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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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