sample mean does not stabilize, and the sample variance always 
increases. This motivates the model of intrinsic random function 
in spatial statistics. 
An intrinsic random function is a random function whose 
increments are second-order stationary. It is characterized by the 
following two equations (Cressie, 1991; Chile, J.P, et al 1999): 
(1) within the whole definition domain: 
E(X(t)-X(t+h))=0 (4) 
(2) the variance function of the increments exists and is 
stationary: 
var(X (6) = X(t + h)) 2 E(X (t) - X (t 5). - (E(X (0) - X(t + h)) } 
- E(X(t) - X(t - 5) = 2y(h) 
(5) 
where y(h) is the semivariogram. 
We now analyze the properties of semivariogram in space 
domain. From the spatial statistics perspective, the data set of a 
digitized image is a realization of a random function with image 
coordinates (columns and rows) as its variables. Once a pair of 
image coordinate (x, y) is given, this random function becomes a 
common random variable: this is the random property of 
semivariogram. On the other hand, two pixels with a distance of 
h pixels in an image is correlative, for there exists correlation 
function: this is the continuity property of semivariogram, which 
hints the structure hidden in the image data set. 
To describe hidden structure in an image is important for global 
image comparison: in fact, global image comparison based on 
global structure is the primary method for a person to perceive 
objects having different structure. From the point of view of 
spatial statistics, the structure of a data set can be described by 
the continuity of the set. In spatial statistics, continuity of a data 
set is defined by (Chiles, J. P. et al, 1999): 
lim E(X(t) - X(t- 5h)? 20 (6) 
compared with equation (5), we immediately have the conclusion 
that semivariogram describes the continuity of a data set, 
therefore, semivariogram can be the tool to reveal the structure 
hidden in the data set. As a matter of fact, semivariogram was not 
firstly proposed in spatial statistics. In stead, it was coined by 
mathematicians around 1920 in order to study the structural 
properties of turbulence and was widely called as structural 
function at that time( Cressie, 1991; Chiles, 1999). 
This conclusion is vitally important for us to define a new 
semivariogram based parameter to describe image similarity, for 
this parameter can reveal the global structure hidden in an image 
data set, which can be the fundamental for global image 
comparison. 
Now, we can define the semivariogram-based parameter to 
describe image similarity. 
Usually, structural analysis for a 2D data set starts with the 
computation of experimental semivariogram. This can be done as 
the following. 
Suppose two points (x, y) and (x+h, y+h) in a continuous image 
have gray level functions: G(x, y) and G(x+h, y+h), then the 
semivariogram can be defined as the below (Carr, J. M. 1997): 
2v(h) = [ f [66.) - GG hy 1] ayax (7) 
we actually use a digitized image, so the discrete format of 
equation (7) can be written as the following, which is the 
operational formulae (Carr, J. M. 1997): 
5 
  
1 N 
y(h) = SS 2 [660 - C+ y+h)] (8) 
1 
where N is the total number of a pair of pixels with the interval of 
h pixels. Usually in spatial statistics, an authentic model should 
be given to fit the experimental semivariogram in order to carry 
out the consequent computation such as kriging interpolation. In 
our case, however, this is not necessary, because our new 
parameter is fully dependent on the semivariogram itself. 
Fig. 1 is a typical variograph: a semivariogram model curve, 
which is the graph of semivariogram versus sample spacing. Key 
y(h) sill 
A 
ae 
HAN 
nugget 
  
  
Fig. 1 Typical variograph showing sill, range and nugget 
parameters include range, nugget and sill. The range of the 
semivariogram indicates a spatial scale of the pattern, the nugget 
is an indication of the level of uncorrelated noise in the data and 
the sill reveals the total variation. Because À is actually a vector: 
different semivariograms of different directions corresponding to 
different directions of vector h can be obtained from equation (8). 
If all the semivariograms in different directions are the same or 
almost the same, the data set from which semivariograms are 
calculated is said to be isotropic, otherwise it is anisotropic. 
If we have two identical images, then all the semivariograms in 
different directions are the same. Otherwise, as shown in Fig. 2, 
we will have two curves corresponding to these images: such two 
different semivariograms distinguishes the two images. 
  
Sill of the standard 
range of the standard 
D B h 
Fig 2. Two variographs of two images along the same direction 
    
  
—506— 
Giv 
con 
rang 
the 
ano 
ima 
inte 
big£ 
the 
then 
aber 
stan 
betv 
stan 
way 
simi 
mea 
para 
diffe 
semi 
oriei 
and 
oriet 
imag 
thres 
has ¢ 
(I) 
Q) 
(3) 
This 
Almo 
check 
whetl 
probl 
once 
proce 
existi: 
have