MEMBERSHIP FUNCTIONS FOR FUZZY SET REPRESENTATION 
OF GEOGRAPHIC FEATURES 
E. Lynn Usery 
University of Georgia 
Department of Geography 
Athens, Georgia, USA 
Commission IV, Working Group 1 
KEY WORDS: Fuzzy set, geographic features, representation, membership functions, data modelling 
ABSTRACT 
A conceptual framework for geographic feature representation has been developed by applying data modelling abstraction levels 
to spatial, thematic, and temporal dimensions of geographic phenomena and structuring attributes and relationships for each 
dimension to build a features knowledgebase. Many aspects of the framework are currently implemented in geographic 
information system (GIS) software directly as classical mathematical sets. However, the strength of the framework appears to be 
in its ability to include multiple representations through feature-based or object-oriented approaches. The theory of fuzzy sets 
provides a basis for multiple representation of non-partitioned geographic features with indeterminate boundaries. Representation 
of geographic phenomena as fuzzy features in a GIS requires the development and implementation of fuzzy operators including 
buffering; overlay supporting union, intersection, and complement; and boundary. These GIS processing operators have: been 
developed (Katinsky, 1994; Usery, 1996a) and implemented as additional functions of the commercial GIS software package, 
Imagine from ERDAS, Inc. 
Using fuzzy set representation within the developed framework and applying the fuzzy set GIS operators requires the 
development of specific fuzzy set membership functions for individual geographic features from multiple data sources. The 
fuzzy set function may model the indeterminacy of spatial position, thematic aitributes, or temporal attributes. This paper 
presents examples of fuzzy set membership functions with either the spatial or thematic dimensions represented. For example, a 
hill may be represented as a fuzzy set by using the elevation values from a digital elevation model (DEM) in the fuzzy 
membership function while a weed patch in an agricultural field can be modelled by using spatial position as the membership 
function in a digitized aerial image. 
1. INTRODUCTION 
The representation of geographic phenomena in 
computer-based processing systems has evolved from purely 
geometric symbolization of points, lines, and areas for 
mapped entities to the inclusion of thematic attributes 
associated with the geometric objects in relational databases 
of current commercial software packages. Recent research 
in feature-based approaches has led to the development of a 
comprehensive framework for representing geographic 
phenomena and provides a mechanism to include both 
thematic and temporal attributes and relationships in 
addition to the spatial or geometric models of current 
systems (Usery, 1993; 1996a; 1996b). This framework 
allows a variety of representations for single features 
including multiple geometries from differing data sources. 
One approach to representing geographic features with 
indeterminate boundaries uses fuzzy set theory (Altman, 
1994; Katinsky, 1994; Usery, 1996a). While this approach 
has been demonstrated to capture the indeterminacy of 
particular features, the membership function of the fuzzy set 
remains problematic for specific feature types. For example, 
one may represent the thematic dimension of a feature as a 
fuzzy set (Burrough, 1989; Wang, 1990). Alternatively, a 
fuzzy boundary based on spatial position may better capture 
the feature’s indeterminacy (Katinsky, 1994). Temporal 
attributes could also be modelled as fuzzy sets. The 
outstanding problem is one of determining an appropriate 
membership function for the specific feature of interest. 
It is the purpose of this paper to present examples of 
geographic features with specific fuzzy set membership 
functions from multiple data sources. Section 2 briefly 
describes the conceptual framework for geographic features 
which allows multiple representations from multiple data 
sources. Section 3 draws from Katinsky (1994) and briefly 
describes the fuzzy operators, union, intersection, buffering, 
and boundary which are necessary for processing fuzzy 
features in a GIS. Section 3 also provides specific examples 
of fuzzy set membership functions based on data source and 
the feature dimension, spatial or thematic, which is the 
controlling parameter of the fuzzy set function. A final 
section draws conclusions and provides a basis for further 
research toward developing a features knowledgebase to 
support geographic analysis. 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996