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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