6K =6K =0K,®e,
K =6K U(K,-5K)
SR OR SRE
--
K-«(8K, HK OK
Fig.6. Space segmentation of fuzzy 9-intersection model
PARK” ORPÄOK:) (Kt)
S. - ue, K:) u(GK, eK) u(&K, K;) [3]
u(K 0K) n(K nek) u(K NK)
where the K, and K, are given two closed sets; the generated
fuzzy sets in consideration of data positional uncertainties are
defined as dR =0K, =X. ©¢ , K,=0K, AK, 5K,), KK)'
UK USK) and K:=(5K,)"(K,-58K,)°; and pu(*) is a
kind of metric functions which are used for deriving the fuzzy
memberships based on the generated sets by logic intersections.
For different purposes, the function u(*) may take the different
forms as used below.
4. SPATIAL RELATIONS BETWEEN UNCERTAIN
SETS
Since the object feature caused uncertainties of spatial relations
can solved by using Hausdorff metrics between sub-sets [Chen
and et. al, 1995, 1996] we only discuss the problems of
conceptual and positional uncertainties of spatial relations in
this paper. For reasons of simplicity the spatial relations
between closed regions discussed in this paper only, related
models for estimation of conceptual and positional fuzzy
membership functions are defined as following sections.
4.1. Conceptual Uncertain Relations
For deriving conceptual fuzzy topologic relations, such as weak
meet and strong meet which were discussed in the section 2.1,
we can use the fuzzy 9-intersection model by selecting
£=0 and K=K, then choice the following u(*) to calculate
related fuzzy memberships from object A to B:
HEA
Line length: u (Gig Cd I [4]
£(6A)
Area size: WARE EICH [5]
ACA?)
0, when *=¢;
is #)— , >
Others: u( ) h when *z; [6]
where " * " means logically intersected sets, £(*) and A(*)are
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996
the line lengths and area sizes. Generally the binary topologic
relations between objects A and B derived by equation [3] are
~
not symmetry , i.e. S usually is not equals to — ,,.
According to the fuzzy set theoretic operations [Zadeh, 1965],
we cans either use union Max[u( 4,B),u(B,A4)] or intersection
Min[u(A,B),u(B,A)] of two fuzzy memberships to integrated
deriving fuzzy topological relations. For example, to calculate
the fuzzy topologic relations of Fig.l(b) by using the union
fuzzy memberships, we can get following results for left and
right figures respectively:
the left figure in Fig.1(b):
9.91 0-0
e soo031l, X908 1|;
1 1
(2,375
ted 8, 0 |
Max(5$,,,5$,,)4 0 008 1
111-4
the right figure in Fig.1(b):
p NEP gel Out 0
eo 10923 1|, 2 «0079.1
1 od |
haul 0:50.01
Max Sa day) 0 0.79 1
à... |
For deriving conceptual fuzzy metric relations, such as near and
far, we can firstly use the fuzzy 9-intersection model to generate
uncertain set K, based on given fuzzy membership functions,
then use the uncertain object K, as instead of the general object
K, to calculate Hausdorff metrics between uncertain sets or
sub-sets [Chen and et. al., 1996]; After that we quantitatively
estimate the conceptual fuzzy membership functions between
uncertain subsets by using the 1(*) in equation [3] as follows:
AA NB) pu, (x,y)
Solid volume: u(4°NB°)= [7]
AA) p(x.)
; _[0, when *=¢;
Others e ve [8]
where "*" means logically intersected sets, u,(x,y)is the
given fuzzy membership function for description of uncertain
metric concepts, A(*) is area size of logically intersected two
objects, and u(A4°NB°) is calculated fuzzy memberships of
derived uncertain spatial metric relations.
4.2. Positional Uncertain Relations
For deriving positional fuzzy topologic relations, we can use the
fuzzy 9-intersection model to generate uncertain set K, based
on uncertain e£ -band, then choice the following u(*) to
108
calculate rel
uA
where +"
the minimu
positional u
topologic r
equation [3]
^
eO s ACCO
n
1965], we «
membershir
Fig.7. Fu
1.0
0.5
Fig.8. Fc
For describ
(such as li
relations, v
positional 1
as K=K¢
distribution
separate tl
between 1
uncertain |
calculate F
Secondly
membersh
measurem
o (A)
Q,(0)=
