The purpose of pattern recognition is to determine which
category or class a given sample belongs. Lets consider
the two-class problem, i.e. each sample belongs to one of
two classes, wı or wo. The conditional density functions
and the a priori probabilities are assumed to be known.
Let X — (z1,72,..,z4)' be an observation vector. The
Bayes decision rule ( 1 or 2 ) can be written as follows:
h(X) = — In p(X|w1) + In p(Xlu) $ t
—w BL xe {2 (3)
where p(w;) - a priori probabilities and p(X |w;) - condi-
tional density functions.
The probability of error evaluates the performance of a
decision rule. The probability of error can be calculated
as follows:
€ = p(w1) - €1 + P(w2) - 2, (4)
where
Too
Ei = lí p(h|w1 )dh, (5)
co / nr (6)
—OO0
Formulae (5,6) do not lead to straightforward calculation
of probability of error, because we need to know the den-
sity function of A(X). But there are some cases when this
can be done.
When the p(X|w;) are normal with expected vectors M;
and covariance matrices X;, the Bayes decision rule (3)
becomes
MX) = 3(X — Mi) E17HX — Mi)
1 15 —1 i [21] <
X MEN ME EN
uU
wa.
(7)
From (7) we see that the discriminant function of A(X)
depends upon the following parameters: mean vectors M;
and covariance matrices X;,7 = 1,2 ( classifier model ).
Let X, which is to be classified, is normally distributed
vector with the true parameters: MT and X7,i — 1,2(
data model ).
3.1. Exact Probability of Error
The probability of error for the first class using Imhof for-
mula can be expressed as
à **? siné(u)
fr J uplu) ^'^ e
el
1^3 T
where
Ig Shs
6(u) = 5 2 [ten " (di ru) v di, ru(1 + di pu) ]-5€».
i=l
p(u) = TJ + & pu?) 5e 22 tton
il
486
The probability of error for the second class is obtained
analogically.
We see that the formula for the probability of error is
rather complicated for computing. In the next Section
the simple formula for the approximation of probability of
error is presented.
3.2. Approximate Probability of Error
When A(X) is a normal random variable (5,6) becomes
ei = @[(-1) —], i= 1,2, (9)
where
mi = B{M(X)lwi}
2
1 .
Net ET
j=1
NS
HMP — MjyE;7 (MT - Mj) 1n = ^
2
0? = EHMX) ni) kei] = FR Y (70/7! 5; 9D!)
4( Y (71) (MJ - Mj) Xj ET
Ve
1
J
(Cn ar - MyyE;19.
Me
1
j
= 2
D(z) = Gi f e-^ dt.
— C0
The accuracy of approximation of the probability of error
is investigated in ( Palubinskas, 1992 ). There we want to
note that the accuracy of approximation is strongly influ-
enced by the concrete structures of the covariance matrices
of classes. Also to calculate the approximate of the prob-
ability of error is much faster than to calculate the exact
probability of error. In the following Section both ana-
lytical methods are used for object classifier performance
evaluation.
4. EXPERIMENTS
In this Section first experimental results on forest classifi-
cation, based on LANDSAT TM data recorded on 30 July
1984, are presented. The aim of this research is to distin-
guish forest types: deciduous forest, coniferous forest and
mixed forest with the help of object classifiers. We have
to note that the same problem on the same data set was
solved in ( Schulz, 1988; Pyka, 1990 ) with the help of
per-pixel classifier. The potential of object classifiers for
forest classification is also of interest.
Defining the training and control fields for supervised clas-
sification is rather difficult task, especially when there is
no possibility to get true ground information. So from vi-
sual analysis of multispectral images and topographic map
1:50 000 ( supplied by IFAG ) two fields were defined for
each class ( forest type ). Only one band: No. 4 was used
for classification. The results of classification are shown
in Table 1. PR is correct classification on training sample
and PK - on control sample.
of t
fielc
peri
of t
redi
exp
TA
In t
acte
Mar
"Tw
of o
erro
Firs
LA?
clas:
han
the
Alf
of ai
nics
105-
Kal:
mod
Trai
461.
Ket!
mul
of h
ics,
Lan
spat
Rec
Lan
mot;
Mar
cat
Met
