Full text: The 3rd ISPRS Workshop on Dynamic and Multi-Dimensional GIS & the 10th Annual Conference of CPGIS on Geoinformatics

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ISPRS, Vol.34, Part 2W2, “Dynamic and Multi-Dimensional GIS", Bangkok, May 23-25, 2001 
Statistically speaking, H(X) tells how much uncertainty the 
variable X has on average. When the value of X is certain, Pp1, 
then H(X)=0. H(X) is at its maximum when all messages have 
equal probability. 
In communication theory, three types of information are 
identified, i.e. syntactic, semantic and pragmatic information. 
Indeed, most of researchers in spatial information science try to 
follow these three types of information for a map. 
2.2 Statistical information of a map: entropy of symbol 
Sukhov (1967, 1970) has adopted the entropy concept for 
cartographic communication. In such a work, only the number of 
each type of symbols represented on a map is taken into 
account. Let N be the total number of symbols on a map, M the 
number of symbol types and Ki is the number of symbols for 
type. Then N = + K 2 + ... + K M . The probability for 
each type of symbols on the map is then as follows: 
where, Pi is the probability for i th symbol type, i =1,2, .... M. 
The entropy of the map can be calculated as follows 
h(x)p u )=-f; /> in(/>) 
The shortcomings of this measure for map information could be 
revealed by Figure 1, which is modified from (Knopfli 1983). 
Both map consist of three types of symbols, i.e. roads, buildings 
and trees and have exactly the same amount of symbols for each 
type. That is, there is a total of 40 symbols, i.e. 7 for roads, 17 
for buildings and 16 for trees. Therefore, according to definitions 
in Equations (2) and (3), both maps shown in Figure 1 have the 
same amount of information, i.e. H=1.5. However, the reality is 
that the distributions of symbols on these two maps are very 
different. In Figure 1(a), the map symbols are mostly located on 
the right side of the diagonal along lower/left to upper/right 
direction and the tree symbols are scattered among buildings. 
Two rows of buildings are along the main road. However, in 
Figure 1(b), there is an area of trees on the left side of diagonal 
along the lower/left to upper/right direction and there is an area of 
buildings on the opposite direction. The roads are almost along 
the diagonal. Indeed, they represent different natures of spatial 
Figure 1. Two maps with the same amount of symbols but with different distribution 
lln other words, the entropy computed in this way only takes into 
account the number of symbols for each type but the spatial 
arrangement of these symbols is completely neglected. Such a 
value is purely statistical and thus is termed as "statistical 
information” in this paper. Indeed, it doesn't mean much in a 
spatial sense. Therefore, the usefulness of such a measure is 
Topological information of a map: entropy of 
the generation of such a dual graph was put forward by 
Rashevsky (1955). 
Figure 2(a) shows a dual graph which consists of seven vertices 
at three levels. There are three types of vertices if classified by 
number of neighbours. There are four vertices with only one 
neighbour, one vertex with two neighbours and two vertices with 
three neighbours. Then, N=7, M=3, thus, the probabilities of 
these three types of vertices are: y, y and y . The entropy 
of this dual graph is then computed using Equation (3) and the 
result is 1.38. 
Neumann (1994) proposed a method to estimate the topological 
information of a map. The method consists of two steps: (a) to 
classify the vertices according to some rules, such as their 
neighbouring relation and so on, to form a dual graph, and (b) to 
compute the entropy with Equations (2) and (3). The method for

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