ibul 2004 
)] (4) 
lassifiers 
compare 
lated to 
| can be 
ation the 
set. The 
decision 
(5) 
(6) 
is total 
r, 1998], 
obability 
y } is the 
(7) 
bility. In 
babilities 
1erman& 
reory of 
3ayesian 
12002, 
dat 
set of N 
about a 
l'as 2°), 
(8) 
istead of 
, assigns 
ombined 
ned as 
(9) 
yperator. 
s can be 
(10) 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
  
where (M represents sets intersection and M, is the 
probability mass function of i" source (classifier). In addition 
M, is a proposition that defined as a combination of the 
elemental hypotheses, 4, and B p 
3. DUAL MEASURE DECISION FUSION (DMDF) 
METHOD ; 
In this section, we introduce new tools including commission 
and omission errors functions, distribution vectors, and matrixes 
based on local classification results. Then, we use commission 
and omission errors measures jointly, and present the algorithm 
of DMDF decision fusion method. 
3.1. Extraction and offering DMDF tools 
We consider the conf, as the confusion matrix obtained from 
the local classification results. In addition, we suppose that 
the N, is the total number of pixels related to the e class and 
we denote the n, as confusion matrix general element. 
The À ÿ is the number of pixels which related to the C class 
that the local classifier is assigned them to the C, class. 
ny My eo Bago yy 
Tor Hy oe Py -4) Py (11) 
confAln,]= RR 
Play Pap c Por nar-n Barn 
My Way 0 Ua) Phy 
= 32 
Ne, > n; (12) 
J 
N° > 13 
= N ( ) 
c, ij 
1 
In which, the N.. is the total number of assigned pixels to the 
=f 
C, class. Now, we define the mc. and mo, measures 
Pi pe 
as the commission and omission errors functions of the i” 
classifier results: 
"eer, =m, E n; | Nc. *i (14) 
i 
Moc = "2 n; = n; | Nc, J EI (15) 
j 
In addition we define the yc’. and i vectors as the 
Mec: Hoc, 
iS 4. . . . ^h 
commission and omission errors distribution vectors of 1 
classifier results. Therefore, we have: 
; n. : 
i Jl T 1 
pee Ame se c h HC eens MC, ] a6) 
C; 
. TJ m Yun 
Loc, ve a A SMO C++ MOG 
In which, 7 is the symbol of matrix transposition. 
Now, we define the commission and omission errors 
. - . . n. ire I j 
distribution matrixes of i" classifier results, Mc'and Mo' , as 
follows: 
Me’ = pel V=Lpel Les [BE duse UD) 
543 
Mo' =[po, 1=[uo, | po, |. | 07, dus (19) 
The Mc'and Mo’ columns are respectively the commission 
and omission errors distribution vectors ( yet and 4/0 c, ) of 
local classifiers. As much as we have lower commission or 
omission errors in classifying results, the Mc// Mo' matrix 
will be more diagonal. Therefore, in order to be aware of 
commission or omission errors of a classifier results, it is just 
sufficient to calculate the Mc'/ Mo‘ matrix and consider the 
diagonal level in different classes. The Heec mo. net : 
ie, i; 5 
Hoi ,Mc'and Mo‘ are DMDF toolbox elements which we 
will use them in the next section. 
Example: For explanation the properties of the DMDF toolbox 
elements (functions, distribution vectors, and matrixes), we 
have used the multispectral scanner data obtained from remote 
sensing related to an agricultural area in Indiana (United State). 
This data was collected by a 12-channel airborne multi-spectral 
scanner system during the 1971. In each local classification, the 
data of 4 bands have been used. (See table 1) 
Table 1. Spectral information for data used in example 1 
  
  
  
  
  
  
  
  
  
Spectral Wavelength Spectral Wavelength 
Bands Cum) Bands Cum ) 
No. No. 
1 0.46-0.49 7 0.61-0.70 
2 0.48-0.51 8 0.72-0.90 
3 0.50-0.54 9 1.00-1.40 
4 0.52-0.57 10 1.50-1.80 
5 0.54-0.60 11 2.00-2.60 
6 0.58-0.65 12 9.30-11.70 
  
  
  
  
By using the confusion matrix of the first local classifier we 
calculate the commission and omission errors matrixes 
( Mc! , Mo"), for this local classifier. The fourth columns of 
the Mc’ and Mo! respectively are the commission and omission 
errors distribution vectors ( HC, and L0, ) of the fourth class. 
(See Figure 2 and Eqs. (20), (21)). 
Cl 
0.5 
C9« 0.4 C2 
  
€ EAE te i 1 1 
Figure 2. The commission and omission errors ( 4/C, , L0, ) 
vector functions. 
Hey = (20) 
[0.18 0.38 0.00 0.18 0.01 0.02 0.01 0.09 0.14]" 
Ho, = (21) 
[0.05 0.06 0.00 0.33 0.01 0.16 0.01 0.36 0.04]" 
As it shown in Figure 2, the commission error distribution of 
the first classifier in assigning the pixels to the fourth class