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of the region R,, i.e. the function identifies the object that 
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COMPIO,, ) Roos dos; 0,,.:J 
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The geometry of the aggregates can be found through the 
geometry of the original objects, for each geometric 
element we can check whether it will be part of an 
aggregated object of type T,. This should be done in two 
steps, which will be explained for the faces of an area 
object O, in relation to an aggregated area object O,,. The 
first step evaluates the function: 
Party, f, O, | O,] = MIN(Part,,[f,,0,], Party, [ O,, 0, 1) 
this function expresses whether the face is related to an 
aggregate through object O,. If that is true then both 
functions in the expression at the right hand side of the 
equation will have the value = 1, and this value is 
assigned to the function at the left hand of the equation. 
If it is not the case then at least one of the functions at 
the left hand side will have the value = O, so that also 
the function at the left hand side will get the value = O. 
The second step is the evaluation of 
Part, [ f,, O,] = MAX, (Party, [ f,, O, | OJ) 
If there is any object through which the face will be part 
of an aggregate then this function will have the value = 
1, otherwise it will be — O. If this function has been 
evaluated for all faces of the map then the geometry of 
the object O,, can be found through their adjacency graph. 
For the edges e, of these faces the function 5/e;, O,] can 
be evaluated and with this function the boundary edges 
can be found (i.e. B/e,O] — 7) and through these the 
topologic relationships with the other objects. 
The geometry of the aggregated area object O, can 
sometimes be simplified by a reduction of the number of 
faces. Therefor the edges e, should be identified for which 
Ble., O,] = 2, that are the interior edges. If these edges 
are not part of some line object so that LO(e;) = © then 
they do not carry any semantic information at this 
aggregation level and could therefor be eliminated. 
The example refers to the situation where a face is related 
through an area object to an aggregated are object, so that 
all involved elements are of dimension 2. Other combina- 
tions of dimensions might occur as well, this could be the 
case when for example an edge is related through a line 
object to an aggregated area object, e.g. it is related 
through a river to a country. 
Itis possible to define aggregation types by means of their 
construction rules. If elementary objects are combined to 
form a compound object, their attribute values are often 
aggregated as well (as in figure 6). We will see in section 
3.2 that farm yield is the sum of the yields per field, and 
the yield per district is the sum of farm yields. The 
desaggregation of such values is usually quite difficult 
because it can only be done if information is added to the 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
system. An aggregation hierarchy has therefore a bottom-up 
character, in the sense that the elementary objects from 
the lowest level are combined to compose increasingly 
complex objects as one ascends in the hierarchy. The 
compound objects inherit the attribute values from the 
objects by which they are composed. 
The PARTOF relations connect groups of objects with a 
certain aggregate and possibly on a higher level with another 
even more complex aggregate, and so on. That means that 
an aggregation hierarchy expresses the relationship between 
a specific aggregated object and its constituent parts at 
different levels. This is different from class hierarchies 
where classes at several generalization levels can be defined 
with their attribute structured and their intentions, but 
where the objects can be assigned to these classes in a 
later stage of a mapping process. 
3. STRATEGIES FOR OBJECT GENERALIZATION 
The formalism of the previous chapter helps us to express 
the structure of spatial datasets. This can be done in an 
abstracted sense, i.e. without any reference to the logic 
model of any implemented spatial data base. Processes 
applied to such datasets could also be expressed through 
this formalism. The four basic operations that will be used 
in generalization processes are: 
- the se/ection of objects to be represented at the 
reduced scale, this selection will be based on the 
attribute data of the objects, 
- the e//mination from the data base of objects that 
should not be represented, 
- the aggregation of area objects that should not be 
represented individually, 
- the rec/assification of the generalized objects. 
For these four operations information about the spatial 
structure of the mapped area will be required. Firstly to 
check which relationships the objects have with their 
environment and what the effect of their eventual 
elimination will be on the spatial structure of that 
environment. Secondly this information is required to 
formulate aggregation rules for the objects that are to be 
merged. Once the process has been formulated one can 
choose how to implement it in any suitable database 
environment. The hydrologic example presented in sections 
2.1 and 3.3 of this paper has been implemented in an 
ArcMnfo environment, but other students of the author 
have made implementations of similar applications in an 
Oracle database, and exercises with Prolog have been made 
as well. 
Several strategies for database generalization can be 
formulated with this formalism and these basic database 
operations .These are: 
- geometry driven generalization: in this strategy it 
is the geometric information that drives the 
aggregation process. A clear example of this case 
is when the geometry of the spatial data has a raster 
structure. If it is then decided that the resolution 
of the raster will be decreased, i.e. when the cell 
size increases, then the original, smaller cells are 
merged into new larger cells. The thematic informa-