ts and objects 
x 
    
dos Man ?d OC 
s and objects 
  
relations in GIS are 
les of object shapes, 
lation between two 
ept and is thus often 
(a)-(b)]. The simplest 
'etween two objects is 
he represented point 
tween two capitals to 
ries. But this method 
ry is big and its shape 
hen and et. al [1995, 
ate metric relations 
is to calculate the 
een their sub-sets of 
he general Hausdorff 
ecial case of its fuzzy 
‚ the object shape and 
of ordering relations 
elations. The ordering 
ed by the method of 
^hen and et. at., 1995, 
GIS are mainly caused 
data. Generally, the 
ire not error-free, they 
the errors of scanning, 
erlaying, and  etal. 
ing the uncertainty in 
nes and areas), we can 
d by Chrisman [1982], 
spatial object K, can 
11996 
  
  
  
  
  
    
  
  
* 
g ef 
  
  
| 
  
Fig.3(b). Positional uncertainties of topologic relations 
  
  
  
  
  
VU ; e 
„BC e, (C 
9 
ac P A. 
Or „9 
Ó d, ed <d Ó d. e sd e 
  
Fig.3(c). Positional uncertainties of ordering relations 
be represented as K -K ®u(e) ,; Where- € is the, buffer 
distance of error distribution and  L(£) is the fuzzy 
membership function derived by £ . As shown in Fig3(a)-(c), 
spatial metric, topologic and ordering relations between 
error £ -band generated objects will cause different kinds of 
uncertainties. 
3. FUZZY 9-INTERSECTION MODEL 
3.1. 9-Intersection 
For driving binary topological relations between sets, Egenhofer 
et al., (1994) developed the 9-intersection model based on the 
usual concepts of point-set topology with open and closed sets, 
in which the binary topological relations between two objects, 
K, and K, sin IR’ is based upon the intersection of K,'s 
interior ( K? ), boundary ( CK, ),and exterior ( K, ) with K,’s 
interior ( K? ), boundary ( CK, ), and exterior ( K; ). A. 3x3 
matrix 3 , called the 9-intersection as follows: 
KK KR Ka 
Sy eK KS CK, CK OK IK. [1] 
Kink" K, "CK Kin 
By considering the values empty (0) and non-empty (1) in 
equation [3], one can distinguish between 2°=512 binary 
topological relations in which only a small subset can be 
realized when the objects of concern are embedded 
in IR’ [Egenhofer and Franzosa, 1991; Mark and et. al., 1995]. 
The beauty and simplicity of 9-intersection model come from 
the fact that it can solve the topologic and geometric problems 
by using the formal logic and algebraic methods. Since present 
digital computers are very strong for logic and arithmetic 
calculations, but they are poor for high level geometric and 
topologic reasoning. So the 9-intersection model has the 
potential abilities for automatically spatial and temporal 
reasoning. 
3.2. Dynamic 9-Intersection 
For integrally deriving different kinds of spatial relations 
between sets, Chen and et al. (1995, 1996) developed the 
dynamic 9-intersection model based on the concepts of the 
metric topology with open and closed sets and the 
morphological dilation, in which the general 9-intersection of 
equation [1] is extended as follows: 
[K,eB(s)^K? [K,9B(s)P^cK, [K9B(e)l'^K; 
San (€) TKOB(ENINKS AK, @B(8)INCK, AKOBENK 
[K,9B(&)] ^K? [KOB(E)TN6K, [KOB(E)T ^K; 
[2] 
where the K, and K, are given two closed sets, the K,®B(e,) 
means relevant morphological dilation by the closed ball B with 
radius £, , and the 5$, (€, ) means dynamic 9-intersection with 
9 
G.j) 
parameter £, from K, to K.. Based on the equation [2], we 
can derive dynamic topological relations by using the different 
parameter £,. In particular case, when €, =0, we have 
K ®B(¢,)=K,®{0}=K,, then the dynamic 9-intersections 
9 
Gj 
and et. al, 1995, 1996]. 
Sen (E,) coincide with the general 9-intersection 3, [Chen 
3.3. Fuzzy 9-Intersection 
For deriving different kinds of spatial relations between 
uncertain sets, we can extend the 9-intersection model to the 
fuzzy 9-intersection model as follows: 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
Fig.5. Space segmentation of dynamic 9-intersection model 
107 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996