database, ie. Um is the universe of M. We will assume that the 
geometry of the objects is represented in a vector format with a 
full topological structure, i.e. the geometry is described in nodes, 
edges and faces defining a geometric partition of the mapped area 
(or O-. 1- and 2-cells). Let Geom(M) be the geometric component 
of M, ie. it is the collection of all geometric elements describing 
the geometry of all objects of the universe. Let Face(M) be the 
collection of all faces in Geom(M); similarly Edge(M) is the 
collection of all edges and Node(M) is the collection of all nodes. 
The function Part, [ f. O] will be introduced to express the 
relation between a face f € Face(M) and an object O € Uy. If this 
function has the value = 7 then the face belongs to the spatial 
extent of the object, if the value = 0 then that is not the case. We 
can now define the set: 
Face(O) = { [1 Party: [ f, O] = 1} 
Face (O) is the spatial extent of O. In this notation the geometric 
description of the objects is organised per object. For each edge e 
we can express its relationship to a face f by the functions: 
Le[ e, f] 7 l if e has fat its left-hand side and 
Le[ e, f] - O otherwise, 
and similarly 
Rif e. f] = I if it has fat its right hand side and 
Ril e, [| = 0 otherwise. 
With these functions the relationship between an edge ¢ and an 
object O can be established: 
Le| e, Ol f] 2 MIN( Le e. f], Party] f. Of) and 
Ril e, Ol f] 2 MIN Ril e, f], Partzsl f, O]) 
Object O Te 
ut / \ 
ee A A 
IT gy 
fp 
PTT 
à Part,[f, O] 2 1 
I 
  
Figure 1: Rclationship between edge, face and Object 
If the edge has the face at its left hand side and the face is part of 
the spatial extent of the object then both functions on the right 
hand side of the first expression have the value = / and therefore 
the first expression gets the value = /. This means that the edge 
has the object at its left-hand side and thus we get Le/ e, O | I4= 
/, otherwise it will be = 0. Similarly if the edge has the object at 
its right hand side then Ri/ e, O | f] = 1 otherwise = 0. If there is 
a face for which Le[ e, O | f] 5 1 then that implies that the edge 
e has object O at its left-hand side, so that Lef e, 0} =l 
Otherwise Le[ e, O] 2 0. and similarly for Rif e. O |. With these 
two functions we define: 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV. Part B4. Istanbul 2004 
Ble. Oj=lefe,O] +Ri[e, Of 
If the edge has the object at its left and right-hand side this function 
has the value 2. If the object is only at one side, so that the edge 
belongs to the boundary of the object, the value will be I. If the 
edge is not related to the object then the value will be 0. The 
boundary of the object is therefore the set of edges for which this 
function has the value 1: 
90-[elB[ e.O] 2 1] 
3. SPATIAL AND THEMATIC PARTITIONS 
The universe U,, is a complete coverage if for every member f of 
Face(M) there is at least one object O so that Parts, [ f, O] = 1. 
That means that objects cover the whole area covered by the 
geometry of M. 
The universe is a spatial partition if it is a complete coverage and 
if the objects do not overlap. i.e. for each face f of Face(M) there 
is exactly onc object O so that Part»; [ f. O] — I. 
| Nat orsid | | forest I agricult | 
Figure 2: A collection of classes forming a thematic partition 
  
Now let P={ CI. C2. ... ,Cn} be a collection of classes so that for 
each i we have Ci © Um. P is a thematic partition of Uy if each 
object is a member of exactly one class. This means that the 
classes are properly specified so that the thematic description of 
the objects is unambiguous, sce Figure 2 and 3. 
If P is a thematic partition of Uy, and Uy, is a spatial partition of M 
then P. generates a. spatial partition of M. That means that the 
classes cover the whole mapped arca and they are spatially 
distinct; see (Molenaar 1998). 
Faces Objects Classes 
2 = Object 1 ay, 
7 os Object 2 = [1 Natgrssind 
9,10,11.... "Object 3 ayy +} 
3,4, == Object 4 «2 Forest 
1; 5,6, 8, 12== Object 5 = Agriculture 
  
Figure 3: Objects and classes Form a dual partition structure, je. 
the combination of a spatial and a thematic partition 
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