New method for feature extraction based on fractal behavior - IDRBT

1072 Y.Y. Tang et al. / Pattern Recognition 35 (2002) 1071–1081
investigations of this new subject. Since Mandelbrot [1]
proposedthis technique, the subject of fractal dimension
has drawn a great deal of attention from mathematicians,
physicists, chemists, biologists, geologists, andelectrical
andcomputer engineers in various disciplines. Specically,
in the area of pattern recognition andimage processing,
the fractal dimension has been used <strong>for</strong> image
compression, texture segmentation and<strong>feature</strong> <strong>extraction</strong>
[2–5], etc. The process of pattern recognition requires
the <strong>extraction</strong> of <strong>feature</strong>s from regions of the image, and
the processing of these <strong>feature</strong>s with a pattern classication
algorithm. Many applications of fractal concepts
rely on the ability to estimate the fractal dimension of
objects. One of the basic characteristics of a fractal is its
dimension. A fractal object can be characterized by its
dimension which is a way of interpretation, determines
how much “space” it occupies between arbitrary m and
m+1 dimensional manifolds. This <strong>feature</strong> has been used
in texture classication, segmentation, shape analysis
andother problems [6–8]. All these approaches face and
attempt to solve two basic problems: (1) accuracy of the
estimate, (2) amount of in<strong>for</strong>mation needed to obtain the
estimate. In this paper, we utilize a <strong>method</strong>called“ box
counting dimension” to estimate the fractal dimension
of an image. All of these <strong>feature</strong>s rely on computing the
mass probability density function <strong>based</strong> on the central
projection trans<strong>for</strong>mation andwavelet decomposition
<strong>for</strong> the fractal set. We analyze the box counting <strong>method</strong>ology
from the standpoint of computing fractal <strong>feature</strong>s.
We focus on fractal dimension <strong>for</strong> computing the fractal
signature. This concept can be useful in the measurement
andclassication of pattern’s <strong>feature</strong>s.
2. Basic theory and algorithm
Fractal theory is <strong>based</strong>on geometry anddimension
theories. Fractals are mathematical sets with a high degree
of geometrical complexity, which can model many
classes of time series data as well as images. The fractal
dimension is an important characteristic of fractals because
it contains in<strong>for</strong>mation about their geometric structure.
It has become an eective tool to study complex
sets. There are many denitions <strong>for</strong> the fractal dimensions
of a fractal set [9,10]. The simplest andmost appealing
way of assigning a dimension to a set that can yield a
fractal dimension to certain kinds of sets is the so-called
box dimension. In the section, the important concept and
algorithm about box dimension and modied algorithm
of fractal dimension will be introduced.
2.1. Fractal dimension
Fundamental to most denitions of dimension is the
idea of measurement at scale . For each , a set can be
measuredin a way that ignores irregularities of size less
than , andwe see how these measurements behave as
→ 0.
Suppose F is a plane curve, the measurement M (F)
denotes the number of sets (with length ) which divide
the set F. A dimension of F is determined by the power
law obeyedby M (F) as → 0. If
M (F) ∼ K −s ; (1)
<strong>for</strong> constants K and s, we might say that F has dimension
s, and K can be considered as “s-dimensional length”
of F.
Taking the logarithm of both sides in Eq. (1) yields
the <strong>for</strong>mula:
log 2 M (F) ≃ log 2 K − s log 2 ;
in the sense that the dierence of the two sides tends
to 0 with , this suggests that the box dimension of F,
denoted by dim(F), shouldsatisfy
log
dim(F) = lim 2 M (F)
→0 −log 2 : (2)
If the limit exists, dim(F) is calledthe fractal dimension
of set F.
2.2. Box counting dimension (BCD)
Box computing dimension or box dimension is one of
the most widely used dimensions. Its popularity is largely
due to its relative ease of mathematical calculation and
empirical estimation.
Let F be a non-empty andboundedsubset of
R n ;= {! i : i =1; 2; 3;:::} be covers of the set F. N (F)
denotes the number of covers, such that
N (F)=|: d i 6 |;
where d i stands <strong>for</strong> the diameter of the ith cover. This
equation means that N (F) is the smallest number of
subsets which cover the set F, andtheir diameters d i ’s
are not greater than .
The upper andlower bounds of the box computing
dimension of F can be dened by the following <strong>for</strong>mulas:
log
dim B F = lim 2 N (F)
→0 −log 2 ;
log
dim B F = lim 2 N (F)
→0 −log 2 ; (3)
where the overline stands <strong>for</strong> the upper bound of dimension
while the underline <strong>for</strong> lower bound.
If both the upper bounddim B F andthe lower bound
dim B F are equal, i.e.
log
lim 2 N (F)
→0 −log 2
= lim log 2 N (F)
→0 −log 2 ;