An intelligent computing and control model of topological relation between spatial objects based on fuzzy theory

For qualitative spatial reasoning, there are various dimensions of objects. A considerable amount of effort has been devoted to 2D representation and analysis of spatial relations. Here we present an exposition for 3D objects. There are three types of binary relations between pairs of objects: topological connectivity, cardinal directions, and distance relations. The combinations of these relations can provide additional useful knowledge. The spatial databases include data and the spatial relations to facilitate end-user spatial querying, it also is important to associate natural language with these relations. Some work has been done in this regard for line-region and region-region topological relations in 2D, and very recent work has initiated the association between natural language, topology, and metrics for 3D objects. However, prior efforts have lacked rigorous analysis, expressive power, and completeness of the associated metrics. Herein we present a detailed study of new metrics required to bridge the gap between topological connectivity and size information for integrating reasoning in spatial databases. The complete set of metrics that we present should be useful for a variety of applications dealing with 3D objects including regions with vague boundaries.

Fuzziness is an internal property of spatial objects. How to model fuzziness of a spatial object is a main task of next generation GIS. This paper proposes basic fuzzy spatial object types based on fuzzy topology. These object types are the natural extension of current non-fuzzy spatial object types. A fuzzy cell complex structure is defined for modeling fuzzy regions, lines and points. Furthermore, fuzzy topological relations between these fuzzy spatial objects are formalized based on the 9-intersection approach. This model can be implemented for GIS applications due to its scientific theory basis.KeywordsCell ComplexSpatial ObjectTopological RelationLower Semicontinuous MappingFuzzy ObjectThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The fuzzy sets theory is observed as an important and powerful tool, which aids numerous researchers and learners in a variety of ways and in countless disciplines by transforming oral information into mathematical data. The majority of our conventional tools for computing, analysis and modeling are based on clear cut, defined, deterministic and precise information (generally crisp in nature) and concentrate on true yes or true no type information rather than more or less type information. It is very easy to deal with the cases, where the statement refers true yes or true no type information, however it is critical to deal with more or less type of data information, which can be easily overcome by fuzzy sets theory. The presented chapter tries to give an idea to the learners and researcher about: the basic concepts of fuzzy, what is fuzzy and fuzzy number, what are the application fields of the developed fuzzy theory, how to import fuzzy data, what is linguistic and importance of linguistic in fuzzy theory, what are the existing developed techniques that integrate the fuzzy theory effectively, etc. The main objective of the presented chapter is to impart fuzzy knowledge to the new learners and readers, so that they augment and disseminate the fuzzy sets theory into new application fields to benefit the society and mankind.

Topological relations are one of the most important aspects in GIS modeling. The topological relations between crisp spatial objects have been well identified. However, the topological relations between fuzzy spatial objects need more investigation. This paper discusses different definitions in fuzzy boundary and their differences and relations. One of the definitions is then selected as the fuzzy boundary for GIS. In order to identify the topological relations between fuzzy spatial objects, three intersection matrices, namely 3*3-intersection matrix, 4*4-intersection matrix and 5*5-intersection matrix, are proposed and formalized in a fuzzy topological space. These matrices are all applicable for identification of topological relations between fuzzy spatial objects. 152 topological relations between two simple fuzzy regions are identified based on the 4×4-intersection approach in R 2 .

Models of spatial relations are a key component of geographical information science (GIS). Efforts have been made to formally define spatial relations. The foundation model for such a formal presentation is the 4-intersection model proposed by Egenhofer and Franzosa (1991). In this model, the topological relations between two simple spatial entities A and B are transformed into pointset topology problems in terms of the intersections of A's interior and boundary with B's interior and boundary. Later, Egenhofer and Herring (1991) extended this model to 9-intersection by addition of another element, i.e. the exterior of an entity, which is then defined as its complement. However, the use of its complement as the exterior of an entity causes the linear dependency between its interior, boundary and exterior. Thus such an extension from 4- to 9-intersection should be of no help in terms of the number of relations. This can be confirmed by the discovery of Egenhofer et al. (1993). The distinction of additional relations in the case where the co-dimension is not zero is purely due to the adoption of definitions of the interior, boundary and exterior of entities in a lower dimensional to a higher dimension of space, e.g. lines in 1-dimensional space to 2-dimensional space. With such adoption, the topological convention that the boundary of a spatial entity separates its interior from its exterior is violated. It is such a change of conventional topological properties that causes the linear dependency between these three elements of a spatial entity (i.e. the interior, boundary and exterior) to disappear, thus making the distinction of additional relations possible in such a case (i.e. the co-dimension is not zero). It has been discussed that the use of Voronoi-regions of an entity to replace its complement as its exterior in the 9-intersection model would solve the problem (i.e. violation of topological convention) or would make this model become more comprehensive. Therefore, a Voronoi-based 9-intersection model is proposed. In addition to the improvement in the theoretical aspect, the Voronoi-based 9-intersection model (V9I) can also distinguish additional relations which are beyond topological relations, such as high-resolution disjoint relations and relations of complex spatial entities. However, high-resolution disjoint relations defined by this model are not purely topological. In fact, it is a mixture of topology and metric.

Directional relations, topological relations and metric relations are the basic spatial constraint in the geometric space. The three kinds of spatial relations are not isolated and restrict each other. In this paper, we focus on the effect of topology and distance on the representation of directional relations and present a description mechanism of directional relations considering topology and distance. In this new model, (1)topological constraint is analyzed as the basis of directional relations categories; (2)distance relations is the other factor of representation of directional relations and distinguish the directional relations of different positions in the same ray line; (3)depending on the analysis of the capability of exiting directional relations models to represent the restraint from the topological relations and distance, a category mechanism based on the topological relations and distance is proposed to improve the accuracy of the directional description. This model combines the restraint from topological and distance relations and consummates the representation of directional relations. Experiments show excellent efficiency in view of directional description.

The development of formal models of spatial relations is a topic of great importance in spatial reasoning, geographic information systems (GIS) and computer vision, and has gained much attention from researchers across these research areas during the past two decades. In recent years significant achievements have been made on the development of models of spatial relations between spatial objects with precisely defined boundaries. However, these models cannot be directly applied to spatial objects with indeterminate boundaries which are found in many applications in geographic analysis and image understanding. This article develops a method for approximately analyzing binary topological relations between geographic regions with indeterminate boundaries based upon previous work on topological spatial relations and fuzzy sets. In addition, examples are given to demonstrate the method and related concepts. It is shown that the eight binary topological relations between regions in a two-dimensional space can be easily determined by the method.

To solve the disadvantages of high difficulty and complexity in studying the spatial topological relation method on different resolution remote sensing images, this paper proposes a spatial topological relationship model based on GeoSOT, which is one of the global subdivision grids. To express spatial objects on different resolution remote sensing images, this paper introduces the spatial objects subdivision location identification. Then, the model calculates spatial topological relation by comparing location identifications in combined table. At last, test shows that it is feasible to analyze the spatial objects topological relations based on GeoSOT and it has high efficiency operation significantly, especially for complicated spatial objects.

Image processing has to deal with many ambiguous situations. Fuzzy set theory is a useful mathematical tool for handling the ambiguity or uncertainty. In order to apply the fuzzy theory, selecting the fuzzy region of membership function is a fundamental and important task. In this paper, a new method of membership function based on fuzzy theory by PSO algorithm optimized was proposed by analyzing the deficiencies of traditional enhancement algorithm. A new entropy definition of a fuzzy set was proposed. The new entropy definition of a fuzzy set was not only related to the membership (fuzzy domain) but also related to the probability distribution (space domain), it can respond to the variety of image input information. In addition, by quoting a novel particle swarm optimization (PSO) algorithm to find the optimization parameters for membership. We have employed the new proposed approach to perform image enhancement and thresholding and obtained satisfactory results.

The ability to describe variables in a health risk model through probability theory enables us to estimate human health risk. These types of risk assessment are interpreted as probabilistic risk assessment (PRA). Generally, PRA requires specific estimate of the parameters of the probability density of the input variables. In all circumstances, such estimates of the parameters may not be available due to the lack of knowledge or information. Such types of variables are treated as uncertain variables. These types of information are often termed uncertainty which are interpreted through fuzzy theory. The ability to describe uncertainty through fuzzy set theory enables us to process both random variable and fuzzy variable in a single framework. The method of processing aleatory and epistemic uncertainties into a same framework is coined as hybrid method. In this paper, we are going to talk about such type of hybrid methodology for human health risk assessment. Risk assessment on human health through different pathways of exposure has been attempted many a times combining Monte Carlo analysis and extension principle of fuzzy set theory. The emergence of credibility theory enables transforming fuzzy variable into credibility distribution function which can be used in those hybrid analyses. Hence, an attempt, for the first time, has been made to combine probability theory and credibility theory to estimate risk in human health exposure. This method of risk assessment in the presence of credibility theory and probability theory is identified as probabilistic-credibility method (PCM). The results obtained are then interpreted through probability theory, unlike the other hybrid methodology where the results are interpreted in terms of possibility theory. The results obtained are then compared with probability-fuzzy risk assessment (PFRA) method. Generally, decision under hybrid methodology is made on the index of optimism. An optimistic decision maker estimates from the $$\alpha$$ -cut at 1, whereas a pessimistic decision maker estimates from the $$\alpha$$ -cut at 0. The PCM is an optimistic approach as the decision is always made at $$\alpha$$ =1.

A weighted inference engine based on interval-valued fuzzy relational theory

Fuzzy difference operators are used in different fields of decision making processes, image analysis and computer vision applications. Change detection in a spatial scene is the basic issue in modeling relative motion between object pair. Many approaches are adopted to change detection in a spatial scene and change in spatial relations is one of them. Separate methodologies are adapted to determine the change in topological, directional and distance relations. In this paper, a methodology for detection of a spatial change based on fuzzy matrix calculus is presented. Difference of combined fuzzy topological and directional relations matrices is determined by fuzzy difference operators. Experiments are performed to validate the proposed method and the promising results are obtained.

Fuzzy sets theory provides a good theoretical basis to represent imprecision of the information, at different levels of representation. It constitutes an unified framework for representing and processing both numerical and symbolic information, as well as structural information (constituted mainly by spatial relationships in image processing). We show that this theory can achieve tasks at several levels, from low level (e.g. grey-level based classification) to high level (e.g. model based structural pattern recognition and scene interpretation). We describe the use of fuzzy set at low level, for processing the basic numerical information contained in the images. Then we go to the object level, and show how to represent objects or structures in the images as fuzzy objects, and we present some operations on such fuzzy objects. At higher level, we are concerned by structural information and spatial relationships between objects, as distances, adjacency, and relative position between fuzzy objects. Finally, we address the problem of combining and fusing information. We show examples at low level for multi-source classification, and at high level for recognition of brain structures based on an atlas.

5 - Geographic Relations

With the increase of XLPE cable we used in power network, the corresponding technology of on-line monitoring becomes increasingly important.For improving the accuracy of on-line monitoring assessment of XLPE cable, we researched an assessment method and device of XLPE cable's insulation state by Fuzzy Theory.According to legitimately distribute between Fuzzy set and membership function in Fuzzy Theory, we solved the problem that how to distribute the membership of XLPE cable's insulation parameters and accomplished the assessment of XPLE cable's insulation state.The experimental operation results showed that the device can operate stably and interactwith background monitoring platform in real-time, which accomplished the on-line monitoring of XLPE cable and assessment of its insulation state.It can effectively ensure the safety of power network's operation.