New method for feature extraction based on fractal behavior - IDRBT

1074 Y.Y. Tang et al. / Pattern Recognition 35 (2002) 1071–1081
The image pixels (m; n) with distance less than
one from (i; j) are taken to be the four immediate
neighbors of (i; j). Similar expressions exist
when the eight-neighborhood is desired. A point
f(x; y) will be included in the blanket <strong>for</strong> when
b (x; y) ¡f(x; y) ¡u (x; y). The blanket denition
uses the fact that the blanket of the surface <strong>for</strong>
radius includes all the points of the blanket <strong>for</strong> radius
−1, together with all the points within radius
1 from the surfaces of that blanket. Eq. (7) ensures
that the new upper surface u is higher by at least
1 from u −1 , andalso at a distance at least 1 from
u −1 in the horizontal andvertical directions.
The volume Vol of the blanket is computedfrom
u and b :
Vol = ∑ (u (i; j) − b (i; j)): (9)
i;j
As the volume Vol of the blanket is measuredwith
radius , the area of a fractal surface can be deduced
as
A = Vol
2
or
A = Vol − Vol −1
: (10)
2
• Let A() be the area of the surface of the blanket.
Basedon the Minkowski dimension, we have
A() ≈ 2−D ; if is suciently small; (11)
where denotes a constant and D stands <strong>for</strong> the fractal
dimension of the image.
Since the dimension can be regarded as a slope on
a log–log scale, only two points are needed to get
the dimension. We use two values of to compute
the fractal dimension, namely, we take = 1 and 2 ,
then A 1 ≈ 2−D
1
and A 2 ≈ 2−D
2
. Thus, we have
A 1 =A 2 ≈ 2−D
1
= 2−D
2
.
Taking the logarithm of both sides yields
2 − D ≈ log 2 A 1
− log 2 A 2
log 2 1 − log 2 2
;
D ≈ 2 − log 2 A 1
− log 2 A 2
: (12)
log 2 1 − log 2 2
It is usedto approximate the fractal dimension of the
signature images, i.e.,
D ≈ 2 − log 2 A() if is suciently small: (13)
log 2
We call D as modied fractal signature of set F. It
will be used to distinguish the dierent handwritten
signatures.
Several points are worth noting:
1. Why do the dierent document images have dierent
fractal dimensions? The essential distinction of
handwritten signature images is their values of A().
2. The value of A() depends on the volume Vol 3 (F )
of thickenedblanket F only.
3. In summary, they can be representedas
D ⇔ A() ⇔ Vol 3 (F ):
Consequently, in this paper, the volume Vol 3 (F )ofthe
thickenedblanket F is applied to identify dierent handwritten
signature, insteadof using the fractal dimension.
We call such technique of approximating the fractal dimension
“Modied Fractal Signature”.
3. Applications
From the specic point of view of shape characterization
of patterns the main aim of the fractal approach is to
nd a “measure” to distinguish between curves with different,
often very complicated, contours. The main idea
is to describe the complexity of the curve through a new
parameter, the fractal dimension, so as to ll in the gap
between one- andtwo-dimensional objects (<strong>for</strong> objects
on a plane). Two applications of the fractal theory are
presentedin this section, namely: (1) the fractal technique
is usedto extract the <strong>feature</strong>s <strong>for</strong> two-dimensional
objects; (2) the fractal signature is employedto identify
dierent handwritten signatures.
3.1. Application of fractal technique <strong>for</strong> <strong>feature</strong>
<strong>extraction</strong>
Feature <strong>extraction</strong> is a crucial step in pattern recognition.
It is responsible <strong>for</strong> measuring the <strong>feature</strong>s of the
objects in an image. Pattern recognition requires the <strong>extraction</strong>
of <strong>feature</strong>s from the regions of an image, andthe
processing of these <strong>feature</strong>s with a pattern classication
technique. We employeda central projection <strong>method</strong>
to reduce the problem of two-dimensional patterns into
that of one-dimensional ones, and thereafter, utilize
the well-establishedone-dimensional wavelet trans<strong>for</strong>m
coupledwith fractal theory to extract the one-dimensional
pattern’s <strong>feature</strong> vectors <strong>for</strong> the purpose of pattern
recognition. The key steps of the experimental procedure
consist of the following:
Step 1: Central projection of two-dimensional
patterns: We denote each of the two-dimensional patterns
in question by p(x; y). Thus, the central projection
of p(x; y) can be expressedas follows:
f(x k )=
M∑
p(x k cos i ;x k sin i ): (14)
i=0
Step 2: Wavelet trans<strong>for</strong>mation of the one-dimensional
patterns: Let f(x k )=c j;k , where k =0; 1;:::;2N −1 and
V 0 = {c j;0 ;c j;1 ;:::;c j;2N−1 }: