Wavelet analysis of Jones-matrix images corresponding to polycrystalline networks of biological crystals in diagnostics of tuberculosis
Determined in this work are analytical interrelations between orientations of optical axes and birefringence of biological crystals as well as characteristic values of elements in the Jones matrix for flat layers of polycrystalline networks, which determine conditions for formation of polariza...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2011
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| Цитувати: | Wavelet analysis of Jones-matrix images correspondingto polycrystalline networks of biological crystals in diagnostics of tuberculosis / N.I. Zabolotna, V.O. Balanetska, O.Yu. Telenga, V.O. Ushenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 2. — С. 228-236. — Бібліогр.: 31 назв. — англ. |
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irk-123456789-1177132017-05-27T03:04:03Z Wavelet analysis of Jones-matrix images corresponding to polycrystalline networks of biological crystals in diagnostics of tuberculosis Zabolotna, N.I. Balanetska, V.O. Telenga, O.Yu. Ushenko, V.O. Determined in this work are analytical interrelations between orientations of optical axes and birefringence of biological crystals as well as characteristic values of elements in the Jones matrix for flat layers of polycrystalline networks, which determine conditions for formation of polarization singularities in laser images. Performed is a complex statistical, correlation and fractal analysis of distributions for the amount of characteristic values inherent to the Jones matrix elements describing layers of saliva taken from a healthy patient and that sick with tuberculosis. The authors have ascertained objective criteria to differentiate optical properties of polycrystalline networks of human saliva in various physiological states. Offered is Jones-matrix diagnostics of tuberculosis. 2011 Article Wavelet analysis of Jones-matrix images correspondingto polycrystalline networks of biological crystals in diagnostics of tuberculosis / N.I. Zabolotna, V.O. Balanetska, O.Yu. Telenga, V.O. Ushenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 2. — С. 228-236. — Бібліогр.: 31 назв. — англ. 1560-8034 PACS 33.50.-j, 34.35.+a, 73.20.Mf, 78.30.-j http://dspace.nbuv.gov.ua/handle/123456789/117713 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
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English |
| description |
Determined in this work are analytical interrelations between orientations of
optical axes and birefringence of biological crystals as well as characteristic values of
elements in the Jones matrix for flat layers of polycrystalline networks, which determine
conditions for formation of polarization singularities in laser images. Performed is a
complex statistical, correlation and fractal analysis of distributions for the amount of
characteristic values inherent to the Jones matrix elements describing layers of saliva
taken from a healthy patient and that sick with tuberculosis. The authors have ascertained
objective criteria to differentiate optical properties of polycrystalline networks of human
saliva in various physiological states. Offered is Jones-matrix diagnostics of tuberculosis. |
| format |
Article |
| author |
Zabolotna, N.I. Balanetska, V.O. Telenga, O.Yu. Ushenko, V.O. |
| spellingShingle |
Zabolotna, N.I. Balanetska, V.O. Telenga, O.Yu. Ushenko, V.O. Wavelet analysis of Jones-matrix images corresponding to polycrystalline networks of biological crystals in diagnostics of tuberculosis Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Zabolotna, N.I. Balanetska, V.O. Telenga, O.Yu. Ushenko, V.O. |
| author_sort |
Zabolotna, N.I. |
| title |
Wavelet analysis of Jones-matrix images corresponding to polycrystalline networks of biological crystals in diagnostics of tuberculosis |
| title_short |
Wavelet analysis of Jones-matrix images corresponding to polycrystalline networks of biological crystals in diagnostics of tuberculosis |
| title_full |
Wavelet analysis of Jones-matrix images corresponding to polycrystalline networks of biological crystals in diagnostics of tuberculosis |
| title_fullStr |
Wavelet analysis of Jones-matrix images corresponding to polycrystalline networks of biological crystals in diagnostics of tuberculosis |
| title_full_unstemmed |
Wavelet analysis of Jones-matrix images corresponding to polycrystalline networks of biological crystals in diagnostics of tuberculosis |
| title_sort |
wavelet analysis of jones-matrix images corresponding to polycrystalline networks of biological crystals in diagnostics of tuberculosis |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2011 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117713 |
| citation_txt |
Wavelet analysis of Jones-matrix images correspondingto polycrystalline networks of biological crystals in diagnostics of tuberculosis / N.I. Zabolotna, V.O. Balanetska, O.Yu. Telenga, V.O. Ushenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 2. — С. 228-236. — Бібліогр.: 31 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
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2025-07-08T12:40:32Z |
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| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 2. P. 228-236.
PACS 33.50.-j, 34.35.+a, 73.20.Mf, 78.30.-j
Wavelet analysis of Jones-matrix images corresponding
to polycrystalline networks of biological crystals
in diagnostics of tuberculosis
N.I. Zabolotna1, V.O. Balanetska2, O.Yu. Telenga3, V.O. Ushenko3
1Vinnytsia National Technical University, Department for Laser and Optoelectronic Technique,
95, Khmelnytske shose, 21021 Vinnytsia, Ukraine.
2Bukovina State Medical University, Department of Biophysics and
Medical Informatics, 2, Teatralnaya Sq., 58012 Chernivtsi, Ukraine.
3Chernivtsi National University, Department for Optics and Spectroscopy,
2, Kotsyubinsky str., 58012 Chernivtsi, Ukraine.
Abstract. Determined in this work are analytical interrelations between orientations of
optical axes and birefringence of biological crystals as well as characteristic values of
elements in the Jones matrix for flat layers of polycrystalline networks, which determine
conditions for formation of polarization singularities in laser images. Performed is a
complex statistical, correlation and fractal analysis of distributions for the amount of
characteristic values inherent to the Jones matrix elements describing layers of saliva
taken from a healthy patient and that sick with tuberculosis. The authors have ascertained
objective criteria to differentiate optical properties of polycrystalline networks of human
saliva in various physiological states. Offered is Jones-matrix diagnostics of tuberculosis.
Keywords: laser, polarization, birefringence, Jones matrix, statistical moments,
autocorrelation, power spectrum, tuberculosis.
Manuscript received 04.07.11; accepted for publication 16.03.11; published online 30.06.11.
1. Introduction
7
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
1.0
Among the methods of optical diagnostics aimed at
biological tissues (BT), the methods of laser polarimetry
of optically anisotropic structures in human tissues have
been already widely spread [1 – 31]. The main
“information product” of using these methods is data for
coordinate distributions inherent to Mueller and Jones
matrixes typical for BT [1 – 5]. Then, these data can be
processed with statistical (statistical moments of the first
to fourth orders [5, 6, 10, 14, 19, 25, 26, 30]), correlation
(auto- and mutual-correlation functions [12, 17, 18, 21,
26]), fractal (fractal dimensionalities [5, 6, 25]), singular
(distributions of amounts of linear and circularly
polarized states), wavelet (sets of wavelet coefficients
for various scales of biological crystals [22, 28])
analyses. As a result, one can determine interrelations
between a set of these parameters and distributions of
optical axis directions as well as the birefringence value
inherent to networks of optically single-axis protein
(myosin, collagen, elastin, etc.) fibrils in optically
anisotropic components of BT layer. Being based on it,
developed was a set of methods for diagnostics and
differentiation of pathological changes in BT structure,
which are related with its degenerative-dystrophic and
oncological changes [4 - 6, 12, 19, 20-22, 27, 29, 31].
Progress in the above methods for studying the matrix
images of BT was reached in [5]. There, the offered new
approach is based on the analysis of coordinate
distributions for the so-called “characteristic values” that
describe conditions for formation of polarization
singularities. Related to these singularities are linear (L-
points) and circularly (C-points) polarized states. In the
case of L-points, the direction of electric field vector
rotation is indefinite (singular). For C-points, indefinite
is the azimuth of polarization for the electric field vector.
Demonstrated in […] was the efficiency of this approach
for Mueller-matrix diagnostics of pathological states in
human BT. At the same time, there is a widely spread
group of optically anisotropic biological objects, for
which the matrix methods of laser polarimetric
diagnostics did not yet acquire any wide application.
These are optically thin (extinction coefficient ≤τ )
layers of various biological liquids (bile, urine, liquor,
joint fluid, blood plasma, saliva, etc.). These objects are
considerably more accessible for direct laboratory
analysis as compared with traumatic methods of BT
biopsy. Being based on these reasons, it seems topical to
adapt the methods of laser polarimetric diagnostics of
optically anisotropic structures observed in BT
polycrystalline networks for medical purposes.
228
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 2. P. 228-236.
Bearing in mind diagnostics of tuberculosis, our
work is aimed at searching possibilities to diagnose and
differentiate optical properties of liquid-crystal networks
in human saliva by using determination of coordinate
distributions for Jones-matrix elements with the
following wavelet analysis of distributions inherent to
their characteristic (singular) values.
2. Main analytical interrelations
As a base for modeling the optical properties of liquid-
crystalline networks in human saliva, we took the
following conceptions [1-4, 7, 9, 14, 16, 23-27, 30]:
• separate (partial) liquid crystals (…) are optically
single-axis and birefringent;
• optical properties of a partial crystal can be
exhaustively described with the Jones operator [5]
{ }
( ) ( )[
( )[ ] ( )
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
] .
;expcossin;exp1sincos
;exp1sincos;expsincos
22
22
2221
1211
δ−ρ+ρδ−−ρρ
δ−−ρρδ−ρ+ρ
==
ii
ii
JJ
JJ
J
(1)
Here, is a direction of the optical axis; ρ
ndΔλ
π=δ 2 – phase shift between orthogonal
components and of the amplitude of
illuminating laser wave with the wavelength
xE yE
λ ; nΔ -
birefringence index of the crystal with the geometric size
. d
Let us consider the possibility to apply the singular
approach in the Jones-matrix images. From the
mathematical viewpoint, a singular value of a matrix
element complex value is defined by the following
conditions:
ikJ
( ) ( )
⎪
⎪
⎩
⎪⎪
⎨
⎧
=
=
=+
.0Im
;0Re
;0ImRe 22
ik
ik
ikik
J
J
JJ
(2)
With account of (2), the analytical expressions (1)
are transformed to interrelations
⎪⎩
⎪
⎨
⎧
=δρ+ρ
=δρ
⇔
.0cossincos
;0sinsin
22
2
11J , (3)
⎪⎩
⎪
⎨
⎧
=δρ+ρ
=δρ
⇔
.0coscossin
;0sincos
22
2
22J (4)
( )
( )⎪⎩
⎪
⎨
⎧
=δ+ρ
=δ+ρρ⇔=
.0cos12sin
;0cos1sincos2 22
2112 JJ (5)
As it follows from (3) to (5), singularities of
complex matrix elements are conditioned by certain
(characteristic) values of orientation and phase
parameters corresponding to liquid-crystalline network
∗
ikJ
∗ρ ∗δ
⎪⎩
⎪
⎨
⎧
±=δ
±=ρ
∗
∗
.180;90;0
;90;45;0
000
000
(6)
On the other hand, the relations (6) determine the
conditions for formation polarization singular states −L
( ) and 00 180;0=δ −C ( ) of the laser beam
by optically single-axis birefringent crystal. Being based
on it, one can find the characteristic values of the Jones
matrix elements that define
090±=δ
∗
ikJ −L and states of
polarization in a laser image of polycrystalline network:
−C
• 02211 == JJ values define states of
polarization;
−L
• 02112 == JJ values define states of
polarization.
−C
It is noteworthy that the analytical consideration of
(1) to (6) is related to a partial optically single-axis
birefringent crystal. Formed in real biological layers are
complex networks of these crystals with different scales
of geometric sizes. Therefore, application of the singular
analysis to the Jones matrix corresponding to this
network requires determining the coordinate
distributions of characteristic values in the
plane of biological liquid layer. These distributions can
be determined by scanning two-dimensional arrays of
elements in horizontal direction
with the step
( yxJ ik ,∗ )
ikJ mx ...,,1≡
pixx 1=Δ . Within the limits of every local
sampling (1pix × npix)(k = 1, 2, …, m), one has to calculate N
characteristic values ( ) 0=kJik , - ( ( )k
ikN ). In this
manner, one finds dependences
( ) )...,,,( )()2()1( m
ikikikik NNNxN ≡ of the amount of
characteristic values for matrix elements. On the other
hand, these multi-scale dependences can be efficiently
analyzed using the wavelet analysis […]. If as a wavelet
function one takes the dependence that possesses a finite
base both in coordinate and frequency spaces, then using
the scaling and shift of this function-prototype the
coordinate distribution of Jones matrix elements can be
expanded into the following series
∑
∞
−∞=
Ψ=
ba
ababik xCxN
,
)()( , (7)
where )()( baxxab −Ψ=Ψ is the base function that is
formed from the function-prototype using the shift b and
scaling a, while the coefficients of this expansion are
defined in the following manner
∫ Ψℵ= dxxxC abikab )()( (8)
In our work, using the analogy with the
investigation […], as a wavelet function we chose the
second derivative of the Gauss function (MHAT
wavelet) that possesses a narrow energy spectrum and
229
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 2. P. 228-236.
two moments equal to zero (zero and first). The
analytical dependence of the MHAT wavelet has the
following look:
2/22/
2
2 22
)1()( zz eze
dz
dz −− −==Ψ . (9)
The result of wavelet transformation (9) is two-
dimensional array of amplitudes or the sp
),( baW that in the space “spatial scale a – spatial
localization b” provides information on relative
contribution of birefringent network in crystals of
different sca the dependence ( )xNik . Thus, it opens
the possibility to realize a scale-selective analysis of real
polycrystalline networks in various bio
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Fig. 1. Optical scheme of the polarimeter: 1 – He-Ne laser; 2 – collimator; 3 – stationary quarter-wave plate; 5, 8 –
mechanically movable quarter-wave plates; 4, 9 – polarizer and analyzer, respectively; 6 – studied object; 7 – micro-objective;
10 – CCD camera; 11 – personal computer.
ectrum
les to
logical layers.
)
3. Optical setup for making Jones-matrix maps
of optically anisotropic biological liquids
Shown in Fig. 1 is the optical scheme of a polarimeter
for measuring the coordinate distributions of elements in
the Jones matrix corresponding to biological layers.
Illumination of bile samples was carried out using a
parallel beam (∅ = 104 µm) of the He-Ne laser (λ =
0.6328 µm, W = 5.0 mW). The polarization illuminator
consists of the quarter-wave plates 3, 5 and polarizer 4,
which provides formation of the laser beam with an
arbitrary polarization state. Using the micro-objective 7
(magnification 4x), images of bile layers were projected
onto the plane of light-sensitive area (800x600 pixels) in
the CCD-camera 10 that provides measurements of
structural elements from 2 to 2000 µm.
The analysis of laser images was performed using
the polarizer 9 and quarter-wave plate 8.
4. Criteria for estimating the spectra of wavelet
coefficients corresponding to the amount of
characteristic values inherent to Jones-matrix images
describing the layers of human saliva
Distributions of the amount of characteristic
values for elements of the Jones matrix
),( baW
( nmJik × are
characterized with the set of statistical moments of the 1-
st to 4-th orders EAM ;;; σ calculated using the
following relations [5, 6, 25, 30]:
( ) ( )
( ) ( ) .),(11,),(11
,),(1,),(1
1
4
4
1
3
3
1
2
1
∑∑
∑∑
==
==
σ
=
σ
=
=σ=
D
j
j
D
j
j
D
j
j
D
j
j
baW
D
EbaW
D
A
baW
D
baW
D
M
(10)
where is the number of pixels for which the
dependence of characteristic values within the
limits of coordinate distribution for Jones-matrix images
of elements is determined.
D
ikN
ikJ
Our analysis of the spectra for wavelet coefficients
in distributions was based on the
autocorrelation method with using the following
function [12, 21, 26]:
),( baW ( )xNik
( ) [ ][∫ Δ−÷==Δ
m
aa dbbbWxbW
m
bG
1
)()1(1 ] . (11)
Here, bΔ is the “step” of changing the
coordinate mbx ÷=≡ 1 .
As to parameters characterizing the
dependences ( )bG Δ , we chose the set of correlation
moments from the 1-st to 4-th orders that are
determined similarly to relations (10).
4;3;2;1=lK
Estimating the degree of self-similarity and
reproducibility for different geometric ( ) scales of the
structure inherent to wavelet coefficients W of the
distributions
d
),( ba
( )xNik corresponding to characteristic
values of the Jones-matrix elements ( )nmJik ×
describing the polycrystalline networks was performed
via calculations of the logarithmic dependences for
power spectra ( )[ ] )log(,log . These
dependences are approximated using the least-squares
method to curves
1−− dbaWJ
( )ηΦ . Straight-line parts of these
curves enable us to determine the slope angles iη and
calculate the values of fractal dimensionality for W
distributions with account of the relations [5, 6, 11, 25]:
),( ba
ii tggD η−= 3)( . (12)
Classification of the distributions W for
dependences of the amount of characteristic values
),( ba
230
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 2. P. 228-236.
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
typical for the matrix elements is carried out
in accord with criteria offered in [5]. If the value of the
slope angle
( nmJik × )
const=η , the dependences for 2 or 3
decades of changing the sizes d , than the distributions
are fractal. Under condition that several
constant slope angles are available in the curve
( )ηΦ
a b
Fig. 2. Polarization images of dendrite polycrystalline networks typical for human saliva in different physiological states. See
explanations in the text.
),( baW
( )ηΦ ,
the sets are multi-fractal. When no stable slope
angles are available over the whole interval of changing
the sizes d , the sets are considered as random.
),( baW
),( baW
To make this comparative analysis of
dependences more objective,
let us use the conception of spectral moments from the
1-st to 4-th orders - the relation (7).
( )[ ] )log(,balog 1−− dWJ
)
4;3;2;1=jS
5. Analysis and discussion of experimental results
As objects of investigation, we used saliva smears of
healthy (18 samples) and sick with tuberculosis (17
samples) patients. Fig. 2 shows laser images of optically-
anisotropic structures typical for the samples of both
types. The images were obtained for crossed
transmission planes of the polarizer 4 and analyzer 9 in
the laser polarimeter (Fig. 1).
It can be seen from a comparative analysis of laser
images corresponding to liquid-crystalline networks of
the studied samples that the level of transmission
(birefringence) grows in the case of small-scale
(d = 10…30 µm) liquid crystals in saliva of the patients
sick with tuberculosis (Fig. 2b). This fact was used as a
basis for our wavelet analysis of the dependences
for distributions of characteristic values
inherent to elements in Jones-matrix
images. Our choice of just these elements is related with
the fact that values characterize the
probability of point formation, which is related
with growth of birefringence in liquid crystals at certain
scales of geometric sizes.
( )xN 21;12
( nmJ ×21;12
021;12 =J
−C
Depicted in Figs 3 and 4 are the results of
determining the dependence (fragment (a)) and
the respective spectrum of wavelet coefficients
( )xN 21;12
( )( )xNW ba, (fragment (b)).
It is seen from the data obtained that the two-
dimensional array ( )( )xNW ba, is a complex coordinate-
inhomogeneous and scale-dependent set of values for
wavelet coefficients. Starting from this analysis, it is
necessary to use a complex statistical, correlation and
fractal analysis of these wavelet spectra for distributions
of characteristic values in the Jones-matrix images of
liquid-crystalline networks.
Shown in Figs 5 and 6 is the set of coordinate
( )( )021;121, =÷== JNW mbconsta , autocorrelation
( )( )( )021;121, =÷== JNWG mbconsta and spectral
dependences that characterize the
spectra of wavelet coefficients for two
scales
1
, log)(log −− dWF ba
( )( xNW ba, )
14=a and 70=a of the wavelet function ba,Ψ
(relation (9)) for the
distributions ( )021;12 =JN corresponding to liquid-
crystalline networks in saliva of healthy patient (Figs 3
and 5) and that sick with tuberculosis (Figs 4 and 6).
It is seen from the data obtained that coordinate,
autocorrelation and spectral dependences that
characterize the sets of wavelet coefficients
( )( )021;12, =JNW ba corresponding to distributions of the
amount of characteristic values in the Jones-
matrix image
021;12 =J
( )nmJ ×21;12 of liquid-crystal network in
the layer of saliva taken from a healthy patient and from
that sick with tuberculosis are individual in every scale
of a MHAT wavelet.
From the quantitative viewpoint, the differences
between coordinate ( )( )021;121, =÷== JNW mbconsta ,
autocorrelation ( )( )( )021;121, =÷== JNWG mbconsta and
spectral dependences characterize 1
, log)(log −− dWF ba
231
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 2. P. 228-236.
Fig. 3. Spectrum of wavelet coefficients Wa,b(N(x)) for the distribution N12;21(x) in the Jones-matrix image J12;21(m × n)
corresponding to liquid-crystalline network of a saliva layer taken from a healthy patient.
Fig. 4. Spectrum of wavelet coefficients Wa,b(N(x)) for the distribution N12;21(x) in the Jones-matrix image J12;21(m × n)
corresponding to liquid-crystalline network of a saliva layer taken from a patient sick with tuberculosis.
the values and ranges for changing the set of statistical,
correlation and spectral moments of the 1-st to 4-th
orders (Table 1).
It is seen from the analysis of the data summarized
in Table 1 that changes in values of statistical moments
of the 1-st to 4-th orders, which characterize the
distributions of wavelet coefficients ( )( )021;12, =ℵNW ba
possess extreme values for the scale of the
MHAT - function.
14=a
We have ascertained the following tendencies of
changes in the statistical moments of the 1-st to 4-th orders:
• mean value ( )( )( )021;1214 == JNWM a increases by
1.9 times;
• dispersion ( )( )( )021;1214 =σ = JNWa increases by
2.85 times;
• skewness ( )( )( )021;1214 == JNWA a increases by 3.2
times;
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
232
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 2. P. 228-236.
a Wa,b(N(J12;21= 0)) G(Wa,b(N(J12;21= 0))) 1
, log)(log −− dWF ba
14=a
70=a
Fig. 5. Coordinate (left column), autocorrelation (central column) and spectral (right column) dependences that characterize the
spectrum of wavelet coefficients Wa,b(N(J12;21= 0)) for the distribution of characteristic values in the Jones-matrix image
J12;21(m × n) of liquid-crystal network in the layer of saliva taken from a healthy patient. See explanations in the text.
a Wa,b(N(J12;21= 0)) G(Wa,b(N(J12;21= 0))) 1
, log)(log −− dWF ba
14=a
14=a
Fig. 6. Coordinate (left column), autocorrelation (central column) and spectral (right column) dependences that characterize the
spectrum of wavelet coefficients Wa,b(N(J12;21= 0)) for the distribution of characteristic values in the Jones-matrix image
J12;21(m × n) of liquid-crystal network in the layer of saliva taken from a patient sick with tuberculosis. See explanations in the text.
• excess ( )( )( )021;1214 == JNWE a increases by 3.85
times.
For the set of correlation moments of the 1-st to 4-
th orders that characterize the autocorrelation
dependences
4;3;2;1=iK
( )( )( )
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
021;1214 == JNWG a
s
of distributions
for wavelet coefficient ( )( )021;12, =JNW ba on the scale
of the MHAT-function for the samples of saliva
taken from the patients sick with tuberculosis, we have
found the following tendencies of changes in values:
14=a
• correlation moment of the 1-st order K1 does not
practically change;
• correlation moment of the 2-nd order K2 increases
by 2.3 times;
• correlation moment of the 3-rd order K3 increases
by 1.8 times;
233
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 2. P. 228-236.
Table 1. Statistical (M; σ; A; E), correlation (Ki = 1;2;3;4), spectral (Si = 1;2;3;4) moments of the 1-st to 4-th orders for various-
scale (a) distributions of wavelet coefficients Wa,b(N(J12;21= 0)) for the amount of characteristic values in Jones-matrix
images corresponding to liquid-crystal network in saliva of healthy (q = 18) and sick with tuberculosis (q = 17) patients.
State Norm Tuberculosis
a 14=a 70=a 14=a 70=a
M 0.086 ± 0.018 0.26 ± 0.051 0.12 ± 0.034 0.37 ± 0.073
σ 0.23 ± 0.057 0.085 ± 0.012 0.11 ± 0.026 0.11 ± 0.017
A 0.37 ± 0.075 0.13 ± 0.029 0.77 ± 0.14 0.21 ± 0.047
E 0.29 ± 0.057 0.19 ± 0.035 0.74 ± 0.17 0.24 ± 0.053
1K 1.34 ± 0.29 0.84 ± 0.19 1.23 ± 0.24 0.91 ± 0.22
2K 0.09 ± 0.02 0.16 ± 0.035 0.17 ± 0.032 0.13 ± 0.027
3K 0.84 ± 0.17 0.58 ± 0.12 1.14 ± 0.25 0.65 ± 0.21
4K 1.18 ± 0.23 0.76 ± 0.15 1.41 ± 0.29 0.87 ± 0.19
1S 0.52 ± 0.14 0.34 ± 0.072 0.49 ± 0.13 0.31 ± 0.065
2S 0.12 ± 0.025 0.11 ± 0.024 0.31 ± 0.075 0.15 ± 0.041
3S 0.41 ± 0.086 0.34 ± 0.077 0.19 ± 0.036 0.31 ± 0.073
4S 0.54 ± 0.12 0.34 ± 0.085 0.26 ± 0.061 0.29 ± 0.063
• correlation moment of the 4-th order increases
by 1.75 times.
4K
For the set of spectral moments of the 1-st to 4-
th orders that characterize distributions of
extremes in the logarithmic dependences
of the power spectra for wavelet
coefficients
4;3;2;1=iS
( ) 1
, log −− dWLogJ ba
( )( )021;12, =JNW ba of the amount of
characteristic values in the Jones-matrix image
corresponding to liquid-crystal network
in saliva of patients sick with tuberculosis, we have
found the following features:
( nmJ ×21;12
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
)
• spectral moment of the 1-st order does not
practically change;
1S
• spectral moment of the 2-nd order increases by
2.1 times;
2S
• spectral moment of the 3-rd order decreases by
2.2 times;
3S
• spectral moment of the 4-th order decreases by
1.9 times.
4S
The experimentally found differences in statistical,
correlation and spectral moments of the 1-st to 4-th
orders in ( )( )021;12, =JNW ba distributions on the certain
scale ( ) of the soliton-like MHAT wavelet
function can be explained by formation of additional
small-scale globulin crystals with an increased level of
birefringence.
14=a
6. Conclusions
Thus, we have demonstrated the diagnostic efficiency of
the wavelet analysis applied to distributions of the
amount of characteristic values in the Jones-matrix
images ( )nmJik × corresponding to liquid-crystalline
networks of human saliva. Besides, we have offered the
following parameters to diagnose tuberculosis:
• statistical moments of the 1-st to 4-th orders that
characterize spectra of wavelet coefficients
( )( )021;12, =JNW ba for distributions of the amount
( )xN 21;12 of characteristic values 021;12 =J in the
Jones-matrix image ; ( )nmJ ×21;12
• correlation moments of the 2-nd to 4-th orders that
characterize autocorrelation functions ( )xG Δ21;12
for the distributions ( )( )021;12, =JNW ba ;
• spectral moments of the 2-nd to 4-th orders that
characterize distributions of extremes in
logarithmic dependences ( )[ ] ( )1
, loglog −− dWF ba
for power spectra of the sets of the wavelet
coefficients ( )( )021;12, =JNW . ba
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