Wavelet analysis for Mueller matrix images of biological crystal networks
Theoretically grounded in this work is the efficiency of using the statistical and fractal analyses for distributions of wavelet coefficients for Mueller-matrix images of biological crystal networks inherent to human tissues. The authors found interrelations between statistical moments and power...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2009
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| Zitieren: | Wavelet analysis for Mueller matrix images of biological crystal networks / Yu.O. Ushenko, Yu.Ya. Tomka, O.G. Pridiy, A.V.Motrich, O.V. Dubolazov, I.Z.Misevitch, V.V. Istratiy // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2009. — Т. 12, № 4. — С. 391-398. — англ. |
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irk-123456789-1188422017-06-01T03:05:29Z Wavelet analysis for Mueller matrix images of biological crystal networks Ushenko, Yu.O. Tomka, Yu.Ya. Pridiy, O.G. Motrich, A.V. Dubolazov, O.V. Misevitch, I.Z. Istratiy, V.V. Theoretically grounded in this work is the efficiency of using the statistical and fractal analyses for distributions of wavelet coefficients for Mueller-matrix images of biological crystal networks inherent to human tissues. The authors found interrelations between statistical moments and power spectra for distributions of wavelet coefficients as well as orientation-phase changes in networks of biological crystals. Also determined are criteria for statistical and fractal diagnostics of changes in the birefringent structure of biological crystal network, which corresponds to pathological changes in tissues. 2009 Article Wavelet analysis for Mueller matrix images of biological crystal networks / Yu.O. Ushenko, Yu.Ya. Tomka, O.G. Pridiy, A.V.Motrich, O.V. Dubolazov, I.Z.Misevitch, V.V. Istratiy // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2009. — Т. 12, № 4. — С. 391-398. — англ. 1560-8034 PACS 87.64.-t http://dspace.nbuv.gov.ua/handle/123456789/118842 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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Theoretically grounded in this work is the efficiency of using the statistical and
fractal analyses for distributions of wavelet coefficients for Mueller-matrix images of
biological crystal networks inherent to human tissues. The authors found interrelations
between statistical moments and power spectra for distributions of wavelet coefficients
as well as orientation-phase changes in networks of biological crystals. Also determined
are criteria for statistical and fractal diagnostics of changes in the birefringent structure of
biological crystal network, which corresponds to pathological changes in tissues. |
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Article |
| author |
Ushenko, Yu.O. Tomka, Yu.Ya. Pridiy, O.G. Motrich, A.V. Dubolazov, O.V. Misevitch, I.Z. Istratiy, V.V. |
| spellingShingle |
Ushenko, Yu.O. Tomka, Yu.Ya. Pridiy, O.G. Motrich, A.V. Dubolazov, O.V. Misevitch, I.Z. Istratiy, V.V. Wavelet analysis for Mueller matrix images of biological crystal networks Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Ushenko, Yu.O. Tomka, Yu.Ya. Pridiy, O.G. Motrich, A.V. Dubolazov, O.V. Misevitch, I.Z. Istratiy, V.V. |
| author_sort |
Ushenko, Yu.O. |
| title |
Wavelet analysis for Mueller matrix images of biological crystal networks |
| title_short |
Wavelet analysis for Mueller matrix images of biological crystal networks |
| title_full |
Wavelet analysis for Mueller matrix images of biological crystal networks |
| title_fullStr |
Wavelet analysis for Mueller matrix images of biological crystal networks |
| title_full_unstemmed |
Wavelet analysis for Mueller matrix images of biological crystal networks |
| title_sort |
wavelet analysis for mueller matrix images of biological crystal networks |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2009 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/118842 |
| citation_txt |
Wavelet analysis for Mueller matrix images of biological crystal networks / Yu.O. Ushenko, Yu.Ya. Tomka, O.G. Pridiy, A.V.Motrich, O.V. Dubolazov, I.Z.Misevitch, V.V. Istratiy // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2009. — Т. 12, № 4. — С. 391-398. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
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2025-07-08T14:46:05Z |
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| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 4. P. 391-398.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
391
PACS 87.64.-t
Wavelet analysis for Mueller matrix images
of biological crystal networks
Yu.O. Ushenko, Yu.Ya. Tomka, O.G. Pridiy, A.V.Motrich, O.V. Dubolazov, I.Z.Misevitch, V.V. Istratiy
Chernivtsi National University, Optics and Spectroscopy Dept., Chernivtsi, Ukraine, 58000
Abstract. Theoretically grounded in this work is the efficiency of using the statistical and
fractal analyses for distributions of wavelet coefficients for Mueller-matrix images of
biological crystal networks inherent to human tissues. The authors found interrelations
between statistical moments and power spectra for distributions of wavelet coefficients
as well as orientation-phase changes in networks of biological crystals. Also determined
are criteria for statistical and fractal diagnostics of changes in the birefringent structure of
biological crystal network, which corresponds to pathological changes in tissues.
Keywords: polarization, Mueller matrix, biological crystal, birefringence, statistical
moment, wavelet analysis, fractal.
Manuscript received 28.05.09; accepted for publication 10.09.09; published online 30.10.09.
1. Introduction
In recent years, laser diagnostics aimed at the structure
of biological tissues efficiently uses the model approach
[1], in accord with which the tissues are considered as
two components: amorphous A and optically
anisotropic M ones. Topicality of this modeling is
related with the possibility to apply the all-purpose
Mueller matrix analysis to changes of polarization
properties, which are caused by transformation of
optical-and-geometric constitution of the anisotropic
component (architectonic network of fibrils) in these
biological objects [2 – 8]. Based on this model, there
developed is the method for polarization differentiation
of optical properties inherent to physiologically normal
as well as pathologically changed biological tissues by
using the wavelet analysis of local features observed in
coordinate distributions of intensities in their coherent
images.
This trend in polarization diagnostics got its
development in investigations of a statistical and self-
similar structure of Mueller-matrix images (MMI) that
are two-dimensional distributions yxfik , [9,10]
describing biological tissues. So, in the approximation of
single light scattering, there found was the interrelation
between a set of statistical moments of the first to fourth
orders 4,3,2,1jZ that characterize orientation ( ) and
phase ( ) structures of birefringent architectonics
inherent to biological tissues as well as a set of
respective statistical moments for MMI [10 – 14]. It is
ascertained that the coordinate distributions of matrix
elements yxfik , describing physiologically normal
biological tissue possess a self-similar, fractal structure.
While MMI of physiologically changed biological
tissues are stochastic or statistical [11].
This work is aimed at studying the efficiency of the
wavelet analysis in application to the local structure of
MMI inherent to biological tissues with using statistical
and fractal analyses of the obtained wavelet-coefficient
distributions for diagnostics of local changes in
orientation-phase structure of their architectonic
networks.
2. Wavelet analysis of Mueller-matrix images of
biological tissues
Wavelet transformation of MMI consisted of its
expansion within a basis of definite scale changes and
transfers of the soliton-like function (wavelet) [5]. The
distribution of values for ikf elements of the Mueller
matrix can be represented in the following form:
lj
jljlik xqxf
,
. (1)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 4. P. 391-398.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
392
Here, xfik distribution belongs to the space
RL2 created by wavelets jl . The basis of this
functional space can be constructed using scale
transformations and transfers of the wavelet xjl with
arbitrary values of basic parameters – the scaling
coefficient a and shift parameter b
.;,; 22
1
RLRba
a
bx
axab
(2)
Being based on it, the integral wavelet
transformation takes a look
.
, 2
1
dxxxf
dx
a
bx
xfabafW
abik
ikik
(3)
Coefficients jlikjl fq , of the expansion (3)
for the function ikf by wavelets can be defined via the
following integral wavelet transformation
jjikjl
l
fWq
2
,
2
1
. (4)
In our work, to analyze MMI we used the most
widely spread soliton-like function MHAT (“Mexican
hat”, [5]) as a wavelet function.
3. Computer modeling the efficiency of the wavelet
analysis to differentiate MMI of birefringent fibrils
Birefringent architectonic networks of BT consist of a
set of co-axial cylinder protein fibrils with a statistical
distribution of optical axis orientations and values of
phase shifts . We considered the most spread case of
pathological changes in BT architectonics – formation of
directions for pathological growth or excrescence of a
tumor. Within mathematical frames, this case was
modeled as a superposition of statistical (equiprobable)
and stochastic (quasi-regular) components in
distributions of orientations of birefringent fibrils as
well as phase shifts that are caused by them
.
2
sin)()(
,
2
sin)()(
2
1
D
BRQ
D
ARQ
(5)
where 2,1iQ are the functions of distributions for and
values; R – random (equiprobable) distribution of
and ; А, В – amplitudes of the stochastic component;
D – mean statistical size of co-axial fibrils.
In accord with distributions of optical-and-
geometrical parameters ),( iQ , the distribution of
values for Mueller matrix elements ikf for such a
territorial matrix can be written in the form
Fig. 1. Wavelet coefficients baW , of statistical-and-stochastic distributions for the matrix element )(bfik (a, b, c).
Commentaries are in the text.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 4. P. 391-398.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
393
),(
2
sin),(),(
ikik f
D
CRf . (6)
Here, С is the amplitude of a stochastic component.
We modeled a superposition of a “background”
),( R and informative signal ),(
2
sin
ikF
D
C for
the following relations between their amplitudes
),(
2
sin6),(
2
sin01.0),(
ikik f
D
Cf
D
CR .
Fig. 1 shows wavelet coefficients baW , for the
respective distributions ),( ikf .
As seen from the data obtained, the distributions of
values for wavelet coefficients baW , of all the types of
signals ),( ikf behave like quasi-harmonic structures.
Even in the case of significant (six-fold) dominance of
the statistical component amplitude (Fig. 1, c), the quasi-
regular structure of coordinate distribution for wavelet
coefficients baW , is preserved in full. This fact confirms
a high efficiency of the wavelet analysis in separation of
the harmonic component in the distribution of ikf
elements of the Mueller matrix.
To make diagnostic possibilities of the wavelet
analysis more objective, we calculated statistical
moments of the first to fourth orders (М, σ, А, Е), which
characterize distributions of the wavelet coefficients
)(, ikba fW for various ratios 601.0 0
A
A
(Fig. 2) and
found their power spectra (Table 1).
Our analysis of the obtained data revealed that the
change of the fourth statistical moment for the
distribution of wavelet coefficients baW , is the most
dynamical from the above viewpoint, as the value of this
moment changes within the range of one order in
dependency of ratios 601.0 0
A
A
.
Our investigation of log-log dependences for power
spectra of distributions describing the wavelet
coefficients baW , of matrix elements ikf allowed
revealing the following regularities:
- all the dependences )/1log(log , dW ba
calculated for various relations between amplitudes of
random and quasi-regular components of distributions
characteristic for the Mueller matrix elements ikf
consist of two parts, namely: the fractal one (with one
slope of the approximating curve within a definite range
d for sizes of birefringent fibrils) and the statistical
one (when a stable value for the slope angle of the
approximating curve does not take place);
Fig. 2. Mean value (a), dispersion (b), asymmetry (c) and excess (d) of distributions inherent to wavelet
coefficients )(, ikba fW . Commentaries are in the text.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 4. P. 391-398.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
394
- when the amplitude of the statistical
component in the ikf distribution grows, the range d
of a linear part in dependences )/1log(log , dW ba is
decreased;
- fractal component of log-log dependences for
the power spectra of wavelet coefficients baW , is
preserved even for significant (six-fold) dominance of
the noise amplitude and comprises the size range
d = 50 – 100 μm.
Thus, the performed computer modeling indicates
the diagnostic efficiency of the wavelet analysis when
detecting local changes in birefringency ( ) of ordered
biological crystals.
Table 1. Log-log dependences of power spectra for the wavelet coefficients Wa, b of statistical-stochastic distributions
for fik elements of the Mueller matrix describing single-axis biological crystals
R )/1log(log , dW ba md , R )/1log(log , dW ba md ,
0.01 1-1000 1 5-100
0.1 5-1000 3 20-100
0.5 5-100 6 50-100
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2009. V. 12, N 4. P. 391-398.
© 2009, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
395
Besides, using the statistical and correlation
analysis of wavelet coefficients baW , in the expansion of
the Mueller matrix elements, we have demonstrated the
possibility to reveal the quasi-harmonic component in
distributions of orientation ( ) and phase ( )
parameters in complex (statistical) architectonic
networks.
4. Diagnostics of local changes in the optical-and-
geometrical structure of architectonic networks
inherent to real biological tissues
We performed comparative investigations of two types
of mounts from connective tissue of a woman matrix:
- healthy tissue (type A) – the set of chaotically
oriented collagen fibrils;
- tissue in the state of displasia (pre-cancer state –
type B) – the set of chaotically oriented collagen fibrils
with local quasi-ordered parts.
From the optical viewpoint, polarization properties
of these tissues (types A and B) are similar to some
extent. For instance, the coordinate distribution of
random values inherent to phase shifts ),( yx , which is
related with the range of changes in geometric sizes of
collagen fibrils, is close in both cases. The main
differences in composition of the set of biological
crystals lie in presence of local parts with quasi-ordered
directions of optical axes in the tissue of the type B.
Being based on this fact, one can assume that the
coordinate distribution of the Mueller matrix element
values for the type A tissue ),( ikf approaches to the
statistical one. The coordinate distribution ),( ikf for
the type B tissue can be represented by a superposition
of the random ),( ikR and quasi-regular
),(
2
sin
ikf
D
C components (6).
As a main element of the Mueller matrix for bio-
tissue of a given type, we chose the “orientation” matrix
element 33f . It is known that the statistical and
correlation analysis of coordinate distributions for this
element is considered as efficient in differentiation of
Fig. 3. MMI of the f33 element (a, b) and coordinate structure (c, d) as well as coefficients of wavelet expansion Wa, b (e, f) for
the f33 element of A and B type connective tissues.
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