ISPRS. Vol.34, Part 2W2. “Dynamic and Multi-Dimensional GIS”, Bangkok, May 23-25, 2001 
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(a) A tree type of dual graph < b > A dual 9 ra P hs with network 
Figure 2 Dual graphs for computation of topological information 
Figure 2(a), i.e. 1.38. But, it is clear that the graph shown in 
Figure 2(b) is more complex than that in Figure 2(a). Thus, such 
topological information may not be able to reflect the true 
complexity of neighbour relations. 
The question arising is "how to form a dual graph for a given 
map?" It is, indeed, a difficult task to produce such a dual graph, 
e.g. for the map given in Figure 1, because most of map features 
are disjoint. River network may be the type of feature convenient 
to form a dual graph. Indeed, in his study, Neumann (1994) 
produced a dual graph for river network. He also tried to produce 
a dual graph for contour lines. This is possibly because contour 
lines are nicely ordered according to their heights. Therefore, the 
usefulness of this method might be limited. Apart from this, the 
entropy computed by this method is only for the distribution of 
vertex type and has little relation with the topological relation. 
2.4 Other types of information for a map 
In fact, the usefulness of such a topological information has also 
been questioned by Bjorke (1996). He provides another 
definition of topological information by considering the topological 
arrangement of map symbols. Instead of one entropy name, he 
used a set. He also introduced some other concepts such as 
positional entropy and metrical entropy. "The metrical entropy of 
a map considers the variation of the distance between map 
entities. The distance is measured according to some metric" 
(Bjorke 1996). He also suggests to "simply calculate the 
Euclidean distance between neighbouring map symbols and 
apply the distance differences rather than the distance values 
themselves". "The positional entropy of a map considers all the 
occurrences of the map entities as unique events. In the special 
case that all the map events are equally probable, the entropy is 
defined as //(x) = ln( jV) , where N is the number of entities. 
3. NEW QUANTITATIVE MEASURES FOR SPATIAL 
INFORMATION ON MAP 
In the previous section, existing measures for information 
contents on a map have been reviewed and evaluated. Their 
limitation should be clear. It is now pertinent to introduce new 
measures in this section, which should be sound in theory. The 
usefulness in practice will be evaluated in Section 4. 
3.1 The line of thought 
The communication theory is about human language, which don't 
have spatial component. Therefore, it could be dangerous to 
follow the line of thought developed in communication theory. 
That is, a completely new line of thought must be followed. 
It is a commonplace that a map contains for following information 
about features: 
(a) (Geo)metric information related to position, size 
and shape; 
(b) thematic information related to the types and 
importance of features; and 
(c) spatial relations between neighbouring features 
implied by distribution 
Therefore, a set of measures needs to be developed, one for 
each of these three types, i.e. metric, topologic and thematic 
information. 
To consider metric information, the position of a feature is not a 
problem. On the other hand, the consideration of size and shape 
is not an easy job. One approach to describe the size is simply 
based on the size of the symbol. However, serious deficiency 
with this absolute approach rest in its ignorance of the following 
facts: 
(a) the size of a point symbol is always smaller than 
an areal symbol; 
(b) the relative space of a feature, i.e. the empty 
space surrounding the feature, makes the feature 
separated from the rest. The larger the empty 
space surrounding the feature has, the more 
easily it can be recognised. 
As map features share empty space surrounding them. It is 
necessary to determine the fair share for each feature. In this 
case, the map space needs to be tessellated by feature-based 
tessellation (Lee et. al., 2000). The Voronoi diagram seems to 
be the most appropriate solution. A Voronoi diagram is 
essentially a partition of the 2-D plane into N polygonal regions, 
each of which is associated with a given feature. The region 
associated with a feature is the locus of points closer to that 
feature than to any other given feature. Figure 3 shows the 
Voronoi diagram of the maps shown in Figure 1. The polygonal 
region associated with a feature is normally called the Voronoi 
region’ (or Thiessen polygon) of that feature and it is formed by 
perpendicular bisectors of the edges of its surrounding triangles. 
Such a Voronoi region is a ‘region of influence’ or ‘spatial 
proximity’ for a map feature. All these Voronoi regions together 
will form a pattern of packed convex polygons covering the whole 
plane (neither any gap nor any overlap). Thus a Voronoi 
diagram of a map feature is its fair share of its surrounding 
space. 
Indeed, Voronoi region is not only adequacy for the 
determination of the fair share of surrounding empty space for a 
map feature but also good for neighbour relationship (Gold, 
1992). This is because the Voronoi region of a feature is