. Both inner and outer product expression of common geometric objects, the mapping relations between geographical objects of different dimensions and their associated basic elements of geometric algbra (Blades) and the unified expression and storage of multidimensional objects based on multivector data structures are given. Based on these, we constructed the main architecture of our multidimesion spatial data model based on geometric algebra. The storage structure, the editing and updating mechanism of multidimensional data are discussed. The application on several standard geometric objects and two 3D geographical sences of different scales suggest that our data model have simple structure and clear geometric meaning, which reduce the space occupation as well as support for mathematical analysis.
(2) The sphere expression under geometric algebra has clear geometric meaning, dynamical updateable parameters and simple geometric relations calculation. Taking advantage of these, the Boundry restrictednon-overlap sphere tree spatial index (BRNO-ST) were proposed. The boundary restruction and the sphere filling algorithm based on non-overlaping spheres, which achieved the blance between granularity of subdivision and expression precision, are proposed. The volume adjusted adaptive batch Neural Gas herichical cluster algorithm (Vol-BGNG-GA), that can quickly, roubustly and relatively uniformly classify the filling sphere, and the level structure of the index trees are then constructed. The query mechanism of the any loction and ranges on and in the solid objects based on the BRNO-ST index are provided. With the help of geometric algebra operators, the hierarcal approximatical nearest linkage distance and dynamical overlapping query under limited time restrictions are also implemented.
(3) Existing geometric algebra operators are filtered and extended to construct the operators and operations that suited for GIS spatial analysis and geographical analysis. The computational mechanism of geometric metric, spatial relations and spatial topology are also discussed. Take advantage of the unified expression of multidimensional geometric objects and metric & spatial relations, the batch multidimensional geometric and topological relations algorithms are proposed. The basic GIS geometric and topological analysis methods are also constructed. We also constructed the unified transformation and expression based on the versor equations for the motion expression and analysis. The motion interpolation algorithm are constructed based on the exponentional expression and their geometric meaning of the even grade versor. The unified multidimensional GIS spatial relation computation framework is proposed based on the unified operator and algorithm libraries and their implementation framework. The case studies on the three dimensional housing estate suggest the framework are simple and efficient in geometric and topology relation computations, which has the potential to support large scale GIS analysis.
(4) The unified multidimensional GIS spatial analysis propo type system is constructed based on the data model, spatial index and spatial relations computation framework proposed above. The main architecture, layer structures and main function moduals are discussed. The core geometric algebra computation engine and the according data interface with other common GIS spatial data files are implementated to fulfill the characteristics of geographical data and the needs of geometric algebra computation. A pluging system is also constructed for implementing the constructionand intergration the multidimensional spatial analysis framework. Main functional moduals, such as the space transformation from geographicalspace in geometric algebra space, data I/O and data management, dynamical sence simulation etc. are impelemented and demonstrated. Finally, a comprehensive application based on the intergrated sea-land-ice coupled dynamical interaction and evolutionary process are proposed, which suggest that our approach can well intergrated the processes from data organization, storage, search and query, algorithem construction and the geographical analysis.
The studies proposed in this paper suggest that geometric algebra can express geometric objects in different dimensions effectively, which also can support simply and effectively computation in multidimesioal space in a coordination free geometric computation. The data model constructed in this paper is simple and of clear geometric meaning, which can also support for unified multidimensional coordinate-free geometric computation. Based on specific problems and requires, we can constructed suited computational space and using basic operators to abstract ad construct the algorithms. Our work can form a basic framework and application platform that will intergrated the underline theories, data models, data analysis and expressions, which will imporove the new GIS analysis technology that are multidimensional unified.
引文
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