Author:  Liu, Kimfung 
Title:  A framework for modeling uncertain relationships between spatial objects in gis based on fuzzy topology 
Degree:  Ph.D. 
Year:  2006 
Subject:  Hong Kong Polytechnic University  Dissertations Fuzzy topology Spatial systems Geographic information systems 
Department:  Dept. of Land Surveying and GeoInformatics 
Pages:  xi, 175 leaves : ill. (some col.) ; 30 cm 
Language:  English 
InnoPac Record:  http://library.polyu.edu.hk/record=b1957957 
URI:  http://theses.lib.polyu.edu.hk/handle/200/1110 
Abstract:  Boundaries of spatial objects in geographical information systems (GIS) may be vague or fuzzy and the classical set theory which is based on crisp boundary. Since the crisp setbased description may not match to what in the real world, and lead a wrong description in GIS and corresponding spatial analysis. As the fuzzy theory gives us another way of representing objects in GIS, we consider fuzzy sets and investigate corresponding fuzzy topological relations. The objectives of the thesis as follows:  To solve the existing problems of 9intersection model by introducing several useful intersection models to describe the relations of point to point, point to line, point to region, line to line, line to region and region to region respectively.  To the handle the vague or fuzzy models by investigating the topological relations in the cases of point to point, point to line, point to region, line to line, line to region and region to region respectively.  To develop models for quantitatively compute uncertainty topological relations between spatial objects in GIS.  To develop qualitative fuzzy topological relations under several invariant properties which including quasicoincidence, connective and others.  To apply the developed fuzzy topology to image processing. The main focus of this thesis is mainly on modeling topological relations between spatial objects in GIS. Several issues on modeling topological relations between spatial objects are addressed, which including (a) defining topological relations and fuzzy GIS elements; (b) proving that topological relations between spatial objects are shape dependent; (c) modeling topological relations between spatial objects by using the concepts of quasicoincidence and quasidifference in fuzzy topology theory; (d) creating the computable fuzzy topology for practically implementing these conceptual topological relations in a computer environment. The first issue is giving a new definition of the topological relations between two spatial objects which actually is an extended model for topological relations between two spatial objects. For this issue, we have found out that the number of topological relations between the two sets is not as simple as finite; actually, it is infinite and can be approximated by a sequence of matrices. Moreover, as point, line and region (polygon) are the basic elements in GIS, thus we define them based on a fuzzy set. Topology is normally considered as shape independent of spatial objects. This may not necessarily be true in describing relations between spatial objects in GIS. We present a proof that the topological relations between spatial objects are dependent on the shape of spatial objects. That is, topological relations of nonconvex sets cannot be deformed to the topological relations of convex sets. The significant theoretical value of this finding is that topology of spatial objects are shape dependent. This indicates that when we describe topological relations between spatial objects in GIS, both topology and the shape of objects need to be considered. There are two theoretical issues on modeling topological relations between spatial objects. The first one is using the concepts of quasicoincidence and quasidifference to distinguish the topological relations between fuzzy objects and to indicate the effect of one fuzzy object on another in a fuzzy topology. Secondly, based on the developed computational fuzzy topology, methods for computing the fuzzy topological relations of spatial objects are proposed in this issue. For modeling the topological relations between spatial objects, the concepts of a bound on the intersection of the boundary and interior, and the boundary and exterior are defined based on the computational fuzzy topology. Furthermore, the qualitative measures for the intersections are specified based on the acut induced fuzzy topology, which are (Aa^A)(x) < 1a and ((Ac)a^A)(x) < 1a. For computing the topological relations between spatial objects, the intersection concept and the integration method are applied. A computational 9intersection model is thus developed. The computational topological relations between spatial objects are defined based on the ratio of the area/volume of the meet of two fuzzy spatial objects to they join of two fuzzy spatial objects. This is a step ahead of the existing topological relations models: from a conceptual definition of topological relations to the computable definition of topological relations. As a result, the quantitative values of topological relations can be computed. 
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