基于几何代数的动态拓扑关系表达与自适应计算模型

Traditional topology analysis methods are over generalized in representation of geometric objects, which leads to the separation of object expression and topological computation. As a result, the representation and analysis of topological relationship turns out to be much more complicated. Based on the geometric algebra (GA) theory system, this project tries to carry on the research of dynamic topological relationship representation and adaptive computing model. The multidimensional unified data representation is firstly constructed with hierarchical structures. And based on which, the topological relationship computing operators can be established with the dimensional adaptability. Then, the formal expression and algebraic calculation of topological relationships between geometric objects can be realized. Secondly, a dynamic topology relation model for the moving objects can be built based on the topological transformation sequences. The parametrization of spatio-temporal topological relationships will be discussed to construct the topological transformation sequences of moving objects based on the idea of solving spatial problems with functions. By using geometric algebra operator which can uniformly express the movements of rigid objects (like transformations and rotations), the expression and computation of dynamic topological relationship can be designed according to the spatio-temporal characteristics embedded in topological transformation sequences. At last, then, the mapping function between the critical state and the critical time of topological transformation sequences will be constructed to realize the evaluation of spatio-temporal topology based on the dynamic topological relationship. The GA-based reasoning and computing algorithms of dynamic topological relationship can then be designed based on the above theoretical study. The project will further promote the researches in reasoning and computing methods of spatio-temporal topological, and offer a new idea for the researches of GIS spatial analysis.

针对现有拓扑关系分析方法中几何对象过度抽象，对象表达和拓扑关系计算分离的问题，引入几何代数理论，进行动态拓扑关系表达与自适应计算模型研究。通过构建多维统一的几何对象层次表达和具有多维自适应性的几何代数拓扑关系算子，实现对几何对象间拓扑关系的形式化表达与代数化计算。进而研究基于拓扑变换序列的动态对象拓扑关系表达与计算模型，借鉴函数化的思想构建基于几何代数的拓扑关系参数化表达模型，进而构建基于几何代数算子的空间拓扑变换序列。利用几何代数算子对刚体对象运动的统一表达特性，并从在空间拓扑变化序列中对时态特征解析的视角，研究对象动态拓扑关系的推理与计算方法。研究拓扑变换序列中临界状态与临界时间的映射函数，实现动态拓扑关系中时空拓扑特征的解析，并设计基于几何代数的动态拓扑关系的推理与计算算法。本项目可进一步促进时空拓扑推理与计算方法研究，为GIS空间关系研究提供新的思路。

本项目针对现有拓扑关系分析中几何对象过度抽象、对象表达和拓扑关系计算分离等问题，从数学理论的本源出发，引入多维统一的几何代数对象层次表达和具有多维自适应性的几何代数拓扑关系算子，进行动态拓扑关系表达与自适应计算模型研究。项目研究获得了如下成果：1）研究了复杂地理对象多维统一表达，探究了地理对象之间的运动变化性质，构建了可支撑多维度运动对象间的通用运动变化映射函数模型；2）设计了基于MVTree数据结构的任意维度几何对象统一表达数据模型；3）构建了动态拓扑序列特征参数化分析模型，设计了拓扑变化序列临界参数计算模型；4）设计了室内行为轨迹的运动语义解析方法，构建了动态环境下基于拓扑约束的人体行为分析模型；5）最后将本项目方法推广应用于室内导航、最佳路线 、轨迹预测与分析等方面。项目研究成果在国内外权威刊物上已发表研究论文3篇，其中SCI/ESCI检索论文2篇，中文核心1篇，申请国家发明专利2项，已接收中文核心1篇，培养博士研究生2名，硕士研究生5名。研究显示利用几何代数对形体和运动过程的多维表达特性，可构建几何对象刚体运动变化模式表达，从而可实现动态拓扑关系的解析式表达。同时基于CNG模型的拓扑关系分层结构，构建了几何对象运动拓扑序列模型，可以更好的实现任意时刻运动地理对象之间拓扑关系的确定，有望推广到多维几何运动对象间拓扑关系表达与计算的统一。以此为基础构建的几何拓扑相统一的MVTree数据模型在基于GA的新一代GIS系统中得到运用，并可用于高动态环境下基于拓扑约束的人体行为分析与模拟，在非结构特征的泛在大数据挖掘应用中具有广阔的前景。