Phase Mapping of Human Biological Tissues: Data Processing Algorithms for Forensic Time of Death Estimation 9819932688, 9789819932689

Citation preview

SpringerBriefs in Applied Sciences and Technology Zhengbin Hu · V. T. Bachinsky · O. Y. Vanchulyak · Iryna V. Soltys · Yu. A. Ushenko · A. G. Ushenko · Igor Meglinski

Phase Mapping of Human Biological Tissues Data Processing Algorithms for Forensic Time of Death Estimation

SpringerBriefs in Applied Sciences and Technology

SpringerBriefs present concise summaries of cutting-edge research and practical applications across a wide spectrum of fields. Featuring compact volumes of 50 to 125 pages, the series covers a range of content from professional to academic. Typical publications can be: • A timely report of state-of-the art methods • An introduction to or a manual for the application of mathematical or computer techniques • A bridge between new research results, as published in journal articles • A snapshot of a hot or emerging topic • An in-depth case study • A presentation of core concepts that students must understand in order to make independent contributions SpringerBriefs are characterized by fast, global electronic dissemination, standard publishing contracts, standardized manuscript preparation and formatting guidelines, and expedited production schedules. On the one hand, SpringerBriefs in Applied Sciences and Technology are devoted to the publication of fundamentals and applications within the different classical engineering disciplines as well as in interdisciplinary fields that recently emerged between these areas. On the other hand, as the boundary separating fundamental research and applied technology is more and more dissolving, this series is particularly open to trans-disciplinary topics between fundamental science and engineering. Indexed by EI-Compendex, SCOPUS and Springerlink.

Zhengbin Hu · V. T. Bachinsky · O. Y. Vanchulyak · Iryna V. Soltys · Yu. A. Ushenko · A. G. Ushenko · Igor Meglinski

Phase Mapping of Human Biological Tissues Data Processing Algorithms for Forensic Time of Death Estimation

Zhengbin Hu School of Computer Science Hubei University of Technology Wuhan, China O. Y. Vanchulyak Bukovinian State Medical University Chernivtsi, Ukraine Yu. A. Ushenko Chernivtsi National University Chernivtsi, Ukraine

V. T. Bachinsky Bukovinian State Medical University Chernivtsi, Ukraine Iryna V. Soltys Chernivtsi National University Chernivtsi, Ukraine A. G. Ushenko Chernivtsi National University Chernivtsi, Ukraine

Igor Meglinski School of Engineering and Applied Science Aston University Birmingham, UK

ISSN 2191-530X ISSN 2191-5318 (electronic) SpringerBriefs in Applied Sciences and Technology ISBN 978-981-99-3268-9 ISBN 978-981-99-3269-6 (eBook) https://doi.org/10.1007/978-981-99-3269-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Acknowledgement

This work received funding from: National Research Foundation of Ukraine, Grant 2020.02/0061 and Scholarship of the Supreme Council for Young Scientists— Doctors of Sciences.

v

Introduction

Currently, forensic practice requires a significant update of the methods for TDE, since the existing methods depend on many environmental factors and the circumstances of death. The existing lack of modern, objective methods for determining the TDE prompts the search and development of new methods for studying the main BT of a person after death [1–19]. Non-invasive optical methods for diagnosing BT structure using a complex of photometric, polarization, spectral and correlation techniques are promising in this direction. The indicated methods of studying the phenomenon of light scattering by BT by their macro inhomogeneities make it possible to search for interrelationships between the dynamics data after mortal changes in the studied body tissues of certain organs with a set of objective photometric, polarization, spectral and correlation parameters of their optical images. On this basis, the possibilities open up for an objective and more accurate determination of the time interval that has passed since the death of a person. This approach is implemented in the spectral photopolarimetric method of laser optics, based on the use of polarization of laser beams of various wavelengths and modeling by matrix operators of the properties of tissues or hematomas of human organs of any type [20–34]. From a forensic medical point of view, it is advisable to carry out a complex spectral photopolarimetric laser study of images of the main types of BT of a human corpse in time dynamics, which is promising for the determination of TDE. So, the relevance of this monograph is due to the need to develop new approaches to the determination of TDE, the search and development of a complex of new methods for objective spectral phase laser diagnostics and monitoring of changes in the parameters of the main types of human BT for the development of objective criteria for the forensic medical definition of TDE.

vii

viii

Introduction

Purpose of the Study The aim was to develop a set of new forensic methods and objective criteria for establishing the antiquity of the onset of death and the time of hematoma formation by means of a phase study of the temporal dynamics after death changes in laser images of histological sections of BT and hematomas of human organs. To achieve this goal, the following tasks should be addressed: 1. Development of the basic principles of phase measurement of microscopic images of the main types of BT of human corpse organs to determine the time of death. 2. Investigation of the possibilities of complex laser spectral-selective phase measurement of images of histological sections of the main types of biological tissues of a human corpse in various spectral regions for establishing the TDE. 3. Development and justification of a set of statistical criteria for the objective determination of TDE. Object of study: changes in the phase properties of laser spectral images of the main types of human BT in the post-mortem period. Subject of study: spectral phase measurement of images of the main types of human BT after death; temporal dynamics after mortal changes in BT of human organs and changes in the phase and statistical parameters of their spectral laser images. Research methods: microscopy (image of histological sections of BT in polarized laser light at different wavelengths); polarimetry (measurement of coordinate distributions of azimuths and polarization ellipticity, parameters of the Stokes vector of BT images and elements of their Mueller matrix); phase measurement (measurement of coordinate distributions of phase shifts of spectral laser images of BT of human organs); statistical processing of research results.

Contents

1 Novel Diagnosis Capabilities and Prospects for Determining Post-mortem Changes in Biological Tissues and the Time of Hematoma Formation in Forensic Medicine . . . . . . . . . . . . . . . . . . . . . 1.1 Novel Possibilities of Principles of Forensic Medical for the Determination of Death Time Detection . . . . . . . . . . . . . . . . . . 1.2 Polarization Mapping of Biological Tissues . . . . . . . . . . . . . . . . . . . . . 1.3 Methods and Means of Phase Measurement of Biological Tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Coordinate Distributions of Phase Shift Values Between Orthogonal Components of the Amplitude of the Laser Radiation Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Method for Measuring Coordinate Distributions of Phase Shift Values Between Orthogonal Components of the Amplitude of the Laser Radiation Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Experimental Scheme of Spectropolarimeter and Phase Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 A Set of Statistical, Correlation and Fractal Criteria for Evaluation of the Phase Structure of Laser Images of Biological Fluids and Tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 A Set of Statistical and Correlation Criteria for Evaluation of the Polarization Properties of Biological Tissues of a Human Corpse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Principles of Phase Measurement of Sections of Biological Tissues of a Human Corpse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 6 7

11

11 13

14

15 17 24

ix

x

Contents

3 Computer Modeling of the Evolution of Statistical Parameters of the Phase Distributions of the Laser Radiation Field Converted by Optically Anisotropic Layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Characteristics of Research Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Image Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Evolution of Phase Distributions of Laser Radiation in Free Propagation Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Diffraction Transformation of Phase Maps of Laser Radiation Converted by a Layer of Ordered Cylindrical Crystals . . . . . . . . . . . . 3.4 Statistical, Correlation and Fractal Parameters Characterizing the Phase Distributions of Laser Radiation Transformed by a Layer of Ordered Cylindrical Crystals . . . . . . . . . . . . . . . . . . . . . . 3.5 Diffraction Transformation of the Statistical, Correlation and Fractal Structure of Phase Maps of Laser Radiation Converted by a Layer of Spherical Crystals . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Spectral Phase Measurement of Laser Images of Sections of Biological Tissues of a Human Corpse for Death Time Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Spectral Phase Measurement of Laser Images of Histological Sections of Structured Tissues of a Human Corpse . . . . . . . . . . . . . . . 4.2 Possibilities of Determination of TDE by the Method of Statistical Analysis of Time Dependences of Power Spectra of Phase Maps of Histological Sections of Human Corpse Tissues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27 27 27 30 32

40

44 49

53 53

68 69

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Abbreviations

BT CCD LP LSPP MT OAL SD TBI TD TDE THF

Biological tissues, biotissue Digital camera Laser polarimetry Laser spectral photopolarimetry Muscle tissue Optically anisotropic layers Skin dermis Traumatic brain injury Time of death Time of death estimation Time of hematoma formation

xi

Chapter 1

Novel Diagnosis Capabilities and Prospects for Determining Post-mortem Changes in Biological Tissues and the Time of Hematoma Formation in Forensic Medicine

1.1 Novel Possibilities of Principles of Forensic Medical for the Determination of Death Time Detection There are various physical, instrumental, biochemical approaches to the diagnosis of TDE by [1–19]: 1. Study of the dynamics of changes in the temperature of individual parts of the corpse and rectal temperature, including taking into account changes in environmental factors, internal factors and various causes of death in order to establish TDE. 2. Determining the time depending on the nature, degree of development, color, which is needed to restore the properties of cadaveric spots after pressing them with a finger. 3. Instrumental examination of cadaveric spots: studies of cadaveric spots with a dynamometer, microdynamometer. 4. Research on rigor mortis. 5. Development of a tonometer—a device for measuring the hardness of skeletal muscles, depending on the degree of development of rigor mortis, special spring scales. 6. Study of the properties of sartorius by their dosed stretching. 7. Determination of the degree of post-death corneal opacity. 8. Study of the entomofauna of the corpse. 9. Comparison of morphological changes in white blood cells after death and stored in vitro blood. 10. Study of the after-mortal distribution of leukocytes in combat injuries or postmortal blood flow. 11. Study of the distribution of eosinophils in the spleen and after mortality indicators of the bone marrow in sudden death and death with a long agonal period. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Z. Hu et al., Phase Mapping of Human Biological Tissues, SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-981-99-3269-6_1

1

2

1 Novel Diagnosis Capabilities and Prospects for Determining …

12. Study of some after mortal increase in the number of leukocytes and the dynamics of their destruction using fluorescence microscopy. 13. Immunological method for determining changes: the value of the immune indicators of the reaction of blast transformation of lymphocytes and T- and B-lymphocytes in the after-mortal period. 14. Use of phagocytic activity of leukocytes and their indicators. 15. Study of the peculiarities of the chemical composition of blood fibrinolysis and fibrinogen, taking into account the degree of blood coagulation; the content of adenosine triphosphoric acid and inorganic phosphorus in cadaveric blood; blood urea, as well as the ratio of free glycogen to bound, depending on the site, blood sampling. 16. Study of electrolytes of potassium and sodium, residual nitrogen in the blood of corpses, the cause of death of which was injuries of the chest and abdomen. 17. Determination of residual nitrogen and proteins in fatal traumatic brain injury. 18. Study of the level of enzymatic activity, macro- and microelements, the activity of potassium ions in the blood with a valinomycin electrode, protein fractions of blood plasma. 19. Study of pericardial fluid, cerebrospinal fluid, skeletal muscles and internal organs of corpses of persons who died due to various reasons. 20. Obtaining data on the after mortality increase in the total protein content, the amount of albumin and globulins, the dynamics of changes in amino acids, uric acid, urea, hypoxanthine and xanthine, after the death dynamics of changes in creatine and creatinine, as well as enzymatic activity at different times after the onset of death. 21. Study of enzyme activity in isolated lymphocytes and granulocytes. 22. Studying after death dynamics of changes in the content of nucleic acids in BT in humans and experimental animals. 23. Study of the activity of malate, lactate, sorbitol, hydroxybutyrate and succinate dehydrogenase, aldolase and other blood enzymes. 24. Use of the vitreous body. 25. Studying the dynamics of changes in the parameters of the acid–base state of the blood, chamber fluid and vitreous body of the eye in corpses of persons whose cause of death was mechanical injury, strangulation asphyxia, cardiovascular insufficiency. 26. Study of the concentration of potassium, sodium, glucose in the eye fluid, vitreous and the content of total protein in the vitreous fluid. 27. Study of the dynamics and level of potassium and sodium pericardial fluid and blood serum of the right half of the heart in the corpses of children and adults who suddenly died. 28. Studying the dynamics of changes in a number of biochemical and chemical indicators in human muscles during the development of rigor mortis. 29. Study of the influence of biochemical factors in the early after death interval on the elasticity and water resistance of three portioned muscles. 30. Determination of creatinine level in human psoas muscles.

1.2 Polarization Mapping of Biological Tissues

3

31. The use of emission spectral analysis to determine changes in the content of macro- and microelement composition and changes in the histological picture of the skin and adjacent soft tissues from the area of cadaveric spots at different times after death. 32. Study of sperm life expectancy during the first 3 days after death. 33. Specific changes in the activity of acid phosphatase, cathepsin, aldolase, lactate dehydrogenase in the testicular tissues of persons who died due to traumatic brain injury were determined. 34. Study of the rate of evacuation of the contents of the gastrointestinal tract, in particular, the storage of undigested substances in it (bones, grain, seeds). 35. Study of agonal and after mortal changes in the nervous system of human and experimental animals. 36. Determination of the hourly reaction of glia and the appearance of macrophages in the corpses of people who have had hemorrhages in the brain substance as a result of craniocerebral trauma. 37. Determination of the concentration of metabolic products of bacteria in the brain tissue, including depending on environmental conditions and conditions of the agonal period. 38. Determination of peptide hydrolase activity and content of free and proteincontaining amino acids in the brain substance of white rats in the period from 0 to 24 h. 39. Study of DNA decay in different organs and tissues of the human body. 40. Measurement of the complex relative dielectric constant and excitability of internal organs and tissues. 41. Use of angiography of the heart and kidneys. 42. Application of magnetic radiospectroscopy, spectrometry and spectrophotometry changes in BT and blood of corpses of children and adults. 43. Application of methods of spectrophotometry and circular dichroism. 44. Study of morphofunctional parameters of cadaveric blood neutrophilic leukocytes using methods of logical and mathematical analysis. 45. Introduction of computer technologies at the level of information and measurement systems. 46. Using physical modeling in real and accelerated timescale.

1.2 Polarization Mapping of Biological Tissues The widespread introduction of modern laser technology in the study of the structure of phase-inhomogeneous objects has stimulated the development of fundamentally new methods in the optics of light scattering. As a result, a group of polarization methods was formed, based on the operation of the “coherence matrix” and the “degree of polarization” [20–27]. In other words, such optical technologies refer to “one-point” methods of obtaining information at each point (r ) of the coherent radiation field about the

4

1 Novel Diagnosis Capabilities and Prospects for Determining …

magnitudes of the orthogonal amplitude components E x (r ); E y (r ), the phase shift δx y (r ) between them, as well as the azimuth α(r ) and ellipticity β(r ) of polarization. The intensive development of optical methods for studying the morphological structure and physiological state of various biological tissues (BT) [21] contributed to the introduction and development of laser polarimetry methods [22] in this area. Methodologically, the set of laser polarimetric techniques is based, in particular, on model concepts of the BT structure, theoretically stated and experimentally tested in a series of scientific works [22–31]. Thus, in [32–36], the geometric architectonics of the BT structural elements was analyzed, and a fractal approach to their analysis was proposed. It is based on the principle of a hierarchical self-similar structure of fibrils of the extracellular matrix of the main types of human tissues—connective and muscle tissue, as well as epithelial and nervous. It was analyzed [33, 34] that the fibrillar elements of the extracellular matrix are discrete in structure and are characterized by large-scale repeatability in a wide range of “optical sizes” (d = 1µm − 103 µm). The main conclusion from the point of view of optical interpretation of the structure of BT of different types is the position that their properties of their extracellular matrices correspond to “frozen” liquid crystals [37]. This approach to the description of the morphological structure of BT was systematized and generalized in further works [38–46]. Here, for the first time, a model of the BT structure was proposed as a twocomponent amorphous-crystalline matrix. The amorphous component of BT is polarization-inactive or isotropic. The crystalline component of BT or the extracellular matrix is formed by filamentous protein (collagen proteins, myosin) fibrils. The properties of each individual fibril are modeled optically by a uniaxial birefringent crystal. The direction of the optical axis of such a biological crystal is determined by the direction of laying the fibril in the BT layer. The birefringence index is determined by the substance of the protein fibrils. The specified network of biological liquid crystals forms the extracellular matrix. This model was used as the basis for the development of methods for polarization mapping of histological sections of BT of various morphological structures and physiological states. Experimentally, the methods for determining the coordinate distributions of polarization states in the plane of laser images of BT layers are based on the CCD cameras used as photodetectors. This technology makes it possible to experimentally realize the operation of discretizing the coordinate distribution of the parameters

1.2 Polarization Mapping of Biological Tissues

5

⎫⎞ ⎛⎧ ⎪ ⎪ E x2 (r ); E y2 (r ); ⎬ ⎨ 2

⎟ ⎜ 2 ⎝ α(r ) = 0, 5π +  E x (r ) + E y (r ) = min ⎠ of the laser image over the set ⎪ ⎪ ⎭ ⎩ β(r ) = ar ctg E x2 (r )/E y2 (r ) ⎞ ⎛ r11 ; ; r1n ⎟ ⎜ ⎟) of the light-sensitive area. of pixels (r ↔ m × n = ⎜ ⎠ ⎝ rm1 ; ; rmn Such discretization made it possible to use a powerful mathematical apparatus for the analysis of coordinate distributions of polarization parameter values—in particular, statistical (statistical moments of the first–fourth orders), correlation (correlation functions, power spectrum), fractal (fractal dimensions, statistical moments of distributions of extrema logarithmic dependencies of power spectra), wavelet (two-dimensional distributions of wavelet coefficients) analysis. Based on the combination of this model, the digital technique of the polarization experiment and a comprehensive analysis of the data obtained, it was possible to explain the mechanisms and substantiate the scenarios of the formation and evolution of the polarization inhomogeneity of laser fields transformed by bone, connective and muscle tissues [41]. Also, for optically thin histological sections of BT, where the conditions of single scattering are implemented, algorithms for the relationship between azimuth α(r ), ellipticity β(r ) of the polarization at a point with the coordinate of the laser image and the direction of the optical axis ρ(r ) and birefringence δ(r ) of a biological protein crystal were determined [44]. On this basis, a method of polarization visualization of the extracellular matrix of various types of BT was developed, and a statistical analysis of laser images was implemented by calculating the average, dispersion, skewness and kurtosis of coordinate intensity distributions. This polarization approach was improved in [39, 43] for the case of multilayer tissues. Here, based on model concepts of the liquid–crystal structure of the extracellular matrix of connective tissue, a method of its polarization mapping was developed taking into account the influence of the surface rough layer. The main result of this method is an analytically substantiated and experimentally verified relationship between a set of statistical moments of first–fourth orders of magnitude characterizing the distributions of slopes of microroughness of the skin surface (surface layer) and directions of optical axes and the magnitude of birefringence of the collagen network of the dermal layer (volume layer) and a set of statistical moments, characterizing the distributions of polarization states (α(r ) and β(r )) of the corresponding microscopic image. The elevated values of skewness and kurtosis observed in the distributions of azimuths and ellipticity of polarization maps in laser images of thin histological sections of human tissue indicate physical mechanisms linked to the disruption of optical axis orientations within the biological crystal network [45]. If for one reason or another of a physiological or pathological type, there is a change in the optical anisotropy (an increase in the dispersion of phase shifts) of biological crystals of the extracellular matrix, then such a process is detected in

6

1 Novel Diagnosis Capabilities and Prospects for Determining …

the form of a transformation of statistical distributions of polarization maps—the statistical moments of the third and fourth orders decrease [34]. These results were used in [23, 24, 30–37] to diagnose the pathological state of BT. In particular, the relationship between the dispersion of the distributions of azimuth values and the ellipticity of polarization of laser images of histological sections of physiologically normal and pathologically altered BT was studied. On this basis, physical criteria for changes in the orientational-phase structure of networks of liquid biological crystals were established and used for statistical polarization differentiation of physiological and pathological states in structured BT.

1.3 Methods and Means of Phase Measurement of Biological Tissues The processes of interaction of electromagnetic radiation of the optical wavelength range with phase-inhomogeneous objects and media are considered, as a rule, within many approximations. Among the most common, traditional ones, such independent directions can be distinguished—photometry and spectrophotometry (light scattering optics) [20– 25]; polarimetry (vector—parametric and matrix optics) [26–38], correlometry (correlation optics, speckle optics) [39–46]. In recent years, on the basis of these directions, new approaches to the study of scattered radiation fields have been formed—fractal optics [32], singular optics [34], which formed the basis for the formation of laser polarimetry of optically anisotropic phase inhomogeneous layers of biological origin [21]. Laser polarimetry methods made it possible to determine the fundamental relationships between a set of statistical (moments of the first−fourth orders), correlation (autocorrelation functions), fractal (fractal dimensions) and singular (networks of points with polarization singular states) parameters characterizing optical anisotropy (distributions directions of optical axes and birefringence values) of optically-thin phase-inhomogeneous layers and polarization parameters (coordinate distributions of azimuths and polarization ellipticity, elements of the Mueller matrix) of their laser images [33–56]. At the same time, under conditions of multiple scattering, polarization information about the optically anisotropic structure of an object is averaged and loses its unambiguous diagnostic meaning. On the contrary, in the processes of conversion of laser radiation by layers with weak phase fluctuations, polarization modulation is practically absent and is considered in the scalar field approximation [57–61]. However, the phase distributions of the fields of scattered coherent radiation remain informative in both cases. Therefore, the further development of new approaches to the analysis of not only the vector but also the phase structure of the fields of laser radiation converted by optically anisotropic layers of various optical thicknesses is urgent.

References

7

Thus, the literature review shows that such a research method as BT laser polarimetry allows one to obtain a number of indicators of the optical properties of tissues, which, according to modern concepts, are a two-component amorphousanisotropic matrix. Elucidation of the possibilities of laser polarimetry for solving problems of forensic medicine formed the basis of our research.

References 1. G. Mall, M. Eckl, I. Sinicina, et al., Temperature-based death time estimation with only partially known environmental conditions. Int. J. Legal. Med. (2004) 2. N. Lange, S. Swearer, W.Q. Sturner, Human postmortem interval estimation from vitreous potassium: an analysis of original data from six different studies. Forensic Sc. Int. 3(66), 159–174 (1994) 3. L.M. Al-Alousi, R.A. Anderson, D.M. Worster et al., Multiple-probe thermography for estimating the postmortem interval: I. Continuous monitoring and data analysis of brain, liver, rectal and environmental temperatures in 117 forensic cases. J. Forensic Sci. 2(46), 317–322 (2001) 4. G.M. Hutchins, Body temperature is elevated in the early postmortem period. Hum. Pathol. 6(16), 560–561 (1985) 5. T. Suzutani, Studies on the estimation of the postmortem interval. 1. The temperature of cadaver (author’s transl). Hokkaido Igaku Zasshi. 3(52), 205–211 (1977) 6. J.L. Melody, S.M. Lonergan, L.J. Rowe et al., Early postmortem biochemical factors influence tenderness and water-holding capacity of three porcine muscles. J. Anim. Sci. 4(82), 1195– 11205 (2004) 7. M.A. Green, J.C. Wright, Postmortem interval estimation from body temperature data only. Forensic Sci. Int. 1(28), 35–46 (1985) 8. G. Mall, M. Hubig, M. Eckl et al., Modelling postmortem surface cooling in continuously changing environmental temperature. Leg. Med. 3(4), 164–173 (2002) 9. L.M. Al-Alousi, R. A. Anderson, D. M. Worster et al., Factors influencing the precision of estimating the postmortem interval using the triple-exponential formulae (TEF). Part II. A study of the effect of body temperature at the moment of death on the postmortem brain, liver and rectal cooling in 117 forensic cases. Forensic Sci. Int. 2–3(125), 223–230 (2002) 10. G. Mall, M. Hubig, G. Beier et al., Determination of time-dependent skin temperature decrease rates in the case of abrupt changes of environmental temperature. Forensic Sci. Int. 1–3(113), 219–226 (2000) 11. S. Sasaki, S. Tsunenari, M. Kanda, The estimation of the time of death by non-protein nitrogen (NPN) in cadaveric materials. Report 3: multiple regression analysis of NPN values in human cadaveric materials. Forensic Sci. Int. 1(22), 11–22 (1983) 12. J. Wiesbock, E. Josephi, E. Liebhardt, Intra-individual changes in potassium in the cerebrospinal fluid after death. Beitr. Gerichtl. Med. 47, 403–405 (1989) 13. J.I. Munoz, J.M. Suarez-Penaranda, X.L. Otero et al., A new perspective in the estimation of postmortem interval (PMI) based on vitreous. J. Forensic Sci. 2(46), 209–214 (2001) 14. A.J. Sabucedo, K.G. Furton, Estimation of postmortem interval using the protein marker cardiac Troponin I. Forensic Sci. Int. 1(134), 11–16 (2003) 15. N. Lynnerup, A computer program for the estimation of time of death. J. Forensic Sci. 4(38), 816–820 (1993) 16. H. Joachim, U. Feldmann, Quantimetric investigations of the time of death by estimating the postmortem threshold (rheobase) of human skeletal muscles to electric stimulus by direct current (author’s transl). Z. Rechtsmed. 1(85), 5–22 (1980)

8

1 Novel Diagnosis Capabilities and Prospects for Determining …

17. M. Shimizu, T. Hayashi, Y. Saitoh et al., Postmortem autolysis in the pancreas: multivariate statistical study. The influence of clinicopathological conditions. Pancreas 1(5), 91–94 (1990) 18. F. Kuroda, K. Hiraiwa, S. Oshida et al., Estimation of postmortem interval from rectal temperature by use of computer (III)-thermal conductivity of the skin. Med. Sci. Law. 4(22), 285–289 (1982) 19. F. Brion, B. Marc, F. Launay, Postmortem interval estimation by creatinine levels in human psoas Muscle. Forensic Sci. Int. 1(52) 113–120 (1991) 20. V. Tuchin, L. Wang, D. Zimnjakov, Optical polarization in biomedical applications (USA. Springer, New York, 2006) 21. Chipman R.: Polarimetry, in Handbook of Optics: Vol I—Geometrical and Physical Optics, Polarized Light, Components and Instruments, ed. by M. Bass (McGraw-Hill Professional, New York, 2010), pp. 22.1–22.37 22. N. Ghosh, M. Wood, A. Vitkin, Polarized light assessment of complex turbid media such as biological tissues via Mueller matrix decomposition, in Handbook of Photonics for Biomedical Science, ed. by V. Tuchin (CRC Press, Taylor & Francis Group, London, 2010), pp. 253–282 23. S. Jacques, Polarized light imaging of biological tissues, in Handbook of Biomedical Optics. ed. by D. Boas, C. Pitris, N. Ramanujam (CRC Press, Boca Raton, London, New York, 2011), pp.649–669 24. N. Ghosh, Tissue polarimetry: concepts, challenges, applications, and outlook. J. Biomed. Opt. 16(11), 110801 (2011) 25. M. Swami, H. Patel, P. Gupta, Conversion of 3×3 Mueller matrix to 4×4 Mueller matrix for non-depolarizing samples. Opt. Commun. 286, 18–22 (2013) 26. D. Layden, N. Ghosh, A. Vitkin, Quantitative polarimetry for tissue characterization and diagnosis, in Advanced Biophotonics: Tissue Optical Sectioning edited by R. Wang, V. Tuchin (CRC Press, Taylor and Francis Group, Boca Raton, London, New York, 2013), pp. 73–108 27. T. Vo-Dinh, Biomedical Photonics Handbook, vol. 3, 2nd edn. (CRC Press, Boca Raton, 2014) 28. A. Vitkin, N. Ghosh, A. Martino, Tissue polarimetry, in Photonics: Scientific Foundations, Technology and Applications, 4th edn., ed. by D. Andrews (Wiley, Hoboken, New Jersey, 2015), pp.239–321 29. V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis, 2nd edn. (SPIE Press, Bellingham, Washington, USA, 2007) 30. W. Bickel, W. Bailey, Stokes vectors, Mueller matrices, and polarized scattered light. Am. J. Phys. 53(5), 468–478 (1985) 31. V. Ushenko, O. Vanchuliak, M. Sakhnovskiy, O. Dubolazov, P. Grygoryshyn, I. Soltys, O. Olar, System of Mueller matrix polarization correlometry of biological polycrystalline layers. Proc. SPIE 10352, 103520U (2017) 32. V. Ushenko, O. Vanchuliak, M. Sakhnovskiy, O. Dubolazov, P. Grygoryshyn, I. Soltys, O. Olar, A. Antoniv, Polarization-interference mapping of biological fluids polycrystalline films in differentiation of weak changes of optical anisotropy. Proc. SPIE 10396, 103962O (2017) 33. O. Dubolazov, L. Trifonyuk, Y. Marchuk, Y. Ushenko, V. Zhytaryuk, O. Prydiy, L. Kushnerik, I. Meglinskiy, Two-point Stokes vector parameters of object field for diagnosis and differentiation of optically anisotropic biological tissues. Proc. SPIE 10352, 103520V (2017) 34. L. Trifonyuk, O. Dubolazov, Y. Ushenko, V. Zhytaryuk, O. Prydiy, M. Grytsyuk, L. Kushnerik, I. Meglinskiy, I. Savka, New opportunities of differential diagnosis of biological tissues polycrystalline structure using methods of Stokes correlometry mapping of polarization inhomogeneous images. Proc. SPIE 10396, 103962R (2017) 35. O. Dubolazov, V. Ushenko, L. Trifoniuk, Y. Ushenko, V. Zhytaryuk, O. Prydiy, M. Grytsyuk, L. Kushnerik, I. Meglinskiy, Methods and means of 3D diffuse Mueller-matrix tomography of depolarizing optically anisotropic biological layers. Proc. SPIE 10396, 103962P (2017) 36. A. Ushenko, A. Dubolazov, V. Ushenko, O. Novakovskaya, Statistical analysis of polarizationinhomogeneous Fourier spectra of laser radiation scattered by human skin in the tasks of differentiation of benign and malignant formations. J. Biomed. Opt. 21(7), 071110 (2016) 37. Y. Ushenko, G. Koval, A. Ushenko, O. Dubolazov, V. Ushenko, O. Novakovskaia, Muellermatrix of laser-induced autofluorescence of polycrystalline films of dried peritoneal fluid in diagnostics of endometriosis. J. Biomed. Opt. 21(7), 071116 (2016)

References

9

38. V. Prysyazhnyuk, Yu. Ushenko, A. Dubolazov, A. Ushenko, V. Ushenko, Polarizationdependent laser autofluorescence of the polycrystalline networks of blood plasma films in the task of liver pathology differentiation. Appl. Opt. 55(12), B126–B132 (2016) 39. A. Ushenko, O. Dubolazov, V. Ushenko, O. Novakovskaya, O. Olar, Fourier polarimetry of human skin in the tasks of differentiation of benign and malignant formations. Appl. Opt. 55(12), B56–B60 (2016) 40. Yu. Ushenko, V. Bachynsky, O. Vanchulyak, A. Dubolazov, M. Garazdyuk, V. Ushenko, Jonesmatrix mapping of complex degree of mutual anisotropy of birefringent protein networks during the differentiation of myocardium necrotic changes. Appl. Opt. 55(12), B113–B119 (2016) 41. A. Dubolazov, N. Pashkovskaya, Yu. Ushenko, Yu. Marchuk, V. Ushenko, O. Novakovskaya, Birefringence images of polycrystalline films of human urine in early diagnostics of kidney pathology. Appl. Opt. 55(12), B85–B90 (2016) 42. M. Garazdyuk, V. Bachinskyi, O. Vanchulyak, A. Ushenko, O. Dubolazov, M. Gorsky, Polarization-phase images of liquor polycrystalline films in determining time of death. Appl. Opt. 55(12), B67–B71 (2016) 43. A. Ushenko, A. Dubolazov, V. Ushenko, Yu. Ushenko, M. Sakhnovskiy, O. Olar, Methods and means of laser polarimetry microscopy of optically anisotropic biological layers. Proc. SPIE 9971, 99712B (2016) 44. A. Ushenko, A. Dubolazov, V. Ushenko, Yu. Ushenko, L. Kushnerick, O. Olar, N. Pashkovskaya, Yu. Marchuk (2016) Mueller-matrix differentiation of fibrillar networks of biological tissues with different phase and amplitude anisotropy. Proc. SPIE 9971, 99712K 45. O. Dubolazov, A. Ushenko, Y. Ushenko, M. Sakhnovskiy, P. Grygoryshyn, N. Pavlyukovich, O. Pavlyukovich, V. Bachynskiy, S. Pavlov, V. Mishalov, Z. Omiotek, O. Mamyrbaev, Laser müller matrix diagnostics of changes in the optical anisotropy of biological tissues Information Technology in Medical Diagnostics II, in Proceedings of the International Scientific Internet Conference on Computer Graphics and Image Processing and 48th International Scientific and Practical Conference on Application of Lasers in Medicine and Biology, vol. 2018, (2019) pp. 195–203 46. M. Borovkova, M. Peyvasteh, O. Dubolazov, Y. Ushenko, V. Ushenko, A. Bykov, S. Deby, J. Rehbinder, T. Novikova, I. Meglinski, Complementary analysis of Mueller-matrix images of optically anisotropic highly scattering biological tissues. J. Eur. Opt. Soc. 14(1), 20 (2018) 47. V.G. Kolobrodov, Q.A. Nguyen, G.S. Tymchik, The problems of designing coherent spectrum analyzers, in Proceedings of SPIE 11th International Conference on Correlation Optics18 September 2013 through 21 September 2013, vol. 2013, p. 9066. Article number 90660N Code 103970 48. V.A. Ostafiev, S.P. Sakhno, S.V. Ostafiev, G.S. Tymchik, Laser diffraction method of surface roughness measurement. J. Mater. Process. Technol. (63), 871–874 (1997) 49. I.G. Chyzh, V. Kolobrodov, A. Molodyk, V. Mykytenko, G. Tymchik, R. Romaniuk, P. Kisała, A. Kalizhanova, B. Yeraliyeva, Energy resolution of dual-channel opto-electronic surveillance system, in Proceedings Volume 11581, Photonics Applications in Astronomy, Communications, Industry, and High Energy Physics Experiments, 115810K (2020) https://doi.org/10.1117/12. 2580338. Event: Photonics Applications in Astronomy, Communications, Industry, and High Energy Physics Experiments (Wilga, Poland, 2020) 50. V.H. Kolobrodov, V.I. Mykytenko, G.S. Tymchik, Polarization model of thermal contrast observation objects. Thermotlectricity (1), 36–49 (2020) 51. V.H. Kolobrodov, M.S. Kolobrodov, G.S. Tymchik, A.S. Vasyura, P. Komada, Z. Azeshova, The output signal of a digital optoelectronic processor, in Proceedings of the SPIE 10808, Photonics Applications in Astronomy, Communications, Industry, and High-Energy Physics Experiments, 108080W (2018) 52. G.S. Tymchik, V.I. Skytsyuk, T.R. Klotchko, H. Bezsmertna, W. Wójcik, S. Luganskaya, Z. Orazbekov, A. Iskakova, Diagnosis abnormalities of limb movement in disorders of the nervous system. Photonics Applications in Astronomy, Communications, Industry, and High-Energy Physics Experiments 104453S–104453S-11 (2017). https://doi.org/10.1117/12.228100

10

1 Novel Diagnosis Capabilities and Prospects for Determining …

53. Z. Hu, M. Ivashchenko, L. Lyushenko, D. Klyushnyk, Artificial neural network training criterion formulation using error continuous domain. Int. J. Mod. Educ. Comput. Sci. (IJMECS) 13(3), 13–22 (2021) https://doi.org/10.5815/ijmecs.2021.03.02 54. Z. Hu, I. Tereikovskyi, D. Chernyshev, L. Tereikovska, O. Tereikovskyi, D. Wang, Procedure for processing biometric parameters based on wavelet transformations. Int. J. Mod. Educ. Comput. Sci. (IJMECS) 13(2), 11–22 (2021). https://doi.org/10.5815/ijmecs.2021.02.02 55. Z. Hu, R. Odarchenko, S. Gnatyuk, M. Zaliskyi, A. Chaplits, S. Bondar, V. Borovik,Statistical techniques for detecting cyberattacks on computer networks based on an analysis of abnormal traffic behavior. Int. J. Comput. Netw. Inf. Secur. (IJCNIS) 12(6), 1–13 (2020). https://doi.org/ 10.5815/ijcnis.2020.06.01 56. Z. Hu, S. Gnatyuk, T. Okhrimenko, S. Tynymbayev, M. Iavich, High-speed and secure PRNG for cryptographic applications. Int. J. Comput. Netw. Inf. Secur. (IJCNIS) 12(3), 1–10 (2020). https://doi.org/10.5815/ijcnis.2020.03.01 57. Z. Hu, I. Dychka, M. Onai, Y. Zhykin, Blind payment protocol for payment channel networks. Int. J. Compu. Netw. Inf. Secur. (IJCNIS), 11(6), 22–28 (2019). https://doi.org/10.5815/ijcnis. 2019.06.03 58. Z. Hu, Y. Khokhlachova, V. Sydorenko, I. Opirskyy, Method for optimization of information security systems behavior under conditions of influences. Int. J. Intell. Syst. Appl. (IJISA) 9(12), 46–58 (2017). https://doi.org/10.5815/ijisa.2017.12.05 59. Z. Hu, S.V. Mashtalir, O.K. Tyshchenko, M.I. Stolbovyi, Video shots’ matching via various length of multidimensional time sequences. Int. J. Intell. Syst. Appl. (IJISA) 9(11), 10–16 (2017). https://doi.org/10.5815/ijisa.2017.11.02 60. Z. Hu, I.A. Tereykovskiy, L.O. Tereykovska, V.V. Pogorelov, Determination of structural parameters of multilayer perceptron designed to estimate parameters of technical systems. Int. J. Intell. Syst. Appl. (IJISA) 9(10), 57–62 (2017). https://doi.org/10.5815/ijisa.2017.10.07 61. Z. Hu, Y.V. Bodyanskiy, Nonna Ye. Kulishova, Oleksii K. Tyshchenko, A multidimensional extended neo-fuzzy neuron for facial expression recognition. Int. J. Intell. Syst. Appl. (IJISA) 9(9), 29–36 (2017). https://doi.org/10.5815/ijisa.2017.09.04

Chapter 2

Coordinate Distributions of Phase Shift Values Between Orthogonal Components of the Amplitude of the Laser Radiation Field

2.1 Method for Measuring Coordinate Distributions of Phase Shift Values Between Orthogonal Components of the Amplitude of the Laser Radiation Field This method is based on the formalism of the Jones matrix, the measurement scheme of which is classical and is presented in detail in [1–18]. Extending this approach to the processes of conversion of the amplitude and phase of laser radiation by an optically uniaxial biological crystal, we consider the following case. Let some optically anisotropic partial crystal of the polycrystalline network of the biological layer be characterized by the direction of the optical axis ρ and the magnitude of the phase shift between the orthogonal components of the laser radiation amplitude δ. Let us consider the process of converting the amplitude of a laser wave that has passed through a biological crystal we have isolated, which is located between the polarizer and the analyzer with mutually orthogonally oriented transmission planes. In matrix form, such a transformation can be written as the following equation. U = {P}{D}{A}U0

(2.1)

Here {P}—polarizer Jones matrix; {D}—Jones matrix of an optically uniaxial birefringent biological crystal; { A}—analyzer Jones matrix; U0 —Jones vector of incident laser wave; U —Jones vector of transformed laser wave. Under these conditions, the expressions for the partial vectors and Jones matrices take the form

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Z. Hu et al., Phase Mapping of Human Biological Tissues, SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-981-99-3269-6_2

11

12

2 Coordinate Distributions of Phase Shift Values Between Orthogonal …

|| || ⎧ || 1 −1 || ⎪ ||; || ⎪ {A} = || ⎪ ⎪ −1 1 || ⎪ ⎪ || || || || ⎪ ⎪ || d11 d12 || || cos2 ρ + sin2 ρexp(−i δ); cosρsin ρ(1 − exp(−i δ)); || ⎪ ⎪ || || || || ⎪ ⎪ {D} = || d d || = || cos ρsin ρ(1 − exp(−i δ)); sin2 ρ + cos2 ρexp(−i δ) ||; ⎪ 21 22 ⎪ ⎪ || || ⎨ || 1 1 || || {P} = || || 1 1 ||; ⎪ ⎪     ⎪ ⎪ ⎪ 1 U 0x ⎪ ⎪ U0 = = ; ⎪ ⎪ U0y exp(−iδ0 ) 1 ⎪ ⎪   ⎪ ⎪ ⎪ Ux ⎪ ⎩ . U= U y exp(−iδ) (2.2) The resulting Jones vector, taking into account expressions (2.1) and (2.2), is determined by the following equation || || || 1 −1 || || || U = 0.25{ A}{D}{P}U0 = 0.25|| −1 1 || || || || || cos2 ρ + sin2 ρ exp[−i δ] cos ρ sin ρ{1 − exp[−i δ]} || || 1 || || || || cos ρ sin ρ{1 − exp[−i δ]} sin2 ρ + cos2 ρ exp[−i δ] || × || 1

|| . 1 || || 1 1 || 1 (2.3)

The solution to matrix Eq. (2.3) is the Jones vector of the following form.  

1 U = 0, 5cos2ρ 1 − exp(−i δ) −1

(2.4)

Based on relation (2.4), it is possible to obtain the value of the intensity I (x, y) of the image of a biological crystal at a point with coordinates x, y /  I (x, y) = U (x, y)U ⊗ (x, y) = I0 cos2 2ρ(x, y)sin2 δ(x, y) 2 ,

(2.5)

where I0 —intensity of the laser beam probing the biological layer. To directly experimentally determine the coordinate distribution of phase shifts δ(x, y) between orthogonal components of the amplitude at the points r ↔ (x, y) of the laser image of the optical-anisotropic layer, it was proposed [13] to place its sample between two crossed polarization filters—quarter-wave plate and polarizer, which plane transmission has angles with axes of the highest speed of propagation of laser radiation +450 and −450 . The amplitude E of the converted laser beam in this experimental arrangement is determined by the equation. |||| || || 1 −1 |||| i |||| || E = 0.25{ A}{ϕ2 }{M}{ϕ1 }{P}E 0 = 0.25|| −1 1 |||| 0

|| 0 || || 1 ||

2.2 Experimental Scheme of Spectropolarimeter and Phase Measurement

|| || || || cos2 ρ + sin2 ρ exp[−i δ] cos ρ sin ρ{1 − exp[−i δ]} || || 1 || || × || × || cos ρ sin ρ{1 − exp[−i δ]} sin2 ρ + cos2 ρ exp[−i δ] || || 0

13

|||| ||  || || 0 || |||| 1 1 || 1 || || i 1 1 || 1 (2.6)

Here {ϕ1 }, {ϕ2 }—Jones matrix quarter-wave plates. The solution to the matrix Eq. (2.6) is the value of the intensity I (δ) at the point with the coordinates (x, y) of the laser image of the biological crystal. The solution to the matrix Eq. (2.6) is the value of the intensity I (δ) at the point with the coordinates (x, y) of the laser image of the biological crystal. / I (δ) = E E ⊗ = I0 sin2 δ 2 .

(2.7)

2.2 Experimental Scheme of Spectropolarimeter and Phase Measurement In Fig. 2.1, a diagram of two-wave (λ1 , λ2 ) measurement of two-dimensional intensity distributions of laser images of histological sections of biological tissues and hematomas of human organs is shown [20–23]. Illumination of sections of biological tissues or hematomas of human organs (10) was carried out with parallel beams of He–Ne laser 1 (λ1 = 0.633 µm, W = 10.0 mW) and He-Cd laser 2 (λ2 = 0.414 µm, W = 10.0 mW). Beams of lasers 1 and 2 were sequentially directed to a beam splitter 3, which set the propagation orientation in the direction of collimator 6, which formed a parallel beam of rays (∅ = 102 µm). The sequence of illumination was determined by blocking the optical path with mechanical diaphragms 4 and 5. The polarizing illuminator for each wavelength consisted

15

2

17

18

5 λ2 λ1 1

4

3

6

7

8

9 10 11 12 13

14

16

Fig. 2.1 Optical scheme of spectral studies of laser images of biological tissues of human corpse organs

14

2 Coordinate Distributions of Phase Shift Values Between Orthogonal …

/ / of plates λ1 4 (7, 15), λ2 4 (9, 17) and polarizer 8 and formed linearly polarized laser beams. Polarized images of human hematoma samples using a microlens 11 were projected into the plane of the light-sensitive plane (800 × 600 pixels) of a CCD camera 14. When performing this work, histological sections of BT were studied from 100 people of different sexes aged 25–90 years who died from chronic ischemic heart disease (93 cases) and coronary heart disease, which was complicated by acute coronary insufficiency (7 cases) of which 40 were experimental cases with known TD and 60 expert cases with unknown TD.

2.3 A Set of Statistical, Correlation and Fractal Criteria for Evaluation of the Phase Structure of Laser Images of Biological Fluids and Tissues [19–34] ⎛

⎞ r11 , ...r1m The most complete two-dimensional coordinate distributions φ ⎝ .......... ⎠ of the rn1 , ...rnm laser radiation field, transformed by layers of biological fluids and tissues, are charφ acterized by a set of statistical moments of the first–fourth orders Ri=1;2;3;4 calculated from the known theoretical relations [15–18] | ∑ N || | i=1 (φ)i ; φ,μ 1 ∑N R2 = N i=1 (φ)i2 ; φ,μ 3 1 ∑N 1 3 R3 =  φ,μ i=1 (φ)i ; N R2 φ,μ 4 1 ∑N 1 2 R4 =  φ,μ i=1 (φ)i , N φ,μ

R1

1 =/ N

(2.8)

R2

where N = 800 × 600—total number of pixels of the CCD camera 10 (Fig. 2.1), which registers the field of laser radiation scattered by biological fluids and tissues. ⎞ ⎛ r11 , ...r1m The analysis of the coordinate structure φ ⎝ .......... ⎠ of the laser radiation field rn1 , ...rnm transformed by layers of biological fluids and tissues is based on the autocorrelation method using autocorrelation functions Wφ (Δr ), the explicit form of which was calculated using the “MATLAB” software package Wφ (Δr ) =

Δr → 0



1 R0

 R0 0

lim Z φ (r )Z φ (r − Δr )



(2.9)

2.4 A Set of Statistical and Correlation Criteria for Evaluation …

15

Statistical parameters (dispersion R2w and kurtosis R4w —relation (2.8)), which characterize autocorrelation dependences Wφ (Δr ), were chosen as the correlation dependences characterizing the laser images of biological tissues. ⎞ ⎛ r11 , ...r1m A fractal analysis of the coordinate structure φ ⎝ .......... ⎠ of the laser radiation rn1 , ...rnm field converted by layers of biological fluids and tissues was carried out as follows [16]: • the power spectra J (φ) of the autocorrelation functions W (Δr ) were calculated and the log–log dependences of the power spectra logJ (q) − log(l −1 ) were l −1 are spatial frequencies that determined by the geometric dimensions () of the structural elements of laser images of biological objects or fields scattered by them; • the dependencies logJ (q) − log(l −1 ) were approximated by the least squares method into curves V (γ ), for the straight sections of which the slope angles γi were determined and the fractal dimensions were calculated using the known relation Fi = 3 − tgγi .

(2.10) ⎛

⎞ r11 , ...r1m The classification of coordinate distributions φ ⎝ .......... ⎠ was carried out rn1 , ...rnm according to the criteria: • coordinate distributions—fractal, provided that the angle of inclination γ = const of the dependence V (γ ) is constant for two to three decades of changes in geometric dimensions l; • distribution coordinate—multifractal in the presence of several constant tilt angles V (γ ); • coordinate distributions—random in the absence of stable tilt angles V (γ ) for the entire range of geometric dimensions l.

2.4 A Set of Statistical and Correlation Criteria for Evaluation of the Polarization Properties of Biological Tissues of a Human Corpse In Fig. 2.2, the phase map of the laser image of a histological section of skeletal muscle is shown. It can be seen from the data obtained that the distribution of the values of the phase shifts Ʌ of the laser image of a histological section of skeletal muscle tissue is coordinated inhomogeneous and occupies a wide range of values. The complexity and

16

2 Coordinate Distributions of Phase Shift Values Between Orthogonal …

Fig. 2.2 Coordinate distributions of phases (a) and corresponding histograms (b) of a laser image of a histological section of skeletal muscle tissue

topological inhomogeneity of the phase map is illustrated by a series of coordinate distributions of discrete samples of the values of phase shifts introduced by optically anisotropic myosin fibrils between the orthogonal components of the laser wave amplitude—Fig. 2.3.

Fig. 2.3 Samples of certain values of phase shifts (0, 0.5 π , π —these values correspond to black dots of fragments (a), (b), (c)) of the laser image of the histological section of the skeletal muscle tissue

2.5 Principles of Phase Measurement of Sections of Biological Tissues … Table 2.1 Statistical moments of the first–fourth orders of the phase distribution of the laser image of a histological section of skeletal muscle tissue (n = 99)

X-dimension of a matrix

480

Y- dimension of a matrix

512

Minimum element

0

Maximum element

1

Average,M1

0.67 ± 0,048

Dispersion, M2

0.43 ± 0,045

Skewness, M3

−0.48 ± 0,039

Kurtosis, M4

2.85 ± 0,21

17

Most objectively, the statistical structure of the phase map of the laser image of a histological section of skeletal muscle tissue is characterized by a set of moments Mi=1,2,3,4 that is shown in Table 2.1. It can be seen from the data obtained that the entire set of statistical moments of the coordinate distribution of the phases of the laser image of a histological section of skeletal muscle tissue has reliable and nonzero values. In other words, the obtained statistical information on the properties of a histological section of the biological tissue of a human corpse can be used to objectively characterize its properties. A similar complex of statistical and correlation studies of polarization and phase properties was carried out for histological sections of other types of parenchymal tissues of human corpse organs. Kidney tissue See Figs. 2.4 and 2.5. Table 2.2 contains data on the statistical moments of the first–fourth orders of the phase distribution of the laser image of the corresponding section. Brain tissue See Figs. 2.6 and 2.7 Table 2.3 contains data on the statistical moments of the first–fourth orders of phase separation of the laser image of the corresponding section.

2.5 Principles of Phase Measurement of Sections of Biological Tissues of a Human Corpse This section shows the coordinate distributions of phase shifts (phase maps) of sections of connective tissue (dermal layer of human skin), myocardium muscle tissue, lung tissue and kidney tissue.

18

2 Coordinate Distributions of Phase Shift Values Between Orthogonal …

Fig. 2.4 Coordinate distributions of phase shifts of the laser image of a histological section of kidney tissue and the corresponding histograms of their values

Fig. 2.5 Samples of specific values of phase shifts (0, 0.5 π , π —these values correspond to black dots) of the laser image of the kidney section Table 2.2 Statistical moments of the first–fourth orders of phase distribution of the laser image of the kidney section (n = 96)

X-dimension of a matrix

480

Y-dimension of a matrix

639

Minimum element

0

Maximum element

1

Average, M1

0.75 ± 0.072

Dispersion, M2

0.35 ± 0.039

Skewness, M3

−0.09 ± 0.086

Kurtosis, M4

0.03 ± 0.0021

2.5 Principles of Phase Measurement of Sections of Biological Tissues …

19

Fig. 2.6 Coordinate distributions of phase shifts of the laser image of a section of brain tissue and the corresponding histograms of their values

Fig. 2.7 Samples of specific values of phase shifts (0, 0.5 π , π —these values correspond to black dots) of the laser image of the brain tissue section

The method of experimental research and analysis of phase images of sections of a human corpse consists of the following sequence of actions: • Each such histological section was located between crossed quarter wave plates (Fig. 2.1). • A phase image of the tissues of a human corpse was formed, which was recorded by a CCD camera.

20

2 Coordinate Distributions of Phase Shift Values Between Orthogonal …

Table 2.3 Statistical moments of the first–fourth orders of the phase distribution of the laser image of the brain tissue section (n = 98)

X-dimension of a matrix

480

Y-dimension of a matrix

511

Minimum element

0

Maximum element

1

Average, M1

0.65 ± 0.049

Dispersion, M2

0.38 ± 0.018

Skewness, M3

0.18 ± 0.013

Kurtosis, M4

0.11 ± 0.073

• Calculated histogram of two-dimensional distribution of random phase values. • According to the obtained histogram, the statistical moments of the phase distributions were found. From the obtained data on the statistical structure of phase images of human tissues with a structured extracellular matrix it follows: • Coordinate distributions of phase values have a complex heterogeneous structure (Figs. 2.8, 2.9 and 2.10). • Histograms of random phase values (Figs. 2.9 and 2.11) indicate the complex statistical nature of the formation of phase images.

Fig. 2.8 Coordinate distribution of phase shifts in the image of muscle tissue (n = 99)

2.5 Principles of Phase Measurement of Sections of Biological Tissues …

21

Fig. 2.9 Histogram of phase shifts in muscle tissue image (n = 99)

Fig. 2.10 Coordinate distribution of phase shifts in the image of the dermal layer (n = 100)

Similar studies of the statistical phase properties of images of histological sections of tissues of parenchymal organs are illustrated by the series Figs. 2.12, 2.13, 2.14 and 2.15.

22

2 Coordinate Distributions of Phase Shift Values Between Orthogonal …

Fig. 2.11 Histogram of the coordinate distribution of phase shifts in the image of the dermal layer (n = 100)

Fig. 2.12 Coordinate distribution of phase shifts in the liver image (n = 96)

2.5 Principles of Phase Measurement of Sections of Biological Tissues …

23

Fig. 2.13 Histogram of the coordinate distribution of phase shifts in the liver image (n = 96)

Fig. 2.14 Coordinate distribution of phase shifts in the image of the lung tissue (n = 98)

An analysis of the data obtained on the dynamics of changes in statistical moments characterizing the phase images of tissues of parenchymal organs indicates the possibility of their use in the tasks of temporal monitoring of morphological and structural changes in biological tissues of a human corpse.

24

2 Coordinate Distributions of Phase Shift Values Between Orthogonal …

Fig. 2.15 Histogram of the coordinate distribution of phase shifts in the image of the lung tissue (n = 98)

Comparative analysis of the set of objective criteria characterizing the parameters z(x, y) of the polarization and phase properties of histological sections of various types of biological tissues of a human corpse revealed their individual dependence on the features of the morphological structure of the extracellular matrix. Therefore, for the tasks of forensic medicine related to the reliable identification of temporal features T of cadaveric changes in biological tissues (the antiquity of the onset of death), or the formation of hematomas of the internal organs of a human corpse, it is relevant to use a new set of criteria.

References 1. P. Xiong, P. Guo, Q. Zeng, The relationship between the postmortem interval and growing of the fly. Fa Yi Xue Za Zhi. 18(2), 71–73 (2002) 2. M.L. Goff, B.H. Win, Estimation of postmortem interval based on colony development time for Anoplolepsis longipes (Hymenoptera: Formicidae). Forensic Sci. 42(6), 1176–1179 (1997) 3. B. Bourel, B. Callet, V. Hedouin et al., Flies eggs: a new method for the estimation of short-term post-mortem interval. Forensic Sci. Int. 135(1), 27–34 (2003) 4. M.S. Archer, The effect of time after body discovery on the accuracy of retrospective weather station ambient temperature corrections in forensic entomology. J. Forensic Sci. 49(3), 553–559 (2004) 5. V. Tuchin, L. Wang, D. Zimnjakov, Optical polarization in biomedical applications (Springer, New York, USA, 2006)

References

25

6. R. Chipman, Polarimetry, in Handbook of Optics vol 1—Geometrical and Physical Optics, Polarized Light, Components and Instruments, ed. by M. Bass (McGraw-Hill Professional, New York, 2010) pp. 22.1–22.37 7. N. Ghosh, M Wood, A. Vitkin, Polarized light assessment of complex turbid media such as biological tissues via Mueller matrix decomposition, in Handbook of Photonics for Biomedical Science, ed. by V. Tuchin (CRC Press, Taylor & Francis Group, London, 2010), pp. 253–282 8. S. Jacques, Polarized light imaging of biological tissues, in Handbook of Biomedical Optics. ed. by D. Boas, C. Pitris, N. Ramanujam (CRC Press, Boca Raton, London, New York, 2011), pp.649–669 9. N. Ghosh, Tissue polarimetry: concepts, challenges, applications, and outlook. J. Biomed. Opt. 16(11), 110801 (2011) 10. M. Swami, H. Patel, P. Gupt, Conversion of 3×3 Mueller matrix to 4×4 Mueller matrix for non-depolarizing samples. Opt. Commun. 286, 18–22 (2013) 11. D. Layden, N. Ghosh, A. Vitkin, Quantitative polarimetry for tissue characterization and diagnosis, in Advanced Biophotonics: Tissue Optical Sectioning. Boca Raton, ed. by R. Wang, V. Tuchin, (CRC Press, Taylor & Francis Group, London, New York, 2013), pp. 73–108 12. T. Vo-Dinh, Biomedical Photonics Handbook: 3 volume Set, 2nd edn. (Boca Raton, CRC Press, 2014) 13. A. Vitkin, N. Ghosh, A. Martino, Tissue polarimetry, in Photonics: Scientific Foundations, Technology and Applications, 4th edn., ed. by D. Andrews ( Wiley, Hoboken, New Jersey, 2015), pp. 239–321 14. V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis, 2nd edn. (SPIE Press, Bellingham, Washington, USA, 2007) 15. O.V. Dubolazov, A.G. Ushenko, Y.A. Ushenko, M.Y. Sakhnovskiy, P.M. Grygoryyshyn, S.V. Pavlov, O. Bakun, O.B. Bodnar, D.L. Kvasnyuk, K. Skorupski, I. Shedreyeva, Degree of local depolarization of laser radiation fields sorted by multi-layer birefringence networks of protein crystals. Proc. SPIE Int. Soc. Opt. Eng. 10808, 108080N (2018) 16. V.A. Ushenko, A.Y. Sdobnov, W.D. Mishalov, A.V. Dubolazov, O.V. Olar, V.T. Bachinskyi, A.G. Ushenko, Y.A. Ushenko, O.Y. Wanchuliak, I. Meglinski, Biomedical applications of Jones-matrix tomography to polycrystalline films of biological fluids. J. Innovat. Opt. Health Sci. 12(6), 1950017 (2019) 17. M. Borovkova, L. Trifonyuk, V. Ushenko, O. Dubolazov, O. Vanchulyak, G. Bodnar, Y. Ushenko, O. Olar, O. Ushenko, M. Sakhnovskiy, A. Bykov, I. Meglinski, Mueller-matrixbased polarization imaging and quantitative assessment of optically anisotropic polycrystalline networks. PLoS ONE 14(5), e0214494 (2019) 18. A.V. Motrich, A.V. Dubolazov, A.G. Ushenko, Analytical modeling of polarization transformation of laser radiation of various spectral ranges by birefringent structures. Proc. SPIE Int. Soc. Opt. Eng. 11105, 111051A (2019) 19. V.G. Kolobrodov, Q.A. Nguyen, G.S. Tymchik, The problems of designing coherent spectrum analyzers. Proc. of SPIE, 2013, vol. 9066, p. Article number 90660N, in 11th International Conference on Correlation Optics 18 September 2013 through 21 September 2013, Code 103970 20. V.A. Ostafiev, S.P. Sakhno, S.V. Ostafiev, G.S. Tymchik, Laser diffraction method of surface roughness measurement. J. Mater. Process. Technol. (63), 871–874 (1997) 21. I.G. Chyzh, V. Kolobrodov, A. Molodyk, V. Mykytenko, G. Tymchik, R. Romaniuk, P. Kisała, A. Kalizhanova, B. Yeraliyeva, Energy resolution of dual-channel opto-electronic surveillance system, in Proceedings Volume 11581, Photonics Applications in Astronomy, Communications, Industry, and High Energy Physics Experiments 115810K (2020) https://doi.org/10.1117/12. 2580338 Event: Photonics Applications in Astronomy, Communications, Industry, and High Energy Physics Experiments (Wilga, Poland, 2020) 22. V.H. Kolobrodov, V.I. Mykytenko, G.S. Tymchik, Polarization model of thermal contrast observation objects. Thermotlectricity (1), 36–49 (2020) 23. V.H. Kolobrodov, M.S. Kolobrodov, G.S. Tymchik, A.S. Vasyura, P. Komada, Z. Azeshova, The output signal of a digital optoelectronic processor, in Proceedings of SPIE 10808, Photonics

26

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

2 Coordinate Distributions of Phase Shift Values Between Orthogonal … Applications in Astronomy, Communications, Industry, and High-Energy Physics Experiments 108080W (2018) G.S. Tymchik, V.I. Skytsyuk, T.R. Klotchko, H. Bezsmertna, W. Wójcik, S. Luganskaya, Z. Orazbekov, A. Iskakova, Diagnosis abnormalities of limb movement in disorders of the nervous system, in Photonics Applications in Astronomy, Communications, Industry, and High-Energy Physics Experiments, pp. 104453S–104453S-11 (2017). https://doi.org/10.1117/12.228100 Z. Hu, M. Ivashchenko, L. Lyushenko, D. Klyushnyk, Artificial neural network training criterion formulation using error continuous domain. Int. J. Mod. Educ. Comput. Sci. (IJMECS) 13(3), 13–22 (2021). https://doi.org/10.5815/ijmecs.2021.03.02 Z. Hu, I. Tereikovskyi, D. Chernyshev, L. Tereikovska, O. Tereikovskyi, D. Wang, Procedure for processing biometric parameters based on wavelet transformations. Int. J. Mod. Educ. Comput. Sci. (IJMECS) 13(2), 11–22 (2021). https://doi.org/10.5815/ijmecs.2021.02.02 Z. Hu, R. Odarchenko, S. Gnatyuk, M. Zaliskyi, A. Chaplits, S. Bondar, V. Borovik,Statistical techniques for detecting cyberattacks on computer networks based on an analysis of abnormal traffic behavior. Int. J. Comput. Netw. Inf. Secur. (IJCNIS) 12(6), 1–13 (2020). https://doi.org/ 10.5815/ijcnis.2020.06.01 Z. Hu, S. Gnatyuk, T. Okhrimenko, S. Tynymbayev, M. Iavich, High-speed and secure PRNG for cryptographic applications. Int. J. Comput. Netw. Inf. Secur. (IJCNIS) 12(3), 1–10 (2020). https://doi.org/10.5815/ijcnis.2020.03.01 Z. Hu, I. Dychka, M. Onai, Y. Zhykin, Blind payment protocol for payment channel networks. Int. J. Comput. Netw. Inf. Secur. (IJCNIS) 11(6), 22–28 (2019). https://doi.org/10.5815/ijcnis. 2019.06.03 Z. Hu, Y. Khokhlachova, V. Sydorenko, I. Opirskyy, Method for optimization of information security systems behavior under conditions of influences. Int. J. Intell. Syst. Appl. (IJISA) 9(12), 46–58 (2017). https://doi.org/10.5815/ijisa.2017.12.05 Z. Hu, S.V. Mashtalir, O.K. Tyshchenko, M.I. Stolbovyi, Video shots’ matching via various length of multidimensional time sequences. Int. J. Intell. Syst. Appl. (IJISA) 9(11), 10–16 (2017). https://doi.org/10.5815/ijisa.2017.11.02 Z. Hu, I.A. Tereykovskiy, L.O. Tereykovska, V.V. Pogorelov, Determination of structural parameters of multilayer perceptron designed to estimate parameters of technical systems. Int. J. Intell. Syst. Appl. (IJISA) 9(10), 57–62 (2017). https://doi.org/10.5815/ijisa.2017.10.07 Z. Hu, Y.V. Bodyanskiy, N.Ye. Kulishova, O.K. Tyshchenko, A multidimensional extended neo-fuzzy neuron for facial expression recognition. Int. J. Intell. Syst. Appl. (IJISA) 9(9), 29–36 (2017). https://doi.org/10.5815/ijisa.2017.09.04 Z. Hu, I. Dychka, Y. Sulema, Y. Radchenko, Graphical data steganographic protection method based on bits correspondence scheme. Int. J. Intell. Syst. Appl. (IJISA) 9(8), 34–40 (2017). https://doi.org/10.5815/ijisa.2017.08.04

Chapter 3

Computer Modeling of the Evolution of Statistical Parameters of the Phase Distributions of the Laser Radiation Field Converted by Optically Anisotropic Layers

3.1 Characteristics of Research Objects We have chosen two types of optically anisotropic layers as optically anisotropic objects for computer modeling of the processes of transformation of the amplitudephase parameters of laser radiation: • a network of birefringent cylinders ordered in one plane; • a set of birefringent spheres whose centers lie in the same plane. This choice of objects is due to considerations of the analysis of the formation of the phase structure of the fields of scattered OAL laser radiation, depending on the influence of the magnitude of the birefringence Δn of the substance of the partial crystals and their geometry [1–7]. On the other hand, such crystal structures are among the most common among real objects—polymers, liquid crystals, suspensions, cells, blood cells, extracellular matrices of plant and animal tissues [8–10]. Let us consider the process of transformation of parameters (amplitude (U ) and phase (δ) of laser radiation by a flat layer (l = const) of a birefringent medium in different diffraction zones.

3.1.1 Image Plane It is known that when a laser wave U0 exp(−i δ0 ) passes through such a layer, it decomposes into two planes, orthogonally polarized partial waves Ux = U0x exp(−i δ0 ) and U y = U0y exp(−i δ0 ). The speed (vx , v y ) propagation of such waves with a length λ due to birefringence (Δn = n x − n y /= 0) of the substance is different. This effect of optical anisotropy leads to the formation of a stationary phase shift (δ(x, y) = const) between orthogonal components Ux , U y within the entire image © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Z. Hu et al., Phase Mapping of Human Biological Tissues, SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-981-99-3269-6_3

27

28

3 Computer Modeling of the Evolution of Statistical Parameters …

plane (x = 0 ÷ m; y = 0 ÷ n) of a flat OAL with a thickness d(x, y) = const.   δ(x, y) = 2π/λ n x − n y d(x, y)

(3.1)

From (3.1), it follows that the main factors that lead to a change in the phase shift δ(x, y) at each point (x, y) of the object are its anisotropy (Δn) and geometric thickness (d). On the other hand, the magnitude of the phase shift δ(x, y) leads to the formation in the general case of an elliptically polarized vibration U exp(−iδ), the polarization trajectory of which is determined by the following equation X2 Y2 2X Y + − cosδ(d) = sin2 δ(d) Ux2 U y2 Ux U y

(3.2)

From (3.2), it follows that a uniformly polarized laser field with the following azimuth α () and ellipticity (β) polarization values will form in the image plane of a flat OAL ⎧  U

tg y/Ux ⎨ ; α = 0, 5ar ctg cos δ (3.3) ⎩ β = 0, 5 arcsin{cos(U y/Ux )tgδ}. If the OAL is characterized by a coordinate change in optical (Δn(x, y) /= const) or geometric (d(x, y) /= const) parameters, then its laser image is transformed into a phase- (δ(x, y) /= const) and polarization (α(x, y) /= const and β(x, y) /= const) inhomogeneous field. Coordinate distributions of azimuth (α(x, y)) and ellipticity (β(x, y)) polarization values in the literature are called OAL polarization maps. Such parameters of the laser radiation field have been well studied for optically thin OAL of histological sections of biological tissues, where the range of changes in phase shifts within one period (0 ÷ 2π) is objectively realized. Moreover, within the thickness of a particular histological section, a certain probabilistic range of changes in phase shifts (φ¯ ± Δφ) is realized, which is determined by the prevailing values of the birefringence index ¯ of protein fibrils and the range of their transverse dimensions (d¯ ± Δd). (Δn) On the other hand, there are objects that introduce weak phase changes (φ ≺ π/20) − various biological fluids, or, conversely, strong phase changes (φ¯ = φ˜ ± 2kπ, k = 0; 1; 2; , ...)—optically thick layers of biological tissues. Let us analyze the effect of phase modulation on the evolution of the structure of polarization maps OAL in two limiting approximations.

3.1 Characteristics of Research Objects

3.1.1.1

29

Weak Phase Modulation

Consider OAL birefringent structures that introduce minor, but reliably measurable, ˜ y) ≤ phase shifts between orthogonal components (Ux ,Uy ) of the laser wave .δ(x, 60 − 100 In such a situation, the following approximations can be used 

˜  sin δ(x, y) ≈ δ(x, y); c os δ(x, y) ≈ 1, 0.

(3.4)

Taking into account, the boundary conditions (3.4), relations (3.2) and (3.3) take the form ∼2 X2 Y2 2X Y + 2− = δ (x, y) 2 Ux Uy Ux U y  α(x, y) ≈ 0, 5(U y/Ux ) = const; β(x, y) ≈ 0.

(3.5)

(3.6)

Thus, we can state that for OAL with weak phase modulations (relation (3.4)), the limiting fiel1 ×d tends to be polarization homogeneous (relations (3.5) and (3.6)). That is, for such phase-inhomogeneous layers, the only relevant information about ˜ y), which we will further call the their anisotropy is the coordinate distribution δ(x, phase map. In addition, we note that the phase map of such an object is to a certain extent an “imprint”, uniquely associated with its optical (anisotropic)—geometric properties.

3.1.1.2

Strong Phase Modulation Δ

Consider OAL birefringent structures which introduce significant (δ (x, y)  2π ), large over a period, phase shifts between the orthogonal components (Ux , U y ) of the laser wave. In this situation, the phase value becomes ambiguously related to the parameters of the optical anisotropy of the polycrystalline network. We can assume the implementation of the following approximations ⎧ ˆ ⎪ y) = δ¯ (x, y) + 2kπ, k = 0; 1; 2; ...N ; ⎨ δ(x, sin δ (x, y) = si nδ(x, ¯ y); ⎪ ⎩ cos δ (x, y) = co s¯ δ(x, y). Δ

Δ

Δ

(3.7)

Provided that (3.7) is fulfilled, the coordinate distribution δ (x, y) of the image of ¯ y) the multiply scattering OAL becomes higher frequency than the distribution δ(x, in the image of an optically thin anisotropic layer.

30

3 Computer Modeling of the Evolution of Statistical Parameters …

This statement is illustrated by the following analytical example. Let us consider the one-dimensional (without reducing the completeness of the analysis) case of harmonious coordinate phase modulation by an optically thin and optically thick layer due to a change in the values of the birefringence index Δn. Taking into account expression (3.1), we can write the following relations

 λ ¯ ; Δn(x) ¯ = sin δ (x) 2π d



    λ k = sin + Δn(x) Δn(x) ˆ = sin 2kπ + δ¯ (x) ¯ 2π d d

(3.8)

From (3.8), it follows that the change in phase shifts between the orthogonal components (Ux ; U y ) of the amplitude of laser radiation, converted by different points of the optically thick layer, occur much faster ( dk ) than the analogous process of phase modulation by an optically thin anisotropic layer. Thus, in the presence of a significant number of acts (N → ∞) of interaction of laser radiation with birefringent structures of such layer, the phase distribution φ(x, y) is ambiguously interconnected with its optical-geometric structure and tends to be equiprobable within the limits 0 ÷ 2π . Q φ = Rect

(3.9)

3.2 Evolution of Phase Distributions of Laser Radiation in Free Propagation Space The amplitude E and phase φ of the field obtained as a result of the propagation of a plane wave U (x, y)exp(−i δ(x, y)) can be determined using the double diffraction integral of the Rayleigh-Somerfeld, separately for the orthogonal components E x and E y z iλ  

E x (ξ, ζ ) = E y (ξ, ζ ) =

z iλ

 

Bx (x, y) × exp{−i[k R + φ(x, y)]}d xd y; R3

(3.10)

B y (x, y) × exp{−i[k R + φ(x, y) + δ(x, y)]}d xd y, (3.11) R3

where Bx (x, √y) and B y (x, y)—aperture input functions of the optically anisotropic layer; R = z 2 + (x − ξ )2 + (y − ζ )2 —is the distance from a point on the object to a point in the observation plane; z—distance from object plane to observation plane; δ(x, y)—wave phase; φ(x, y)—phase difference between orthogonal components.

3.2 Evolution of Phase Distributions of Laser Radiation in Free Propagation …

31

Having received the real and imaginary parts of the complex amplitude E i (ξ, ζ ), we can calculate the amplitude modulus E(x, y), phase φi (x, y) (i = x, y) and phase difference φ(x, y) = φ y (x, y) − φx (x, y) (Fig. 3.1). The calculation of the input functions Bi (x, y), δ(x, y) and φ(x, y) was carried out in accordance with Fig. 3.2 by the following formulas

Fig. 3.1 A model analysis of the application of diffraction integrals

Fig. 3.2 The analysis of the calculation of the phase difference between ordinary and extraordinary beams

32

3 Computer Modeling of the Evolution of Statistical Parameters …

B(x, y) = [|U |cosθ sinθ (cosφ2 − cosφ1 )]2 + [|U |cosθ sinθ (sinφ1 − sinφ2 )]2 ; (3.12) 2  2  B y (x, y) = |U |(cos2 θ cosφ2 + sin2 θ cosφ1 ) + |U |(cos2 θ sinφ2 + sin2 θ sinφ1 ) ; (3.13)   sinφ1 − sinφ2 ; (3.14) δ(x, y) = arctan cosφ2 − cosφ1     sinφ1 − sinφ2 cos2 θ sinφ2 + sin2 θ sinφ1 − arctan , (3.15) φ(x, y) = arctan cosφ2 − cosφ1 cos2 θ cosφ2 + sin2 θ cosφ1 where θ ≡ θ (x, y) is the angle between the 0Y axis (with which the direction of polarization of the incident beam coincides) and 0Y “ − the optical axis of the crystal, and δi ≡ δi (x, y)—the phase difference of the ordinary (i = 1) and extraordinary rays (i = 2), which was calculated in accordance with Fig. 3.2. Thus, the use of algorithm (3.11)–(3.15) makes it possible to estimate the change in the phase distributions of laser radiation converted by a layer of optically anisotropic crystals in different diffraction zones.

3.3 Diffraction Transformation of Phase Maps of Laser Radiation Converted by a Layer of Ordered Cylindrical Crystals A set of cylinders with the following parameters was considered as the object of research: • birefringence index Δn = 1, 5 × 10−2 (analogous to the value of birefringence of a substance of optically anisotropic structures of most biological tissues); • the optical axes are parallel (the orientation angle relative to the axes O x and O y is the magnitude ξ = 450 ) and lie in the same plane; • radius of an individual crystal cylinder R = 50 μm. Considered two types (A– and B–) crystal networks. The first (A-type) is a network of cylinders with the same optical-geometric parameters. The second (B-type) is a crystalline optical-anisotropic network with an ensemble (N ∗ = 10) of central cylinders with an “increased” level of birefringence (Δn ∗ = 7, 5 × 10−2 ). Such optically anisotropic networks were irradiated with a plane-polarized laser beam with the wavelength λ = 0, 63 μm and the azimuth of polarization α0 = 450 . The results of calculations (relation (3.1), (3.10)–(3.15)) of the coordinate (lefthand sides) and probabilistic (right-hand sides) dependences of the phase distribution φ(x = 1 ÷ m, y = 1 ÷ n) for an array with m × n = 800 × 600 (analog of a CCD camera photosensitive area) laser radiation transformed by virtual objects in different

3.3 Diffraction Transformation of Phase Maps of Laser Radiation …

33

Fig. 3.3 Phase maps φ(x, y) of laser radiation converted by a grid of rectilinear birefringent cylindrical crystals of A (fragment “a”) and B (fragment “c”) types, and histograms of distributions of phase shift values (fragments “b” and “d”, respectively) in the boundary diffraction zone z = 100 μm

zones diffraction patterns (z = 100 μm; 1000 μm; 10000 μm) are illustrated in Figs. 3.3, 3.5 and 3.7. In the series in Figs. 3.4, 3.6 and 3.8, the autocorrelation functions (left fragments) W (Δx) (see Chap. 2, paragraph 2.3, relation (2.9)) and logarithmic dependences (right fragments) logJ − logl −1 of the power spectra (see Chap. 2, paragraph 2.3, relation (2.10)) of the phase distributions of the laser radiation field transformed by crystal networks of ordered A-and B-type cylinders in different diffraction zones are shown. Analysis of the data of computer modeling of the diffraction transformation of the coordinate distributions of the phases of the field of laser radiation transmitted through an ordered network of cylindrical crystals revealed the following features. Phase maps φ(x, y) in all diffraction zones (Figs. 3.4, 3.6 and 3.8) are complex coordinate-inhomogeneous distributions formed by a set of “phase domains” (φ(x ± Δx, y ± Δy) ≈ const). The sizes of such topological structures increase from 2 ÷ 5 μm (boundary diffraction zone) to 15 ÷ 50 μm (Fraunhofer diffraction zone). The higher-frequency coordinate modulation of the values of phase shifts φ(x, y) in the boundary diffraction zone, in our opinion, is associated with the

34

3 Computer Modeling of the Evolution of Statistical Parameters …

Fig. 3.4 Autocorrelation functions W (Δx) (fragments “a” and “c”) and logarithmic dependences of power spectra LogG − logl −1 (fragments “b” and “d”, respectively) of the phase distributions of laser radiation transformed by a grid of rectilinear birefringent cylindrical crystals A and B in the boundary diffraction zone z = 100 μm

predominant influence of the phase modulation (δ(x, y)) of laser radiation by an optically anisotropic substance of crystalline cylinders. In such a diffraction zone, a close order plays a decisive role—the formation of the distribution of phase shifts (δ ∗ (x, y)) within one or two cylinders with subsequent interference of a limited number of partial wave fronts Uk exp(−i (δ ∗ (x, y) + φk (x, y))). Let us analyze the patterns of the formation of coordinate changes in the values of phase shifts in more detail. In the boundary zone for one crystal cylinder in the direction√ perpendicular to the axisO x, due to a change in the geometric dimensions 4R 2 − x 2 , the change in the phase is described by the relation.d(x) = d(x) = √ 2 2 4R − x √ (3.16) δ ∗ (x) = (2π/λ)Δn 4R 2 − x 2 Let us analyze the result of the interference of partial wave fronts Uk exp(−i (δ ∗ (x) + φk (x))), which are formed when radiation passes through one cylindrical crystal. For this purpose, let us single out three partial wave fronts—one

3.3 Diffraction Transformation of Phase Maps of Laser Radiation …

35

Fig. 3.5 Phase maps φ(x, y) of laser radiation transformed by a grid of rectilinear birefringent cylindrical crystals of A (fragment “a”) and B (fragment “c”) types, and histograms of distributions of phase shift values (fragments “b” and “d”, respectively) in the Fresnel diffraction zone z = 1000 μm ∗(d=0,1R) ∗(d=2R) central (δmax ) and two “side” (δmin ). Let us consider their interference at a distance z = 2R corresponding to the boundary diffraction zone

         ∗ ∗12 ∗13 U1 exp −i δmin () []2 exp −i δmax () []3 exp −i δmin () []

 |   3 ∑ 3     ⊗  || ∑ |   |= |× U1 exp −i δ ∗ () []2 exp −i δ ∗12 () []3 exp −i δ ∗13 () [] U U cos(ϕ − ϕ ) i k i k max min min | | i=1 k=1

(3.17) Here U1 = U2 = U3 ≡ 1, 0; φ12 = 2π R



√ 4− 5 /λ; φ

13

∗0,2π RΔn/λ ∗8π RΔn/λ;min

→ 0; δmax

(3.18)

;

(3.19)

36

3 Computer Modeling of the Evolution of Statistical Parameters …

Fig. 3.6 Autocorrelation functions W (Δx) (fragments “a” and “c”) and logarithmic dependences of the power spectra LogG −logl −1 (fragments “b” and “d”, respectively) of the phase distributions of laser radiation transformed by a grid of rectilinear birefringent cylindrical crystals A and B in the Fresnel diffraction zone z = 1000 μm

Analysis of relation (3.17), taking into account the amplitude (3.18) and phase (3.19) conditions, revealed a significant advantage of the “object” phase modulation (δ ∗ (x)) over the “field” (φ(x)). Quantitatively, this fact illustrates the relationship between the depths of the corresponding phase modulations. Δδ ∗ (x) δ = φ −φ 12 Δφik (x) max∗∗ 13  1, 0

(3.20)

min

With distance from the plane of the layer of anisotropic cylinders, the depth of low-frequency phase modulationΔφik (x) increases due to an increase in the number of wave fronts from various partial cylinders. In accordance with this, the size of the “phase domains” in the Fresnel diffraction zone grows (Fig. 3.6). In the Fraunhofer diffraction zone (Fig. 3.8), the relationship dominates Δφik (x)  Δδ ∗ (x), which manifests itself in the coordinate “stabilization” of the topological structure of the phase maps of laser radiation.

3.3 Diffraction Transformation of Phase Maps of Laser Radiation …

37

Fig. 3.7 Phase maps φ(x, y) of laser radiation transformed by a grid of rectilinear birefringent cylindrical crystals A (fragment “a”) and B (fragment “c”) types, and histograms of distributions of phase shifts values (fragments “b” and “d”, respectively) in the Fraunhofer diffraction zone z = 10000 μm

Quantitatively, the specificity of the coordinate distributions of the values of the phase shifts of laser radiation in different diffraction zones is illustrated by histograms M(φ),—the right fragments of Figs. 3.4, 3.6 and 3.8. Analysis of the distributions M(φ) in all diffraction zones for both types of crystal networks revealed the widest possible range of variation from -900 to + 900 random phase values φ. Comparison of the histograms M(φ) for the fields formed by crystal networks showed that the A-type layer is characterized by the presence of local, rather “blurred” extrema in the vicinity φ = ±900 . For similar dependences characterizing the phase-shifting ability of the B-type crystal network, such extrema are absent. Therefore, quantitative statistical analysis is relevant for their differentiation (see Chap. 2, paragraph 2.3, relation (2.9)). Within the framework of the correlation approach, it was revealed that in the boundary diffraction zone, the autocorrelation functions W (Δx) (Fig. 3.4) are rapidly decreasing dependences with the presence of periodic fluctuations ΔW (0, 0 ≤ W (xi + Δx) − W (xi ) ≤ 1, 0) of the eigenvalues.

38

3 Computer Modeling of the Evolution of Statistical Parameters …

Fig. 3.8 Autocorrelation functions W (Δx) (fragments “a” and “c”) and logarithmic dependences of power spectra LgG − lgl −1 (fragments “b” and “d”, respectively) of the phase distributions of laser radiation transformed by a network of rectilinear birefringent cylindrical crystals A and B in the Fraunhofer diffraction zone z = 10000 μm

In the Fresnel diffraction zone (Fig. 3.6), the rate of “falling” of the dependences W (Δx) somewhat slows down (the half-width increases L = x1 (W = 0, 5) − x j (W = 0, 5)). Fluctuations ΔW persist. However, the frequency (ϑ = 2π/X = 2π/ x (ΔW )−x (ΔW ) ) of their repetition increases. [ i+1 ] i The Fraunhofer diffraction zone (Fig. 3.8) is characterized by an increase in the half-width L of autocorrelation functions with a significant (several times) decrease in the amplitude of oscillations of the eigenvalues of such functions calculated for phase maps. The revealed features of the correlation structure of phase maps can be explained as follows. First, in the boundary diffraction zone, the coordinate modulation of phase shifts is high-frequency and is determined mainly by the optical (Δn)—geometric (d(x)) parameters of the crystalline cylinders. Therefore, when the phase maps are shifted by an amount x ∗ = R, there is an almost complete decorrelation of the phase distributions,δ(x) − W (Δx∗) → 0. With further displacement by stepsΔx ∗ = k R, periodic oscillations (fluctuations) of the relative values of the autocorrelation functions are observed.

3.3 Diffraction Transformation of Phase Maps of Laser Radiation …

39

Analytically, this effect can be illustrated as follows. Let the law of phase change in the boundary diffraction zone be harmonious δ(x) = G sin(2π x/λ), where G = 2πΔn/λ is the proportionality coefficient. In this case, the phase autocorrelation function (see Chap. 2, paragraph 2.3, relation (2.10)) takes the following form. W (Δx) = lim

X →0

G2 x ∫ sin(2π x/λ) sin(2π (x−Δx)/λ)d x X0 0

(3.21)

From (3.21) follows the presence of harmonic fluctuations ΔW of the values of the autocorrelation function with the spatial frequency ϑ = 2π/R . Second, in the Fresnel diffraction zone, the frequency of phase modulation of laser radiation due to interference low-frequency phase modulation Δφik (x) decreases (relation 3.20). Therefore, the half-width of the autocorrelation W (Δx) functions increases (Fig. 3.6), which is determined by the average size of the phase domains in the laser radiation field. In addition, the distance between adjacent fluctuations ϑ = 2π/X = 2π/[xi+1 (ΔW )−xi (ΔW )] increases. Third, in the Fraunhofer diffraction zone, the prevailing contribution to the formation of the resulting phase at the point of the laser field is made by the crossinterference of partial fronts from all birefringent cylinders Δφik (x)  Δδ ∗ (x). As a result, the autocorrelation functions of such distributions (Fig. 3.8) are characterized by a much smoother (statistical) fall of the eigenvalues. Fourth, a comparative analysis of the correlation structure of phase maps of laser radiation for a B-type network revealed the following features. Autocorrelation functions W (Δx) are characterized by a smaller half-width L, higher frequency ϑ, and lower fluctuation amplitude ΔW . From a physical point of view, this can be associated with an increase in the frequency ϑ(δ ∗ ) and depth Δδ ∗ of modulation of phase shifts in the boundary zone due to an increase in the birefringence index Δn ∗ . Fractal analysis (see Chap. 2, paragraph 2.3) for a limited number of birefringent elements in planar networks is reduced to the analysis of the transformation of the power spectra of phase distributions. In this sense, the following features are revealed. In the boundary diffraction zone for the logarithmic dependences lgJ − lgl −1 of the power spectra of the phase distributions, there are three expressive slopes of the approximating curve V (γ ). This indicates the presence of several predominant frequency regions in the power spectra of the coordinate phase distributions φ(x, y) of laser fields formed by type A and B crystal networks. In the Fresnel diffraction zone, the logarithmic dependences lgJ − lgl −1 are transformed. So, due to the growing influence of low-frequency phase modulation φ(x, y) (relation (3.21)–(3.25)), extrema are formed in the region of medium (l ∼ 100 ÷ 500 μm) and large((l ∼ 500 ÷ 1000 μm)) spatial frequencies (Fig. 3.7, right fragments). In addition, a self-similar set of phase distribution φ(x, y) values is formed, which fall within the interval of small sizes of structural elements of the phase map. As a result, the number of slopes of the approximating curve increases and the difference between the values of the slope angles decreases. In the Fraunhofer diffraction zone, the approximating curves to the logarithmic dependences lgJ − lgl −1 are characterized by fairly stable values of the tilt angles

40

3 Computer Modeling of the Evolution of Statistical Parameters …

(Fig. 3.9, right-hand sides). This multifrequency of the phase map can be associated with the fact that, along with harmonious phase distributions δ(x), the depth of lowfrequency phase modulation Δφik (x) grows due to an increase in the number of wave fronts propagating from various partial cylinders. Therefore, along with a self-similar set of phase shifts that are formed in the high-frequency region of spatial frequencies l −1 (l = 1 ÷ 100 μm), a harmoniously distributed set of phase shifts arising as an effect of spatial low-frequency modulation (l = 100 ÷ 1000 μm) is formed. A comparative analysis of the logarithmic dependences lgJ − lgl −1 of the power spectra of the phase distributions of laser radiation fields in all diffraction zones formed by the A- and B-type grids revealed a smaller modulation depth for the distributions formed by the B-type grids. Physically, such differences in power spectra can be associated with the fact that due to the growth of birefringence (Δn ∗ ) of the sub-assembly of ordered cylinders (N ∗ ), the spatial frequency of change of “object”   phase shifts (δ ∗ ) increases. Therefore, the number of extrema of dependences J l −1 increases, which manifests itself in a decrease in their amplitudes. The next step of the study was to determine the objective quantitative criteria for assessing and differentiating the phase distributions of coherent laser radiation fields.

3.4 Statistical, Correlation and Fractal Parameters Characterizing the Phase Distributions of Laser Radiation Transformed by a Layer of Ordered Cylindrical Crystals [11–26] This paragraph contains the results of the study of changes in statistical (Tables 3.1 and 3.2), correlation (Tables 3.3 and 3.4) and fractal (Tables 3.5 and 3.6) parameters characterizing the phase structure of the laser radiation field in different diffraction zones. From the analysis of the data obtained it follows: ϕ • In different diffraction zones, the set of all statistical moments Ri=1−4 is nonzero, which indicates a complex structure of the phase distributions φ(x.y) of the laser radiation field. • With increasing distance z from the object, the dispersion R2ϕ increases (by 2.2– φ 3 times). Statistical moments of higher orders Ri=3;4 that characterize φ(x.y) φ ϕ decrease (R3 by 2.5–3 times; R4 by 2 times). This fact illustrates the tendency towards “normalization” of phase distributions due to the growing influence in their formation of partial waves from all points of the object. • The presence of local areas of increased anisotropy (type B network) does not manifest itself in significant differences in the values of dispersion R2ϕ (15–20%). φ ϕ skewness R3 (15–25%) and kurtosis R4 (10–25%) of the corresponding phase

3.4 Statistical, Correlation and Fractal Parameters Characterizing the Phase …

41

Fig. 3.9 Coordinate structure and histograms of phase maps φ(x.y) of laser radiation transformed by a layer of birefringent spherical A-type crystals in the boundary (z = 100 μm) zone (top row). Fresnel diffraction zone z = 1000 μm (central row) and Fraunhofer diffraction zone z = 10000 μm (bottom row)

42

3 Computer Modeling of the Evolution of Statistical Parameters … φ

Table 3.1 Statistical moments Ri=1−4 that characterize the distributions φ(x, y) of laser radiation transformed by a layer of cylindrical A-type crystals z φ R1 φ R2 φ R3 φ R4

z = 100 μm

z = 1000μm

z = 10000μm

0.56

0.47

0.39

0.14

0.23

0.31

0.21

0.14

0.09

0.38

0.27

0.18

φ

Table 3.2 Statistical moments Ri=1−4 that characterize the distributions φ(x.y) of laser radiation transformed by a layer of cylindrical B-type crystals z φ R1 φ R2 φ R3 φ R4

z = 100μm

z = 1000μm

z = 10000μm

0.56

0.42

0.31

0.21

0.29

0.38

0.18

0.14

0.06

0.31

0.22

0.14

w that characterize the autocorrelation functions of the distriTable 3.3 Statistical moments Ri=1−4

butions φ(x.y) of laser radiation transformed by a layer of A-type cylindrical crystals in different diffraction zones z

z = 100μm

R2w R4w

z = 1000μm

z = 10000μm

0.02

0.034

0.043

7.98

6.11

5.63

w that characterize the autocorrelation functions of the distriTable 3.4 Statistical moments Ri=1−4

butions φ(x.y) of laser radiation transformed by a layer of B-type cylindrical crystals in different diffraction zones z

z = 100μm

R2w R4w

z = 1000 μm

z = 10000 μm

0.045

0.069

0.094

3.19

2.61

1.46

Table 3.5 Fractal dimensions Fk and dispersion D f of the logarithmic dependences of the power spectra of the distributions φ(x.y) of laser radiation transformed by a layer of A-type cylindrical crystals z φ F1 φ F2

Df

z = 100 μm

z = 1000 μm

z = 10000 μm

Stat

Stat

2.45

Stat

1.87

1.96

0.45

0.27

0.11

3.4 Statistical, Correlation and Fractal Parameters Characterizing the Phase …

43

Table 3.6 Fractal dimensions Fk and dispersion D f of the logarithmic dependences of the power spectra of the distributions φ(x.y) of laser radiation transformed by a layer of B-type cylindrical crystals z φ F1 φ F2

Df

z = 100 μm

z = 1000 μm

z = 10000 μm

Stat

Ctat

2.34

Stat

1.96

2.06

0.37

0.22

0.09

distribution with similar statistical parameters map φ(x.y) of the laser radiation field converted by an optical-anisotropic A-type layer. The local area of increased anisotropy (Δn ∗ ) due to the higher-frequency modulation of the object phase shifts of the grid of B-type crystalline cylinders is more expressively manifested in the specifics of the topographic structure of the phase distributions of laser radiation fields. w This was quantitatively assessed by determining the statistical moments Ri=1−4 of the autocorrelation distribution functions φ(x.y) of laser radiation and transformed by a layer of A-and B-type cylindrical crystals in different diffraction zones—Tables 3.3 and 3.4. Analysis of the data on the correlation structure of the phase distributions of the laser radiation field made it possible to establish a high sensitivity of dispersion R2ϕ and kurtosis R4w , which characterize ϕ(x, y) to the differentiation of changes in birefringence in all diffraction zones. This manifested itself in significant differences ϕ in the values of dispersion R2 (2 - 2.5 times) and kurtosis R4w (2.5–4 times), which characterize the corresponding phase distributions. The results of fractal analysis of sets φ(x.y) are shown in Tables 3.5 and 3.6. Comparative analysis of the obtained data revealed: • Random distributions ϕ(x, y) (there is no stable slope of the approximating curves V (γ )) in the boundary diffraction zone are transformed into multifractal (there are two angles γ1 = const; γ2 = const) for the far diffraction zone. • The dispersion D f of the distribution of the values of the logarithmic dependences of the power spectra logJ − logl −1 of the distributions φ(x.y) turned out to be sensitive to the local change in the birefringence (Δn ∗ ) of the optically anisotropic layer. The differences between the values of this parameter for A- and B-type networks are from 15% (Fraunhofer diffraction zone) to 45% (boundary diffraction zone).

44

3 Computer Modeling of the Evolution of Statistical Parameters …

3.5 Diffraction Transformation of the Statistical, Correlation and Fractal Structure of Phase Maps of Laser Radiation Converted by a Layer of Spherical Crystals The object of the study was a flat layer consisting of a set N = 50 of crystal spheres. The centers of which lie in the same plane with the following parameters: the value of birefringence Δn = 2.5 × 10−2 ; the centers of all spheres lie in the same plane; radius of a single sphere R = 50 μm. Considered two types (A– and B–) crystal networks. The first (A-type) is a network of crystal spheres with the same optical and geometric parameters. The second (Btype) is a crystalline optically anisotropic network with an ensemble (N ∗ = 10) of central spheres with an “increased” level of birefringence (Δn ∗ = 7.5 × 10−2 ). φ The results of calculating the coordinate (φ(x.y)). statistical M(φ); Ri=1;2;3;4 . w ) and fractal (logJ − logl −1 ; Fk ; D f ) structures of the correlation (W (Δx); Ri=1;2;3;4 dependences of the phase distribution φ(x = 1 ÷ m.y = 1 ÷ n) of the laser radiation field in different diffraction zones (z = 100 μm; 1000 μm; 10000 μm) are illustrated in Figs. 3.9, 3.10, 3.11 and 3.12. φ Tables 3.7, 3.8 and 3.9 show the values of the statistical moments Ri and Riw as well as the values of the fractal dimension Fk and dispersion D f of the distributions of the logarithmic dependences of the power spectra φ(x.y) in different diffraction zones. An analysis of the data obtained (Figs. 3.10 and 3.11 and Table 3.7) revealed similar patterns as for the layers of crystalline cylinders (Figs. 3.4, 3.6 and 3.8), which manifested themselves in the trend of changes of the statistical (histograms of the distribution of random values and a set of statistical moments of the first–fourth orders) structure: • average R1φ , dispersion R2φ , skewness R3φ , and kurtosis R4φ characterizing the distribution of phase shifts. are nonzero; φ

Table 3.7 Statistical moments Ri=1−4 that characterize the distributions φ(x.y) of laser radiation transformed by a layer of spherical crystals of A- and B-types in different diffraction zones z

z = 100 μm

Type

A

φ R1 φ R2 φ R3 φ R4

z = 1000 μm B

A

z = 10000 μm B

A

B

0.52

0.59

0.45

0.55

0.41

0.49

0.19

0.14

0.26

0.21

0.39

0.28

0.52

0.11

0.28

0.06

0.16

0.04

0.33

0.09

0.21

0.08

0.07

0.05

3.5 Diffraction Transformation of the Statistical, Correlation and Fractal …

45

Fig. 3.10 Coordinate structure and histograms of phase maps φ(x.y) of laser radiation transformed by a layer of birefringent spherical B-type crystals in the boundary (z = 100 μm) zone (top row). Fresnel diffraction zone z = 1000 μm (central row) and Fraunhofer diffraction zone z = 10000 μm (bottom row)

46

3 Computer Modeling of the Evolution of Statistical Parameters …

Fig. 3.11 Autocorrelation functions W (Δx) (left column) and logarithmic dependences of power spectra LgG − lgl −1 (right column) of phase distributions of radiation transformed by a layer of spherical A-type crystals in the boundary (z = 100 μm) zone (top row). Fresnel diffraction zone z = 1000 μm (central row) and Fraunhofer diffraction zone z = 10000 μm (bottom row)

3.5 Diffraction Transformation of the Statistical, Correlation and Fractal …

47

• with an increase in the distance z from the plane of the optically—anisotropic layer. The dispersion increases by 1.5–2 times and the skewness and kurtosis decrease, respectively, by 3 and 5 times; • for a crystalline layer of type B, the conditions for the formation of a normal distribution in the far diffraction zone are more quickly realized—the corresponding values of the statistical moments of the third and fourth orders are close to zero. The features of the change in the topographic structure of the phase maps of laser radiation fields in different diffraction zones are illustrated in Figs. 3.12 and 3.13 as well as Tables 3.8 and 3.9. A comparative analysis of the correlation and fractal structure of phase maps of optically anisotropic layers formed by birefringent crystals of various geometric configurations revealed the following general features: • the presence of statistical and quasi-regular components (Figs. 3.11 and 3.12. left columns) of autocorrelation functions W (Δx) of coordinate distributions of phase shifts φ(x.y); • greater sensitivity to changes in birefringence in all diffraction zones in comparφ ison with phase statistical moments Ri statistical moments Riw (Table 3.8) characterizing autocorrelation functions W (Δx); • transformation of random phase distributions φ(x.y) of the boundary field of laser radiation into multifractal ones for the Fraunhofer diffraction zone. The performed computer simulation made it possible to reveal the main scenarios of the diffraction transformation of the coordinate distributions of the phase shifts φ(x.y) of the laser radiation field converted by networks of optically anisotropic crystals of various geometric configurations and birefringence: φ

1. Statistical moments of the first–fourth orders (Ri=1−4 ) that characterize the coordinate distributions φ(x.y) of the values of phase shifts formed by optically anisotropic layers of all types experience the following changes. Dispersion increases average, skewness and kurtosis of the corresponding phaseinhomogeneous fields of laser radiation, on the contrary, decrease. w ), which charac2. Correlation moments of the second and fourth orders (Ri=1−4 terize the autocorrelation functions W (Δr ) of the distributions φ(x.y), turned out to be more sensitive to changes in the registration area, the magnitude of birefringence of the crystal substance and a change in their geometric configuration. 3. Fractal dimensions (Fk ) and dispersion (D f ) of the distribution of the values of the logarithmic dependences of the power spectra (J ) of phase maps φ(x.y) on spatial frequencies (l −1 ), which are determined by the range of geometric dimensions (l) of laser radiation in different diffraction zones, are interrelated with changes in the registration area, birefringence and crystal shape. 4. With an increase in the distance from the object, the statistical sets of values of the phase shifts of the laser radiation field in the boundary diffraction zone are transformed into multifractal ones. In this case, the value of the dispersion of the

48

3 Computer Modeling of the Evolution of Statistical Parameters …

Fig. 3.12 Autocorrelation functions W (Δx) (left column) and logarithmic dependences of power spectra LgG − lgl −1 (right column) of phase distributions of radiation transformed by a layer of spherical B-type crystals in the boundary (z = 100 μm) zone (top row). Fresnel diffraction zone z = 1000 μm (central row) and Fraunhofer diffraction zone z = 10000 μm (bottom row)

References

49

w Table 3.8 Statistical moments Ri=1−4 that characterize the autocorrelation functions of the

distributions φ(x.y) of the laser radiation field in various diffraction zones z

z = 100 μm

Type

A

B

A

z = 1000 μm B

A

z = 10000 μm B

R2w R4w

0.023

0.052

0.045

0.091

0.064

0.12

5.14

2.51

3.24

1.32

2.21

0.72

Table 3.9 Fractal dimensions Fk and dispersion D f of the logarithmic dependences of the power spectra of the coordinate distributions φ(x.y) of the laser radiation field in different diffraction zones z

z = 100 μm

Type

A

φ F1 φ F2

Df

z = 1000 μm B

A

z = 10000 μm B

A

B

2.17

2.13

Stat

Stat

Stat

2.11

2.09

2.21

2.14

0.22

0.19

0.15

0.11

0.38

0.34

distribution of the values of the logarithmic dependences lgJ − lgl −1 decreases both with increasing distance and with the values of birefringence of substances of crystal structures. 5. Certain relationships between changes in the set of statistical correlation and fractal parameters characterizing the phase distributions of the laser radiation field and changes in the geometry and birefringence of crystal networks were used as the basis for differentiating the parameters of the phase distributions of laser radiation fields scattered by real optically anisotropic layers.

References 1. V. Tuchin. Tissue optics: Light Scattering Methods and Instruments for Medical Diagnosis. 2nd edn. (SPIE Press. Bellingham. Washington. USA. 2007) 2. W. Bickel. W. Bailey. Stokes vectors. Mueller matrices. and polarized scattered light. Am. J. Phys. 53(5). 468–478 (1985) 3. A. Doronin. C. Macdonald. i. Meglinski. Propagation of coherent polarized light in turbid highly scattering medium. J. Biomed. Opt. 19(2). 025005 (2014) 4. A. Doronin. A. Radosevich. V. Backman. I. Meglinski. Two electric field Monte Carlo models of coherent backscattering of polarized light. J. Opt. Soc. Am. A. 31(11). 2394 (2014) 5. P. Arun Gopinathan. G. Kokila. M. Jyothi. C. Ananjan. L. Pradeep. S. H. Nazir. Study of collagen birefringence in different grades of oral squamous cell carcinoma using picrosirius red and polarized light microscopy. Scientifica 2015. 802980 (2015). 6. L. Rich. P. Whittaker. Collagen and picrosirius red staining: a polarized light assessment of fibrillar hue and spatial distriburuon. Braz. J. Morphol. Sci. (2005)

50

3 Computer Modeling of the Evolution of Statistical Parameters …

7. S. Bancelin. A. Nazac. I. Bicher Haj. at. al.. Determination of collagen fiber orientation in histological slides using Mueller microscopy and validation by second harmonic generation imaging. Opt. Express 22(19). 22561–22574 (2014) 8. Ushenko A. Pishak V. Laser polarimetry of biological tissue: principles and applications. in Handbook of Coherent-Domain Optical Methods: Biomedical Diagnostics. ed. by V. Tuchin (Environmental and Material Science. 2004). pp. 93–138 9. O. Angelsky. A. Ushenko. Y. Ushenko. V. Pishak. A. Peresunko. Statistical. correlation and topological approaches in diagnostics of the structure and physiological state of birefringent biological tissues. Handbook of Photon. Biomed. Sci. (2010). pp. 283–322 10. Y. Ushenko. T. Boychuk. V. Bachynsky. O. Mincer. Diagnostics of structure and physiological state of birefringent biological tissues: statistical. correlation and topological approaches. in Handbook of Coherent-Domain Optical Methods. ed. by V. Tuchin (Springer Science+Business Media. 2013) 11. Kolobrodov V.G., Nguyen Q.A., Tymchik G.S. The problems of designing coherent spectrum analyzers.- Proc. of SPIE, 2013, vol. 9066, p. Article number 90660N, 11th International Conference on Correlation Optics18 September 2013 through 21 September 2013, Code 103970. 12. Ostafiev V.A., Sakhno S.P., Ostafiev S.V., Tymchik G.S. Laser diffraction method of surface roughness measurement.- Journal of Materials Processing Technology, 1997, N63, pp.871–874. 13. I. Chyzh, V. Kolobrodov, A. Molodyk, V. Mykytenko, G. Tymchik, R. Romaniuk, P. Kisała, A. Kalizhanova, B.Y. Energy, resolution of dual-channel opto-electronic surveillance system.Proceedings Volume 11581, Photonics Applications in Astronomy, Communications, Industry, and High Energy Physics Experiments 2020; 115810K (2020) doi: 10.1117, 12.2580338 Event: Photonics Applications in Astronomy, Communications, Industry, and High Energy Physics Experiments, 2020 (Wilga, Poland, 2020) 14. Kolobrodov V. H., Mykytenko V. I., Tymchik G.S. Polarization model of thermal contrast observation objects. - Thermotlectricity, 2020, № 1, p. 36–49. 15. Kolobrodov V. H., Kolobrodov M. S., Tymchik G.S., Anatoliy S. Vasyura, Paweł Komada, Zhanar Azeshova The output signal of a digital optoelectronic processor.- Proc. SPIE 10808, Photonics Applications in Astronomy, Communications, Industry, and High-Energy Physics Experiments 2018, 108080W (1 October 2018). 16. Tymchik G.S., Skytsyuk V.I., Klotchko T.R., Bezsmertna H., Wójcik W., Luganskaya S., Orazbekov Z., Iskakova A. Diagnosis abnormalities of limb movement in disorders of the nervous system.- Photonics Applications in Astronomy, Communications, Industry, and HighEnergy Physics Experiments 2017, 2017/8/7, pp. 104453S-104453S-11. DOI https://doi.org/ 10.1117/12.228100 17. H.U. Zhengbing, M. Ivashchenko, L. Lyushenko, D. Klyushnyk, Artificial Neural Network Training Criterion Formulation Using Error Continuous Domain. International Journal of Modern Education and Computer Science (IJMECS) 13(3), 13–22 (2021). https://doi.org/10. 5815/ijmecs.2021.03.02 18. H.U. Zhengbing, I. Tereikovskyi, D. Chernyshev, L. Tereikovska, O. Tereikovskyi, D. Wang, Procedure for Processing Biometric Parameters Based on Wavelet Transformations. International Journal of Modern Education and Computer Science (IJMECS) 13(2), 11–22 (2021). https://doi.org/10.5815/ijmecs.2021.02.02 19. H.U. Zhengbing, R. Odarchenko, S. Gnatyuk, M. Zaliskyi, A. Chaplits, S. Bondar, V. Borovik, Statistical Techniques for Detecting Cyberattacks on Computer Networks Based on an Analysis of Abnormal Traffic Behavior. International Journal of Computer Network and Information Security (IJCNIS) 12(6), 1–13 (2020). https://doi.org/10.5815/ijcnis.2020.06.01 20. H.U. Zhengbing, S. Gnatyuk, T. Okhrimenko, S. Tynymbayev, M. Iavich, High-Speed and Secure PRNG for Cryptographic Applications. International Journal of Computer Network and Information Security (IJCNIS) 12(3), 1–10 (2020). https://doi.org/10.5815/ijcnis.2020. 03.01 21. H.U. Zhengbing, I. Dychka, M. Onai, Y. Zhykin, Blind Payment Protocol for Payment Channel Networks. International Journal of Computer Network and Information Security (IJCNIS) 11(6), 22–28 (2019). https://doi.org/10.5815/ijcnis.2019.06.03

References

51

22. H.U. Zhengbing, Y. Khokhlachova, V. Sydorenko, I. Opirskyy, Method for Optimization of Information Security Systems Behavior under Conditions of Influences. International Journal of Intelligent Systems and Applications (IJISA) 9(12), 46–58 (2017). https://doi.org/10.5815/ ijisa.2017.12.05 23. H.U. Zhengbing, S.V. Mashtalir, O.K. Tyshchenko, M.I. Stolbovyi, Video Shots’ Matching via Various Length of Multidimensional Time Sequences. International Journal of Intelligent Systems and Applications (IJISA) 9(11), 10–16 (2017). https://doi.org/10.5815/ijisa.2017. 11.02 24. H.U. Zhengbing, I.A. Tereykovskiy, L.O. Tereykovska, V.V. Pogorelov, Determination of Structural Parameters of Multilayer Perceptron Designed to Estimate Parameters of Technical Systems. International Journal of Intelligent Systems and Applications (IJISA) 9(10), 57–62 (2017). https://doi.org/10.5815/ijisa.2017.10.07 25. Zhengbing Hu, Yevgeniy V. Bodyanskiy, Nonna Ye. Kulishova, Oleksii K. Tyshchenko, “A Multidimensional Extended Neo-Fuzzy Neuron for Facial Expression Recognition”, International Journal of Intelligent Systems and Applications (IJISA), Vol.9, No.9, pp.29–36, 2017. DOI: https://doi.org/10.5815/ijisa.2017.09.04 26. H.U. Zhengbing, I. Dychka, Y. Sulema, Yevhen Radchenko,"Graphical Data Steganographic Protection Method Based on Bits Correspondence Scheme". International Journal of Intelligent Systems and Applications (IJISA) 9(8), 34–40 (2017). https://doi.org/10.5815/ijisa.2017.08.04

Chapter 4

Spectral Phase Measurement of Laser Images of Sections of Biological Tissues of a Human Corpse for Death Time Detection

4.1 Spectral Phase Measurement of Laser Images of Histological Sections of Structured Tissues of a Human Corpse Given the high sensitivity of phase shifts in cadaveric changes, at the beginning of the study, data are presented selective experimental data on the temporal dynamics of changes in phase images at time intervals of 5–10% of the total interval for the determination of TDE [1–15]: – Phase maps (λ1 ) and (λ2 ) defined for TDE 6 h. – Statistical moments of distributions of phases (λ1 ) and (λ2 ) determined for TDE 6 h. – Autocorrelation functions of phase maps (λ1 ) and (λ2 ) defined for TDE 6 h. – Power spectra of phase maps (λ1 ) and (λ2 ) determined for TDE 6 and 12 h. – Statistical moments of distributions of the distribution of the extrema of the power spectra of the phase maps (λ1 ) and (λ2 ) determined for TDE 6 and 12 h [16–26]. – Time dependences of statistical moments of phase distributions (λ1 ) and (λ2 ). – Time dependences of statistical moments of distributions of distribution of extrema of power spectra of phase maps (λ1 ) and (λ2 ). Myocardium tissue In Fig. 4.1, the results of measuring the experimental distributions of phase shifts of a laser image of a histological section of myocardium tissue (TDE 6 h) using red λ1 = 0.632μm (fragments “a”. “c”) and blue λ2 = 0.414μm (fragments “b”. “d”) radiation are shown. It can be seen from the data obtained that, in all spectral regions, the coordinate distributions of phase shifts (λ1 ) and (λ2 ) have a complex inhomogeneous structure with a maximum range of variation of the phase shift value (Table 4.1). © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Z. Hu et al., Phase Mapping of Human Biological Tissues, SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-981-99-3269-6_4

53

54

4 Spectral Phase Measurement of Laser Images of Sections of Biological …

Fig. 4.1 Phase maps (λ1 ) and (λ2 ) of the myocardium tissues determined for TDE 6 h

Table 4.1 Statistical moments of phase distributions (λ1 ) and (λ2 ) of the myocardium tissue (n = 98)

(λ1 )

(λ2 )

M1

0.40 ± 0.29

M1

0.62 ± 0.56

M2

0.36 ± 0.27

M2

0.27 ± 0.16

M3

9.16 ± 0.65

M3

23.14 ± 1.72

M4

7.35 ± 0.61

M4

28.46 ± 2.1592

A comparative analysis of the statistical structure of phase maps (λ1 ) and (λ2 ) of histological section of myocardium tissue revealed: – A set of statistical moments of the first–fourth orders for the same TDE value in different spectral ranges have individual values for each phase map. – Average and dispersion values differ by 30–45%. – The skewness and the kurtosis of the phase distributions in the blue region of the spectrum prevail over the analogous values of the moments of the third and fourth orders by 2.5 and 4 times, respectively. Thus, it can be stated that the statistical analysis of phase maps (λ2 ) of a histological section of myocardium tissue in the short-wavelength region of the spectrum λ2 = 0.414μm is promising for increasing the interval and improving the accuracy of determining TDE (Tables 4.2 and 4.3).

4.1 Spectral Phase Measurement of Laser Images of Histological Sections …

55

Table 4.2 Time dynamics of changes in the statistical moments of the distribution of phase shifts in the image of a section of myocardium tissue (λ1 = 0.632µm) (n = 98) T

12

24

36

48

60

72

84

M1

0.36

0.25

0.19

0.17

0.14

0.12

0.12

M2

0.32

0.21

0.14

0.11

0.09

0.07

0.08

M3

8.36

7.04

6.35

4.49

3.18

2.24

2.25

M4

6.65

6.03

5.34

4.75

3.49

2.29

1.17

Table 4.3 Time dynamics of changes in the statistical moments of the distribution of phase shifts in the image of a section of myocardium tissue (λ2 = 0.414µm) (n = 98) T

12

24

36

48

60

72

84

M1

0.56

0.51

0.43

0.37

0.24

0.22

0.21

M2

0.24

0.18

0.12

0.09

0.05

0.06

0.05

M3

21.03

17.56

13.54

9.78

7.14

4.57

4.81

M4

24.92

22.16

17.43

14.25

11.78

9.65

8.98

The results of correlation studies illustrate the autocorrelation functions (Fig. 4.2) and power spectra (Fig. 4.3) of phase maps (λ1 ) and (λ2 ) of the myocardium tissue, which are determined for TDE 6 and 12 h. A comparative analysis of the correlation parameters in different parts of the laser radiation spectrum revealed that the half-widths of the autocorrelation functions in the red and blue parts of the laser radiation spectrum practically do not differ. The largest differential differences are observed in the statistical moments of the third and fourth orders, which characterize the distribution of the extrema of the power spectra of the phase maps (λ1 ) and (λ2 ) of the histological section of the myocardium tissue of the human corpse, which are determined for different TDE intervals—Table 4.4. From the analysis of the data obtained it follows: – The value of the statistical moments of the distributions of the extrema of the power spectra in the red (λ1 ) and blue (λ2 ) parts of the spectrum is individual and differs from 1.5 times (M1 .M2 ) to 3 times (M3 .M4 ). – Time dynamics of changes in the set of statistical moments of the distributions of the extrema of the power spectra, the highest in the red (λ1 ) and the lowest in the blue (λ2 ) parts of the spectrum. The identified different falling rates Mi=1.2.3.4 (T ) can be used to improve accuracy and increase the interval definition of TDE. Figures 4.4 and 4.5 show the time dependences Mi=1.2.3.4 (T ) for a histological section of myocardium tissue in different spectral ranges λ1 = 0.632µm and λ2 = 0.414µm.

56

4 Spectral Phase Measurement of Laser Images of Sections of Biological …

Fig. 4.2 Autocorrelation functions of phase maps (λ1 ) and (λ2 ) of the myocardium tissue determined for TDE 6 h

Fig. 4.3 Power spectra of phase maps and myocardium tissue determined for TDE at 10 and 20 h

4.1 Spectral Phase Measurement of Laser Images of Histological Sections … Table 4.4 Statistical moments of distributions of extrema of power spectra of phase maps (λ1 ) and (λ2 ) of the myocardium tissue (n = 98)

TDE 10 h.λ1

TDE 10 h.λ2

M1

0.26 ± 0.012

M1

0.28 ± 0.013

M2

0.45 ± 0.037

M2

0.44 ± 0.029

M3

4.75 ± 0.36

M3

12.9 ± 1.05

M4

9.25 ± 0.74

M4

17.2 ± 2.04

TDE 20 h.λ1

57

TDE 20 h.λ2

M1

0.22 ± 0.014

M1

0.26 ± 0.017

M2

0.40 ± 0.024

M2

0.42 ± 0.031

M3

4.01 ± 0.34

M3

11.61 ± 1.02

M4

8.9 ± 0.72

M4

15.01 ± 1.23

Fig. 4.4 Time dependences of the average (“M1 ”), dispersion (“M2 ”), skewness (“M3 ”) and kurtosis (“M4 ”) of the power spectra of phase images of a section of myocardium tissue in different spectral ranges—λ1 = 0.632µm

Based on the data given in Tables 4.3, 4.4, 4.5 and 4.6, the following intervals and accuracy of determining TDE for a histological section of myocardium tissue are established: – λ1 = 0.632µm: T = 120 h. T = 3 h – λ2 = 0.414µm: T = 160 h. T = 2.5 h

58

4 Spectral Phase Measurement of Laser Images of Sections of Biological …

Fig. 4.5 Time dependences of the average (“M1 ”), dispersion (“M2 ”), skewness (“M3 ”) and kurtosis (“M4 ”) of the power spectra of phase images of a section of myocardium tissue in different spectral ranges—λ2 = 0.414µm Table 4.5 Time dynamics of changes in the statistical moments of the distribution of the extrema of the power spectra of the phase maps (λ1 ) of the myocardium tissue section (n = 98) T

1

10

20

30

40

50

60

70

9.501

9.254

8.907

7.753

5.505

3.036

1.258

0.403

S(i) S(4) S(3)

5.004

4.756

4.014

2.705

1.703

1.012

0.501

0.254

S(2)

0.503

0.457

0.404

0.355

0.312

0.255

0.215

0.154

S(1)

0.303

0.262

0.224

0.185

0.144

0.126

0.111

0.084

T

80

90

100

110

120

130

140

0.212

0.159

0.157

0.155

0.153

0.152

0.151

S(i) S(4) S(3)

0.155

0.119

0.117

0.115

0.113

0.109

0.105

S(2)

0.094

0.078

0.079

0.076

0.074

0.072

0.071

S(1)

0.068

0.059

0.057

0.055

0.054

0.053

0.052

Thus, it can be stated that the use of the statistical analysis of the power spectra of the phase maps of the histological section of myocardium tissue provided the maximum range for the methods of statistical and correlation polarimetry for the methods of statistical and correlation polarimetry while increasing the accuracy its

4.1 Spectral Phase Measurement of Laser Images of Histological Sections …

59

Table 4.6 Time dynamics of changes in the statistical moments of the distribution of the extrema of the power spectra of the phase maps (λ2 ) of the myocardium tissue section (n = 98) 1

10

20

30

40

50

60

70

S(4)

19.606

17.204

15.013

12.605

10.023

7.311

4.812

2.909

S(3)

13.204

12.905

11.612

7.513

2.754

1.408

0.804

0.509

S(2)

0.47

0.44

0.4

0.36

0.3

0.2

0.16

0.13

S(1)

0.3

0.28

0.26

0.22

0.18

0.15

0.12

0.09

T

80

90

100

110

120

130

140

S(4)

1.312

0.705

0.403

0.212

0.158

0.109

0.054

S(3)

0.4

0.3

0.2

0.15

0.1

0.05

0.03

S(2)

0.11

0.09

0.07

0.05

0.03

0.02

0.015

S(1)

0.07

0.05

0.03

0.02

0.015

0.01

0.005

T S(i)

S(i)

definition is almost twice. In the blue region of the spectrum, the maximum values for the interval and accuracy of determining the TDE are achieved. A similar complex of studies was carried out for histological sections of other types of biological tissues of a human corpse. Spleen See Figs. 4.6, 4.7, 4.8, 4.9 and 4.10 and Tables 4.7, 4.8 and4.9). Power spectra of phase maps (λ1 ) and (λ2 ) of spleen tissue determined for TDE at 6 and 12 h (Tables 4.10, 4.11 and 4.12). From the analysis of the data obtained, it follows: – The values of the statistical moments of the distributions of the extrema of the power spectra in the red (λ1 ) and blue (λ2 ) regions of the spectrum are individual and differ from 1.6 times (M1 .M2 ) to 1.7 times (M3 .M4 ). – Time dynamics of changes in the set of statistical moments of the distributions of the extrema of the power spectra, the highest in the red (λ1 ) and the lowest in the blue (λ2 ) regions of the spectrum. Based on the data given in Tables 4.7, 4.8, 4.9, the following intervals and accuracy of determining TDE for a histological section of spleen tissue are established: – λ1 = 0.632µm: T = 22 h. T = 1.5 h – λ2 = 0.414µm: T = 32 h. T = 1 h Large intestine (colon) In Fig. 4.11, the coordinate distributions of phase shifts in laser images of histological sections of the large intestine are shown. Table 4.13 illustrates the sensitivity of the method of phase measurement of statistical moments of laser images of histological sections of the large intestine (Tables 4.14 and 4.15).

60

4 Spectral Phase Measurement of Laser Images of Sections of Biological …

Fig. 4.6 Phase maps (λ1 ) and (λ2 ) of the spleen tissues determined for TDE 6 h

Fig. 4.7 Autocorrelation functions of phase maps (λ1 ) and (λ2 ) of spleen tissue determined for TDE 6 h

4.1 Spectral Phase Measurement of Laser Images of Histological Sections …

61

Fig. 4.8 Power spectra of phase maps (λ1 ) and (λ2 ) of spleen tissue determined for TDE at 6 and 12 h

Fig. 4.9 Time dependences of the average (“M1 ”), dispersion (“M2 ”), skewness (“M3 ”) and kurtosis (“M4 ”) of the power spectra of the phase images of the spleen tissue section in different spectral ranges—λ1 = 0.632µm

62

4 Spectral Phase Measurement of Laser Images of Sections of Biological …

Fig. 4.10 Time dependences of the average (“M1 ”), dispersion (“M2 ”), skewness (“M3 ”) and kurtosis (“M4 ”) of the power spectra of the phase images of the spleen tissue section in different spectral ranges—λ2 = 0.414µm Table 4.7 Statistical moments of the distributions of phases (λ1 ) and (λ2 ) of the spleen tissue (n = 98) (λ2 )

(λ1 ) M1

0.29 ± 0.017

M1

0.34 ± 0.027

M2

0.30 ± 0.019

M2

0.44 ± 0.038

M3

4.29 ± 0.026

M3

6.24 ± 0.56

M4

7.32 ± 0.052

M4

8.35 ± 0.81

Table 4.8 Time dynamics of changes in the statistical moments of the distribution of phase shifts of the image of a histological section of the spleen tissue (λ1 = 0.632µm) (n = 98) T

3

6

9

12

15

18

M1

0.38

0.29

0.21

0.17

0.11

0.13

M2

0.43

0.35

0.27

0.22

0.14

0.11

M3

5.84

4.29

3.12

2.63

1.97

1.26

M4

9.37

7.32

5.76

3.31

2.59

2.14

4.1 Spectral Phase Measurement of Laser Images of Histological Sections …

63

Table 4.9 Time dynamics of changes in the statistical moments of the distribution of phase shifts in the image of a histological section of the spleen tissue (λ2 = 0.414µm) (n = 98) 3

T

6

9

12

15

18

M1

0.45

0.34

0.29

0.14

0.13

0.10

M2

0.49

0.44

0.36

0.22

0.18

0.16

M3

7.99

6.24

4.32

2.18

1.07

0.87

M4

11.58

8.35

6.27

4.88

2.31

1.05

Table 4.10 Statistical moments of the distributions of the extrema of the power spectra of the phase maps (λ1 ) and (λ2 ) of the spleen tissue (n = 98) TDE 6 h.λ2

TDE 6 h.λ1 M4

4.12 ± 0.21

M4

22.74 ± 1.85

M3

3.28 ± 0.26

M3

4.96 ± 0.38

M2

0.69 ± 0.058

M2

0.60 ± 0.048

M1

0.36 ± 0.025

M1

0.23 ± 0.013

TDE 12 h.λ1

TDE 12 h.λ2

M4

1.23 ± 0.12

M4

7.68 ± 0.57

M3

0.82 ± 0.071

M3

1.68 ± 0.14

M2

0.47 ± 0.038

M2

0.41 ± 0.037

M1

0.22 ± 0.015

M1

0.11 ± 0.08

Table 4.11 Time dynamics of changes in the statistical moments of the distribution of the extrema of the power spectra of the phase maps (λ1 ) of the spleen tissue section (n = 98) T

1

2

4

6

8

10

12

19.874

18.425

10.847

4.121

3.012

2.685

1.232

S(i) S(4) S(3)

7.855

6.623

4.754

3.286

2.113

1.384

0.821

S(2)

0.854

0.812

0.745

0.698

0.633

0.574

0.478

S(1)

0.512

0.467

0.411

0.365

0.323

0.286

0.225

T

14

16

18

20

22

24

0.584

0.385

0.264

0.221

0.195

0.178

S(i) S(4) S(3)

0.384

0.321

0.274

0.218

0.176

0.138

S(2)

0.414

0.365

0.301

0.247

0.169

0.099

S(1)

0.188

0.134

0.103

0.078

0.053

0.033

64

4 Spectral Phase Measurement of Laser Images of Sections of Biological …

Table 4.12 Time dynamics of changes in the statistical moments of the distribution of the extrema of the power spectra of the phase maps (λ1 ) and (λ2 ) of the spleen tissue (n = 98) 1

2

4

6

8

10

12

S(4)

29.932

30.988

27.842

22.745

17.111

11.978

7.687

S(3)

19.984

16.008

9.142

4.968

3.325

2.366

1.684

S(2)

0.775

0.717

0.666

0.608

0.542

0.488

0.417

S(1)

0.387

0.301

0.265

0.233

0.192

0.154

0.118

T

14

16

18

20

22

24

S(4)

4.489

2.976

1.865

1.023

0.627

0.326

S(3)

1.027

0.821

0.689

0.421

0.307

0.194

S(2)

0.367

0.289

0.211

0.169

0.123

0.089

S(1)

0.098

0.074

0.061

0.043

0.033

0.021

T S(i)

S(i)

Fig. 4.11 Phase maps (λ1 ) and (λ2 ) of the large intestine tissues identified for TDE 6 h

4.1 Spectral Phase Measurement of Laser Images of Histological Sections …

65

Table 4.13 Statistical moments of the phases (λ1 ) and (λ2 ) distributions of large intestine tissue (n = 94) (λ2 )

(λ1 ) 0.19 ± 0.016

M1

0.30 ± 0.024

M1

M2

0.25 ± 0.017

M2

0.44 ± 0.036

M3

5.73 ± 0.43

M3

11.13 ± 1.04

M4

3.25 ± 0.19

M4

4.01 ± 0.36

Table 4.14 Time dynamics of changes in the statistical moments of the distribution of phase shifts of the image of a section of large intestine tissue (λ1 = 0.632µm) (n = 94) 3

T

6

9

12

15

18

M1

0.25

0.19

0.15

0.11

0.09

0.08

M2

0.31

0.25

0.18

0.13

0.11

0.12

M3

8.42

5.73

3.67

2.14

1.07

1.12

M4

4.65

3.25

2.43

2.11

1.76

1.65

Table 4.15 Time dynamics of changes in the statistical moments of the distribution of phase shifts of the image of a section of large intestine tissue (λ2 = 0.414µm) (n = 94) T

3

6

9

12

15

18

M1

0.43

0.31

0.25

0.21

0.15

0.14

M2

0.53

0.44

0.37

0.24

0.19

0.16

M3

14.78

11.13

8.34

4.78

2.19

1.81

M4

8.09

4.83

2.87

2.11

1.32

0.97

From the analysis of the data obtained, it follows: – The value of the statistical moments of the distributions of the extrema of the power spectra in the red (λ1 ) and blue (λ2 ) regions of the spectrum is individual and differs from 1.8 times (M1 .M2 ) to 2.1 times (M3 .M4 ). – Time dynamics of changes in the set of statistical moments of the distributions of the extrema of the power spectra is the highest in the red (λ1 ) and the lowest in the blue (λ2 ) regions of the spectrum (Figs. 4.12, 4.13, 4.14 and 4.15 and Tables 4.16 and 4.17). Based on the data given in Tables 4.12, 4.13, 4.14, the following intervals and accuracy of determining TDE for a histological section of large intestine tissue have been established: – λ1 = 0.632μm: T = 20 h. T = 1.5 h – λ2 = 0.414µm: T = 32 h. T = 1 h

66

4 Spectral Phase Measurement of Laser Images of Sections of Biological …

Fig. 4.12 Autocorrelation functions of phase maps (λ1 ) and (λ2 ) of large intestine tissue determined for TDE 6 h

Fig. 4.13 Power spectra of phase maps (λ1 ) and (λ2 ) of large intestine tissue determined for TDE at 6 and 12 h

4.1 Spectral Phase Measurement of Laser Images of Histological Sections …

67

Fig. 4.14 Time dependences of the average (“M1 ”), dispersion (“M2 ”), skewness (“M3 ”) and kurtosis (“M4 ”) of the power spectra of the phase images of the large intestine tissue section in different spectral ranges—λ1 = 0.632µm

Fig. 4.15 Time dependences of the average (“M1 ”), dispersion (“M2 ”), skewness (“M3 ”) and kurtosis (“M4 ”) of the power spectra of the phase images of the large intestine tissue section in different spectral ranges—λ2 = 0.414µm

68

4 Spectral Phase Measurement of Laser Images of Sections of Biological …

Table 4.16 Statistical moments of the distributions of the extrema of the power spectra of the phase maps (λ1 ) and (λ2 ) of the large intestine tissue (n = 94) TDE 6 h.λ2

TDE 6 h.λ1 M4

25.98 ± 1.75

M4

32.86 ± 2.37

M3

2.48 ± 0.14

M3

7.01 ± 0.52

M2

0.80 ± 0.052

M2

0.65 ± 0.042

M1

0.46 ± 0.025

M1

0.31 ± 0.012

TDE 12 h.λ2

TDE 12 h.λ1 M4

9.12 ± 0.75

M4

7.50 ± 0.51

M3

0.82 ± 0.057

M3

0.74 ± 0.061

M2

0.61 ± 0.048

M2

0.43 ± 0.023

M1

0.27 ± 0.015

M1

0.17 ± 0.08

Table 4.17 Time dynamics of changes in the statistical moments of the distribution of the extrema of the power spectra of the phase maps (λ1 ) of a section of the colon tissue (n = 94) T

1

2

4

6

8

10

12

S(4)

39.875

38.025

33.354

25.985

19.689

13.897

9.125

S(3)

9.375

8.026

3.453

2.484

1.891

1.819

0.827

S(2)

1.256

1.089

0.921

0.802

0.752

0.684

0.618

S(1)

0.672

0.601

0.536

0.465

0.398

0.332

0.271

T

14

16

18

20

22

24

S(4)

5.775

3.249

1.851

0.984

0.621

0.327

S(3)

5.775

3.249

1.851

0.984

0.621

0.327

S(2)

0.543

0.456

0.386

0.301

0.234

0.184

S(1)

0.217

0.173

0.127

0.081

0.064

0.042

S(i)

S(i)

4.2 Possibilities of Determination of TDE by the Method of Statistical Analysis of Time Dependences of Power Spectra of Phase Maps of Histological Sections of Human Corpse Tissues The results of the proposed method for analyzing the power spectra of the entire complex of polarization parameters of laser images of histological sections of parenchymal tissues of a human corpse were tested in laboratory conditions with a priori known time of death (myocardium infarction) and are shown in Table 4.18. Thus, the use of the correlation analysis of phase maps of sections of the main types of structured and parenchymal tissues of a human corpse in different spectral regions made it possible to obtain the maximum intervals for TDE with high accuracy for the blue laser wavelength λ2 = 0.414μm.

References

69

Table 4.18 Time efficiency T and deviation interval T of TDE by statistical methods of correlometry of power spectra of parenchymal tissues of a human corpse Biological tissue

λ1 = 0.632µm

λ2 = 0.414µm

T

T

T

T

Myocardium tissue

1–120

2

160

1.5

Skin dermis

1–110

2

140

1.5

Liver

1–20

1.5

1–28

1

Lung tissue

1–19

1.5

1–27

1

Large and small intestine wall

1–20

1.5

1–32

1

Spleen

1–22

1.5

1–32

1

Kidney

1–20

1.5

1–30

1

Brain

1–12

1.2

1–19

0.7

References 1. G. Mall, M. Eckl, I. Sinicina, et al., Temperature-based death time estimation with only partially known environmental conditions. Int. J. Legal. Med. (2004) 2. N. Lange, S. Swearer, W.Q. Sturner, Human postmortem interval estimation from vitreous potassium: an analysis of original data from six different studies. Forensic Sci. Int. 66(3), 159–174 (1994) 3. L.M. Al-Alousi, R.A. Anderson, D.M. Worster, et al., Multiple-probe thermography for estimating the postmortem interval: I. Continuous monitoring and data analysis of brain. Liver, rectal and environmental temperatures in 117 forensic cases. J. Forensic Sci. 46(2), 317–322 (2001) 4. G.M. Hutchins, Body temperature is elevated in the early postmortem period. Hum. Pathol. 16(6), 560–561 (1985) 5. T. Suzutani, Studies on the estimation of the postmortem interval. 1. The temperature of cadaver (author’s transl). Hokkaido Igaku Zasshi. 52(3), 205–211 (1977) 6. J.L. Melody, S.M. Lonergan, L.J. Rowe, et al., Early postmortem biochemical factors influence tenderness and water-holding capacity of three porcine muscles. J. Anim. Sci. 82(4), 195–11205 (2004) 7. M.A. Green, J.C. Wright, Postmortem interval estimation from body temperature data only. Forensic Sci. Int. 28(1), 35–46 (1985) 8. G. Mall, M. Hubig, M. Eckl, et al., Modelling postmortem surface cooling in continuously changing environmental temperature. Leg Med. 4(3), 164–173 (2002) 9. L.M. Al-Alousi, R.A. Anderson, D.M. Worster, et al., Factors influencing the precision of estimating the postmortem interval using the triple-exponential formulae (TEF). Part II. A study of the effect of body temperature at the moment of death on the postmortem brain. liver and rectal cooling in 117 forensic cases. Forensic Sci. Int. 125(2–3), 223–230 (2002) 10. G. Mall, M. Hubig, G. Beier, et al., Determination of time-dependent skin temperature decrease rates in the case of abrupt changes of environmental temperature. Forensic Sci. Int. 113(1–3), 219–226 (2000) 11. V.G. Kolobrodov, Q.A. Nguyen, G.S. Tymchik, The problems of designing coherent spectrum analyzers, in Proc. of SPIE, 2013, vol. 9066, p. Article number 90660N, 11th International Conference on Correlation Optics18 September 2013 through 21 September 2013, Code 103970 12. V.A. Ostafiev, S.P. Sakhno, S.V. Ostafiev, G.S. Tymchik, Laser diffraction method of surface roughness measurement. J. Mater. Proc. Technol. N63, 871–874 (1997)

70

4 Spectral Phase Measurement of Laser Images of Sections of Biological …

13. I. Chyzh, V. Kolobrodov, A. Molodyk, V. Mykytenko, G. Tymchik, R. Romaniuk, P. Kisała, A. Kalizhanova, B. Yeraliyeva, Energy resolution of dual-channel opto-electronic surveillance system, in Proceedings Volume 11581, Photonics Applications in Astronomy, Communications, Industry, and High Energy Physics Experiments (Wilga, Poland, 2020) 14. V.H. Kolobrodov, V.I. Mykytenko, G.S. Tymchik, Polarization model of thermal contrast observation objects. Thermotlectricity 1, 36–49 (2020) 15. V.H. Kolobrodov, M.S. Kolobrodov, G.S. Tymchik, A.S. Vasyura, P. Komada, Z. Azeshova, The output signal of a digital optoelectronic processor, in Proc. SPIE 10808, Photonics Applications in Astronomy, Communications, Industry, and High-Energy Physics Experiments 2018, 108080W (1 October 2018). 16. G.S. Tymchik, V.I. Skytsyuk, T.R. Klotchko, H. Bezsmertna, W. Wójcik, S. Luganskaya, Z. Orazbekov, A. Iskakova, Diagnosis abnormalities of limb movement in disorders of the nervous system, in Photonics Applications in Astronomy, Communications, Industry, and High-Energy Physics Experiments (2017), pp. 104453S-104453S-11. https://doi.org/10.1117/12.228100 17. H. Zhengbing, M. Ivashchenko, L. Lyushenko, D. Klyushnyk, Artificial neural network training criterion formulation using error continuous domain. Int. J. Modern Educ. Comp. Sci. (IJMECS) 13(3), 13–22 (2021). https://doi.org/10.5815/ijmecs.2021.03.02 18. H. Zhengbing, I. Tereikovskyi, D. Chernyshev, L. Tereikovska, O. Tereikovskyi, D. Wang, Procedure for processing biometric parameters based on wavelet transformations. Int. J. Modern Educ. Comp. Sci. (IJMECS) 13(2), 11–22 (2021). https://doi.org/10.5815/ijmecs.2021.02.02 19. H. Zhengbing, R. Odarchenko, S. Gnatyuk, M. Zaliskyi, A. Chaplits, S. Bondar, V. Borovik, Statistical techniques for detecting cyberattacks on computer networks based on an analysis of abnormal traffic behavior. Int. J. Comp. Netw. Inform. Sec. (IJCNIS) 12(6), 1–13 (2020). https://doi.org/10.5815/ijcnis.2020.06.01 20. H. Zhengbing, S. Gnatyuk, T. Okhrimenko, S. Tynymbayev, M. Iavich, High-speed and secure PRNG for cryptographic applications. Int. J. Comp. Netw. Inform. Sec. (IJCNIS) 12(3), 1–10 (2020). https://doi.org/10.5815/ijcnis.2020.03.01 21. H. Zhengbing, I. Dychka, M. Onai, Y. Zhykin, Blind payment protocol for payment channel networks. Int. J. Comp. Netw. Inform. Sec. (IJCNIS) 11(6), 22–28 (2019). https://doi.org/10. 5815/ijcnis.2019.06.03 22. H. Zhengbing, Y. Khokhlachova, V. Sydorenko, I. Opirskyy, Method for optimization of information security systems behavior under conditions of influences. Int. J. Intell. Syst. Appl. (IJISA) 9(12), 46–58 (2017). https://doi.org/10.5815/ijisa.2017.12.05 23. H. Zhengbing, S.V. Mashtalir, O.K. Tyshchenko, M.I. Stolbovyi, Video shots’ matching via various length of multidimensional time sequences. Int. J. Intell. Syst. Appl. (IJISA) 9(11), 10–16 (2017). https://doi.org/10.5815/ijisa.2017.11.02 24. H. Zhengbing, I.A. Tereykovskiy, L.O. Tereykovska, V.V. Pogorelov, Determination of structural parameters of multilayer perceptron designed to estimate parameters of technical systems. Int. J. Intell. Syst. Appl. (IJISA) 9(10), 57–62 (2017). https://doi.org/10.5815/ijisa.2017.10.07 25. H. Zhengbing, Y.V. Bodyanskiy, N.Y. Kulishova, O.K. Tyshchenko, A multidimensional extended neo-fuzzy neuron for facial expression recognition. Int. J. Intell. Syst. Appl. (IJISA) 9(9), 29–36 (2017). https://doi.org/10.5815/ijisa.2017.09.04 26. H. Zhengbing, I. Dychka, Y. Sulema, Y. Radchenko, Graphical data steganographic protection method based on bits correspondence scheme. Int. J. Intell. Syst. Appl. (IJISA) 9(8), 34–40 (2017) https://doi.org/10.5815/ijisa.2017.08.04

Conclusions

The monograph proposes and substantiates a complex of new forensic methods and objective criteria for establishing the antiquity of the onset of death by spectral photopolarimetric and phase research of temporal dynamics after death changes in laser images in histological sections of biological tissues of a human corpse based on statistical and correlation analysis of the data obtained. The following conclusions were made: 1. The relationships between the phase (distribution of phase shifts) parameters of laser images of histological sections of the main types of biological tissues and organs of the corpse in various spectral ranges and the temporal dynamics of their post-mortem morphological changes have been investigated and analyzed. 2. A method of two-wave phase measurement was developed, based on the statistical and correlation analysis of the temporal dynamics of cadaveric changes in phase maps of laser images of histological sections of biological tissues in different spectral ranges. (λ1 = 0.632 µm and λ2 = 0.414µm). On its basis, the maximum TDE determination interval was realized for structured (1 h–120 h, myocardium tissue) and parenchymal tissues (1 h–32 h, spleen tissue).

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 Z. Hu et al., Phase Mapping of Human Biological Tissues, SpringerBriefs in Applied Sciences and Technology, https://doi.org/10.1007/978-981-99-3269-6

71