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would produce 256*256 dimensional feature 
vectors. However, the estimates would be 
highly unreliable. The normal procedure 
to compress the number of gray levels, 
leads to 8*8 or 16*16 dimensional vectors, 
which are still quite high. In the classi 
cal paper of /HaShDi73/ a set of fourteen 
features are derived from the cooccurrence 
matrix. Usually only a few of them have 
been used, the famous five being energy, 
entropy, contrast, correlation, and homo- 
genity. However, the features that Hara- 
lick et al. extract from the cooccurrence 
matrix, reduce the amount of information. 
E.g., Conners and Harlow /ConHar80/ de 
monstrated by two different textures, ha 
ving different cooccurrence matrices, that 
the five features were the same, even when 
using different separation parameters d. 
Often, the use of just one separation para 
meter is not sufficient. For each separa 
tion parameter, its own matrix can be com 
puted and the features extracted can be 
averaged or concatenated. We have chosen 
to use the cooccurrence matrix as a feature 
vector, without computing any ad hoc featu 
res out of it. To reduce the dimensionali 
ty, we have used normal feature extraction 
methods (see e.g. /DevKit82/), and trans 
formed the autocorrelation matrix to an 
orthogonal subspace. This strategy was 
originally applied in /0jaPar87/. 
The second problem comes from the parameter 
tuning; which are the separation parameters 
to be chosen for our purpose? Zucker and 
Terzopoulos considered the cooccurrence 
matrix as a contingency table, and used x _ 
statistics to analyze periodicities in 
the texture for finding the right separa 
tion parameter /ZucTer80/. However, this 
statistics have been criticized by SelkSin- 
aho, Parkkinen and Oja (/SePaOj87/ and /Se- 
PaOj88/). They demonstrated that this 
statistics does not properly discriminate 
among the types of dependencies indicated 
by the cooccurrence matrix. They also sug 
gested a new statistic, the /c-statistics, 
which they demonstrated to work better 
than the x _s tatistics. Also the computa 
tional complexity is much lower. We have 
applied their methodology in tuning the 
parameters of our texture descriptors. 
The following algorithm summarizes our 
textural feature extractor in the case of 
cooccurrence statistics: 
(1) For each class, compute all possib 
le (using all possible separation 
parameters suitable for the window 
size) cooccurrence matrices, form 
their /c-statistics, and take the 
one with highest value to present 
the cooccurrence statistics of that 
class. Compress the number of gray 
levels to 8 (dimension of the cor 
responding feature vector is 64). 
(2) Combine the resulting matrices to 
a feature vector, compute the auto 
correlation matrix, perform a KL- 
transformation (See /DevKit82/) 
to the resulting matrix, and take 
the part which statistically de 
scribes 99% of the information to 
represent the cooccurrence featu 
res. For the ALSM-classifier this 
is done separately for each class. 
2.2 The power spectral method 
The power spectrum of the 2D Fourier trans 
formation is another widely used textural 
descriptor. Since specific components in 
the frequency domain representation contain 
explicit information about the spatial 
distribution, useful features are obtained. 
It was first applied by Bajcsy /Bajcsy73/, 
who derived several features from the spec 
trum and showed its power in the problem 
of texture analysis. The traditional tex 
ture features, after Weszcka et al. /We- 
DyRo76/, extracted from the spatial fre 
quency domain are usually limited to an 
array of summed spectral energies within 
ring and wedge shaped regions. This re 
sults in a good texture discrimination 
only, if the chosen ring or wedge energies 
happen to be measured from correct loca 
tions of the power spectrum. D'Astous 
and Jernigan /DasJer84/ used a more intel 
ligent methodology by measuring the dist 
ributions of the frequency components. 
They concluded to use five descriptors 
for each peak in the power spectrum and 
three global measures for the whole spect 
rum. Later Liu and Jernigan tried to find 
"still better texture measures" from the 
Fourier domain /LiuJer90/. They extracted 
a total of 28 features from the power 
spectrum and from the phase spectrum. 
We have used here the same strategy as we 
used in the context of the second order 
statistics. The whole power spectrum is 
used as such, without any special feature 
extraction. For small windows the dimen 
sionality of the spectrum is quite low, 
and the final reduction is done with the 
help of the orthogonal transformation. 
For comparison we have included the method 
described in /DasJer84/. 
2.3 The fractal descriptors 
The appealing concept of fractals by Man 
delbrot /Mandel77/ has also been applied 
to the problem of texture analysis. A 
theoretical fractal object is self-similar 
to all magnifications, meaning that each 
segment of the object is statistically 
similar and invariant over scale transfor 
mations. The only description of the met 
ric properties of an ideal fractal comes 
from the fractal dimension, which is usual 
ly higher than the topological dimension. 
The applicability in texture analysis is 
due to the fact, that the fractal dimension 
of a surface corresponds quite closely to 
our intuitive notion of roughness. The 
more wiggling is the object, the higher 
fractal dimension it has. 
Most real world objects are not ideal frac 
tals (e.g. /Goodch80/). Instead, the frac 
tal dimension varies along scale. Rather 
than using the fractal dimension in the 
strict sense, the changes of the fractal 
dimension can be registered along the sca 
le. This should give more power to the 
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