{"paper_id":"381f53bc-3032-453a-adaf-cfbd84ba88c9","body_text":"System of multifunctional Jones matrix tomography of phase \nanisotropy in diagnostics of endometriosis  \n \nV.O. Ushenkoa, G.D. Kovalb , Yu. O. Ushenkoa, L.Y. Pidkamina, M.I. Sidora, O. Vanchuliakb,   \nA.V.  Motricha, M.P.  Gorskya,  I.Meglinskiyc,  \n \na Chernivtsi National University, 2 Kotsyubinsky Str., Chernivtsi, 58012, Ukraine \nb Bukovinian State Medical University, Chernivtsi, 58000, Ukraine \nc University of Oulu, P.O. Box 4500, Oulu, Finland \nABSTRACT \nThe paper presents the results of Jones-matrix mapping of uterine wall histological sections with second-degree and \nthird-degree endometriosis. The technique of experimental measurement of coordinate distributions of the modulus and \nphase values of Jones matrix elements is suggested. Within the statistical and cross-correlation approaches the modulus \nand phase maps of Jones matrix images of optically thin biological layers of polycrystalline films of plasma and \ncerebrospinal fluid are analyzed. A set of objective paramete rs (statistical and generalized correlation moments), which \nare the most sensitive to changes in the phase of anisotropy, associated with the features of polycrystalline structure of \nuterine wall histological sections with second-degree and third-degree endometriosis are determined. \nKeywords: Jones matrix, tomography, phase anisotropy, diagnostics \n1. INTRODUCTION \nIn recent years in the field of biomedi cal optics a new basic approach in a well- tested diagnostic direction has been \nformed - Mueller-matrix polarimetry of biological tissues 1 - 13.  \nHowever, for non-depolarizing biological layers 5,6,14-28 more appropriate to apply the Jones-matrix formalism.  \nOur work consists in the study of the modulus and phase dist ributions of the Jones-matrix  elements characterizing the \noptically anisotropic structure of uterine wall histological sections with second-degree and third-degree endometriosis \nusing the statistical and correla tion approaches. The aim of the research is to  determine the objective criteria that ensure \nreliable differentiation of such objects and can be treated as the basis for the development of Jones-matrix diagnostics of \nnon-depolarizing biological layers. \n2. BRIEF THEORY \nIt is obtained an analytical expression for the resulting Jones matrix of a phase anisotropic transparent layer by modeling \nas a sequence of linearly and circularly birefringent layers in 18-28.  \n {}\nU\nUiUU\nU\nU\nU\nU\nUiU\njj\njjJ\n2\nsincossin\nsin\n2\nsincos\n2221\n1211\nδζ\nζδ\n+\n−−\n== . (1) \n \n2\n2222\n2 ⎟\n⎠\n⎞⎜\n⎝\n⎛+==\nδζθNU . (2) \nExpression (1) is an accurate analytical record of Jones-matrix elements of an optically anisotropic layer with linear ( δ ) \nand circular ( ζ ) birefringence. \nIn the approximation of weak phase fluctuations (1) can be rewritten as follows \nApplications of Digital Image Processing XL, edited by Andrew G. Tescher, Proc. of SPIE\nVol. 10396, 103962M · © 2017 SPIE · CCC code: 0277-786X/17/$18 · doi: 10.1117/12.2273764\nProc. of SPIE Vol. 10396  103962M-1\nDownloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 10/10/2017 Terms of Use: https://spiedigitallibrary.spie.org/ss/TermsOfUse.aspx\n\n0.8\n0.6\n0.4\n0.2\nS(x,Y)\n(1)\n200\n150\n100\n50\n0\n0\nN(S)\nI\n50010001500\n \n \n ( )[] [ ]\n[] ()[] 22\n2\n21\n1211\n2\n2221\n1211\n12\n21\nδζζδζ\nζδζδζ\nii\nii\njj\njj\n+++\n+−+= . (3) \nFrom Jones matrix (3) we obtain the analytical expressions for linear (δ )  and circular ( ζ )birefringence \n ( )21;122 jtgArg=δ . (4) \n ( )( )\n()() 22;11\n21;12\n5,01 jArgtg\njArgtg−=ζ . (5) \nMeasurements of the elements of Jone s matrix distributions were carried ou t according to the classical technique \npresented in 16.  \n3. ANALYSIS AND DISCUSSION OF EXPERIMENTAL DATA  \nTwo groups of histological sections of the endometriosis uterine wall second-degree (sample 31 - group 1) and third-\ndegree (sample 31) were studied. \nThe series of Fig. 1 - Fig. 4 represent maps (fragments 1), histograms (fragments (2)), autocorrelation functions \n(fragments (3)) and logarithmic dependences of power spectra (fragments (4)) extreme values distributions of Linear \n(\nδ )and circular ( ζ )birefringence of the group 1 (Figure 1, Figure 3) endometrium histological sections and group 2 \n(Fig. 2, Fig. 4). \n \nFigure 1. Statistical, correlation and fractal parameters of the group 1 of endometrium sample linear birefringence \ndistributions. \nProc. of SPIE Vol. 10396  103962M-2\nDownloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 10/10/2017 Terms of Use: https://spiedigitallibrary.spie.org/ss/TermsOfUse.aspx\n\n0.8\n0.6\n0.4\n0.2\nalX,Yl\n(1)\nK(S)\n5001000\n(2)\n1500\n0.8\n0.6\n0.4\n0.2\nSIx5Y/\n(1)\n500100015002000\n600\nNG)\n500-\n400-\n300\n200\n0 50010001500\n(2)\n \n \n \nFigure 2. Statistical, correlation and fractal parameters of the group 2 of endometrium sample linear birefringence \ndistributions. \n \nFigure 3. Statistical, correlation and fractal parameters of the group 1 of  endometrium sample circular birefringence \ndistributions. \nProc. of SPIE Vol. 10396  103962M-3\nDownloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 10/10/2017 Terms of Use: https://spiedigitallibrary.spie.org/ss/TermsOfUse.aspx\n\n0.8\n0.6\n0.4\n0.2\n4(X,Y)\n(1)\n700\n650\n600\n550\n500\n450-\n4000\nN (4)\nI\nN\n5001000\n(2)\n1500\n \n \n \n  \nFigure 4. Statistical, correlation and fractal parameters of the group 2 of  endometrium sample circular birefringence \ndistributions. \nIn order to identify the sensitivity of the Jones matrix met hod of planar polycrystalline n on-depolarizing layers mapping \nto changes of linear and circular birefringence the results obtained (Fig. 1 - Fig. 4) were compared by determining the set \nof statistical (statistical moments of the 1st - 4th orders 14), correlation (correlation moments 2nd and 4th orders 15) and \nfractal 16 parameters that characterize the distribution δ  и ζ – Table 1. \nTable 1. Statistical, correlation and fractal moments of linear and circular birefringence distributions  \niZ  Group 1 Group 2 \nδ  ζ δ  ζ\n1Z  0,055 0,06 0,029 0,052 \n2Z  0,029 0,08 0,017 0,06 \n3Z  0,22 1,52 0,02 1,23 \n4Z  1,35 3,17 0,77 1,94 \n2kZ  0.11 0.008 0.14 0,12 \n4kZ  1.88 2.44 1.02 3.89 \nfD  0.19 0.25 0.23 0,33 \n \nThe comparative analysis of data presented in Table 1 showed diagnostic sensitivity of a number of objective \nparameters: \n• ()\n() ;86,1\n;05,2\n4\n4\n=Δ\n=Δ\nζ\nδ\nZ\nZ  \nProc. of SPIE Vol. 10396  103962M-4\nDownloaded From: https://www.spiedigitallibrary.org/conference-proceedings-of-spie on 10/10/2017 Terms of Use: https://spiedigitallibrary.spie.org/ss/TermsOfUse.aspx\n\n• ()\n() ; 66 , 1\n; 79 , 1\n4\n4\n= ΔΖ\n= ΔΖ\nζ\nδ\nk\nk\n• \n()\n() . 63 , 1\n; 55 , 1\n= Δ\n= Δ\nζ\nδ\nf\nf\nD\nD\nCONCLUSION \nThe Jones matrix model of differentiating the weak changes of phase anisotropy of the endometrium histological sections \nwith different pathologies is suggested. \nThe coordinate distributions of the endometrium tissues linear and circular birefringence are experimentally determined. \nWithin the statistical, correlation and fractal analysis of coordinate distributions of linear and circular birefringence the \nobjective parameters are determined  that are most sensitive (statistical moments of the 3rd and 4th orders, generalized \ncorrelation moment of the 4th order) to changes of the phase anisotropy of optically thin endometrium biological tissue \nof different endometriosis pathology. \nREFERENCES \n[1] Müller G. et al., Eds., [Medical Op tical Tomography: Functional Imaging and Monitoring] Vol. IS11, SPIE\nPress, Bellingham, Washington (1993).\n[2] Wang L. 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