Biomedical Engineering Reference
In-Depth Information
Fig. 1.3
An example of
repetitive pattern at the origin
of fractals
(e.g. respiratory, circadian, cardiac, electroencephalogram, etc.) represents a pow-
erful tool for transient detection. Several algorithms are available; here we discuss
only two: (i) the Katz algorithm and (ii) the box-counting method.
The definition introduced by Katz is given as[
50
]
log
(L)
log
(d)
F
d
=
(1.18)
where
L
is the total length of the curve or sum of distances between successive
points, and
d
is the diameter estimated as the distance between the first point of the
sequence and the most distal point of the sequence. Hence,
d
can be expressed as
max
x(
1
)
x(i)
,
=
=
∀
i.
(1.19)
The
F
d
compares the actual number of units that compose a curve with the min-
imum number of units required to reproduce a pattern of the same spatial extent.
Consequently,
F
d
depends on the measurement units. Naturally, if units will be dif-
ferent, so will
F
d
values. The solution is to create a general unit, e.g. the average
step or average distance between successive points, denoted by
a
. Normalization
applied to (
1.18
) results in a new definition:
d
log
(L/a)
log
(d/a)
F
d
=
(1.20)
There is also a relationship between the length, area or volume of an object and
its diameter. If one tries to cover the unit square with little squares (i.e. boxes) of
side length
ε
FD
, then one will need 1
/ε
FD
boxes. To cover a segment of length 1,
there is need only for 1
/ε
FD
boxes. If we need to cover a 1
×
1
×
1 cube, then we
need 1
/ε
FD
boxes. The general rule emerges as
1
/ε
FD
,
0 (1.21)
where
ε
FD
is the length of the box,
S
is the full data set,
N(ε
FD
)(S)
is the minimum
number of
n
-dimensional boxes needed to cover
S
entirely and
d
is the dimension
of
S
. Using this, one can define
F
d
as
N(ε
FD
)(S)
≈
for
ε
FD
→
ln
N(ε
FD
)(S)
ln
ε
FD
F
box
d
=−
lim
ε
FD
(1.22)
→
0
Usually, for systems whose dynamics is intrinsic fractal, the graphic representa-
tion of
F
d
will be a line and its slope denotes the value of the fractal dimension.
