基于方向关系矩阵的定性空间方向关系模型及相似性研究
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摘要
空间推理是人工智能的一个重要分支，是自治机器人导航、机器视觉、图像检索、智能家居等高新前沿领域的基础理论问题之一。空间关系是空间推理的主要研究内容，在空间推理中占有重要地位，受到广泛关注。
方向关系是最基本且应用十分广泛的空间关系之一，也是近年来空间推理的研究热点。定性空间方向关系的研究方法主要包括代数法和逻辑法，代数法因表达能力强、算法效率高而广受青睐，其中尤以方向关系矩阵最具代表性。目前关于方向关系矩阵的研究主要集中于表达能力、推理与相似性度量，由于缺少复杂度较低且可实现的推理算法和有效的相似性度量算法，以及方向关系矩阵无法处理现实世界的不确定性等原因，导致其研究仍停留在理论阶段，降低了该模型的实用性。
本文基于方向关系矩阵，紧紧围绕方向关系模型和相似性这两个问题展开研究，提出了方向关系形式化模型和一系列简单易行的推理算法与相似性度量算法。进行了以下工作：在定性空间方向关系模型方面：针对主方向关系复合方法不可实现的问题，提出了时间复杂度为O(1)的方向关系矩阵复合算法；提出了时间复杂度为O(n~2)的互为参考对象时方向关系矩阵相容性验证算法，较已有时间复杂度为O(n~3)的验证算法更高效；提出了能够对不确定区域间方向关系进行表示与推理的形式化模型DRMRB.在定性空间方向关系相似性方面：提出了可有效应用于空间场景查询与匹配的定性空间方向关系相似性度量算法。
具体研究结果如下：
（1）针对缺少复杂度较低且可实现的方向关系推理算法的问题，本文分别对关系推理中的复合算法和互为参考对象时相容性验证算法进行研究，提出了方向关系矩阵复合算法和互为参考对象时方向关系矩阵相容性验证算法。
基于矩阵的可运算性质，定义了方向关系矩阵的投影、析取、求幂及Most运算，通过上述运算对主方向关系复合思想进行细化与符号化表示；在此基础上，证明了方向关系矩阵复合定理，据此实现了复杂度为O(1)的方向关系矩阵复合算法Composing并证明了算法的正确性。通过定义矩形矩阵、水平矩阵与垂直矩阵，根据矩形代数与最小边界矩形的对应关系建立了36个坐标矩阵与矩形代数的关联表，从而将矩形代数良好的运算性质应用于方向关系矩阵的推理过程，提出并证明了方向关系矩阵求逆定理，据此实现了复杂度为O(n~2)的互为参考对象时方向关系矩阵的相容性验证算法PC-Checking。
（2）针对方向关系矩阵无法对不确定区域间方向关系进行表示与推理的问题，提出了一种不确定区域间方向关系的形式化建模方法——宽边界方向矩阵DRMRB.
采用宽边界区域来表示不确定区域，利用宽边界区域的最小边界矩形将空间平面划分为25部分，从而对方向关系矩阵进行扩展提出了能够表达不确定方向关系的4-值55宽边界方向矩阵，根据矩形代数与投影关系的一一对应给出了原子宽边界矩阵的形式化描述；为得到合理的不确定方向关系，本文给出且证明了合理宽边界方向矩阵的约束规则，并据此实现了宽边界方向矩阵合理性判定算法Rationality-Con().由DRMRB与四元组模型的关系对应表可知：DRMRB的表达能力不弱于四元组模型；对DRMRB及四元组模型的合理性判定算法分别进行实现，由实验结果对比可知：DRMRB模型的效率更高、更简单易行。随后基于DRMRB，提出了互为参考对象时不确定区域间方向关系的相容性验证算法。我们对13种基本区间关系进行扩展得到26种扩展的区间关系，根据宽边界区域的性质将这26种关系划分为15组，并分别给出15组区间关系对应的投影关系，据此建立了225个矩形宽边界方向矩阵与矩形代数的关联表；通过定义水平矩阵与垂直矩阵之间的按位析取与扩展运算以及宽边界方向矩阵与矩形代数关联表，实现了DRMRB与13种基本区间关系的相互转化，继而提出并证明了宽边界方向矩阵求逆定理；据此实现了互为参考对象时不确定区域间方向关系的相容性验证算法PC-Checking(DRMRB).
（3）针对方向关系矩阵的相似性计算方法不能对定性方向关系和不确定方向关系的相似性进行度量的问题，提出了一种方向关系相似性度量方法。
由方向关系矩阵的4-邻居距离图及方向关系4-连通定义，构建了DRMRB的25个原子矩阵的4-邻居距离图，根据方向关系矩阵和DRMRB的4-邻居距离图求解出方向关系矩阵相互转化的最大代价；分别证明了方向关系矩阵相互转化最小代价的求解问题与均衡运输、分配问题的等价性；据此分别提出了确定、不确定方向关系的相似性计算方法，并实现了算法SA-DRM和SA-DRMRB.以Goyal定量方向关系相似性测试场景中目标对象的位置和比例尺这两种基本的空间变化作为benchmark对上述算法进行测试，分别由实验结果和实例说明了算法在空间场景查询与匹配中的有效性。
Spatial reasoning is an important branch of artificial intelligence, it is also one of themost basic theoretical issues in the high-tech fields of machine vision, autonomous robotnavigation, image retrieval and smart home etc.. Qualitative spatial relation is the mainresearch content of spatial reasoning.
Direction relation is one of the most basic and the most applicable spatial relations; it isalso a hot issue of spatial reasoning in recent years. Modeling and similarity assessment aretwo very important research contents of direction relations. At present, methods of research onqualitative direction relations mainly include algebra and logic, the algebraic method is verypopular because of its expressive ability and the high efficiency of its algorithms, the directionrelation matrix is the most representative model. Existing models mostly concentrated in howto express the relationships, reasoning is rare and lack of formal reasoning methods; most ofthe existing research is for the ideal spatial objects, we always ignore the complexity anduncertainty of the real world; related studies are all carried out at the theoretical level, but lackof practical application-oriented research etc.
Based on direction relation matrix, this paper studies the modeling and similarityassessment of direction relations, promotes formal direction relation model and a series ofsimple reasoning, similarity assessment algorithms. The main results of this dissertation aresummarized as follows.
(1) Aim at the low level of formality of the reasoning algorithms of direction relationmatrix, and the problem of these algorithms cannot be implemented. We refine the directionrelation matrix to expand its expressive ability and establish the corresponding relationsbetween direction relation matrix and rectangular algebra to improve its level of formality. Onthis basis, reasoning algorithms of direction relation matrix are proposed: the compositionalgorithm Composing and the pairwise-consistency checking algorithm PC-Checking. Theimplementation of these two algorithms improves the practicality of the theory of directionrelation reasoning.
(2) Aim at the limitation of expressiveness of direction relation matrix and to model thedirection relations between uncertain regions, based on the idea of expanding the directionrelation matrix given by Cicerone, the direction relation matrix of regions with broad boundaries DRMRB is proposed. The constraint rules of DRMRB are also presented andcertified, based on these rules, we implement the constraint algorithm Rationality-Con().Comparing with the4-tuples model, we find that DRMRB is as expressive as the4-tuplesmodel, but the constraint algorithm of DRMRB is much simpler and easier to be achieved.Based on DRMRB, the pairwise-consistency checking method is proposed. This problem isstill an open problem, and the traditional method cannot be used for uncertain directionrelations. We establish the corresponding relations between the rectangle DRMRB and therectangle algebra, and then we complete the mutual conversion of the DRMRB with the basicinterval relationships. On this basis, the algorithm of pairwise-consistency checking betweenuncertain direction relations is implemented. The accomplishment of this algorithm solves theproblem of the direction relation matrix cannot handle uncertain direction relations.
(3) Aim at existing methods cannot deal with the similarity assessment of qualitativedirection relations and uncertain direction relations, a new approach is proposed. Based on the4-neighbourhood distance graphs of direction relations, the transportation algorithm and theassignment algorithm, the similarity assessment of qualitative direction relations and theuncertain direction relations are given, the assessment algorithms SA-DRM and SA-DRMRBare implemented. The experimental results are used to verify the effectiveness of them, andthe instances can explain the practical value of this work.
方向关系是最基本且应用十分广泛的空间关系之一，也是近年来空间推理的研究热点。定性空间方向关系的研究方法主要包括代数法和逻辑法，代数法因表达能力强、算法效率高而广受青睐，其中尤以方向关系矩阵最具代表性。目前关于方向关系矩阵的研究主要集中于表达能力、推理与相似性度量，由于缺少复杂度较低且可实现的推理算法和有效的相似性度量算法，以及方向关系矩阵无法处理现实世界的不确定性等原因，导致其研究仍停留在理论阶段，降低了该模型的实用性。
本文基于方向关系矩阵，紧紧围绕方向关系模型和相似性这两个问题展开研究，提出了方向关系形式化模型和一系列简单易行的推理算法与相似性度量算法。进行了以下工作：在定性空间方向关系模型方面：针对主方向关系复合方法不可实现的问题，提出了时间复杂度为O(1)的方向关系矩阵复合算法；提出了时间复杂度为O(n~2)的互为参考对象时方向关系矩阵相容性验证算法，较已有时间复杂度为O(n~3)的验证算法更高效；提出了能够对不确定区域间方向关系进行表示与推理的形式化模型DRMRB.在定性空间方向关系相似性方面：提出了可有效应用于空间场景查询与匹配的定性空间方向关系相似性度量算法。
具体研究结果如下：
（1）针对缺少复杂度较低且可实现的方向关系推理算法的问题，本文分别对关系推理中的复合算法和互为参考对象时相容性验证算法进行研究，提出了方向关系矩阵复合算法和互为参考对象时方向关系矩阵相容性验证算法。
基于矩阵的可运算性质，定义了方向关系矩阵的投影、析取、求幂及Most运算，通过上述运算对主方向关系复合思想进行细化与符号化表示；在此基础上，证明了方向关系矩阵复合定理，据此实现了复杂度为O(1)的方向关系矩阵复合算法Composing并证明了算法的正确性。通过定义矩形矩阵、水平矩阵与垂直矩阵，根据矩形代数与最小边界矩形的对应关系建立了36个坐标矩阵与矩形代数的关联表，从而将矩形代数良好的运算性质应用于方向关系矩阵的推理过程，提出并证明了方向关系矩阵求逆定理，据此实现了复杂度为O(n~2)的互为参考对象时方向关系矩阵的相容性验证算法PC-Checking。
（2）针对方向关系矩阵无法对不确定区域间方向关系进行表示与推理的问题，提出了一种不确定区域间方向关系的形式化建模方法——宽边界方向矩阵DRMRB.
采用宽边界区域来表示不确定区域，利用宽边界区域的最小边界矩形将空间平面划分为25部分，从而对方向关系矩阵进行扩展提出了能够表达不确定方向关系的4-值55宽边界方向矩阵，根据矩形代数与投影关系的一一对应给出了原子宽边界矩阵的形式化描述；为得到合理的不确定方向关系，本文给出且证明了合理宽边界方向矩阵的约束规则，并据此实现了宽边界方向矩阵合理性判定算法Rationality-Con().由DRMRB与四元组模型的关系对应表可知：DRMRB的表达能力不弱于四元组模型；对DRMRB及四元组模型的合理性判定算法分别进行实现，由实验结果对比可知：DRMRB模型的效率更高、更简单易行。随后基于DRMRB，提出了互为参考对象时不确定区域间方向关系的相容性验证算法。我们对13种基本区间关系进行扩展得到26种扩展的区间关系，根据宽边界区域的性质将这26种关系划分为15组，并分别给出15组区间关系对应的投影关系，据此建立了225个矩形宽边界方向矩阵与矩形代数的关联表；通过定义水平矩阵与垂直矩阵之间的按位析取与扩展运算以及宽边界方向矩阵与矩形代数关联表，实现了DRMRB与13种基本区间关系的相互转化，继而提出并证明了宽边界方向矩阵求逆定理；据此实现了互为参考对象时不确定区域间方向关系的相容性验证算法PC-Checking(DRMRB).
（3）针对方向关系矩阵的相似性计算方法不能对定性方向关系和不确定方向关系的相似性进行度量的问题，提出了一种方向关系相似性度量方法。
由方向关系矩阵的4-邻居距离图及方向关系4-连通定义，构建了DRMRB的25个原子矩阵的4-邻居距离图，根据方向关系矩阵和DRMRB的4-邻居距离图求解出方向关系矩阵相互转化的最大代价；分别证明了方向关系矩阵相互转化最小代价的求解问题与均衡运输、分配问题的等价性；据此分别提出了确定、不确定方向关系的相似性计算方法，并实现了算法SA-DRM和SA-DRMRB.以Goyal定量方向关系相似性测试场景中目标对象的位置和比例尺这两种基本的空间变化作为benchmark对上述算法进行测试，分别由实验结果和实例说明了算法在空间场景查询与匹配中的有效性。
Spatial reasoning is an important branch of artificial intelligence, it is also one of themost basic theoretical issues in the high-tech fields of machine vision, autonomous robotnavigation, image retrieval and smart home etc.. Qualitative spatial relation is the mainresearch content of spatial reasoning.
Direction relation is one of the most basic and the most applicable spatial relations; it isalso a hot issue of spatial reasoning in recent years. Modeling and similarity assessment aretwo very important research contents of direction relations. At present, methods of research onqualitative direction relations mainly include algebra and logic, the algebraic method is verypopular because of its expressive ability and the high efficiency of its algorithms, the directionrelation matrix is the most representative model. Existing models mostly concentrated in howto express the relationships, reasoning is rare and lack of formal reasoning methods; most ofthe existing research is for the ideal spatial objects, we always ignore the complexity anduncertainty of the real world; related studies are all carried out at the theoretical level, but lackof practical application-oriented research etc.
Based on direction relation matrix, this paper studies the modeling and similarityassessment of direction relations, promotes formal direction relation model and a series ofsimple reasoning, similarity assessment algorithms. The main results of this dissertation aresummarized as follows.
(1) Aim at the low level of formality of the reasoning algorithms of direction relationmatrix, and the problem of these algorithms cannot be implemented. We refine the directionrelation matrix to expand its expressive ability and establish the corresponding relationsbetween direction relation matrix and rectangular algebra to improve its level of formality. Onthis basis, reasoning algorithms of direction relation matrix are proposed: the compositionalgorithm Composing and the pairwise-consistency checking algorithm PC-Checking. Theimplementation of these two algorithms improves the practicality of the theory of directionrelation reasoning.
(2) Aim at the limitation of expressiveness of direction relation matrix and to model thedirection relations between uncertain regions, based on the idea of expanding the directionrelation matrix given by Cicerone, the direction relation matrix of regions with broad boundaries DRMRB is proposed. The constraint rules of DRMRB are also presented andcertified, based on these rules, we implement the constraint algorithm Rationality-Con().Comparing with the4-tuples model, we find that DRMRB is as expressive as the4-tuplesmodel, but the constraint algorithm of DRMRB is much simpler and easier to be achieved.Based on DRMRB, the pairwise-consistency checking method is proposed. This problem isstill an open problem, and the traditional method cannot be used for uncertain directionrelations. We establish the corresponding relations between the rectangle DRMRB and therectangle algebra, and then we complete the mutual conversion of the DRMRB with the basicinterval relationships. On this basis, the algorithm of pairwise-consistency checking betweenuncertain direction relations is implemented. The accomplishment of this algorithm solves theproblem of the direction relation matrix cannot handle uncertain direction relations.
(3) Aim at existing methods cannot deal with the similarity assessment of qualitativedirection relations and uncertain direction relations, a new approach is proposed. Based on the4-neighbourhood distance graphs of direction relations, the transportation algorithm and theassignment algorithm, the similarity assessment of qualitative direction relations and theuncertain direction relations are given, the assessment algorithms SA-DRM and SA-DRMRBare implemented. The experimental results are used to verify the effectiveness of them, andthe instances can explain the practical value of this work.
引文
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