基于几何代数的GIS计算模型研究
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摘要
空间分析是地理信息系统(GIS)的核心功能,也是GIS深化应用与服务的重要支撑。随着物联网、对地观测等技术的发展,地理数据愈发丰富,出现海量的高维度、多要素的密集型GIS空间数据。但与之相对的GIS空间数据的计算规则与分析方法却相对滞后,使得海量数据资源难以得到有效利用。现有空间数据分析方法在多维对象的自适应表达、空间数据的统一分析以及多维统一分析框架的构建方面仍显不足。引入几何代数的多维表达结构及统一运算结构,从底层理论上对现有表达与计算方法进行创新,设计面向多维度、多要素复杂数据的计算模型,是突破传统GIS分析方法不统一、构建效率低下等问题,应对目前GIS应用瓶颈的有效途径。
论文以研究组前期基于几何代数的GIS理论、方法与应用的探索研究为基础,开展基于几何代数的GIS计算模型研究。分析了几何代数空间的多维度表达特征与地理空间的表达模式,开展了基于几何代数的GIS计算空间构建研究,对基于几何代数GIS计算方法与计算模板的构建进行了探索。在解析传统算法基础上,系统研究了基于几何代数GIS算法的构建方法与计算策略,最后以系统实现与案例示范的方式对上述理论加以验证。论文主要研究内容与取得的成果如下：
(1)利用几何代数的多维度表达特性设计了地理空间向几何代数空间的嵌入模式,构建了基于几何代数的GIS运算空间,利用blade和多重向量结构实现了GIS基本对象和多要素融合对象运算结构构建；针对不同几何代数模型的分析特性,定义了相应的多重向量运算规则与算子算法库,并将其向多重向量扩展；以几何积的可反性为基础,构建了几何代数框架下问题求解的一般策略,设计了GIS计算从空间定义—对象表达—问题形式化表达—几何代数求解的一般流程；
(2)基于几何代数空间中几何对象的直接可运算性及计算规则的统一性,构建了多维统一的分析框架,并在传统方法解析基础上构建算法模板。为多维矢量、高维场和网络数据设计相应的维度嵌入与算法求解策略。提出了基于多重向量的多维矢量特征融合表达,给出了多维矢量数据计算规则与计算模板并进行了案例示范；设计了基于几何代数多维统一的场空间构建与表达方法,利用几何代数微分运算与特征空间投影实现场特征参数的统一计算,进而从特征维度、特征结构和特征参数三个角度出发构建了多维场数据的统一分析方法；利用几何代数维度运算构建了GIS网络中节点、边与路径的统一表达结构与延拓方法,设计不同类别路径约束嵌入方法,实现节点型和混合型约束网络最优路径的求解。
(3)设计了基于几何代数的GIS计算引擎,为几何代数算法的计算机实现提供基础。构建了几何代数空间和GIS空间中的存储类结构和计算类结构,并详细论述了运算接口设计及数据流结构的构建；基于计算空间、算子库和算法求解三层架构思路进行了计算引擎研发,并设计了几何代数空间算子向GIS空间算法的扩展方法；制定空间计算算法的流程模板,进而构建了算法功能的插件式嵌入机制。
(4)设计了“基于几何代数的多维空间计算系统”,面向多元数据的维度融合建模和动态关系计算进行了案例验证。系统实现了CityGML、dxf、Shape file等多维矢量数据,三维场数据以及多约束网络数据的统一表达与融合分析；设计了多元场景结构中的空间关系动态求解算法,场景污染物结构特征分析算法和顾及污染物分布的最优路径规划算法,验证了系统对多元混合、多维度、动态场景的分析能力。
本论文研究显示：基于几何代数构建的GIS计算模型可应对多维度、多要素及具有复杂结构的GIS空间数据分析需求,在所设计的几何代数算子、算法库的支撑下,可构建简明、直观、可扩展的GIS空间求解模板。通过基于几何代数GIS计算引擎的设计,实现复杂、动态场景中GIS问题统一分析与代数化求解。基于几何代数的GIS计算模型有望为复杂的GIS空间分析问题提出一套完整的运算框架与求解模式,促进以多元融合分析为特征的新一代GIS的发展。
Spatial analysis is the core function of geographic information science as well as the basic for ensuring better progress in geographic information system application. As the development of observation technologies including Internet of things, air survey, satellite remote sensing, and so on, the geographic data have become more and more rich. Lots of high-dimension and multi-factor intensive GIS spatial data have appeared. At the same time, the corresponding calculating regulations of GIS spatial data lag behind. It results in the problem that existing analyzing methods can't analyze effectively the above data. Most of existing spatial data analyzing methods have deficiencies in self-adaptive expression of multi-dimension objects, unified analysis of spatial data and construction of multi-dimension unified analyzing framework. The multi-dimension expression structure and unified calculating structure of geometric algebra (GA) are introduced to innovate upon the expression and calculation methods from underlying theory, and to define calculating and analyzing regulations oriented to multi-dimension and multi-factor complex data. It is an effective way of breaking the bottle neck in current GIS application and overcoming the drawbacks of conventional GIS analyzing methods.
This thesis researches the GA-based GIS spatial calculating model which is founded on early studies of our research group on GA-based GIS theory, methods and applications. This paper researches the multi-dimension expression features of GA and the expression models of geographic space, studies the construction of GIS calculating space based on GA, and then explores the build of GA-based GIS spatial calculating methods and models. In this thesis, the constructing methods and calculating strategies of GA-based GIS algorithms are studied systematically, and finally the above theories are validated by system realizing and cases demonstrating. The main research achievements are as follow:
1. The embedding model from geographic space into GA space is designed by using the multi-dimension expression features of GA and the GA-based GIS calculation space is constructed. The build of calculation structure of GIS basic objects and multi-factor syncretic objects is realized using blade and multivector. The corresponding multivector calculating regulations and operators and algorithms library aimed at different characters of different GA models are defined, and are expanded to multivector. On the basic of reversibility of geometric product, the general strategies of solving problems in GA framework are structured and the general process of GIS spatial calculating from spatial definition to objects expression to problem formulation to GA solution are designed.
2. On the basic of direct computability of geometric objects and unity of calculating regulations in GA space, the multi-dimension unified analyzing framework is structured, and the algorithm models are constructed which is founded on the analysis of conventional methods. It designs the corresponding dimension-embedding and algorithm solving strategies for multi-dimension vectors, high-dimension field and internet data. The synthetic expression of multivector and multi-dimension features is proposed. The computing regulations and models of multi-dimension data are provided in the paper and then the cases are demonstrated. The construction and expressing methods of multi-dimension unified field space are designed. It realizes the unified calculation of field feature parameters using differential operation and feature space projection of GA and further builds the unified analyzing methods of multi-dimension field data. The unified expression structures and expanding methods of nodes, edges and routes in GIS internet are constructed by using the dimensional computation of GA. And then embedding methods of different kinds of route constraints are designed to realize the solution of optimal path in node-type and mixed type constraints internet.
3. The GA-based GIS computation engine are designed which provides the basic theories of the implementations of GA algorithms. The data structures of storage class and calculating class are constructed in GA space and GIS space, and then the operation interface and the data flow of the computation engine are accomplished. The code and system implementation is developed in the three-tier architecture which includes computation space, operator library and algorithm solving. The process templates of GIS computation are formulated to realize the particular GIS applications, and finally an embedded plug-in mechanism for GIS algorithms is proposed.
4. At last, the GA-based multidimensional space computation system is implemented with the execution of integrated modeling instance and dynamic calculation instance, which can verify the main conclusions of the thesis. In the system we realize the unified expression and integrated analysis of the multidimensional vector data, the three dimensional field data and multi-constrained network data. Under the condition of multicomponent geography scene, the GIS analysis algorithms including computation of the dynamic space relationship, structural analysis of the pollution status and planning of the optimal path are designed which consider the distribution of pollutants. All the case studies suggest that GA-based GIS computational model is qualified the computation of multicomponent, multidimensional and dynamic geographic scene.
The studies in this thesis show that the GA-based GIS spatial computation methods can meet the analyzing need of multi-dimension, multi-factor and structure complex GIS spatial data. With the support of designed GA operators and algorithms library, simple, intuitive and expandable GIS spatial solution templates can be constructed. The unified analyzing and algebraic solving of GIS problems in complicated and dynamic scenes are realized by the design of GIS computation engine based on GA. The GA-based GIS spatial computing methods are expected to provide complicated GIS spatial computing problems with a complete calculating framework and solving model and promote the development of a new generation of GIS which is characterized by multi-element synthetic analysis.
论文以研究组前期基于几何代数的GIS理论、方法与应用的探索研究为基础,开展基于几何代数的GIS计算模型研究。分析了几何代数空间的多维度表达特征与地理空间的表达模式,开展了基于几何代数的GIS计算空间构建研究,对基于几何代数GIS计算方法与计算模板的构建进行了探索。在解析传统算法基础上,系统研究了基于几何代数GIS算法的构建方法与计算策略,最后以系统实现与案例示范的方式对上述理论加以验证。论文主要研究内容与取得的成果如下：
(1)利用几何代数的多维度表达特性设计了地理空间向几何代数空间的嵌入模式,构建了基于几何代数的GIS运算空间,利用blade和多重向量结构实现了GIS基本对象和多要素融合对象运算结构构建；针对不同几何代数模型的分析特性,定义了相应的多重向量运算规则与算子算法库,并将其向多重向量扩展；以几何积的可反性为基础,构建了几何代数框架下问题求解的一般策略,设计了GIS计算从空间定义—对象表达—问题形式化表达—几何代数求解的一般流程；
(2)基于几何代数空间中几何对象的直接可运算性及计算规则的统一性,构建了多维统一的分析框架,并在传统方法解析基础上构建算法模板。为多维矢量、高维场和网络数据设计相应的维度嵌入与算法求解策略。提出了基于多重向量的多维矢量特征融合表达,给出了多维矢量数据计算规则与计算模板并进行了案例示范；设计了基于几何代数多维统一的场空间构建与表达方法,利用几何代数微分运算与特征空间投影实现场特征参数的统一计算,进而从特征维度、特征结构和特征参数三个角度出发构建了多维场数据的统一分析方法；利用几何代数维度运算构建了GIS网络中节点、边与路径的统一表达结构与延拓方法,设计不同类别路径约束嵌入方法,实现节点型和混合型约束网络最优路径的求解。
(3)设计了基于几何代数的GIS计算引擎,为几何代数算法的计算机实现提供基础。构建了几何代数空间和GIS空间中的存储类结构和计算类结构,并详细论述了运算接口设计及数据流结构的构建；基于计算空间、算子库和算法求解三层架构思路进行了计算引擎研发,并设计了几何代数空间算子向GIS空间算法的扩展方法；制定空间计算算法的流程模板,进而构建了算法功能的插件式嵌入机制。
(4)设计了“基于几何代数的多维空间计算系统”,面向多元数据的维度融合建模和动态关系计算进行了案例验证。系统实现了CityGML、dxf、Shape file等多维矢量数据,三维场数据以及多约束网络数据的统一表达与融合分析；设计了多元场景结构中的空间关系动态求解算法,场景污染物结构特征分析算法和顾及污染物分布的最优路径规划算法,验证了系统对多元混合、多维度、动态场景的分析能力。
本论文研究显示：基于几何代数构建的GIS计算模型可应对多维度、多要素及具有复杂结构的GIS空间数据分析需求,在所设计的几何代数算子、算法库的支撑下,可构建简明、直观、可扩展的GIS空间求解模板。通过基于几何代数GIS计算引擎的设计,实现复杂、动态场景中GIS问题统一分析与代数化求解。基于几何代数的GIS计算模型有望为复杂的GIS空间分析问题提出一套完整的运算框架与求解模式,促进以多元融合分析为特征的新一代GIS的发展。
Spatial analysis is the core function of geographic information science as well as the basic for ensuring better progress in geographic information system application. As the development of observation technologies including Internet of things, air survey, satellite remote sensing, and so on, the geographic data have become more and more rich. Lots of high-dimension and multi-factor intensive GIS spatial data have appeared. At the same time, the corresponding calculating regulations of GIS spatial data lag behind. It results in the problem that existing analyzing methods can't analyze effectively the above data. Most of existing spatial data analyzing methods have deficiencies in self-adaptive expression of multi-dimension objects, unified analysis of spatial data and construction of multi-dimension unified analyzing framework. The multi-dimension expression structure and unified calculating structure of geometric algebra (GA) are introduced to innovate upon the expression and calculation methods from underlying theory, and to define calculating and analyzing regulations oriented to multi-dimension and multi-factor complex data. It is an effective way of breaking the bottle neck in current GIS application and overcoming the drawbacks of conventional GIS analyzing methods.
This thesis researches the GA-based GIS spatial calculating model which is founded on early studies of our research group on GA-based GIS theory, methods and applications. This paper researches the multi-dimension expression features of GA and the expression models of geographic space, studies the construction of GIS calculating space based on GA, and then explores the build of GA-based GIS spatial calculating methods and models. In this thesis, the constructing methods and calculating strategies of GA-based GIS algorithms are studied systematically, and finally the above theories are validated by system realizing and cases demonstrating. The main research achievements are as follow:
1. The embedding model from geographic space into GA space is designed by using the multi-dimension expression features of GA and the GA-based GIS calculation space is constructed. The build of calculation structure of GIS basic objects and multi-factor syncretic objects is realized using blade and multivector. The corresponding multivector calculating regulations and operators and algorithms library aimed at different characters of different GA models are defined, and are expanded to multivector. On the basic of reversibility of geometric product, the general strategies of solving problems in GA framework are structured and the general process of GIS spatial calculating from spatial definition to objects expression to problem formulation to GA solution are designed.
2. On the basic of direct computability of geometric objects and unity of calculating regulations in GA space, the multi-dimension unified analyzing framework is structured, and the algorithm models are constructed which is founded on the analysis of conventional methods. It designs the corresponding dimension-embedding and algorithm solving strategies for multi-dimension vectors, high-dimension field and internet data. The synthetic expression of multivector and multi-dimension features is proposed. The computing regulations and models of multi-dimension data are provided in the paper and then the cases are demonstrated. The construction and expressing methods of multi-dimension unified field space are designed. It realizes the unified calculation of field feature parameters using differential operation and feature space projection of GA and further builds the unified analyzing methods of multi-dimension field data. The unified expression structures and expanding methods of nodes, edges and routes in GIS internet are constructed by using the dimensional computation of GA. And then embedding methods of different kinds of route constraints are designed to realize the solution of optimal path in node-type and mixed type constraints internet.
3. The GA-based GIS computation engine are designed which provides the basic theories of the implementations of GA algorithms. The data structures of storage class and calculating class are constructed in GA space and GIS space, and then the operation interface and the data flow of the computation engine are accomplished. The code and system implementation is developed in the three-tier architecture which includes computation space, operator library and algorithm solving. The process templates of GIS computation are formulated to realize the particular GIS applications, and finally an embedded plug-in mechanism for GIS algorithms is proposed.
4. At last, the GA-based multidimensional space computation system is implemented with the execution of integrated modeling instance and dynamic calculation instance, which can verify the main conclusions of the thesis. In the system we realize the unified expression and integrated analysis of the multidimensional vector data, the three dimensional field data and multi-constrained network data. Under the condition of multicomponent geography scene, the GIS analysis algorithms including computation of the dynamic space relationship, structural analysis of the pollution status and planning of the optimal path are designed which consider the distribution of pollutants. All the case studies suggest that GA-based GIS computational model is qualified the computation of multicomponent, multidimensional and dynamic geographic scene.
The studies in this thesis show that the GA-based GIS spatial computation methods can meet the analyzing need of multi-dimension, multi-factor and structure complex GIS spatial data. With the support of designed GA operators and algorithms library, simple, intuitive and expandable GIS spatial solution templates can be constructed. The unified analyzing and algebraic solving of GIS problems in complicated and dynamic scenes are realized by the design of GIS computation engine based on GA. The GA-based GIS spatial computing methods are expected to provide complicated GIS spatial computing problems with a complete calculating framework and solving model and promote the development of a new generation of GIS which is characterized by multi-element synthetic analysis.
引文
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