2. A FRAMEWORK FOR GEOGRAPHIC FEATURES 
Since all geographic phenomena can be modelled along the 
three dimensions of space, theme, and time (Berry, 1964; 
Sinton, 1978) and each phenomenon possesses unique 
characteristics and interactions with other phenomena, 
feature representation must include attributes of individual 
features and relationships among features (Usery, 19942). 
The data modelling levels of concept, model, and structure 
for each of the three feature dimensions are documented 
elsewhere within a conceptual framework which supports 
multiple representations (Usery, 1996a; 1996b). That 
framework is constructed with the geographic feature as the 
real world entity with its object representation including 
attributes of space, theme, and time. Thus, the space- 
dominant model of current software packages and current 
GIS is avoided and the actual locational coordinates and 
topology of a geographic phenomenon become attributes and 
relationships of the feature in a manner similar to the 
thematic and temporal attributes and relationships. 
The strength of this representational framework is to allow 
geometry of the feature to vary with the data source. For 
example, a terrain feature, such as a hill, can be represented 
in a DEM as a set of pixel values defined in a Boolean 
operation. Alternatively, the geometry of the hill may be 
defined as a fuzzy set with some pixels possessing partial 
membership values in the hill feature. A third geometric 
representation from the same source is to draw a vector line 
around the spatial extent of the hill, again defining a 
Boolean set of pixels but this time only using the vector line 
as the hill boundary rather than the actual elevation values 
of the pixels as in the first case. If one examines the hill 
from a different data source, a raster scanned topographic 
map or digital raster graphic (DRG) for example, the 
geometry is likely to vary from the DEM representation. 
Finally, if various scales of topographic maps are used, then 
the representational geometry of the hill changes with scale. 
The feature-based framework allows all of these 
representations of the hill to be equally valid, 
simultaneously available, and any one of them may be used 
for analytical purposes. For a more detailed presentation of 
the framework, see Usery (1994a; 1994b; 1996a; 1996b). 
3. FUZZY GIS OPERATORS AND FUZZY FEATURES 
Representation of geographic phenomena as fuzzy features 
in a GIS requires operators which perform standard GIS 
functions, such as overlay and buffering, on the fuzzy 
representations. Katinsky (1994) developed a suite of fuzzy 
operators including fuzzy overlay with union, intersection, 
and complement, fuzzy spatial buffering, and fuzzy 
boundary. The mathematical model and GIS data model for 
these operations are detailed in Katinsky (1994). An 
implementation of these functions as an extension of the 
Imagine GIS and image processing software has been 
developed (ERDAS, 1995). The examples below are 
developed from the Katinsky mathematical theory and the 
Imagine implementation. 
Using spatial position to determine the fuzzy extent of a 
geographic feature requires the following definitions: 
© Definition 1: Given a universe, V, of objects, a 
fuzzy set A* C V is a mapping, denoted f. 
from V to the unit interval, /0,7] where f,.(x) is 
the membership value of x in A* for any xeV. 
© Definition 2: A map space V is a bounded 
subset of R?. 
© Definition 3: A fuzzy feature is a fuzzy set 
whose universe is a map space. 
© Example 1: Let V — [0, 100mP be a map space 
and let A* C V be a fuzzy feature representing 
a weed patch in a cotton field centered at 
(15m,20m). Define the feature with the 
following membership function: 
1 
ve, Iv -(10,20)|<10 
10-lv (15,201; 10<iv -(10,20)j<20 (1) 
T : 20«lv -(1020)] 
fa*- 
where the dual vertical bars indicate the 
Euclidean distance between the points. 
The example defines any location within 10 metres as 
definitely within the weed patch. Any location farther than 
20 metres from the center of the weed patch is definitely 
outside the feature. Locations greater than 10 metres and 
less than 20 metres have membership values which linearly 
relate to distance from the weed patch center. While this is 
a simplistic model using a linear interpolation formula, it 
illustrates the concept of a fuzzy feature defined by the 
spatial feature dimension. 
Note that once the spatial extent of the fuzzy feature is 
defined using spatial position as the defining parameter in 
the fuzzy set function, that function can be used with any 
data set which has corresponding geometry. For example, 
the formula above can be used to define the weed patch in a 
DEM and generate the elevations over which the weed 
patch occurs. Similarly, the same formula can be used to 
define the pixels in a Landsat Thematic Mapper image to 
determine the spectral reflectance in multiple image bands 
for the weed patch with a possibility value associated with 
each pixel position indicating the strength of that pixel as a 
part of the weed feature. 
An example defined on the basis of thematic attribute data 
uses a similar spatial extent formula; however, the 
membership function determining the spatial extent uses the 
thematic attribute as the defining parameter of the fuzzy set 
function. An example with elevation values defining a hill is 
presented in Usery (19962). Following is an example 
defining the extent of moisture based on measured rainfall. 
¢  Example2: Let V — [0, 1000? be a map space 
and let A* C V be a fuzzy feature representing 
rainfall amount with a peak value of 5 cm. 
Define the feature with the following 
membership function: 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B4. Vienna 1996 
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