Biomedical Engineering Reference
In-Depth Information
the enlarged pieces are different from those in the other direc-
tion, then the fractal is self-affine. This distinction is, however,
often smeared and for the purpose of being more expressive, self-
similarity is used when indeed self-affinity is meant
(7)
.
2.2. Power Law
Scaling Relationship,
Fractal Dimension
The power law scaling relationship gives the mathematical basis
for calculating a fractal parameter. It can mathematically be
derived from self-similarity
(8)
. Briefly, when a quantitative prop-
erty,
q
, is measured on scale
s
, its value will be dependent on
s
according to the following scaling relationship:
q
=
f
(
s
)
(2.1)
For fractals,
q
does not converge if
s
is decreasing, but, instead
exhibits a power law scaling relationship with respect to
s
,
whereby with decreasing
s,
it increases (because newer details
become visible) without any limit.
ps
ε
q
=
(2.2)
where
p
is a factor of proportionality (prefactor) and
ε
isaneg-
ative number, the scaling exponent. The value of
can be easily
determined as the slope of the linear regression fitted to the data
pairs on the plot of log
q
versus log
s
:
ε
=
+
ε
log
q
log
p
log
s
.
(2.3)
The scaling exponent (or its simple derivative) is itself the most
essential fractal parameter, the fractal dimension. Because its value
usually is non-integer, this is the eponym of fractals (i.e., fractus
=
broken)
(9)
. Instead of the fractal dimension, another fractal
descriptor, the Hurst exponent, is widely used
(8)
because the
most essential scaling properties of
temporal
signals can be com-
pared on a scale of 0 to 1. The fractal dimension can describe
the highly dimensional structure of the process; its value can vary
between 0 and theoretically infinitum, however, in practice rarely
higher than 3. If the scaling exponent is described in the fre-
quency domain of the signal, then it is referred to as spectral index
(
β
). The spectral index is, by convention, the negative value of the
scaling exponent.
The most remarkable visual feature of a fractal object is its scale
invariance, which refers to the fact that the physical size of the
object cannot be judged by its perceived image. Mathematically
speaking, the ratio of two estimates of
q
measured at two different
scales,
s
1
and
s
2
,
q
2
/
2.3. Scale Invariance
and Scaling Range
q
1
depends only on the ratio of scales (relative
