ISPRS, Vol.34, Part 2W2, “Dynamic and Multi-Dimensional GIS", Bangkok, May 23-25, 2001 
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determined by two factors, i.e. (a) the size of feature and (b) the Voronoi regions to describe spatial relations between map 
neighbouring features. Indeed, Chen et al. (2001) have used features. 
Figure 3 Voronoi diagrams of the maps shown in Figure 1 
For these reasons, the authors try to relate spatial information of 
map features to their Voronoi regions to develop a set of new 
quantitative measures. However, detailed discussion of the 
formation of Voronoi regions is outside the scope of this paper. 
Algorithms for generation of Voronoi region in vector mode have 
been presented by Okabe, et al. (1992) and a raster-based 
algorithm has recently been proposed by Li et al. (1999). 
Therefore, no further discussion on this topic will be presented in 
this paper. 
3.2 Metric information on map: entropy of Voronoi regions 
Metric information here considers the space occupied by map 
symbols only. In this case, an analogy to the entropy of binary 
image is used. That is, if the space occupies by each symbols is 
similar, the map has larger amount of information. If the variation 
is very large, the amount is smaller. This can be achieved by 
using the ratio between Voronoi-region of a map system over the 
enclosed area of the whole map as the probability used in the 
entropy definition. Let S be the whole area and and it is 
tessellated by Si, i=1, 2, ... N. Then, such a probability can be 
defined as follows: 
The entropy of the metric information, donated as H(M), can then 
be defined as follows: 
W(M) = H(P,, P 2 P, ) = -£ ^(In S,- - In S) (5) 
(=1 
H(M) has its maximum when R has the same value for all i=1, 2, 
.. .N. In other word, when the area S,is equal. Mathematically, 
H(M) m =H(.P„P 2 ,...,P. = (6) 
For example, the two maps shown in Figure 4 have different 
amount of metric information although both are tessellated by 9 
polygons. The map in Figure 4(b) has the maximum H(M) for 
any tessellation into 9 polygons. 
In the case of map, it is clear that for the same number of 
feature, the entropy will be larger is the symbols are more evenly 
distributed. However, it is clear that such entropy is related to 
the number of map symbols and thus it would be not convenient 
to compare two maps with different number of symbols. In order 
to overcome this shortcoming, the entropy could be normalised 
as follows: 
H(M)„ = 
h(m) 
H{M) m 
(?) 
Another possible measure is the ratio, Rm, between mean of the 
areas (m A ) and the standard deviation a A . 
m 
l n 
,—L4 
n 
i=l 
I 
n -1 
(8) 
(9) 
R 
M 
(10) 
Figure 4 Two different tessellation of an area, 
resulting in two different amount of metric information