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