New method for feature extraction based on fractal behavior - IDRBT

Y.Y. Tang et al. / Pattern Recognition 35 (2002) 1071–1081 1073
the common value is called box computing dimension or
box dimension of F, namely
log
dim B F = lim 2 N (F)
→0 −log 2 : (4)
Further discussions on fractal theory can be found in
Refs. [11,12,10].
The modied box computing dimension <strong>method</strong> gives
a very goodestimate of fractal dimension. It can be easily
shown that computation complexity of other approaches,
including the original box counting dimension, is much
higher than that of this approach. Thus it has the advantages
of simplicity in computation andimprovement in
eciency.
2.3. Minkowski dimension and modied fractal
signature
There is an important equivalent denition of box
counting dimension of a rather dierent <strong>for</strong>m that is the
-parallel body F of F.
F = {x ∈ R n : |x − y| 6 ; <strong>for</strong> y ∈ F}: (5)
If F is a subset of R n and, <strong>for</strong> some d, if the limit of
[Vol n (F )]= n−d tends to be positive and nite as → 0,
then it makes sense to regard F as d-dimension. The
limiting value is calledthe Minkowski dimension of F,
nameddim M F. Vol n (F ) is Lebesgue measure of F.
Even if this limit does not exist, we may be able to extract
the critical exponent of andthis turns out to be
relatedto the box dimension. The initial andkey step <strong>for</strong>
Minkowski dimension is to evaluate the Vol(F ) under
dierent .
The relationship between the box counting dimension
and Minkowski dimension can be provided by the following
equation:
log
dim B F = n − lim 2 Vol n (F )
;
→0 −log 2
log
dim B F = n − lim 2 Vol n (F )
; (6)
→0 −log 2
where F stands <strong>for</strong> -parallel body of F, and
Vol n (F ) denotes n-dimension area or volume of F .
For a nonempty andboundedset F in R n , we have
dim M F =dim B F.
Let F = {X i;j };i=0; 1;:::;K; j=0; 1;:::;L be the
image of a pattern with multi-gray level, and X i;j be
the gray level of the (i; j)th pixel. In a certain measure
range, the gray level surface of F can be viewedas a
fractal. The surface area can be usedto approximate its
fractal dimension.
In particular, the gray level function F is a nonempty
andboundedset in R 3 <strong>for</strong> either text areas, graphics
areas or backgroundareas in the pattern. We use a
Fig. 1. Surface andits blankets.
technique that is referredto as Blanket Technique [13]
to thicken the function F. It leads to a set F , which is
still a nonempty andboundedset in R 3 . According to the
denition of Minkowski Dimension andRef. [5], we can
conclude that if
Vol 3 (F
lim )
= ¿0;
→0 3−D
then
D =dim M F =dim B F;
where denotes a constant, Vol 3 (F ) stands <strong>for</strong> the volume
of the blanket F .
The idea of the blanket technique is <strong>based</strong> on the
equivalent denition of the box computing dimension
shown in Eq. (5), i.e. the -parallel body. In the blanket
technique, all points of the three-dimensional space at
distance from the gray level surface are considered.
These points construct a “blanket” of thickness 2 covering
this surface. A graphical illustration is shown in
Fig. 1. The image is representedby a gray-level function
g(i; j). The covering blanket is dened by its upper
surface u (i; j) andits lower surface b (i; j).
• Initially, = 0 andgiven the gray-level function
equals the upper andlower surfaces, namely:
g(i; j)=u 0 (i; j)=b 0 (i; j):
• For =1; 2;:::, the blanket surfaces are dened as
follows:
u (i; j)
{
= max u −1 (i; j)+1;
b (i; j)
{
= min b −1 (i; j) − 1;
}
max u −1 (m; n) ;
|(m;n)−(i;j)|61
(7)
}
min
|(m;n)−(i;j)|61
b −1 (m; n) :
(8)