# Full text: Close-range imaging, long-range vision

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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

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