. (6)
ning
that
991)
rete
(8)
r)
To
OW,
in
(9)
ren
(10)
within its range of scales . The difference
between the BLANKET method and the SAVR method
is, the former method provides the FD value
of the whole image window , in which only the
thick of 'blanket' is varied with different r,
the latter method takes the variation of r in
all directions into account, that is,not only
the thick but also the size of area are all
related to r. The comparative studies which
descriped in the next section show the SAVR
method has many good properties for image
analysis. It should be noted that if the
average processing step is omitted, the SAVR
method can provide the FD value related to a
single point of image , called single point
SAVR (PSAVR) method and if n of Eqs. (D) is
equal to 2 , the SAVR method can provide the
FD value of a profile of image in such case,
2,3 The other methods
Fractional Brownian Increase Random Field
(FBIRF) can be applied to model the surface
of image (Petland, 1984), FBIRF based methods,
we called FBM method and FBV method, can be
written as following:
logE[ | GCi, j) -G(k, 1) | ]=Hlogr+C ...... (12)
| (i, j) - (k, 1} | =i
logVar [ | G(i, j) -G(k, D | 1=2Hlogr+C..... (13)
| (1, j- (k, 1) | ET
where D-n-H, In Eqs. (12) the mean values are
used (FBM) and the variance values are
applied in Eqs. (13) (FBV). the FBM method has
described and applied in many papers (Petland,
1984).
Box measuring method ( BOXM) developed by
Mandelbrot and Voss (Peitgen, 1988) has been
given a detail description in Keller's paper
(Keller, 1989). We just adopted the original
method not modified by Keller to use for the
comparative experiments.
Density Correlation Function based method
(DCF) developed by Tao (Tao, 1992) to apply to
estimating the FD from grey level image. The
FD values are estimated according to relation
between the density correlation function C
and r:
Cüa:k. 1" =... oo (14)
where, Cír) is obtained by box covering
technique which has been applied to BOXM
method,
53
3. COMPARATIVE STUDIES ON FRACTAL DIMENSION
ESTIMATION METHODS
In this section, the above FD estimation
methods first are compared in many aspects.
For the purpose of comparing the methods
correctly, the simulated fractal images with
kown FD values are generated by recursive sub
-division approach (Amanatides, 1987).
3.1 Correction of FD estimation
The above six methods are all tested on the
simulated images (FD ranging from 2.1 to 2.9).
It is shown from Tab. 1 that SAVR, BLANKET and
FBM three methods acquired good results,
Which behave in such two cases: first, the
estimates of FD are approximated to the
original FD values, second, the linear
relation between the estimates and original
of FD values is explicit,
3.2 Scale limits
The FD values can be estimated correctly only
within its scale limits. In application, the
problem is that the scale limit is difficult
to determined. To overcoming the problem, we
should select the method which is not
sensitive to scale limits, that is, the scale
limit of this method is relative long, since
the different methods used in FD estimation
may produce different scale limits. Fig. 3
illustrates the experiments results of
testing the six methods mentioned above on
simultated images whose FD is 2.6. As can be
seen, BLANKET method and SAVR method behave
good straight linearities and have long scale
limits.
8.8 Characteristic with multiresulotion
À theoretical fractal object is self-similar
under all magnification and the FD is stable
with changing resolution, however, it is
obvious that this property will change with
using different methods. To test the charac-
teristics of multiresolution of different
methods , the four levels multiresolution
images of the simulated fractal images are
genernted by averaging processing . Three
methods are carried out to test their charac-
teristics of multiresolution and the results
are shown in Tab. 2. We can find that both
SAVR method and BLANKET method have good
results, especially, the former is more sta-
ble in FD estimates. FBM method behaves not
