The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B3b. Beijing 2008 
617 
a Gibbs fields. For this approach the image is characterised by 
the conditional probability. This probability distribution denotes 
the probability for accepting a grey level gj by a pixel Zj, 
assuming the neighbouring pixels Z 1 have the grey levels g 1 (see 
e.g. Gimel’farb, 1999). 
e Mp L ~ argmax P 0 {g) 
with the approximation (3) 
p e (.g)~Y\ p e(gi\g , ) 
ieiR 
P(Z,=g,\Z' =g , ) = Te™"‘' ) 
with (1) 
S = ^ e ~ H(g ’ gi) 
g =o 
S is called partition sum, while H stands for the energy function 
characterizing the Markov random field. In most cases, this 
energy function is parameterised by a vector 0 = [b 0 , b x ] T . 
This parameter vector characterizes the texture and can be used 
for segmentation as a multidimensional texture feature. The 
used autobinomial model has the following energy function 
(Schroeder et al., 1998): 
Different local maxima may exist due to the complexity of this 
probability distribution function. The simulated annealing 
method was used to avoid local maxima and to find the global 
maximum. 
As second method for parameter determination, the conditional 
least square (CLS) estimation was used (Schroeder et al.,1998). 
This approach attempts to minimise the error between the given 
picture and the expected image. By means of the least square 
adjustment calculus the parameter vector can be calculated 
as follows: 
Ocls =argrnin£(g,.-g,.) 2 
i 
and (4) 
H d (g) = H e (g i ,g i ) = - In 
with 
T 
\Si ) 
-gi-V 
r i=b 0 +Yj b j 
j 
Sj+Sj 
G 
(3) 
G is the maximum grey value. In the first term each grey value 
gi of the pixel Z\ is weighted separately. The middle grey levels 
are stronger weighted than the bright or dark grey levels of a 
pixel, because of the special structure of the autobinomial 
coefficients. In the second term the grey values gj are applied 
together with the grey values gj of their neighbouring pixels.. In 
doing so different neighbourhood systems (see e.g. GimeTfarb, 
1999) can be used (see Figure 3). 
T44 
3*32 
*41 
2\13 
*22 
2"12 
*21 
*42 
2-31 
In 
*1 
*11 
*31 
X42 
121 
*12 
*22 
*43 
141 
*32 
*44 
-C22 
*12 
*21 
*11 
*, 
*11 
121 
*12 
*22 
Xi2 
*11 
*. 
*11 
*12 
Figure 3. Frequently used neighbouring systems 
The parameter vector 0 has to be determined in order to 
segment a given image. For this purpose two different methods 
were used. The first method is to determine the parameter using 
the maximum pseudo likelihood (MPL) estimation. Through 
this approach the overall probability for the image is maximised. 
Unfortunately, the overall probability for the autobinomial 
model is unknown. For this reason, Besag’s suggestion to 
approximate the overall probability by the product of all 
conditional probabilities was used (Besag, 1986): 
6 - (G T G)~'G T d 
While g j represent the expected grey values, G represents a 
matrix containing grey values of neighbouring pixels and d 
represents a logarithmic vector of these grey values. Both 
methods achieve the same level of quality, but the calculus of 
CLS is significantly faster and was therefore used for further 
calculations. 
The investigation of MRFs for texture segmentation was carried 
out in several steps. Firstly, the validity of the parameter 
extraction was analysed to examine whether the parameter 
extraction is reproducible, and to find conditions under which 
the algorithm is stable. Synthesised MRF images were used in 
order to compare the extracted with the correct parameters. The 
images were synthesised using a Markov chain and the Gibbs 
sampler from a given parameter vector (thus from a given 
MRF). This iterative procedure is explained in (Paget & 
Longstaff, 1998). Typical parameter sets were extracted from 
Brodatz textures (Brodatz, 1966). The effects of noise, image 
scaling and neighbourhood system size as well as the parameter 
normalisation were also examined. 
In order to achieve a quasi-continuous scaling and to avoid 
aliasing effects, the bicubic resampling method has been 
applied. Examinations of the fourier spectrum of the scaled 
pictures have shown that low-frequency information is 
preserved by this method. 
Having investigated the parameter extraction the segmentation 
of Brodatz textures has been researched. Using Brodatz textures 
various influencing factors like the sunset were avoided. The 
segmentation has been realised using the point-based method of 
support vector machines. This segmentation system is 
appropriate, because it facilitates fast segmentation in a 
multidimensional feature space. 
All segmentations were made in the quasi-continuous scaling 
space for quantifying the influence of scaling under different