International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
  
( C,) is not desirable. For example, in this classification, the 
chance of assigning the pixels of other classes to the C, class is 
higher than the chance of assigning the C, pixels to the 
C, class (bad classification). So, in an exact expression, we can 
write: 
me,, -l-mc,., -1—0.18z0.82 -eg (22) 
j*4 C€4€4 
€ j*€4 
Also, Figure 2 shows that the omission error distribution of the 
first classifier in assigning the C, pixels is not desirable. 
mo, =1—mo, ; =1—0.33=0.67 (23) 
4C j 4€4 
Cj *€4 
Eq. (24), shows that only ?633 of the C, pixels allocated to 
the C,. In other words, after classification, the most pixels of 
the fourth class (%67) allocated to the other classes (bad 
classification). In next section, we will present the DMDF 
method for decision fusion, by applying the relevant tools and 
measures, which we have extracted in this section. 
3.2. The formulation of DM DF method 
We can apply each of the mc. andmo_ measures as the 
dui ej 
probability mass function that indicated in Eq. (9). Therefore, 
for fusing the results of the two local classifiers, considering the 
commission or omission errors matrix of each classifier, we can 
deploy the new form of Eq. (10) as follow: 
a®h 
uL =p - 
; ; (24) 
m(l)ym (1) m(2).m (2) m( M ).m (M) T 
QU X K K 
1j T id 
; ; > ARIA a®b . 
In which, ®is symbol of classifiers fusion, "is 
4 
commission or omission errors distribution vector obtained 
^ ~ . * * * . 
from fusion. Also, M and m are respectively commission or 
omission errors distribution functions of a and b classifiers and 
Qt, ,, coefficient is equal to: 
m(1 ym (1) m(2).m (2) m(M ym (M) 
i K + K + .t K | (25) 
> ij ij i.j 
where: 
m(),m (s) x 
Kx" Am m. (26) 
Lio CL 0. aC 
The selected class as the result of fusion is a class, which 
a®b 
7 (fusion output). In other 
/ 
includes the maximum value of H 
words, we have: 
n @/ 
X € C, if max (u^ (e) - 
e (27) 
ml) am (1) mk) am (k) 
Kl SU A 
i.j 
ab gl 
ij 
a.b 
max {a 
[i 
The new measures provide the advantages of considering the 
important features of local decisions, i.e. commission or 
omission errors, and they depend on only to the local classifier 
performances. Therefore, the calculation of these new tools 
compare with the basic DS method probability masses is very 
easier. If we use the commission or omission errors tools for 
improving the DS method, then the improved method name is 
DS (PM) or DS (CM). Although, using these new measures, we 
can considerably improve the fusion results in DS method, in 
544 
this section; we present the DMDF algorithm, which is based on 
the parallel usage of these measures. Supposing the decision 
fusion results, using commission and omission errors measures 
are as follows: 
192 192 
uem, (c,) = max {MC (6,)j 28) 
k 
pom,” (c,) 7? maxt4o; (c,)) (29) 
Cr , 
; 1@2 2 
In which, Hem, (CC) and Lom? 
7 ey 
(c,) respectively are 
1®2 2 | 
the values of HC. and uoc. at C=C, and C= Cr , and 
j 
they are equal to: 
a 12 a » s 
uem, (e,) À ok; me, mc, (30) 
1254 102. > 
Hom," (c,) À oo,; mo. mo, ,. (31) 
In which, QC; j and Qo; ; are @ coefficient that defined in 
Eq. (25). In the DMDF method, the final decision for selected 
(winner) class is considered as follow: 
Xe 
Q, if Locnl^ (c,) » poeni? (c,) 
ey df pocnl e)» oen e.) 
ala, if ocn (c,)- pocni (c,) 
; 192 ; . 
In which, 4/ocm 2%.) and oem” (c;) respectively 
J 
are the extracted joint measures from fusion results, by applying 
the commission and omission errors vectors, and defined as 
follows: 
20cm > eo, Y' A pom le, ) + pein" e.) (33) 
Locm de te) À Lom. (c,)* HM (c ,) (34) 
Figure 3, illustrates the decision fusion procedure in DMDF 
algorithm. 
  
„A ®2 
He, max (uci? (c, 
A 441928.ON 
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s 
    
  
  
| ON 108U9S 
  
  
mT ZO 
  
  
      
  
"ME 
192 
max iuo (ce, )} 
C, 
CON 408UON 
  
  
  
  
  
  
  
N UN 4U5u3S 
  
  
Figure 3. The multiple classifiers decision fusion scheme in 
DMDF method. 
4. Deployments and Comparing the Methods 
In this section, we have deployed the decision fusion methods 
including MVF, rank based, Bayesian inference, Dempster- 
Shafer and DMDF for fusing the local classifier decisions. Then 
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