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Зміст 4. Diagnostics of local changes in the opticalandgeometrical structure of architectonic networks inherent to real biological tJ. Biomed. Opt. 
Wavelet analysis for Mueller matrix images of biological crystal networks Yu.O. Ushenko, O.G. Pridiy, A.V.Motrich, O.V. Dubolazov, I.Z.Misevitch, V.V. Istratiy Abstract. Theoretically grounded in this work is the efficiency of using the statistical and fractal analyses for distributions of wavelet coefficients for Muellermatrix 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 orientationphase 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; accepted for publication; published online. 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 and optically anisotropic ones. Topicality of this modeling is related with the possibility to apply the allpurpose Mueller matrix analysis to changes of polarization properties, which are caused by transformation of opticalandgeometric 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 selfsimilar structure of Muellermatrix images (MMI) that are twodimensional distributions [9] 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 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 describing physiologically normal biological tissue possess a selfsimilar, fractal structure. While MMI of physiologically changed biological tissues are stochastic or statistical [ ? ]. 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 waveletcoefficient distributions for diagnostics of local changes in orientation phase structure of their architectonic networks. 2. Wavelet analysis of Muellermatrix images of biological tissues Wavelet transformation of MMI consisted of its expansion within a basis of definite scale changes and transfers of the solitonlike function (wavelet) [5]. The distribution of values for elements of the Mueller matrix can be represented in the following form: . (1) Here, distribution belongs to the space created by wavelets . The basis of this functional space can be constructed using scale transformations and transfers of the wavelet with arbitrary values of basic parameters – the scaling coefficient a and shift parameter b (2) Being based on it, the integral wavelet transformation takes a look (3) Coefficients of the expansion (3) for the function by wavelets can be defined via the following integral wavelet transformation . (4) In our work, to analyze MMI we used the most widely spread solitonlike 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 coaxial 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 (quasiregular) components in distributions of orientations of birefringent fibrils as well as phase shifts that are caused by them (5) where 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 coaxial fibrils. In accord with distributions of opticalandgeometrical parameters , the distribution of values for Mueller matrix elements for such a territorial matrix can be written in the form . (6) Here, С is the amplitude of a stochastic component. We modeled a superposition of a “background” and informative signal for the following relations between their amplitudes . Fig. 1 shows wavelet coefficients for the respective distributions . Fig. 1. Wavelet coefficients of statisticalandstochastic distributions for the matrix element. Commentaries are in the text. As seen from the data obtained, the distributions of values for wavelet coefficients of all the types of signals behave like quasiharmonic structures. Even in the case of significant (sixfold) dominance of the statistical component amplitude (Fig. 3, c), the quasiregular structure of coordinate distribution for wavelet coefficients is preserved in full. This fact confirms a high efficiency of the wavelet analysis in separation of the harmonic component in the distribution of 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 for various ratios (Fig. 4) and found their power spectra (Table 1). Fig. 2. Mean value (a), dispersion (b), asymmetry (c) and excess (d) of distributions inherent to wavelet coefficients . Commentaries are in the text. Our analysis of the obtained data revealed that the change of the fourth statistical moment for the distribution of wavelet coefficients 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 . Table 1. Loglog dependences of power spectra for the wavelet coefficients of statisticalstochastic distributions for elements of the Mueller matrix describing singleaxis organic crystals
Our investigation of loglog dependences for power spectra of distributions describing the wavelet coefficients of matrix elements allowed revealing the following regularities:
Thus, the performed computer modeling indicates the diagnostic efficiency of the wavelet analysis when detecting local changes in birefringency () of ordered biological crystals. Besides, using the statistical and correlation analysis of wavelet coefficients in the expansion of the Mueller matrix elements, we have demonstrated the possibility to reveal the quasiharmonic component in distributions of orientation () and phase () parameters in complex (statistical) architectonic networks. ^ 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 (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 , 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 quasiordered 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 (3.4) approaches to the statistical one. The coordinate distribution for the type B tissue can be represented by a superposition of the random and quasiregular components. As a main element of the Mueller matrix for biotissue of a given type, we chose the “orientation” matrix element. It is known that the statistical and correlation analysis of coordinate distributions for this element is considered as efficient in differentiation of optical properties characterizing collagen networks in healthy and pathologically changed skin [???]. Shown in Fig. 3 are MMI for the element (f_{33} ???)(fragment a – type A, fragment b – type B) and respective wavelet expansions (fragments e and f for the linear crosssection of , fragments c and d for healthy and pathologically changed connective tissue of the uterus neck. Fig. 3. MMI of the (???) element and coordinate structure (a, b, c, d) as well as coefficients of wavelet expansion (e, f) for the (???) element of A and B type connective tissues. Our comparative analysis of the obtained data shows a complex statistical structure of twodimensional distributions for the matrix element (?) (see fragments a and b) as well as its linear crosssections (fragments c and d) for MMI of biotissues for their both types. Аналогічно можна сказати про розподіли вейвлеткоефіцієнтів (фрагменти (д), (е)). The same can be stated for the distributions of wavelet coefficients (fragments e and f). With account of the above observations, it seems actual to verify the efficiency of statistical (the set of the first to fourth moments for distributions) and correlation (features of the power spectra for wavelet coefficients of ) methods of analysis for diagnostics of formation of local parts containing quasiordered () birefringent fibrils. With this aim, we performed stepbystep “screening” the pictures for wavelet coefficients (Fig. 3, e and f) using the scale change for the wavelet function . For each value of the wavelet function , we found the dependences and calculated the loglog dependences for the scale of their power spectra. As a result of this linear scanning, we obtained the array of data summarized in Table 2. Table 2. Loglog dependences of спектрів потужності вейвлеткоефіцієнтів залежностей кількості сингулярних значень матричного елементу для ДШ шкіри (???) людини «А» і «В» типів.
Coordinate distributions for the matrix element corresponding to a mount of the sample of healthy myometrium tissue (Table 2)(???) are characterized by statistical distributions for the wavelet coefficients in all the range of scale а changes of the wavelet function μ. It is confirmed by the absence of any stable slope of the approximating curve for the considered set of dependences . Another picture can be observed for the sample of changed myometrium tissue (Fig. 3). If the scale coefficient а of the wavelet function μ possesses some definite dimension, then there exist a linear part in the respective loglog dependence of the power spectrum characterizing the distribution of wavelet coefficients . In accord with performed computer modeling, this fact indicates the availability of a quasiharmonic component in distributions, which is caused by respective geometry of biological crystals. In our case, for the mean statistical size of a pathological creation a = 28 μm one can observe a quasiordered part of collagen fibrils with sizes lying within the range 10 to 50 μm. Thus, one can state that the correlation approach in the analysis of coefficients valid for the wavelet distribution in MMI of “orientation” element for connective tissue of the uterus neck is rather efficient for differentiation of its healthy and pathologically changed samples. Some additional information for differentiation of these objects was obtained using the statistical analysis of coordinate distributions . Shown in Fig. 4 are the dependences of statistical moments of the 1^{st} to 4^{th} orders for coordinate distributions of wavelet coefficients in MMI of, which corresponds to different scales of the wavelet function μ. Fig. 4. Dependences of statistical moments of the 1^{st} to 4^{th} orders on the scale of the wavelet function μ for myometrium tissue of A and B types. The obtained data show:  statistical moments of the 1^{st} and 4^{th} orders for distributions of wavelet coefficients , like to the case of computer modeling (Fig. ???) the element describing the samples of myometrium of both types, suffer insignificant changes (0 to 0.2) within the whole range of scales for the wavelet function μ (Fig. ???);  the range of changes in values of asymmetry (A) and excess (E) for distributions lie within the range of two orders (Fig. ???);  main differences between myometrium tissues of A and B types are found in the vicinity of the scale a = 28 μm where dependences between А(а) and Е(а) reach two or threefold level. Thus, we found that the differences between values of statistical moments of higher orders for a definite range of scales a of the wavelet function can be also used to differentiate local changes in orientation changes of optical axes inherent to territorial matrix crystals. References
