where D is the FD value, Now the problem is 
how to estimate the A(r) from a image. In the 
BLANKET method developed by Peleg (Peleg, 1984), 
the image intensity surface is covered with 
a 2r ihick 'blanket'. thus the A(r) can be 
calculated by the blanket volume divided by 
2r. The blanket volume is considered by its 
upper and lower surfaces u ard b: 
V(r)=E ,, ; [u (r, i, j) -b(r, L, 33] t9 t n n (3) 
where the blanket surfaces are defined as 
following: 
U(r, i, j) =max{u(r-1, i, j) *1, max [u (r-1, m, n) ]) 
(m, n) € 8 
b (r, i, j) -min (b (r-1, i, j) -1, nin [b (r-1, m, n)]) 
(m n) € s 
u (0, i, j) =b (0, i, j) =G (i, j) 
p (3) 
in which s={ (mn) | | m 2) -(i, j) | =1 } 
hence we have obtained the measured A(r) at a 
distance r: 
An=yin fz, — 3 $ .... (4a) 
A(ns(Vii-Ya-D]/s —. ... (4b) 
Peleg considered that the Eqs. (4b) provides 
reasonable results for both fractal and quasi 
-fractal surfaces, however, it has been shown 
from our computation results that when the 
Eqs. (4b) is used, the scale limit is so small 
that the reliability of the FD values will be 
influenced. Fig. 1 gives the illustration of 
the BLANKET method in one dimension 
In fact , the calculation of u and b can 
be implemented by erosion and dilation 
transformation in mathematic morphology: 
u (r)=G® rk=max {G (i-m, j-n) +r - k (m, n) } 
m nC k 
i-m j-nc H 
b (1) G9 rk-min(G (i^m, j4n) -r - k (m, n)} 
m nC k 
; i-m j-ac H 
LAS (6) 
where k is the unit structure element with a 
sphere shape, K,H is the project sets onto 
the plane from G and k ( Haralick, 1987). 
Essentically, Eqs. (5) is consistent with 
Eqs. (3). 
2.2 SAVR method 
The relation between the superficial area and 
volume has been given by  Mandelbrot 
32 
(Mandelbrot, 1982): 
gu» w^ qua. v — 8 (6) 
But Mandelbrot has not explained the meaning 
of the above expression explicitly, so that 
many papers have quoted the relation in a 
simple way: S'/^P - K. V!/*, Dong (Dong, 1991) 
pointes out the above equation between S 
and V is not correct and derives the concrete 
equation from (6) written as following: 
g1/Da-ı = k - pf (8-1-Da-1) /Dn-1, yi/n os (7) 
where n denotes the Euclidean dimension which 
is generally greater than the FD. The above 
equation (7) has been proved theoretically, 
however, it has not been so far used in 
application, since both the S and V can not 
be calculated easily. 
In this paper, a new method called SAVR method 
( Superficial Area-Volume Relation method). 
is proposed, Fig. 2 illustrates the SAVR method 
in a profile of image. In a rXr area centered 
by an arbitrary point of a image window, u. (r), 
b.(r), V.(r) and A, ( r) can be computed 
according to Eqs. (2), Eqs. (3) and Eqs. (4a). 
Then we can estimate the supperficial area S 
and volume V by a virtual way: 
S (r) -2n - A. (r) 
V (r) n - V. (r) - (0-1) (2011) ? 
boe (8) 
If we select point every certain distance in 
the image window, the average S(r) and V (r) 
are obtained for the whole image window, To 
estimating the FD value of this image window, 
Eqs. (7) can be written as following (n-3 in 
this case): 
S(r)79k + 12-970 vias sqam (9) 
the logarithm of both sides of (9) are taken 
to yield: 
log8 (r) -21ogr-Dlogk* (1/3) DlogV (r) -Dlogr... (10) 
where D and k are both constants, so we have : 
log (S (r) /1?) -Dlog (V (1) */9/p) aC ...... (11) 
which is in the form of a linear equation, 
This question can be used as the basis for a 
linear regression, taken at different r. The 
parameter D is the slope of the best-fit line 
within it 
between | 
is, the 
of the wl 
thick of 
the latte 
all direi 
the thicl 
related | 
descriped 
method | 
analysis, 
average | 
method ci 
single p: 
SAVR (PS; 
equal to 
FD value 
2.3 The 
Fraction: 
(FBIRF) « 
of image 
we call 
written : 
logE[ | 
| (i, jj 
logVar 
| (ij) 
where D-i 
used (FBI 
applied 
describe 
1984). 
Box meas: 
Mandelbr 
given a 
(Keller, 
method n 
comparat 
Density 
(DCF) de 
estimatii 
FD value 
between 
and r: 
where, | 
techniqu 
method,