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The 3rd ISPRS Workshop on Dynamic and Multi-Dimensional GIS & the 10th Annual Conference of CPGIS on Geoinformatics
Chen, Jun

ISPRS. Vol.34, Part 2W2. “Dynamic and Multi-Dimensional GIS”, Bangkok, May 23-25, 2001
(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.
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
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
(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
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