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Identification
of typical GDV-grams on the basis of analysis of their fractal
dynamics
Introduction
It
is well known [1,2] that electrical self-activity of the organism
generated by complex nonlinear and nonequilibrium dynamic
systems are fractal by its nature, i.e. possessing the scale
invariance property. The same nature have the processes of
induced electrical activity and, in particular, the processes
which enable the GDV-images of fingers. The fractal structures
of this kind processes may be investigated in different ways.
May be studied Statical Images, i.e. the resulting picture
of Gas Discharge Visualization as a photograph or a television
shot. However, another way may be used - "dynamic"
investigation, i.e. examination of "genesis" of
the same image, by means of fragmentation of this shot into
a number of lines-ovals (pic.1) enclosed one into another,
and investigation of an energy component of each discrete
value of the corresponding oval (its brightness) and, what
is especially important, analysis of its information component,
a certain length of radius-vector crossing some discrete values
(pic.2). Here the length is the number of illuminated pixels
- image elements, crossed by the radius-vector, drawn at a
fixed angle to a chosen oval [3]. The length defined in such
a way would depend upon the angle of radius-vector, drawn
from a fixed image center. This line-oval image dissector
gives an opportunity to examine dynamics of the process of
image formation in time and space. We used the decomposition
of GDV-image into 8 lines-ovals, each of which was also digitized
according to 0,35156 0 degree of rotation of radius-vector.
As a result, two vectors F1<j> and F2<j>, sized
1024 x 1 each, are formed for each of the ovals. At that,
F1 - characterizes the energy component, and F2 - the information
component, j - is the number of oval evaluated within the
limits 0…7 (j Î 0,1,..7). Using the operation of
columns connection, the two data matrixes F1 and F2, sized
1024 x 8 each, containing the information about the GDV-image,
may be created. These data will be converted and compressed
according to the method of fractal dynamics analysis.
The
method of analysis of fractal dynamics
This method was developed for the EEG analysis [4] and later
applied for the analyze of the GDV-images of fingers [5].
The core of the method is the investigation of the dynamic
power spectra of the initial process's fragments. For the
purpose of their description the two-parameter mathematical
models are constructed, where one of the parameters is the
energetic,
and
the other - informational. The estimation of the parameters
on the basis of nonlinear regression is constructed and the
dynamics of their change from one fragment to the other is
monitored. These changes are evaluated by means of singular
decomposition of the corresponding matrixes. The final result
of these conversions and estimates may be compactly presented
in terms of a certain resulting vector sized 6 x 1. The two
vectors of this kind are formed as a result of F1 and F2 matrixes
conversion and compression respectively.
Let
us discuss the method's realization in more details.
1. Conversion and structuring of the initial data.
F1 and F2 matrix columns undergo the operation of folding
into a matrix sized 128 x 8. For example, F1<0> null
column of F1 matrix fold into A0 matrix:
A0 = fold (F1<0>)
Schematically
the fold operation is shown on the diagram:
F1<0>
= 
= A0
As
a result of this procedure instead of F1, F2 matrixes we get
A0, A1, A2,…A7 and B0, B1,…B7 matrixes respectively.
2.
Fast Fourier transformation (FFT) of the columns of A and
B matrixes.
C0 <i> = FFT (A0 <i>) , i = 0..7,
D0 <i> = FFT (B0 <i>) , i = 0..7
Here
the symbol <i> means the number of the corresponding
column of the matrixes A0, B0.
These
conversions result in vectors with the complex components,
which determine spectral constituents of the processes. It
is worth mentioning that in case we take up such a procedure
in MathCad program, because of the property of complex conjugacy
of the constituents MathCad system deduce in fact only a half
of their number. This means that with the A0 <i> vector
dimension equal to 128, C0 <i> will include 64 constituents
only.
3.
Calculation of the power spectra.
To get the required power spectrum of the segment of A0 <i>,
or B 0 <i> oval, it is necessary determine respectively:
CM <i> = | C0 <i> | 2 , i = 0..7,
DM <i> = | D0 <i> | 2 , i = 0..7
Examples
of the specific power spectra of the required processes are
shown on fig.3,4. It is obvious from the pictures that the
power spectra change considerably from one segment of an oval
to the other and from an oval to another oval, respectively.
In order to describe these changes mathematically within fractal
approach it is necessary to use the model of spectral envelope
which is usually applied in such cases.
4. Evaluation of parameters of the
model of power spectral envelope »
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