Istanbul 2004 
is function 
it the edge 
e |. If the 
be 0. The 
which this 
IONS 
ember f of 
f 0] = 1. 
red by the 
verage and 
'e(M) there 
| 
partition 
; so that for 
U,, if each 
ns that the 
scription of 
tition of M 
ns that the 
¢ spatially 
icture, Le. 
lon 
This has been illustrated by Figure 3 where: 
The set of faces is 
Fy (fl, f2. 3, 4. f5, f6, [7. f8. f9, fIO, fI, f12] 
The universe of the map is 
Un = 01, 02, O3, O4, 05] 
The collection of classes is 
P z (Natural Grassland, Forest, Agriculture] 
We see in Figure 2 that P is a thematic partition because each 
object belongs to exactly one class. Uy forms a spatial partition 
because each face of F4 belongs to exactly one object from Uy. 
We can see that this implies indeed that P generates a spatial 
partition, i.e. the classes cover the whole mapped space and there 
are no spatially overlapping classes. This data set has a dual 
partition structure. 
Hierarchical partitions 
If several thematic partitions {P,, P,, ...,P, } have been specified 
for Uy so that for any combination P, and P,; the relation be- 
tween the classes of P, and the classes of P,,, is n:1 ( many to 
one), then these thematic partitions form a hierarchy. This can be 
expressed as follows: 
let IT= {P,. P2, …P7 } be a collection of partitions 
then ITis a hierarchy of classes if 
(VC;e P,Ik « (JC; e P1.) (Cic) 
and Iis a strict hierarchy if 
(VC;e Pk kx m3C;e P) (Cic) 
These definitions imply that the partitions of the collection /7 are 
ordered. so that each partition contains the classes of a particular 
level of this class hierarchy, P, represents the highest level of the 
hierarchy and P, the lowest level. Because every P, is a partition 
each object of Jy is always a member of exactly one class of 
each level of the hierarchy, see Figure 4. 
fete eomm n 
HANA | 
os] jai pep 
C4 C5 
Ci2C Ci Cis CioCr Cia C902) Cot Gz2075 P, 
Figure 4: Classification hierarchies (Molenaar, 1998) 
The definition states that in a hierarchy every class of each level 
(with exception of the highest level) is always a subset of some 
class at the next higher level, in a strict hierarchy it is always a 
proper subset of some class at the next higher level. Consequently 
every class of a level (with exception of the lowest level) in a 
strict hierarchy contains always two or more subclasses at the 
next lower level. If the hierarchy is not strict then there may be 
classes that contain exactly one class at the next lower level. 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B4. Istanbul 2004 
4. THEMATIC OBJECT AGGREGATION 
4.1 Class driven aggregation 
Suppose that a database contains the situation of Figure 5.a; this is 
a detailed description of a terrain situation with different types of 
land use (Liu, 2002). A less detailed spatial description can then 
be obtained. if the original objects area aggregated to form larger 
spatial regions per major land use class. This less detailed 
description can be obtained in two steps: 
I First the objects are assigned to more general classes 
representing the major fand use types. 
Then mutually adjacent objects are combined per class to 
form aggregated objects.. 
^ consequence of this procedure is that there can be no two 
adjacent aggregated objects that are of the same type, i.e. that 
belong to the same land use class. This has been illustrated in the 
aggregation step from Figure 5.a to 5.b. 
The original objects form a geometric partition of the mapped 
arca and the classes form a thematic partition of the universe of 
the map. The relationships between the classes at different levels 
of class generalization form a hierarchy, so that the classes at each 
level of this hierarchy form a thematic partition according to 
Section 3. We saw in the previous section that when a collection 
of objects forms a geometric partition before aggregation, then the 
new collection after aggregation will also form a geometric 
partition. The combination of these two observations implies that 
when thc aggregation procedures of this section are applied then 
n2 
the dual partition structure will be maintained. In the terms of 
(Molenaar, 1989) we can say tat this procedure transfers a single 
valued vector map into a new single valued vector map (Molenaar 
1989. 1998). 
4.2 Similarity driven aggregation 
Generalization 
This aggregation procedure is quite different from map 
generalization processes because it does not necessarily eliminate 
all small objects. This class driven aggregation process generates 
objects at a higher (more general) thematic class level; so the 
thematic content of the data set is driving the process not the 
resolution or scale of the (graphical) representation. 
If small objects should be removed then a special step is required 
to identify these objects. A size criterion has been applied to the 
objects of Figure 5.b. the results are shown in Figure 5.c. These 
sclected objects have no neighbours with a common super class, 
therefore a criterion can be formulated to measure the thematic 
similarity of these objects and their neighbours (Yoalin, 2002), 
(Bregt and Bulens. 1996). In the step from Figure 5.c to 5.d these 
objects have been merged with the neighbours that were most 
similar according to such a criterion. This similarity driven 
procedure is in fact a modification of the class driven approach: 
the strict. requirement for the aggregation of objects with a 
common super class has been relaxed by the use of a similarity 
measure thus allowing a wider range of applications. But in both 
approaches it is the thematic similarity (or thematic 
generalization) that drives the process so that spatial resolution 
depends on thematic specification. 
Image analvsis 
The similarity driven approach has also been applied to the data 
set of Figure 6.a. (Gorte, 1998) This figure gives an example how