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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 is theoretically grounded in this work. 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 the criteria for statistical and fractal diagnostics of changes in the birefringent structure of biological crystal network, which corresponds to pathological changes in tissues.

In recent years, laser diagnostics aimed at the structure of biological tissues efficiently use the model approach [

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 [

This trend in polarization diagnostics got its development in investigations of a statistical and self-similar structure of Mueller-matrix images (MMIs) that are two-dimensional distributions

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.

Wavelet transformation of MMI consisted of its expansion within a basis of definite scale changes and transfers of the soliton-like function (wavelet) [

Here,

Being based on it, the integral wavelet transformation takes a look at

Coefficients

In our work, to analyze MMI we used the most widely spread soliton-like function MHAT (“Mexican hat”, [

Birefringent architectonic networks of BT consist of a set of coaxial cylinder protein fibrils with a statistical distribution of optical axis orientations

The analysis of (

That is why, one can assume that coordinate distributions

For simplicity we consider one-dimensional coordinate distribution in the following form:

Here,

We modeled a superposition of a “background”

Figure

Wavelet coefficients

As seen from the data obtained, the distributions of values for wavelet coefficients

To make diagnostic possibilities of the wavelet analysis more objective, we calculated statistical moments of the first to fourth orders

Log-log dependences of power spectra for the wavelet coefficients

0.01 | 1 | ||||

0.1 | 3 | ||||

0.5 | 6 |

Mean value (a), dispersion (b), the skewness (c), and the kurtosis (d) of distributions inherent to wavelet coefficients

Our analysis of the obtained data revealed that the change of the fourth statistical moment for the distribution of wavelet coefficients

Our investigation of log-log dependences for power spectra of distributions describing the wavelet coefficients

all the dependences

when the amplitude of the statistical component in the

fractal component of log-log dependences for the power spectra of wavelet coefficients

Thus, the performed computer modeling indicates the diagnostic efficiency of the wavelet analysis when detecting local changes in birefringency (

Besides, using the statistical and correlation analysis of wavelet coefficients

Figure

Optical scheme of polarimeter 1:He-Ne laser; 2:collimator; 3:stationary quarter-wave plates; 5, 8:mechanically movable quarter-wave plates; 4, 9:polarizer and analyzer correspondingly; 6:object of investigation; 7:micro-objective; 10:CCD camera; 11:personal computer.

The parallel (^{4}

The optical thin (the absorption coefficient

The technique of obtaining such objects is convenient: biological tissue is freezing to nitrogen temperature with the following obtaining, by means of medical microtome, the histological sections (from 10

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 dysplasia (precancer state (type B))—the set of chaotically oriented collagen fibrils with local quasiordered 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

As a main element of the Mueller matrix for biological tissue of a given type, we chose the “orientation” matrix element

Shown in Figure

MMI of the

Our comparative analysis of the obtained data shows a complex statistical structure of two-dimensional distributions for the matrix element

With account of the above observations, it seems actual to verify the efficiency of statistical (the set of the first to fourth moments for

With this aim, we performed step-by-step “screening” of the pictures for wavelet coefficients

Log-log dependences of power spectra for the wavelet coefficients

a | “A” | “B” |
---|---|---|

14 | ||

42 | ||

70 |

Coordinate distributions for the matrix element

Another picture can be observed for the sample of changed connective tissue (Figure

In our case, for the mean statistical size of a pathological creation

Thus, one can state that the correlation approach, in the analysis of

Some additional information for differentiation of these objects was obtained using the statistical analysis of coordinate distributions

Shown in Figure

Dependences of statistical moments of the 1st to 4th orders on the scale of the wavelet function

The obtained data show that

statistical moments of the 1st and 4th orders for distributions of wavelet coefficients

the range of changes in values of skewness (

main differences between connective tissues of A and B types are found in the vicinity of the scale

Thus, we found that the differences between values of statistical moments of higher orders for a definite range of scales