共形几何代数框架下时空拓扑关系的统一表达与计算方法研究

Spatio-temporal topological analysis is an important fundamental issue of the research topics in geographic information science. Currently, the temporal topological relationship is expressed with qualitative methods to meet with the people's cognitive habits, it's difficult to metric analysis the spatio-temporal topology. It's need design relatively independent topological analysis algorithms to response different types and dimensions of geographic objects, lack of uniformity of spatial structure and computation framework. The project intends to introduce conformal geometric algebra which integrate representation and computation of objects at different dimensions provides potations to express space and time in an unified framework. Achieving mutual conversion of complex geo-object between Euclidean space and conformal space, providing mapping method between spatio-temporal objects and spatio-temporal topological relation, to construct a new type of spatio-temporal data model supporting the representation and computation of unified multi-dimensional geographical objects based on multidimensional vector. Design and implement an unified spatio-temporal topology computing framework, including spatio-temporal topology operator, as well as spatio-temporal topology computing algorithms and engines. Finally, test and verify the feasibility and effectiveness of the proposed method with the examples of land use temporal topological analysis.

时空拓扑分析是地理信息科学中一个重要的基础研究方向。目前时空拓扑关系表达普遍采用定性的方法以符合人们的认知习惯，无法直接对时空拓扑关系进行定量计算。针对不同类型、不同维度的地理对象往往需要构建相对独立的时空拓扑分析算法，缺乏空间结构与计算结构的统一性。本项目拟基于具有多维统一表达与几何计算能力的共形几何代数理论，实现复杂地理时空与几何代数空间的相互转换，构造基于多重向量的多维时空对象统一表达模型；通过时空对象与时空拓扑关系之间的映射，构建多维统一、语义明确的时空拓扑关系形式化表达与描述方法；设计并实现时空拓扑关系统一计算框架，包括完备的时空拓扑关系算子、算法集以及时空拓扑计算引擎与运算接口；最后结合土地利用时空拓扑分析实例，对所提出模型、方法的可行性与有效性进行验证。

时空拓扑分析是地理信息领域的重要基础性研究。受底层数学理论基础的限制，主流时空对象的表达与时空拓扑关系的表达彼此分离，较难定义相对完备的时空拓扑分析算子对时空拓扑关系进行统一表达与计算。本研究从多维时空统一视角出发，引入共形几何代数理论，将不同维度的复杂时空对象分解为若干简单几何形体，利用外积和内积进行时空构建与时空对象表达，构造基于多重向量的时空对象统一表达模型。设计时空对象表达模型，探讨这一数学框架下的时空拓扑形式化表达方法与时空拓扑关系计算。.使用定量与定性相结合的多分支决策树方法对时空拓扑关系进行形式化表达，利用拓扑关系判断规则集推导简单时空对象之间的拓扑关系，并进一步对复杂对象之间的时空拓扑关系进行推演，实现时空对象表达与时空拓扑关系表达的有机统一。建立共形几何代数支持下的时空拓扑关系统一计算框架，研发欧氏空间中常用拓扑关系计算方法在共形几何代数框架下的算法结构与实现流程，设计时空拓扑关系算子、算法集，支撑时空拓扑关系计算。.本研究以具有典型离散特征的土地利用时空数据为例，实现了土地利用现状数据拓扑检查、历史回溯等应用示范，验证本研究的可行性、有效性及其计算结果的正确性。