papers to each other, but contradictory 
results have been achieved. This certainly 
comes from the difficulty involved in their 
usage and parameter tuning. Both methods, 
as such, produce high dimensional feature 
vectors and the usual approach is to compu 
te some ad hoc features from the original 
descriptors. This reduces the original 
information and makes the comparison dif 
ficult. We have tried to avoid this prob 
lem by careful parameter tuning and by 
utilizing standard feature extraction met 
hods (/DevKit82/) which do not reduce dras 
tically the amount of information but only 
the dimensionality of the feature vectors. 
In addition to these two popular texture 
descriptors, we have included in the compa 
rison three other texture measures. First 
ly, a simple first order statistic in the 
form of local variance, serves as a kind 
of reference. Secondly, the appealing 
fractal based descriptors, the fractal 
dimension and the fractal signature, are 
included. Thirdly, a new method, called 
the amplitude varying rate statistical 
approach after Zhuang and Dunn /ZhuDun90/, 
is included in the comparison, because of 
the most promising results achieved in 
/ZhuDun90/. 
When comparing the performance of texture 
descriptors in the context of classifica 
tion, attention has to be paid, not only 
to the descriptors, but also to the clas 
sifier itself. It has to be chosen to 
properly work with the features chosen. 
The usual brute-force application of an 
"optimal" maximum likelihood classifier 
assuming multi-normal probability densi 
ties, has been, for the writers' opinion, 
distorting many comparative studies. Espe 
cially, when using textural descriptors, 
the decision boundaries can be highly non 
linear. In these instances, a non-para- 
metric classifiers would be the only rea 
sonable choice. A simple, but computa 
tionally heavy, k-NN classifier, has been 
proven to have a large sample size error 
rate that decreases monotonically to the 
optimal Bayesian error rate /CovHar67/. 
Its computational complexity can also be 
thoroughly improved by the so called edi 
ting and condensing techniques (see Chap 
ter 3). Because a k-NN classifier can pro 
duce highly non-linear decision boundaries, 
it is extensively compared with the ML- 
classifier in the present paper. The non 
linearity problem is widely addressed in 
Artificial neural network classifiers. 
Such an adaptive classifier is the Average 
Learning Subspace Method (ALSM) developed 
by Oja in /Oja83/. Because of its reported 
suitability to texture classification, 
especially in the context of cooccurrence 
statistics and power spectral methods, it 
is the third classifier adopted in this 
context. 
In Chapter 2 we will review the texture 
descriptors, their technical implementa 
tion, and the feature extraction methods 
utilized in this project. Chapter 3 is 
concentrated on the description of the 
classifiers used, and Chapter 4 gives a 
summary of the results. Finally Chapter 
5 draws some conclusions. 
2. TEXTURE DESCRIPTORS 
Haralick defines texture as consisting of 
two basic dimensions /Harali79/. The first 
one consists of the image texture elements 
itself, and the second one of the spatial 
dependencies between these elements. This 
spatial organization may be random or may 
have dependencies between its primitives. 
This dependence may be structural, proba 
bilistic, or functional. Texture can be 
described with such words as fine, coarse, 
smooth, granular, regular, irregular, ran 
dom, or structural. 
Even today there is no exact mathematical 
definition of texture and we still rely 
on those loose descriptions. A large part 
of the texture analysis techniques are in 
fact ad hoc and many statistical approaches 
to the measurement and characterization 
of image texture exist. In statistical 
methods, the pixels are supposed to have 
spatial distribution having some statis 
tical characteristics and the analysis 
techniques try to determine corresponding 
parameters. The statistical characteris 
tics, which one measures, make the dif 
ference between the methods. For a good 
survey see e.g. /Harali79/, /Harali86/, or 
/GoDeOo85/. 
In the underlying comparison we have chosen 
four texture descriptors which are reported 
to own good discriminative characteristics, 
namely the second order cooccurrence sta 
tistics /HaShDi73/, the 2D power spectrum 
/Bajcsy73/, the fractal descriptors /Pent- 
la83/, and the amplitude varying rate app 
roach /ZhuDun90/. These methods, the prob 
lems involved and their technical implemen 
tation will be addressed in Chapters 2.1- 
2.4. 
2.1 Second order (cooccurrence) statistics 
There has been psychovisual evidence that 
two textures with identical second order 
statistics are not separable from each 
other /Julesz62/. Later it was pointed 
out by Gagalowicz and Tournier-Lasserve, 
that for non-homogeneous textures this 
does not hold /GagTou86/. Gagalowicz and 
Tournier-Lasserve also claim that natural 
textures are usually inhomogeneous. How 
ever, in practice it seems to be a good 
approximation for texture distinguishabili- 
ty. 
The cooccurrence matrix (often referred as 
the Gray Tone Spatial Dependence Matrix) 
is an estimate of the second order joint 
conditional probability density function, 
and is defined by /DyHoRo80/ as follows: 
Cooccurrence matrix is a G*G matrix, where 
each entry (i,j) is the number of times 
gray levels i and j occur at separation d 
in the picture, which has been quantized 
to G levels. 
The high dimensionality of the cooccurrence 
matrix produces the first problem. In 8 
bit images, a straightforward application 
334