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SPATIAL RELATIONS BETWEEN UNCERTAIN SETS
Xiaoyong CHEN, Takeshi DOIHARA and Mitsuru NASU
AAS Research Institute,
Asia Air Survey Co., LTD. ,
8-10, Tamura-Cho, Atsugi-Shi, Kanagawa 243, JAPAN
Commission III, Working Group III/IV
KEY WORDSS: GIS Theory, Fuzzy Set, Data Uncertainty, Spatial Relations, Spatial and Temporal Reasoning.
ABSTRACT:
As a part of our serial researches, this paper presents methodologies for modeling spatial relations between uncertain sets. The
uncertainty of spatial relations may arise through the fuzzily defined concepts or linguistics, the presence of varying shapes and
features of complicated spatial objects, and the imprecise measurements of spatial data. By using fuzzy set theory, Mathematical
Morphology, and the dynamic 9-intersection model for integrally representing spatial relations [Chen, et. al. 1995, 1996], a fuzzy 9-
intersection model is developed in which the spatial relations are defined in terms of the intersections of the boundaries, interiors
and exteriors of two dynamically generated uncertain sets. Then, the presented models are extended for quantitatively deriving the
spatial relations between sets in consideration of conceptual and positional uncertainties. Finally, some potential applications of
presented theories and the ideas for spatial and temporal reasoning in Geographical Information Systems (GIS) are also suggested.
1. INTRODUCTION
Geographical Information Systems (GIS) have evolved from
tools for spatial data management and cartography into
sophisticated decision support systems that utilize variety of
spatial and tabular analysis to derive new information. These
systems are finding a wide variety of applications including:
urban and regional planning; environmental and resource
management; facilities management; archaeology; and market
research. In the field of GIS research and application, one of the
most fundamental requirements is to modeling and
communicating error in spatial databases. With increased
research into error modeling over the past few years, there has
been a considerable body of models and techniques available
for measurement spatial and temporal database error from
researching to real applications [Goodchild, 1989; Hunter and
Goodchild, 1995; Shibasaki, 1994; Vergin, 1994]. Spatial
relationships (such as distance, direction, ordering, and
topology) between spatial objects, as very useful tools for
spatial and temporal reasoning in GIS, may be strongly
influenced by the uncertainties of original data. The practical
needs in GIS have led to the investigation of formal and sound
methods for driving spatial relations and their variations with
uncertainties [Chen and et.al., 1995, 1996; Egenhofer and
Franzosa, 1991; Frank, 1992; Kainz, et.al., 1993; Peuquet and
Zhang, 1987]. However, how to derive spatial relations between
uncertain sets based on an mathematically well-defined algebra
framework is still an open problem up to now. The lack of this
comprehensive theory has been a major impediment for solving
many sophisticated problems in GIS, such as formally deriving
spatial relations between complicated spatial objects, spatial
and temporal reasoning in GIS with multiple representations,
and generation of the formal standards for transferring spatial
relations.
105
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
As a part of our serial researches, this paper presents
methodologies for modeling spatial relations between uncertain
sets. The uncertainty of spatial relations may arise through the
fuzzily defined concepts or linguistics, the presence of varying
shapes and features of complicated spatial objects, and the
imprecise measurements of spatial data. By using fuzzy set
theory, Mathematical Morphology, and the dynamic 9-
intersection model for integrally representing spatial relations
[Chen, et. al. 1995, 1996], a fuzzy 9-intersection model is
developed in which the spatial relations are defined in terms of
the intersections of the boundaries, interiors and exteriors of
two dynamically generated uncertain sets. Then, the presented
models are extended for quantitatively deriving the spatial
relations between sets in consideration of conceptual and
positional uncertainties. Finally, some potential applications of
presented models and the ideas for spatial and temporal
reasoning in GIS are also suggested.
This paper is structured into three main sections that follow this
introduction. Section 2 contains a review of related definitions
concerning uncertainty and imprecision for deriving spatial
relations. In section 3, after the brief introduction of some
fundamental theories, the fuzzy 9-intersection model is
developed for integrally deriving spatial relations between
uncertain sets. Section 4 contains the extensions of the
presented theories for deriving conceptual and positional
uncertainties between sets. In the last section conclusions and
outlook for further research are given.
2. UNCERTAINTIES OF SPATIAL RLATIONS
Uncertainty and imprecision refer to the degree of knowledge
(or ignorance) which we have concerning some domain of
interest. Uncertainty is an assessment of our belief (or doubt) in
