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where 
and z' is the !th feature of the pixel z. The group of 
adjacent pixels defines an object of an image. 
Let us denote the object of an image by X, where 
Ka an 
is a vector of size Nqx1 and N is the number of pixels in 
the object. 
Suppose that the pixels of an object have Gaussian distri- 
bution, i.e. 
X ~ N(pi, Zs), 
where 
pi = (ulia, ud) 
is the mean of the pixel x of the ith class and Xj is the 
covariance matrix of x. 
The aim of image recognition is to classify each pixel of 
an object or the whole object of an image into one of m 
possible classes. 
The classical procedure is à per-pixel classifier whichs as- 
signs pixel x to the class 7, when 
p(zle;) — max p(ale;), 9 
where p is the class-conditional density function for the 
class w;. This decision rule classifiers pixels alone using 
only the spectral characteristics of pixels of an image. 
The decision rule for the object classifier is the following 
EE. Je 2 
MX kei) = max PX is) (2) 
In Gaussian case p(X|w;) is characterized by K - covari- 
ance matrix of an object X of size Nqx Nq and M = 
(tty ..., pt)" - mean of X of size Nqx1. 
The usage of object classifier (2) in the general case is very 
restricted because for rather large N and q it is difficult to 
estimate matrix K because of a limited size of the learning 
sample. To overcome these difficulties, one has to make 
certain assumptions about the structure of K. 
The solution of this problem is based on the two following 
assumptions about the structure of matrix K. 
First, it is assumed that the correlation between the pixels 
of an object does not depend on g¢, i.e. K = R®Y, where 
R is the spatial correlation matrix of size N x N, ® is the 
Kroneker product. 
Secondly, assumptions about the structure of matrix R are 
made. Often, it is assumed that the pixels inside the ob- 
ject are independent ( Ketting, 1976; Landgrebe, 1980). 
In this case R is the identity matrix and the spatial char- 
acteristics are employed indirectly. Another popular as- 
sumption is that an object of an image is a Markov ran- 
dom field ( Switzer, 1980; Mardia, 1984; Guyon and Yao, 
1987; Kalayeh and Landgrebe, 1987; Palubinskas, 19882) 
which is represented by causal autoregressive model. In 
this case matrix R is characterized by few parameters, the 
number of which depends on the order of Markov model. 
All object classifiers are based on the popular separable 
correlation function 
[i-k| 
= 
corr(z;, eg) pl" o9, 
485 
where p; and p; are spatial correlation coefficients between 
adjacent pixels of an object in the horizontal and vertical 
directions, respectively. 
So we see that depending on the structure of matrix R 
there can be a wide variety of object classifiers. The prob- 
lem of selection of object classifier is actual. From Section 
l we see that there are a lot theoretical investigations of 
object classifiers, but there are only few results on real 
data. 
So object classifiers must be investigated more thoroughly 
on real data. For this purpose IMAX - image analysis 
and classification system was developed in Data Analysis 
Department, Institute of Mathematics and Informatics ( 
Palubinskas, Cibas, Repsys, 1991 ) on PC computer. 
IMAX offers the choice between 10 classifiers: one conven- 
tional per-pixel maximum likelihood classifier ( PIX ), one 
conventional per-pixel minimum distance classifier, 6 ob- 
ject maximum likelihood classifiers and 2 object minimum 
distance classifiers. Object classifiers classify the central 
pixel of an object. Then the window is moved by one pixel. 
Four object classifiers are for cross-shaped block and other 
four are for square-shaped block. In each group there are 
the following object classifiers: 
- object classifier incorporating the spatial characteristics 
of an image directly on the base of causal Markov ran- 
dom field model of the first and third order, respectively 
( OMARK1, OMARK3 ), 
- object classifier based on the assumption that the pixels 
inside the block are independent ( OIND1, OIND3 ), 
- object classifier which classify the mean of the block un- 
der the assumption that the pixels inside the block are 
independent ( OMEANIND1, OMEANIND3 ), 
- object classifier which classify the mean of the block un- 
der the assumption that the pixels inside the block are in- 
dependent and with covariance matrix equal identity ma- 
trix ( OMEAN1, OMEANG3 ). 
These 8 object classifiers were investigated in this work. 
For detailed description of object classifiers see ( Palubin- 
skas, 19882 ). 
3. ANALYTICAL METHOD FOR EVALUATION OF 
CLASSIFIER PERFORMANCE 
The conventional method for evaluation of classifier per- 
formance ( the probability of error ) is mathematical mod- 
eling. First, one has to programm the classifier, then to 
run it on test data in order to calculate the probability of 
error. But it is rather time consuming way. 
The analytical way to calculate the probability of error 
requires only calculation of statistical characteristics of 
data ( mean and covariance matrix ). So it can be much 
more faster. In ( Fukunaga, 1972 ) two analytical meth- 
ods of calculating the probability of error are proposed. 
First calculates the exact probability of error of discrimi- 
nant function using Imhof formula or characteristic func- 
tion. Second approximates the probability of error under 
assumption of discriminant function normal distribution. 
But these formulae are in the case of classifier model and 
data model equality. In practice, usually, these models 
are different. In ( Palubinskas, 1988b; 1992 ) these formu- 
lae are extended for the case of classifier model and data 
model inequality. So these formulae allow to compare 
several classifiers in the same conditions. Now we shall 
present these formulae in more details.