Advanced System Development Technologies I (Studies in Systems, Decision and Control, 511) [1st ed. 2024] 3031443462, 9783031443466

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Studies in Systems, Decision and Control 511

Mykhailo Bezuglyi · Nadiia Bouraou · Volodymyr Mykytenko · Grygoriy Tymchyk · Artur Zaporozhets   Editors

Advanced System Development Technologies I

Studies in Systems, Decision and Control Volume 511

Series Editor Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences, Warsaw, Poland

The series “Studies in Systems, Decision and Control” (SSDC) covers both new developments and advances, as well as the state of the art, in the various areas of broadly perceived systems, decision making and control–quickly, up to date and with a high quality. The intent is to cover the theory, applications, and perspectives on the state of the art and future developments relevant to systems, decision making, control, complex processes and related areas, as embedded in the fields of engineering, computer science, physics, economics, social and life sciences, as well as the paradigms and methodologies behind them. The series contains monographs, textbooks, lecture notes and edited volumes in systems, decision making and control spanning the areas of Cyber-Physical Systems, Autonomous Systems, Sensor Networks, Control Systems, Energy Systems, Automotive Systems, Biological Systems, Vehicular Networking and Connected Vehicles, Aerospace Systems, Automation, Manufacturing, Smart Grids, Nonlinear Systems, Power Systems, Robotics, Social Systems, Economic Systems and other. Of particular value to both the contributors and the readership are the short publication timeframe and the worldwide distribution and exposure which enable both a wide and rapid dissemination of research output. Indexed by SCOPUS, DBLP, WTI Frankfurt eG, zbMATH, SCImago. All books published in the series are submitted for consideration in Web of Science.

Mykhailo Bezuglyi · Nadiia Bouraou · Volodymyr Mykytenko · Grygoriy Tymchyk · Artur Zaporozhets Editors

Advanced System Development Technologies I

Editors Mykhailo Bezuglyi Igor Sikorsky Kyiv Polytechnic Institute National Technical University of Ukraine Kyiv, Ukraine

Nadiia Bouraou Igor Sikorsky Kyiv Polytechnic Institute National Technical University of Ukraine Kyiv, Ukraine

Volodymyr Mykytenko Igor Sikorsky Kyiv Polytechnic Institute National Technical University of Ukraine Kyiv, Ukraine

Grygoriy Tymchyk Igor Sikorsky Kyiv Polytechnic Institute National Technical University of Ukraine Kyiv, Ukraine

Artur Zaporozhets General Energy Institute National Academy of Sciences of Ukraine Kyiv, Ukraine

ISSN 2198-4182 ISSN 2198-4190 (electronic) Studies in Systems, Decision and Control ISBN 978-3-031-44346-6 ISBN 978-3-031-44347-3 (eBook) https://doi.org/10.1007/978-3-031-44347-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Paper in this product is recyclable.

Preface

The monograph is dedicated to the 125th anniversary of the Igor Sikorsky Kyiv Polytechnic Institute. Over the course of more than 100 years of its existence, this university has formed dozens of globally renowned scientific schools and developed hundreds of scientific directions. Research devoted to the creation of precise systems and technologies occupies a prominent place among them. The monograph reflects some of the results obtained at the Igor Sikorsky Kyiv Polytechnic Institute in this field. The chapters of the monograph are grouped into three directions: optoelectronic technologies, medical technologies, and technological support. The first chapter is dedicated to some relevant problems in the development of optoelectronic technology. It covers issues related to the modeling and design of infrared systems, multi-channel optoelectronic observation systems, automation of optical system calculation, and ophthalmological systems. Modern thermal imaging systems are widely used for object detection and recognition. In developed countries, the intensive development of vehicle design theory and methods continues, which is connected not only with the improvement of their elemental base but also with the need to increase the amount of information processed in real time. The quality of the image displayed on the screen is determined by the spatial and energy resolution of the thermal imager. Therefore, developing mathematical models to calculate the spatial and energy resolution of thermal imagers is an important task. This chapter examines the vehicle model and its components, including the optical system, detector, display, and operator’s visual analyzer. Based on this model, methods for calculating the spatial and energy resolution of thermal imagers are proposed. Examples of calculating the modulation transfer function, equivalent to the noise of the temperature difference, and the minimum resolvable temperature difference are considered. Improving the performance of electro-optical surveillance systems (EOSS) with image fusion requires the solution of several scientific and technical problems. The main challenges include the establishment of scientific bases and the application of methods for analysis, synthesis, and adjustment of information channels on a uniform methodological basis. The methodological bases for improving the EOSS power consumption with image fusion are scientifically substantiated. They include v

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Preface

the methods of constructive harmonization of the EOSS’s main unit’s characteristics, the mechanism of the adaptive selection of the best method of spectral image merging, and the means of experimentally determining the EOSS’s main characteristics, which allow for an increase in the performance of the system. Designing an arbitrary multi-element optical system is a complex and multifaceted task. It cannot be solved analytically and requires significant experience and effort from the designer. It is shown that with the help of the developed specialized software with implemented modern global optimization algorithms, it is possible to automate the process of optical design. In particular, this approach has been successfully tested in the design of various optical systems with constant and variable parameters. The results obtained confirm the high image quality achieved in the developed systems, which include an eyepiece for a microdisplay, a classic fast prime lens, a SWIR lens, a zoom optical sight, and an aspherical catadioptric lens. The second chapter is devoted to systems and methods of analysis of the state of biological objects and their modeling. The photometric analysis method for simultaneously considering the optical properties of biological media in reflected and transmitted light within the total solid angle was presented in this chapter and given scheme-technical versions of the informationmeasurement system for spatial photometry. The setup element synthesis, exploitation, and calibration features are described. The light scattering indicatrix analysis results by different thickness muscle tissue samples with the transverse and longitudinal fibers placement obtained from experimental studies at goniometric type setup were shown. The identity of experimentally determined values of the diffuse reflection and total transmission coefficients was confirmed by the Monte Carlo simulation results by sections of the single scattering anisotropy factor average values. Preconditions of ellipsoidal reflectors manufacturing and use in the light scattering detecting systems, researching, and testing of new constructive, functional, and biomedical solutions are presented in this chapter. The method and equipment with ellipsoidal reflector theoretical and experimental features for light scattering photometry by biological media at the detection of their optical properties and identification of the physical and physiological state are presented. The fundamental principles of ray tracing in ellipsoidal reflectors are presented. They interconnect the parameters of the radiative source, biological media, ellipsoidal mirror, and CCD camera in the information and measurement system of the biomedical photometer, which allows solving the direct and inverse problems of light propagation in tissue by Monte Carlo simulation. This chapter presents the theoretical foundations and research principles for creating automated systems for testing variable magnetic field devices. The research includes the principles of precise technical sources, systems, and devices that generate a magnetic field, methods, and tools for increasing their efficiency, determination, and evaluation of magnetic field systems parameters. The principal designs and calculating methods parameters of inductors are considered. Special attention is paid to the structural features of the construction of automated systems for testing variable magnetic field devices and complexes, increasing the efficiency and accuracy reproduction of magnetic field metrological parameters.

Preface

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The creation features of a myographic system of the prosthetic wrist with advanced movements and gesture capabilities based on the myoelectric signals recognition by a neural network interface and the contact surface of the fingers touch optical identifier are presented in this chapter. The method of prosthesis movement coordination based on simultaneous registration and recognition of physiological and optical signals is presented. The method of recognizing categories of physiological movements by analyzing electro- and force-myographic signals on the base theoretical and experimental features of multilevel artificial neural networks is presented. The method of recognizing the contact surface by a prosthesis finger with an optical energy mirror concentrator has been shown, which made it possible to increase the identification reliability of the manipulation object’s structure. In the third chapter, issues related to various aspects of automating the determination of the state of complex objects and their management are considered. Condition monitoring systems of complex rotating objects use diagnostic information obtained directly during the operation. These are data on the parameters and characteristics of dynamic processes, among which vibrational and vibroacoustic processes occurring in rotating systems during operation should be highlighted. The trends in the development of vibrodiagnostic diagnostic methods involve the improvement of both their means (sensors, transducers, etc.), as well as methodological and algorithmic support for the analysis of vibration and acoustic signals. The Chapter presents the results of the development of a new methodology for vibroacoustic monitoring and diagnosis of initial damage of elements of rotating systems at the steady-state and non-steady-state modes of operation. Crack-like damage to rotor blades and shafts is considered. To increase the reliability of diagnostics, it is proposed to perform multi-level processing of the signals radiated by the rotating rotor, with the step-by-step application of various analysis methods for feature detection of initial damage. The effectiveness of the proposed approach is confirmed by the results of mathematical and physical experiments. Modern approaches to the synthesis of algorithmic support for a strapdown inertial attitude system are considered. The general characteristics of navigation systems and the composition of their inertial measurement unit are given. The methods of initial alignment of the system on a stationary base are described. Particular attention is paid to the attitude kinematic parameters of the body frame and methods of their numerical integration. Picard’s methods for integrating the Bortz and Poisson kinematic equations are shown. An algorithm for a strapdown inertial attitude system based on using real signals of high-precision laser gyroscopes is proposed. System simulation was carried out using the proposed algorithmic methods.

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Preface

The relevance of the control system created for the processing device parts in the conditions of automated manufacturing is substantiated. Theoretical studies are presented, and the relation between electrical signals, the level of tool wear, and the main reasons for generating electrical signals are identified. A mathematical model of cutting tool wear control was developed based on measuring the variable component of cutting electromotive force. A control system for processing device parts on computer numerical control machines in automated production conditions has been developed. It allows for recording critical wear and breakage of the cutting tool, performing its dimensional adjustment directly on the device, and carrying out its industrial approval in flexible production systems. Kyiv, Ukraine June 2023

Mykhailo Bezuglyi Nadiia Bouraou Volodymyr Mykytenko Grygoriy Tymchyk Artur Zaporozhets

Contents

Electro-optical Technologies Mathematical Models for Calculating the Spatial and Energy Resolution of Thermal Imagers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Valentin Kolobrodov

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Electro-Optical Surveillance Systems for Unmanned Ground Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Volodymyr Mykytenko

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Automated Design of Multi-element Optical Systems for Various Purposes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vyacheslav Sokurenko

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Biomedical Technologies Ellipsoidal Reflectors for Biological Media Light Scattering Photometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Mykhailo Bezuglyi Biological Tissues Axial Anisotropy Spatial Photometry . . . . . . . . . . . . . . . 155 Natalia Bezugla Myographic System of the Bionic Wrist with Surface Type Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 Kostiantyn Vonsevych Automated Devices and Methods for Reproducing an Alternating Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Mykola Tereshchenko and Grygoriy Tymchyk

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Technological Support Information Provision for Monitoring the Current State of Electric Power Facilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Vitalii Babak, Artur Zaporozhets, Svitlana Kovtun, Yurii Kuts, Mykhailo Fryz, and Leonid Scherbak Methodology of Vibroacoustic Monitoring and Diagnosis of Initial Damage of Elements of Rotating Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 Nadiia Bouraou Details Processing Control System at the Automated Manufacturing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 Vadym Shevchenko Developing of Strapdown Inertial Attitude System . . . . . . . . . . . . . . . . . . . . 381 Oleksandr Sapehin

About the Editors

Prof. Mykhailo Bezuglyi received the B.Sc. and M.Sc. degrees in laser and optoelectronic technic from the Kyiv Polytechnic Institute in 2000 and 2002, respectively. He received his Philosophy Doctoral and Science Doctoral degrees in Biological and Medical Devices and Systems from the National Technical University of Ukraine, “Igor Sikorsky Kyiv Polytechnic Institute,” in 2008 and 2020, respectively. He obtained the Associate Professor and Professor titles at the Department of Manufacturing Technologies of Devices in 2012 and 2022, respectively. From 2022 he will be the chairman of the Department of Computer-Integrated Manufacturing Technologies of Devices at Igor Sikorsky Kyiv Polytechnic Institute. He has authored over 200 published papers and patents on Optics, Biophotonics, and Information Systems. His research interests include designing, producing, and controlling optoelectronic and automated systems for biomedical areas. Prof. Nadiia Bouraou is working as Head of the Department of Computer-Integrated Optical and Navigation Systems of the National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”. She graduated from the Department of Instruments and Systems of Orientation and Navigation of the Kyiv Polytechnic Institute in 1981. She obtained her Ph.D. in 1991 and Dr. Sc. (Eng.) in 2006. The title of Associate Professor she obtained in 1998, and the title of Professor she obtained in 2008. Her professional activity is associated with the National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute” (1981–till now). Since 2008 she led 15 research projects funded by the Ukrainian Ministry of Education and Science and by various Ukrainian Scientific and Technical Enterprises. Since 2000, she taught the following courses: undergraduate courses in Mathematical Models of Physical Processes and Theory of Control Systems; master’s courses in Diagnostics and Reliability of Devices and Systems and Basics of the Scientific Research; doctoral course in Methodology of Scientific Research for students of the Faculty of Instrumentation Engineering of the National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”. She has published more than 230 scientific works, over 30 articles in international peer-reviewed journals and conference proceedings. Her current research interests include Structure Health Monitoring, xi

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About the Editors

Signal Processing and Decision Making, Intelligent Moving Objects, Orientation, Navigation, and Control Systems. Prof. Volodymyr Mykytenko graduated from the Optical Devices Department of the Kyiv Polytechnic Institute in 1985. In 1991, he received his Ph.D., and in 2020, he obtained a Dr. Sc. (Eng.) degree. He was granted the title of Associate Professor in 2004 and the title of Senior Researcher in Optics in 2006. In 2022, he was appointed as a Professor at the Department of Computer-Integrated Optical and Navigation Systems of the National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute,” where he currently holds the position of a professor. In addition to his teaching activities, he also provides scientific consulting services to industrial enterprises. He has participated in the implementation of several dozen scientific projects commissioned by Ukrainian and foreign companies and organizations, including NATO. He has around 200 publications, including 9 monographs, 19 patents, and 13 articles in publications indexed in the Scopus and Web of Science scientific databases, over 70 articles in Ukrainian peer-reviewed journals. The main directions of his research are computer-integrated optoelectronic remote sensing systems, methods of information fusion in multi-channel surveillance systems, and methods for evaluating image quality in optoelectronic surveillance systems. Prof. Grygoriy Tymchyk is working as Dean of the Faculty of Instrumentation of the National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Doctor of Technical Sciences, Professor. Specialist in the field of optical instrumentation and laser technology. Under the scientific guidance of Prof. Grygoriy Tymchyk and with his participation, 47 contractual and state budget research projects were carried out aimed at creating specialized laser information-measuring systems for monitoring the technical condition and parameters of dynamic systems and objects. He created a new class of laser spectral-correlation systems based on passive Fabry– Pierrot resonators for operational control of quality parameters of parts and the state of technological equipment in precision instrumentation. The results of the research carried out were published in 654 scientific papers and 48 monographs and textbooks, 115 inventions. Under the scientific guidance of Prof. Grygoriy Tymchyk defended 14 dissertations for the degree of technical sciences, two postgraduate students are studying and two applicants for scientific degrees are working. Dr. Artur Zaporozhets is working as Deputy Director in the General Energy Institute of the National Academy of Sciences of Ukraine. He graduated from the Applied Physics Department of the National Aviation University in 2013. He obtained his Ph.D. in 2017 and Dr. Sc. (Eng.) in 2022. The title of Senior Researcher in Metrology and Information-Measuring Technology he obtained in 2019. He is a Laureate of the Medal “For work and achievement” (2022), Award of the NAS of Ukraine for Young Scientists (2022), Award of the Verkhovna Rada of Ukraine to Young Scientists (2021), Award of the President of Ukraine for Young Scientists (2019), and others. His main professional activity is associated with institutions of the NAS of Ukraine: G. V. Kurdyumov Institute of Metal Physics (2009–2011), Institute of Engineering

About the Editors

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Thermophysics (2013–2021), General Energy Institute (2022–till now). Since 2013, he took part in 22 research projects funded by the NAS of Ukraine and the Ukrainian Ministry of Education and Science. In 2014–2016, he taught undergraduate courses in Advances in Information Systems, Software Engineering and Programming Technologies, Information and Measuring Systems, Metrology, Biomedical Methods, and Laboratory workshops for students of the Applied Physics Department of the National Aviation University. He has published more than 200 scientific works, among them 14 books in Springer, over 50 articles in international peer-reviewed journals, and 60 conference proceedings. His current research interests include Energy Informatics, Power Equipment Diagnostics, Environmental Monitoring, Algorithms and Data Structures, Big Data, and Data Processing.

Electro-optical Technologies

Mathematical Models for Calculating the Spatial and Energy Resolution of Thermal Imagers Valentin Kolobrodov

1 Introduction Thermal imagers and thermal imaging systems for various applications are devices and systems that belong to critical technologies. The level of their development is directly related to the economic potential and defense capability of the state. Modern thermal imagers (TI) are widely used for searching, detecting and recognizing objects in space at a limited range, and measuring their parameters, such as spatial coordinates, speed of movement and dimensions [1–4]. In recent years, TI have received a new development, which has a revolutionary character and is associated with the creation of uncooled infrared detectors [5–8]. In many developed countries, intensive development of the theory and methods of design TI, which is connected not only with the improvement of their elemental base, but also with the need to take into account a set of new factors. This is, for example, a sharp increase in the amount of information that needs to be processed in real time, features of high-resolution spatial and temporal resolution, etc. These countries are constantly increasing research costs in the field of thermal imaging technology for various applications. For example, the US Department of Defense annually invests more than a billion dollars in the development of TI. This chapter is devoted to issues of design TI, namely the development of a mathematical model for calculating the spatial and energy resolution of thermal imagers. The chapter presents the results of the author’s many years of research during the performance of research works on the development of design methods for new thermal imaging systems for military, space and civilian applications commissioned by various ministries and organizations. The research results were embodied in unique textbooks on thermal imaging with the seal of the Ministry of Education of V. Kolobrodov (B) National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 M. Bezuglyi et al. (eds.), Advanced System Development Technologies I, Studies in Systems, Decision and Control 511, https://doi.org/10.1007/978-3-031-44347-3_1

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Ukraine (V. G. Kolobrodov, N. Schuster (Germany) Thermal imaging systems (physical foundations, design and control methods, application) and V. G. Kolobrodov, Lyholit M. I. Design of thermal imaging and television surveillance systems). These textbooks were recognized as the best textbooks of NTUU “KPI”. As well as foreign publications [9, 10]. A scientific school on thermal imaging was created, within which 5 doctoral theses and 9 candidate theses were defended under the author’s supervision. Thermal imaging systems consist of the main elements: an optical system (OS) in the form of an infrared lens, a detector, analog and digital video signal processing units and a display. The complexity of transforming information from objects and backgrounds to the operator’s eyes makes it necessary to consider TI at the level of a common system approach using the methods of analysis of linear invariant optical-electronic systems described in textbooks and monographs [2, 5, 8, 11]. One of the most important characteristics of a TI is the maximum operating range, which depends on many factors of both the system itself and the environment. The quality of the thermal image formed on the display screen is determined by spatial and energy resolutions. The process of image perception by the operator has a probabilistic nature. Therefore, the TI can also be characterized by the maximum detection range and the maximum recognition range of objects at a given probability or the probability of detection and recognition. The monograph will be useful for students, masters and postgraduates of optical and radio technical specialties and scientific and engineering workers of optical and electronic instrumentation. Monitoring the process of fuel combustion is reduced to controlling the content of flue gases, while the objects of research are the boiler and the air-fuel path [1]. The block diagram of monitoring the process of burning fuel is shown in Fig. 1.

Fig. 1 Formation of the image in the TI system

Mathematical Models for Calculating the Spatial and Energy Resolution …

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2 Generalized Model of the Thermal Imager General Provisions and Definitions Generalized characteristics of thermal imagers (TI) or quality criteria are a measure of successful performance of tasks facing the system and take into account various tactical-technical, operational and technical-economic requirements. In most cases, quality criteria are used in practice, which can be divided into the following groups [2]: – – – –

quality criteria of the image formed by the TI; technical parameters and characteristics of the system; indicators of the effectiveness of achieving the goal for which the TI was created; technical and economic indicators.

Image quality is determined by indicators such as geometric (spatial) resolution, or angular and linear resolution, modulation transfer function (MTF). Tactical and technical characteristics determine the possibility of solving the tasks facing the TI in various operating conditions. The main ones are: noise equivalent temperature difference (NETD) and minimum resolvable temperature difference (MRTD). These indicators also include: sensitivity; the dynamic range of radiance perceived by the TI; operating spectral range; angular field of view; instantaneous field of vision; image format (frame); the number of resolution elements and some others. Efficiency indicators of TI reflect the statistical nature of solving the problems facing these systems. These indicators include, first of all, the maximum range of action, detection, recognition, identification or the probability of detection, recognition, identification. The technical and economic indicators include the estimated cost of the system, its manufacturability, weight and dimensions, time to failure, power consumption, and some others. The above classification of quality indicators is conditional, since many of these indicators depend on each other. Some of these indicators depend not only on the TI, but also on the operating conditions of the system and the observer (operator). Considering the variety of tasks solved by TI, spatial, energy, temporal and spectral resolution is used to characterize their quality. For most vehicles operating in static mode, it is important to provide high spatial and energy resolution. For systems operating in dynamic mode, in addition to high spatial and energy resolution, it is also important to provide temporal resolution. Generalized characteristics and properties are required to characterize the efficiency of the TI as a whole. They are determined by the following factors: 1. Modulation transfer function of the TI without system noise. 2. The limit value of the noise of the TI. 3. Combination of image quality with system noise.

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For a TI, the main generalized characteristics are the maximum detectable range (MDR) and the maximum recognizable range (MRR) with given detection and recognition probabilities. These characteristics depend on the spatial resolution; energy resolution; signal transfer functions; transfer function; illumination equivalent to noise; noise-equivalent temperature difference; minimum detective temperature difference; minimum resolution temperature difference; spectral operating range; dynamic range. Maximum Detectable Range Rd (MDR) is the maximum distance between the TI and the standard test object at which the test object is detected on the display screen with a given probability Pd in the case of unlimited observation time. Maximum Recognizable Range Rr (MRR) is the maximum distance between the TI and the standard test object at which the test object is recognized on the display screen with a given probability Pr in the case of unlimited observation time. For thermal imager systems, the quality of the formed image, which is subjectively determined by the observer, is of great importance. The perceptual quality of the same image differs significantly among different observers, and also changes over time for the same observer. Therefore, the quality of the image cannot be evaluated in absolute values. There are numerous formulas for evaluating image quality, each of which is empirically derived for specific observation conditions. All of them to one degree or another use two main characteristics—spatial and energy resolution [2].

Generalized Model of Thermal Imaging Monitoring System The principle of forming a thermal image is as follows. Radiation from the object of observation and the background passes through the atmosphere and enters the entrance pupil of the infrared lens of the thermal imager. The optical system forms an image of the object and the background in the detector plane. The detector converts the radiation flux that forms the image into an electrical video signal, which, after amplification, is sent to analog and digital processing devices. After the necessary transformations, the video signal is sent to the display, on the screen of which a visible analogue of the object and the background perceived by the operator is formed. The main task of the TI is to transform the invisible infrared radiation of thermal contrast objects of observation into an analogue of the visible image on the display screen for their detection, recognition and research. Since the thermal image is formed mainly due to the own radiation of objects and backgrounds, which have a different distribution of temperature and radiation coefficient on the surface, it is significantly different from its counterpart in the visible region of the spectrum. For example, a thermal (infrared) image has no shadows, which complicates the spatial perception of objects. In addition, for most objects and backgrounds, there is no relationship between the reflectance coefficient and the temperature field and the radiation coefficient. However, the thermal field usually corresponds to the shape and size of the object, which allows you to recognize objects by their contours.

Mathematical Models for Calculating the Spatial and Energy Resolution …

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Although the considered generalized model of the TI is obvious to specialists of opto-electronic instrumentation, insufficient consideration of information transformation processes in individual elements of the system can cause serious errors during the design of the TI. It is also important to note that the creation of a sufficiently high-quality system is possible only if the parameters of all TI chains are taken into account, including the parameters and characteristics of the objects of observation, obstacles, backgrounds, atmosphere, as well as the observer operator. It is proposed to use the theory of linear systems and spatial-temporal spectral analysis to research the spatial and energy resolutions of TI. For this purpose, mathematical models of individual components of the generalized vehicle model are proposed. Objects and backgrounds. Any energy value of radiation can be used as an energy parameter of objects and backgrounds (radiant flux Φe , radiant intensity Ie , emittance Me , irradiance E e , radiance L e ), brought either to the plane of objects, or to the plane of the entrance pupil of the lens, or to the image plane. At the same time, both spectral and integral values are used. For the quantitative assessment of the background and target situation, the following are most often used: Absolute contrast of energy radiance ΔL e = L et − L eb ,

(1)

{λ {λ where L et = λ12 L et (λ)dλ) and L eb = λ12 L eb (λ)dλ) are the integral energy radiance of the object and the background. Contrast of energy illumination in the entrance pupil of the lens ΔE ep = E ept − E epb ,

(2)

{λ {λ where E et = λ12 E ept (λ)dλ), E epb = λ12 E epb (λ)dλ) are the integral illuminances in the entrance pupil of the optical system, which are formed by the surfaces of objects and backgrounds contained within the immediate field of vision. Temperature contrast ΔT = Tt − Tb ,

(3)

where Tt , Tb are the temperature of the object and background surfaces that exist within the TI systems field of view. Sometimes, for a point object, its energy spectral or integral power of radiation is specified {λ2 Iet =

Iet (λ)dλ), λ1

W . sr

(4)

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At the same time, the illumination of the entrance pupil will be equal E ept =

W τ A (R) τ A (R) Iet , , Iet = R2 cm2 Ao Ωo

(5)

where τ A (R) is the atmospheric transmittance coefficient; Ωo is the solid angle within which the radiation from the object reaches the entrance pupil of the lens; Ao the area of the entrance pupil of the lens; R is the distance from the object to the TI system (Fig. 1). Establish a relationship between the energy parameters ΔL e , ΔE ep , ΔT provided that the object and the background radiate (reflect) according to the Lambert law and are “gray” bodies. Let’s express the energy radiance contrast ΔL e in terms of the temperature contrast ΔT : 1 ΔL e = π =

1 π

{λ2 λ1

1 ΔMe (λ)dλ = π

{λ2 [Mλt (Tb + ΔT ) − Mλb (Tb )]dλ = λ1

(6)

{λ2 [εt (λ)Mλ (Tb + ΔT ) − εb (λ)Mλ (Tb )]dλ. λ1

Let’s make some simplifications in the general Formula (6), assuming that the object and the background have close radiation coefficients εt ≈ εb and low temperature contrast ΔT ≪ Tb . Then Formula (6) can be written in the form εt εt ΔL e = ΔMe = ΔT π π

{λ2 λ1

∂ Mλ (T ) dλ. ∂T

(7)

The calculation according to Formula (7) for blackbody, if Tb = 300 K in the spectral range λ1 …λ2 = 8…12 μm gives the result: W . cm2 sr ( ) c2 → 1, therefore Since in the region of maximum radiation exp λT b ΔL e = 6.3 × 10−5 ΔT

C2 λTb2

∂ Mλ (λ,Tb ) ∂T

≈

Mλ (λ, Tb ) [9]. Therefore, Formula (7) can be written in an approximate form

ΔL e = ΔT

εt c2 π Tb2

{λ2 λ1

1 Mλ (Tb )dλ. λ

(8)

Mathematical Models for Calculating the Spatial and Energy Resolution …

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Determine the illumination of the entrance pupil of the lens E ep . If the surface of the emitting object is located perpendicular to the axis of observation, then the spectral flux of radiation enters the entrance pupil of the optical system. Φλ (λ) = τ A (λ)L λt (λ, )Ωo At =

W 1 τ A (λ)Mλ (λ, Tt )Ωo At , m, π μ

(9)

where τ A (λ) is spectral transmittance coefficient of the atmosphere; At is the area of the object of large dimensions, which is contained within the immediate field of vision of the TI. Then the integral energy illumination of the entrance pupil

E ep

Ωo Φe At = = Ao Ao

{λ2 λ1

εt At τ A (λ)L λt (λ)dλ = π R2

{λ2 τ A (λ)Mλ (Tt )dλ.

(10)

λ1

From Formulas (8), (9) and (10) we establish the relationship between the energy illumination contrast ΔE ep in the entrance pupil of the OS and the energy brightness contrast ΔL e ΔL e ΔE ep

At τ A = 2 R =

At τ A R2

{λ2 [L t (λ, Tb + ΔT ) − L b (λ, Tb )]dλ λ1

{λ2 ΔL λ (λ)dλ = λ1

At τ A ΔL e , R2

(11)

where τ A is the average atmospheric transmittance coefficient in the operating spectral range λ1 …λ2 . From Formulas (2), (3) and (11) we can establish a connection between the contrast of energy illumination ΔE ep at the entrance of the lens and the temperature contrast ΔT ΔE ep

At = π R2

{λ2 τ A (λ)[εt (λ)Mλ (λ, Tb + ΔT ) − εb (λ)Mλ (λ, Tb )]dλ

(12)

λ1

If Formula (8) is valid, then relation (12) will have a simpler form:

ΔE ep = ΔT

εt c2 At π Tb2 R 2

{λ2 λ1

1 τ A (λ) Mλ (Tb )dλ. λ

(13)

When using Formulas (11) and (13), it is necessary to remember that they are valid for large objects. From Fig. 1 we have the ratio At : R 2 = A D : f o , 2, where

10

V. Kolobrodov

A D is the area of the sensitive element of the detector; f o, is a focal length of the lens. Then Formulas (11) and (13) are written in the form ΔE ep =

ΔE ep

ADτA ΔL e ; fo ,2

εt c2 A D = ΔT π Tb2 f o , 2

{λ2 λ1

1 τ A (λ) Mλ (Tb )dλ. λ

(14)

(15)

During the design of optical-electronic surveillance systems (OESS), not absolute values of contrasts play an important role, but relative ones. In the general case, the contrast in the observation plane is defined as the ratio of the difference ΔL e of the integral energy brightness of the object L et and the background around L eb to their sum: CL =

L et − L eb . L et + L eb

(16)

For objects of observation, the spatial distribution of brightness L et (x, y) is the deterministic function, and for backgrounds L eb (x, y) is the random function. The division of the brightness of objects and backgrounds can be considered both in the area of spatial coordinates x, y and in the area of spatial frequencies νx , ν y . The spatial spectrum of the object brightness distribution is defined as [ L et (νx , ν y ) =

{∞ {∞

{ ( )] L et (x, y) exp − j2π νx x + ν y y d xd y.

(17)

−∞ −∞

Optical system. Optical systems (OS) of the TI perform the functions: 1. Form an image in the detector plane. At the same time, the necessary spatial resolution (image quality) and energy resolution (luminosity) must be ensured. 2. Provide spatial analysis in a given field of view. The first function is performed by the lens, and the second by the scanning system. The main requirement for OS is to have a large (sufficient) transmittance coefficient in the operating spectral range λ1 …λ2 . The OS design process consists of overall, energy and aberration calculations. The main task of dimensional calculation is the determination of focal lengths, light diameters and mutual distances between individual components (elements) of the OS [10]. Formulas of the ideal operating system are used for preliminary overall calculation. The ability of the lens to concentrate radiation energy is characterized by energy , , which is formed on the sensitive surface of detector at the specified illumination E e0 energy brightness of the object L et . For the axial point of the axisymmetric optical

Mathematical Models for Calculating the Spatial and Energy Resolution …

11

system of the lens, the spectral illuminance is determined by the formula π τ A (λ)τ O (λ)Iλt (λ) , 4ke2f f

E e0 (λ) =

(18)

where τ A (λ) and τ O (λ) are the spectral transmittance coefficient of the atmosphere and OS; Iλt (λ) is the spectral power of the object’s radiation; ke f f is the effective f-number of the lens. The limit of spatial resolution is determined by the size of the sensitive detector area VD × W D and the focal length f O, . The smallest angle divided in the direction of the axes are equal to: δωx =

VD ; f O,

δω y =

WD . f O,

(19)

Formula (19) allows you to find the minimum possible values of spatial resolution of vehicles. A more complete characteristic of the spatial resolution of the lens is determined by the modulation transfer function (MTF). The application of the theory of linear systems to OS that create images formed the concept of the optical transfer function (OTF), as a normalized Fourier transform from the point spread function (PSF). The point spread function P S F(x , , y , ) represents the distribution of illumination in the image of an ideal point object. In the general case, the OTF is defined by an expression { {∞ HO (νx, , ν y, )

=

−∞

)] { ( P S F(x , , y , ) exp − j2π νx, x , + ν y, y , d x , dy , { {∞ , , , , , −∞ P S F(x , y )d x dy

(20)

where νx, , ν y, are the spatial frequencies of the image plane, mm−1 . | | The OTF module is named a modulation transfer function | HO (νx, , ν y, )| = M O (νx, , ν y, ) Only for a diffraction limited OS with and without central shielding can the OTF be accurately calculated [12]. The modulation transfer function of a diffraction limited lens with a circular entrance pupil diameter without central shielding is defined by the function M O (νx, )

{ =

√ − x 1 − x 2 ), i f 0 ≤ x ≤ 1; 0, i f x > 1,

2 (arccos x π

(21)

where x =λ

f O, . Dp

(22)

The maximum value of the spatial frequency is found from the equation M O (νx, ) = 0 The solution of this equation is

12

V. Kolobrodov , νx,max =

Dp . λ f O,

(23)

The normalized PSF of a diffraction limited system is determined by the function P S Fn,dl (x , , y , ) = P S Fn,dl (r , ) =

( ) D 2J1 π λ f p, r , O

D

π λ f p, r ,

.

(24)

O

The distribution of illumination in the image of a point source, which forms a diffraction limited OS, is named an Airy picture. It is a light round spot called Airy’s circle, around which light concentric rings are placed. The radius of the Airy circle, as follows from formula (24), is equal to r E = 1.22λ

f O, . Dp

(25)

The graph of the MTF of the diffraction limited OS is shown in Fig. 2. The graph of the PSF of diffraction limited OS is shown in Fig. 3. The Airy circle defines the minimum possible distance between the images of two points that are observed separately. For example, for a lens with a relative aperture D p : f O, = 1:1 for λ = 10 μm, the Airy circle has a diameter of 24.4 μm, and for λ = 0.55–1,34 μm. For off-axis points, MTF is most often determined in meridional and sagittal sections. To assess the quality of the image created by the lens, it is necessary to determine the mathematical form of the MTF. Modern computer programs allow you to calculate the MTF of a real system. The MTF of a specific OS can be measured at the appropriate installation. Fig. 2 The modulation transfer function of the diffraction limited OS (graph 1) and its linear approximation (graph 2) at the ηdi = 1

Mathematical Models for Calculating the Spatial and Energy Resolution …

13

Fig. 3 Normalized PSF of diffraction limited OS

Analytical approximations of the MTF are used for mathematical modeling of OS lenses. The author suggests that the MTF of infrared lenses be approximated by a function Mo (νx, ) = 1 − 1.22λ

f O, νx, D p ηdi

(26)

where the parameter ηdi < 1 is a very important characteristic caused by diffraction, since the appearance of the MTF depends on it. An important parameter of the OS that creates the image is the resolution νr, es , that is, the ability of the system to create a separate image of two point sources of radiation of the same intensity, placed next to each other. Spatial resolution is defined as the inverse of twice the smallest distance between two image points that are formed separately. Detector. The detector is the most important element of the TI, where it transforms weak radiation flux with low contrast into an electrical signal. Sensitivity R D is the ratio of the electrical signal (voltage u s or current i s ) to the optical signal (radiation flux Φ,e , illumination E e, , exposure He, ), which falls on the sensitive area of detector and creates an electrical signal: RD =

us . Φ,e

(27)

Spectral sensitivity R D (λ) is determined, first of all, by the material of the sensitive element of the detector. Most detectors have protective windows that act as an additional spectral filter. Frequency characteristic of the sensitivity R D ( f ) is the dependence of the sensitivity on the modulation frequency f of the radiation flux that falls on the detector. This characteristic depends on the time constant of the detector t D . Noise Power Spectrum G n ( f ) or N P S( f ) depends on the frequency f and is defined as

14

V. Kolobrodov

Gn( f ) =

d Pn ( f ) , df

(28)

where Pn ( f ) is the power of the noise signal at the output of detector. Sometimes the function N P S( f ) is defined as the normalized value of the noise spectrum N P S( f ) =

Gn( f ) . G n,max

(29)

If the function G n ( f ) = G n = const does not depend on the frequency f , then such noise is named “white”. Noise Equivalent Power (NEP) or the threshold flux Φ,e,min is such a value of the radiation flux that creates a signal at the detector output that is equal to the root mean square noise value u n in the specified frequency band: N E P = Φ,e,min =

un RD

(30)

Effective noise band Δ f characterizes the ability of the amplifier to convert the signal and the noise of the detector. It is determined by the ratio Δf =

1

{∞ 2 G n ( f )M El ( f )d f ,

G n,max

(31)

0

where M El ( f ) is the MTF of the amplifier (normalized frequency gain coefficient). The specific detection ability D ∗ characterizes the noise properties of the detector and is determined by the formula √ An Δ f cm Hz , , D = NEP W √

∗

(32)

where A D is the sensitive area of the detector, cm2 . The specific detection ability D ∗ depends on the temperature of the object Tt , the operating temperature of the detector TD , the wavelength λ and the radiation modulation frequency f. In the general case, it is impossible to approximate the function D ∗ (Tt , TD , λ, f ). Therefore, D∗ is determined for specific conditions. Usually, D ∗ (Tt ) is determined relative to blackbody. Always with an increase in the operating temperature TD , the detector noise increases. From Formulas (27) and (32) it is possible to establish a relationship between the sensitivity R D and the specific detection ability D ∗ : R D = D∗ √

un . An Δ f

(33)

Mathematical Models for Calculating the Spatial and Energy Resolution …

15

It follows from this formula that spectral D ∗ (λ) and frequency D ∗ ( f ) are the specific detection capabilities are proportional to the corresponding sensitivities R D (λ) and R D ( f ) of the detector. The modulation transfer function M D of the detector is described by the product of two functions: the spatial MTF M Ds determined by the geometric shape of the sensitive area, and the temporal MTF M Dt , determined by the inertia of the detector. The total MTF of the detector is determined by the product M D = M Ds M Dt .

(34)

The spatial MTF is the module of the normalized Fourier transform of the sensitivity distribution R D (x , , y , ) within the sensitive area of the detector. For a singleelement of the detector, the sensitivity of which is constant within the rectangular area VD × W D , the spatial MTF is determined by the function M Ds (νx, , ν y, ) =

sin(π VD νx, ) sin(π W D ν y, ) . π VD νx, π W D ν y,

(35)

, , Such MTF has the first zero value for νx01 = 1/VD and ν y01 = 1/W D . At higher spatial frequencies, the MTF becomes negative, i.e. contrast reversal occurs. During the design of the TI, the size of the sensitive site of the detector must satisfy the condition ( ) ( ) 1/νx, max VD < , (36) 1/ν y, max WD

where νx, max and ν y, max are the maximum spatial frequencies of the formed image in the detector plane along the corresponding coordinates. In most cases, stricter conditions are observed during the design of TI: (

νx, max νx, max

(

)
0.

(44)

where x = λ Depep νx ,, ; Dep are the focal length and diameter of the exit pupil of the eyepiece With the one shown in Fig. 4 of the generalized scheme of the thermal imaging monocular, it can be seen that the MTF of the lens and MBM are determined in the rear focal plane of the lens, and the MTF of the display and the eyepiece are determined in the plane of the display screen. It should also be noted that in most cases the spatial frequency νx is determined in the space of objects and measured in mrad−1 . In addition, the time MTF of the MBM and the electronic unit depend on the time frequency f . The relationship between spatial νxa and temporal f frequencies is determined by the relation [8]

18

V. Kolobrodov

Fig. 5 The relationship between angular spatial frequencies in the observation space νxa ,, and the object space νxa

f =

αD νxa , to

(45)

where α D is the angular size of the matrix pixel; to is the signal formation time of one pixel. Let’s establish a connection between the angular spatial frequency in the observation space νxa ,, and the spatial frequency in the object space νxa using Fig. 5 [9]. Suppose a Foucault gauge with a linear period of Vt p is located at a distance of R in the object plane. Then the angular period and spatial frequency are defined as αt p =

Vt p R 1 and νxa = = . αt p Vt p R

(46)

The lens forms a Foucault measure image with a linear period Vt,p and an angular spatial frequency , νxa =

f o, Vt,p

=

R = νxa . Vt p

(47)

The MBM forms on the display screen the image of the Foucault measure with the period Vt p ,, , which is observed by the operator through the eyepiece with the , . The angular period of this image and the angular spatial frequency focal length f ep are defined as αt p ,, =

, f ep Vt p ,, 1 ,, = . and ν = xa , αt p ,, Vt p ,, f ep

(48)

Taking into ratios (47) and (48), we establish a connection between the angular spatial frequencies in the observation space νxa ,, and the space of objects νxa : νxa ,, = V

,,

1 αt p ,,

=

, f ep

Vt p ,,

=

, f ep

Vt,p βel

νxa ,

(49)

where βel = Vt p, is the electronic magnification of the TI. tp The angular magnification of the “TI-operator” system is determined as (Fig. 5)

Mathematical Models for Calculating the Spatial and Energy Resolution …

Γs =

tgα t p ,, αt p ,, Vt p ,, R Vt p ,, f , f , βel ≈ = , = , o = o . , tgα t p αt p f ep Vt p f ep Vt,p f ep

19

(50)

Ratio (49) can be presented in the form νx ,, =

νxa . Γs

(51)

Operator. The main consumer of information obtained with the help of a thermal imager is the human visual analyzer (eyes of the operator). Therefore, the laws of visual image perception are decisive in the calculations of spatial and energy resolution. Visual perception is the process of holistic display of objects, backgrounds and situations that arise during the action of the light field on the organs of vision. This multi-level process includes the stages of detection, recognition and identification [10]. Detection is the selection of an object from the background by the operator by establishing only the presence of the object in the field of view without evaluating its shape and features. Recognition is the selection by the operator of some features of the object and its identification with the standard stored in memory. Identification is the selection by the operator of essential features of the object and assigning it to a certain class. Often the stages of recognition and identification are considered as one process of recognition. In the thermal imagers, the observer perceives images of objects and backgrounds from the display screen. Therefore, it is important for TI developers to know how the operator perceives the image and to what extent it should differ from the real one, so that the operator can perform the most effective detection, recognition and identification. The basis of visual perception and decision-making is the assessment (or perception) of three main parameters of the image: the size of the object, its brightness and contrast. Let’s first consider only those characteristics of visual perception that play an important role during the research of TI. The field of vision has the following sections: – very good vision—diameter 1…2° within the central fovea of the retina; – good vision—a diameter of about 9° within the yellow spot of the retina; – satisfactory vision—30° in elevation and 40° in azimuth. Good vision determines the size of the vehicle display screen, which in most cases should not exceed 9°. Satisfactory vision is explained by the 4:3 frame format adopted in television. For a normal eye, the distance of best vision is 250 mm. The range of image brightness perceived by the operator is from 3 × 10–6 to 3 × 4 10 cd/m2

20

V. Kolobrodov

Visual acuity is the simplest measure of visual resolution, which is equal to the inverse angular size of the image element perceived by the eye, expressed in angular minutes. For a normal eye, the lens with a pupil diameter D E = 2...3 mm has no aberrations and the angular resolution is determined by diffraction effects and is one angular minute. The spectral sensitivity of the eye is determined by the curve of relative visibility and exists in the range from 0.38 to 0.77 μm. The maximum of the spectral sensitivity corresponds to the wavelength of radiation λm = 0, 55 μm (yellow-green color). At low brightness, when the light flux affects only the rods, the sensitivity maximum shifts towards shorter wavelengths (λ,m = 0, 51 μm). This phenomenon is called the Purkinje effect. Threshold contrast Cv,th is the smallest contrast perceived by the eye. At the same time, the object with a size greater than 1° is placed on a background with a brightness L vb = 100 cd/m2 . For a normal eye Cv,th =

L vt − L vb ΔL th = 0.02...0.03, = L vb L vb

(52)

where ΔL th is the threshold brightness. The inverse value 1/Cv,th is named contrast sensitivity. The contrast value Cv,th is a function of the brightness of the adaptation background and the angular size δξt ,, of the detection element of the image layout. If the background brightness L Sb = 10 cd/m2 , Cv,th reaches the smallest values and changes little with further increase of L Sb . The dependence of Cv,th on δξt ,, is explained by the effect of spatial integration in the visual analyzer. Blackwell’s research showed that Cv,th sharply decreases with an increase in δξt ,, up to 10…20, and then changes insignificantly. The minimum threshold contrast value Cv,th = 0.027 is observed if the image element size δξt ,, = 12, . If the size of the image element δξt ,, = 1, , which corresponds to the angular resolution of the eye, Cv,th = 0.9. To approximate the threshold contrast, the function proposed by Schulz [8] is sometimes used Cv,th (νx ) =

CE , exp(−c1 νx ) − exp(−c2 νx )

(53)

where C E = 0.01033 is a constant; c1 = 0.1138°; c2 = 0.325°; νx is the spatial frequency, deg−1 . With low contrasts of the image in a limited field of view (for example, within the central fovea of the retina), the perception of the image by the eye is considered as a linear invariant process. In this case, the MTF M E (νx , ν y ) can be used for the image perception process. In the general case, the MTF of the eye is strongly influenced by various characteristics of the organs of vision, the main of which are diffraction at the entrance pupil, lens aberrations, the final sizes of photosensitive receptors, and others. In addition, the operation of the visual analyzer depends on the external conditions of

Mathematical Models for Calculating the Spatial and Energy Resolution …

21

observation, for example, the average brightness of the display screen, the duration of observation, etc. However, the MTF of the eye does not take into spatial noise, background illumination, the position of the observer relative to the display, and the exposure time. Each of these parameters affects the quality of image perception, and therefore the empirical dependences that characterize the MTF of the eye are only an approximation of the real MTF. At the moment, a sufficient number of different approximations of the MTF of the eye have been proposed, among which the Shultz approximation has become the most widespread M E (νx ) = 2.71[exp(−0.11νx ) − exp(−0.325νx )].

(54)

3 Spatial Resolution of Thermal Imagers Spatial resolution of TI characterizes the ability of the system to distinguish (separate) the minimum spatial dimensions of objects and their elements. In the general case, it is limited by the diffraction of the lens, the dimensions of the sensitive area (pixel) and the pitch of the detector, the width of the frequency bandwidth of the electronic path, and the resolution of the monitor. Each of the TI subsystems is characterized by its own resolution. So, the resolution of the lens is estimated by the Rayleigh criterion or the point spread function (PSF). The resolution of the detector is determined by the angular size of the pixel, the electronic tract by the Nyquist frequency, the display by the size of pixels or the number of TV lines. The modern approach to the evaluation of the TI resolution is based on the theory of linear spatial filtering, in the framework of which the observation process is considered as the spatial filtering of the object of observation by a multi-chain two-dimensional filter. The limit resolution is determined by the spatial frequency at which the MTF of the thermal imaging monitoring system (TIMS) decreases to a certain level, for example, to the level of 0.5.

Transfer Function of the Thermal Imager The transfer function can be applied only to linear invariant systems. Thermal imaging surveillance systems with small brightness contrasts within a limited field of view can be considered linear invariant systems. At the same time, it is believed that the object and the background emit incoherently, and each element of the generalized TI has its own transfer function and MTF. The modulation transfer function (MTF) of the TI is determined by the product of the MTF of its individual elements: optical system, detector, electronic unit and display. For the one-dimensional case along the x-axis, which coincides with the scanning direction, the MTF of the TIMS is defined

22

V. Kolobrodov

as Ms (νx ) = Mo (νx ) · M Ds (νx ) · M Dt ( f ) · M El ( f ) · M S (νx )Mep (νxa ),

(55)

where Mo (νx ) is the MTF of the OS, which can be approximated by Formula (25); M Ds (νx ) = sin c(α D νx )

(56)

– spatial MTF of the detector with a rectangular sensitive area; 1 M Dt ( f ) = / 1 + 4π 2 t D2 f 2

(57)

– time MTF of the detector, where t D —time constant of the detector; M El ( f ) is the MTF of the electronic unit; Ms (νx ) is the MTF of the display, which can be described by the function (43); Mep (νx ) is the MTF of the eyepiece, which will be approximated by the function (44). It should be noted that in most cases the angular spatial frequency is determined in the space of objects and measured in mrad−1 . The relationship between the spatial frequency in the space of observation and the space of objects is determined by Formula (49). The relationship between the spatial and temporal f frequencies is determined by the relation (45), which increase of the lens β will have the form f =

αD νx . (1 − β)to

(58)

At the same time, the MTF of the entire vehicle is well approximated by a Gaussian function [11] ) ( Ms (νx ) = exp −2π2 rs2 νx2 ,

(59)

where rs is the radius of the image of the point radiation source on the display screen, mrad. It follows from the last expression that the PSF of the TI the form {∞ h s (ωx ) = −∞

) ( ω2x Ms (νx )exp(2π ω x νx )dνx = √ exp − 2 , 2rs 2πrs 1

(60)

where ωx is the variable field of view angle, mrad. In some cases, the function (59) is used only within 0 ≤ νx ≤ 1/α D or 0 ≤ νx ≤ 1/2α D . The frequency νxc = 1/α D is named the cutoff frequency at which M Ds (νxc ) = sin c(α D νxc ) = 0. The frequency ν N = 1/2α D is named the Nyquist frequency.

Mathematical Models for Calculating the Spatial and Energy Resolution …

23

Spatial Resolution Criteria Consider the classical definitions of spatial resolution. Rayleigh’s criterion, according to which two image points are observed separately if the center of the diffraction image of one point coincides with the first minimum of the diffraction image of the second point (Fig. 3). It follows from this criterion that the distance between the images of two points observed separately is equal to the radius of the Airy circle (25). From this it follows that the resolution of the lens is determined by the formula νrEes =

1 . 2r E

(61)

Resolution νrees corresponds to a certain fate of the concentration of light energy in the image of a point source with a radius re . If the distribution of illumination in the image of a point source is described by the function E , (r , ), then the radius re is determined from the integral equation { re { 0∞ 0

E , (r , )r , dr , E , (r , )r , dr ,

= ke ,

(62)

where ke is the proportion of light energy concentrated in the image of a point source within a circle with a radius of re . At the same time, the resolution of the lens is determined as νrees =

1 . 2re

(63)

Resolution νrCes , which corresponds to a decrease in image contrast ( )to a certain value C0 . The lens be characterized by a one-dimensional MTF M O νx, . If the test object has a contrast equal to unity, then the resolution will be determined from equal ( ) Mr es νxC = C0 .

(64)

At the same time, the radius of the OS spread circle will be equal to re =

1 . 2νrCes

(65)

The disadvantages of these resolution definitions are: Rayleigh’s criterion is valid only for diffraction-limited optical systems in which there are no aberrations. In addition, the images of two adjacent points must have the same intensity of monochromatic light.

24

V. Kolobrodov

The resolution νrees = 2r1e , which is determined by Formula (63), has a certain uncertainty in the selection of the coefficient ke . In some cases, it is considered that ke = 0.5 The resolution νrCes , which is determined by Eq. (64), also has a certain uncertainty in the choice of the contrast value C0 . In most cases, it is considered that C0 = 0.5.

Geometric Noise Bandwidth of the Thermal Imager In electronic information systems that process video signals to create an image, the effective noise band Δ f , has been used with great success for many years. The electronic system have MTF M El ( f ), which is determined by its amplitude-frequency characteristic. If a noise signal with energy density G n ( f ) enters the input of such a system, then the effective noise band is defined as Δf =

1

{∞ 2 G n ( f )M El ( f )d f .

G n,max

(66)

0

In most cases, optoelectronic devices operate in the frequency range where the noise can be considered “white”, i.e. G n ( f ) = G 0 = const. Then the expression (66) will have the form {∞ Δf =

2 M El ( f )d f .

(67)

0

The ratio (67) means that the ability of the electronic system with the MTF M El ( f ) to convert a noise signal is equivalent to the electronic system with the MTF { M El,e f f ( f ) =

1, i f 0 ≤ Δ f ; 0, i f f > Δ f,

(68)

that is, such an MTF has a rectangular shape. ( ) Use a similar approach for the lens with MTF Mo νx , ν y [14, 15]. At the same time, it is necessary to take into account some features of this approach: 1. Unlike the one-dimensional MTF of the electronic system, the MTF of the lens is a two-dimensional function of spatial frequencies νx , ν y , measured in mm−1 . 2. Unlike the temporal frequency f , which varies from 0 to ∞, the spatial frequencies νx , ν y vary from –∞ to ∞. By analogy with Formula (67), we define the geometrical noise bandwidth (GNB) as

Mathematical Models for Calculating the Spatial and Energy Resolution …

25

{∞ {∞ GB =

M O2 (νx , ν y )dνx dν y .

(69)

−∞ −∞

In Formula (69), the right part defines the volume of the figure bounded by the surface M O2 (νx , ν y ), and the left part is the volume of the parallelepiped, which has a base with an area and a height equal to one. This approach is used in television as the Schade criterion for determining resolution [8]. The parameter G B is beginning to be widely used to determine the quality of the image that forms the OES. This approach is based on the fact that all detectors have a sensitive area of certain sizes and therefore represent integral filters in the area of spatial frequencies. In addition, detector always forms an output signal together with a noise signal. An analogue of geometric noise in a lens can be a spread circle. ( ) ν = If the MTF is a function with separable variables, that is M , ν o x y ( ) Mox (νx )Moy ν y , then the geometric noise band (69) is defined as {∞ G B = Δνx · Δνx =

{∞ M O2 (νx )dνx

−∞

M O2 (ν y )dν y .

(70)

−∞

( ) If the MTF Mo νx , ν y is a function that has an axis of symmetry, i.e. ( ) Mo νx , ν y = Mo (νr ),

(71)

/ where νr = νx2 + ν y2 is the radius of the spatial frequency in the polar coordinate system, then the geometric noise bandwidth is defined as {∞ G B = 2π

M O2 (νr )νr dνr .

(72)

0

For an axisymmetric MTF, the resolution νr es can be given as the radius of a cylinder with a height equal to one, the volume of which is equal to the volume under the surface M O2 (νr ), i.e. G B = π νr2es . Where / νr es =

[ | {∞ | GB | =|2 M02 (νr )νr dνr . π

(73)

0

As an example of calculating the geometric noise bandwidth and resolution, consider an OS whose MTF is approximated by the formula: Mo (νr ) = exp(−2π 2 ro2 νr2 ),

(74)

26

V. Kolobrodov

where ro is the radius of the spread circle at the level of 0.606 from the maximum illuminance value. The radius ro can be determined from equation Mo1 (νr 1 ) = exp(−2π 2 ro2 νr21 ) = 0.606.

(75)

√ where νr 1 = ( 2πro )−1 . The point spread function of such OS has the form [ ( , )2 ] r 1 1 exp − P S F(r , ) = . 2 2πro 2 ro

(76)

Substitute (74) into (72): {∞ G B = 2π

exp(−4π 2 ro2 νr2 )νr dνr = 0

1 . 4πro2

(77)

Then the OS resolution according to (44) is equal to νr es =

1 . 2πro

(78)

At the same time, instead of linear resolution in image space, angular resolution in object space is often used δωo =

1 . 2νr es f o,

(79)

The geometric noise bandwidth is a generalized parameter that determines the resolution of optical and optoelectronic imaging devices. This strip allows you to examine both the optical and electronic elements of the thermal imager from the same positions.

Agreement of Modulation Transfer Functions of the Lens and Detector An important characteristic of thermal imagers is the spatial resolution, which is determined by the modulation transfer function (MTF) of the lens and detector. A significant number of works [16, 17] have been devoted to the problem of matching lens aberrations and geometric parameters of detector. But these works do not consider the issue of choosing the spatial frequency or value of MTF, at which the aberrations of the lens and the period of the pixel structure of the MTF are matched.

Mathematical Models for Calculating the Spatial and Energy Resolution …

27

This subsection considers the relationship between the radius of the lens spread circle and the period of the detector pixel matrix when matching the modulation transfer functions of the lens and detector [18]. The modulation transfer function can be used only for linear invariant systems. Thermal imagers that observe objects with a small temperature contrast within a limited field of view can be considered linear invariant systems. The modulation transfer function of such systems is determined by the product of the MTF of its individual elements: lens, detector, electronic unit and display. In most practical cases, the electronic unit and display do not distort the image created by the thermal imager [11]. Therefore, it is believed that the MTF of the electronic unit, display and eyepiece are equal to unity within the working spatial spectral range. Criteria for matching the MTF of the lens and the detector. To simplify the calculations use the one-dimensional case along the x-axis, which coincides with the scanning direction. Under such conditions, the MTF of the thermal imager (55) is defined as Ms (νx ) = Mo (νx ) · M Ds (νx ),

(80)

where M S (νx ) is the MTF of the “lens-detector” system; M O (νx ) is the MTF of the lens;M Ds (νx ) is the spatial MTF of the detector; νx is the spatial frequency, mm−1 . Modern infrared lenses have aberrations that are limited by the diffraction of radiation at the lens aperture. The MTF of such lenses can be conveniently approximated by the function (21), which will be presented in the form { ( M O (νx ) =

2 π

) √ arccos x − x 1 − x 2 , i f 0 ≤ x ≤ 1; 0 i f x > 1,

(81)

where x = 1.22λko = r E νx ; ko is the aperture number of the lens; r E is the radius of the Airy circle. For the practical application of MTF (81), it is approximated by a function { M O (νx ) =

1 − ηxdi , i f 0 ≤ x ≤ ηdi ; 0 i f x > ηdi ,

(82)

where the parameter ηdi determines the relative deviation of the MTF of the lens with aberrations from the diffraction-limited MTF. For example, 80% of the diffraction limit of the image quality corresponds to the value of the parameter ηdi = 0.8. For a diffraction-limited lens ηdi = 1.0. The spatial MTF of the microbolometric matrix is determined by the periods of the matrix and has the form for the one-dimensional case M Ds (νx ) =

sin(π v D νx ) . π v D νx

(83)

28

V. Kolobrodov

Thus, the MTF of the thermal imager depends on the radius of the spread circle ro of the lens (or the radius of the Airy circle r E ) and the period VD of the matrix detector. To achieve a high spatial resolution of the thermal imager, it is necessary to coordinate the parameters ro and VD . Agreement parameters of lens ro and detector VD . For the MTF of the lens and the detector, two criteria for the parameters ro and VD are proposed: 1. The first criterion is the equality of the MTF of the lens and the detector at the Nyquist frequency ν N = 1/2VD , when the condition is fulfilled: M O (ν N ) = M Ds (ν N ) = M1 .

(84)

2. The second criterion is the equality of the values of the MTF of the lens and the detector at the spatial frequency νx2 , when the condition is fulfilled: M O (νx2 ) = M Ds (νx2 ) = M2 .

(85)

Consider several options for image formation with the “lens-detector” system: 1. Normal quality detector and high quality lens (small aberration circle). In this case, no matter how good the lens is, the image quality will be limited by detector. 2. Lens of normal quality and detector of high quality (small matrix period). In this case, similar to the previous one, no matter how high-quality the matrix we take, the quality of the image will be limited by lens aberrations. 3. The detector and the lens are of the same quality. In this case, to obtain a highquality image, it is necessary to coordinate the parameters of the lens ro and the detector VD . To match the system parameters according to the first criterion, we first establish the dependence of the radius of the lens spread circle on the period of the matrix. Substitute (82) and (83) into (84) ( 1−

1.22λke f f = 2VD ηdi

sin

π VD 2VD

π VD 2VD

) = M1 =

2 . π

Dependence of the radius of the Airy circle r E on the period of the matrix VD with the agreed MTF of the lens and detector at the Nyquist frequency r E = 1.22λke f f

) ( 2 2VD ηdi = 0.726VD ηdi . = 1− π

(86)

From these conditions, the resulting MTF of the system at the Nyquist frequency ( )2 have the value M1s π2 = 0.406 i.e., the contrast of the image is reduced to 40%. Figure 6 shows the MTF of the lens, detector and thermal imager when matching at the Nyquist frequency, which are determined by Formulas (82), (83) and (80), when

Mathematical Models for Calculating the Spatial and Energy Resolution …

29

Fig. 6 Agreement parameters of lens ro and detector VD according to the first criterion: 1—MTF of lens M O (νx ) for ηdi = 0,672; 2—spatial MTF M Ds (νx ) of the detector; 3—MTF of the thermal imager M S (νx )

k o = 1, λ = 10 μm, V D = 25 μm. Under such conditions, the relative deviation of the MTF of the lens with aberrations from the diffraction-limited MTF is ηdi = 0.672, which indicates the possibility of using a simple lens with relatively large aberrations. To match the system parameters according to the second criterion substitute (82) and (83) into (85) 1−

r E νx2 sin(π v D νx2 ) = = M2 . ηdi π v D νx2

(87)

Solve the system of Eq. (87) to establish the dependence of the radius of the Airy circle of the lens r E on the period of the matrix VD at a given contrast M2 . The first find from the second equation the dependence vx2 of the spatial frequency on the MTF M2 : sin(π VD νx2 ) = sin c(VD νx2 ) = M2 , π VD νx2 where νx2 =

1 sin c−1 (M2 ), VD

(88)

where sin c−1 (z) is the inverse function of sin c(z). Substitute the frequency (88) into the first Eq. (87): 1−

rE 1 sin c−1 (M2 ) = M2 , ηdi VD

where rE =

1 − M2 ηdi VD . sin c−1 (M2 )

(89)

30

V. Kolobrodov

Fig. 7 Alignment of lens parameters ro and the detector VD according to the second criterion: 1—MTF of a diffraction limited lens M O (νx ); 2—spatial MTF of the detector M Ds (νx ); 3—MTF of the thermal imager Ms (νx )

To check the reliability of the obtained Eq. (89), we will consider the use of this equation when matching according to the first criterion, when M2 = π2 : rE =

1− sin c

2

(π ) ηdi VD = 0.726ηdi VD . −1 2 π

which coincides with Formula (86). In Fig. 7 on the given MTF of the lens, detector and thermal imager when matching according to the second criterion (85), which are determined by Formulas (82), (83) and (85), when ko = 1, l = 10 mm, VD = 25 mm, ηdi = 1.0. The analysis of the obtained results indicates this: 1. When matched according to the first criterion, the contrast of the image decreases to 0.406. 2. For a diffraction-limited lens, the resulting MTF of the system at the Nyquist frequency is 0.482, indicating an increase in image contrast by 7.6% relative to the image obtained when the first criterion is matched. Under these conditions, matching takes place with frequency νx2 = 13 mm−1 at the contrast M2 = 0.88. It is suggested that the effectiveness of matching the aberrations of the lens and the period of the matrix structure of detector is determined by the product of the Nyquist spatial frequency Ms (ν N ) and the MTF of the thermal imager at this frequency, which will be named the effective spatial bandwidth of the thermal imager Δνe f = ν N Ms (ν N ).

(90)

According to this criterion, thermal imagers whose MTF are presented in Figs. 6 and 7 have the corresponding bandwidths Δνe f 1 = 20 · 0,406 = 8.12 mm−1 and Δνe f 2 = 20 · 0.482 = 9.64 mm−1 . This means that the quality (informativeness) of the image when matching according to the second criterion increased by 18.8%. Recently, significant scientific and technological work has been carried out to create a microbolometric matrix with a period of VD = 25 μm. Therefore, there was a need to determine the MTF of a thermal imager Ms (νx ) using just such a matrix.

Mathematical Models for Calculating the Spatial and Energy Resolution …

31

Fig. 8 Alignment of lens parameters ro and the detector VD according to the first criterion: 1—MTF of a diffraction limited lens M O (νx ); 2—spatial MTF of the detector M Ds (νx ), when VD = 17 mm; 3—MTF of the thermal imager Ms (νx )

Fig. 8 shows the modulation transfer functions of the lens, detector and thermal imager, which are determined by Formulas (82), (83) and (85) when ko = 1, λ = 10 μm, VD = 17 μm, ηdi = 1.0. An important conclusion emerges from the graphs obtained, that the matching of a diffraction limited lens for a wavelength λ = 10 μm with a matrix having a pixel period VD = 17 μm occurs at a spatial frequency νxr = 28 mm−1 that is close = 29 mm−1 . Complete agreement occurs for to the Nyquist frequency ν N = 1000 2·17 the wavelength VD = 17 μm. At the same time, the effective bandwidth of such a thermal imager is Δνe f = 29 · 0.406 = 11.8 mm−1 . This means that the image quality of a thermal imager that uses a microbolometric matrix with a pixel period VD = 17 μm and a diffraction limited lens has increased by 22.1% compared to a thermal imager that has a matrix with a period of VD = 25 μm. The conducted research testify to the following: 1. Two criteria for agreement of parameters the MTF of the lens and detector were considered, which allowed to evaluate the influence of the radius of the lens spread circle and the period of the matrix of the detector on the quality of the thermal image. 2. It is proposed to use the effective spatial bandwidth of the thermal imager to compare the quality of the thermal image. According to this criterion, the agreement of using a diffraction-limited lens with a spread circle r E instead of an aberration lens with a spread circle radius 1.49r E in a thermal imager will improve image quality by 18.8%, provided that the detector has a matrix period of 25 μm. 3. It was established that the use of a microbolometric matrix with a pixel period of 17 μm in a thermal imager allows to improve image quality by 22% compared to a thermal imager that has a matrix with a period of 25 μm.

32

V. Kolobrodov

4 Energy Resolution of the Thermal Imagers Energy resolution refers to the ability of a detecting system to distinguish signals from large background objects based on the available contrast. It depends on the energy characteristics of the detection system, the sensitivity of the detector, and the noise of the system. However, it is independent of the spatial resolution of the system. Sensitivity is determined by the smallest detectable signal that can be registered by the receiving system, which typically corresponds to a detected signal that forms a signal-to-noise ratio equal to unity at the detector output. In thermal imager (TI), the operator’s perceived signal-to-noise ratio S N R E from the display screen may exceed the ratio S N Rs , created by the system on the monitor. As a result, modern TI technology can produce a satisfactory image on the monitor screen, even if the values S N Rs are less than one. The Maximum Observation Range (MOR) is the farthest distance at which the thermal imaging surveillance system can detect or recognize an object with a specified level of probability. The observation range is determined by both spatial and energy resolution capabilities of the system. If the TI is limited by spatial resolution, then the MOS Rm is limited only by the dimensions of the object Vt and the limiting angular resolution δωs , Rm =

Vt . 2δωs

(91)

In turn, the limiting (minimum) resolution is constrained by the diffraction-limited resolution of the OS, the pixel pitch of the detector and the display resolution of the monitor. In thermal imaging systems with sampling, the minimum resolution is typically constrained by the spatial sampling frequency (or Nyquist frequency). In another limiting case, the MOS is constrained by the thermal imaging systems energy resolution. This situation occurs when observing large low-contrast objects, i.e., when the object’s angular size greatly exceeds the system’s angular resolution Vt ≫ δωs , R

(92)

For such systems, the main characteristic is the signal-to-noise ratio, which is defined as zero spatial frequency ΔL , SN R ∼ = exp(−κ A R) Ln

(93)

where ΔL is the absolute contrast of the object, L n is the brightness equivalent to the noise signal of the system, reduced to the plane of the object. In this case, the MOS increases indefinitely as the signal-to-noise ratio increases The spatial resolution does not unambiguously determine MOS or image quality, since it does not take into account the sensitivity of the system. Indeed, low-contrast

Mathematical Models for Calculating the Spatial and Energy Resolution …

33

objects of large sizes cannot be detected at a distance that provides spatial resolution if the level of the informative signal is below the noise level of the system.

Characteristics of the Thermal Imager that Affect Its Temperature Resolution Signal Transfer Function (STF) u s (L t ) is the dependence of the electrical signal at the output of the electronic unit on the brightness of the observed object. Noise Equivalent Irradiance (NEI) is the illuminance in the input pupil plane of the optical system that generates an electrical signal u s at the output of the detector equivalent to a noise signal u n , N E I = Ep

un , us

(94)

where E p is the illuminance of the input pupil, which creates the us signal us . Otherwise, the NEI defines the minimum detectable illuminance. Noise Equivalent Temperature Difference (NETD) is the temperature difference between the standard test object and the background emission as an black body. It is defined as the temperature at which the ratio of the peak value of the signal at the output of the standard reference filter TI which takes into account the test object to the noise is equal to 1 [11]. The test object should have angular dimensions ξt p,x × ξt p,y that are several times larger than the angular size of the sensitive area of the detector α D × β D in order to remove the influence of spatial resolution on the measurement results (Fig. 9a). To remove the influence of the electronic system and identify the measurement results, a standard reference filter with a transfer function is used [ H f,t p ( f ) = 1 +

(

f fo

)2 ]−0.5 ,

(95)

where f o = 1/2to ; t o is the scanning time of one image element.

Methods of Calculating Generalized Characteristics of Thermal Imagers Let’s consider the algorithms for calculating the generalized characteristics of thermal imagers, which play an important role in the design and use of TI. These include the signal transfer function, modulation transfer function, noise-equivalent temperature difference and minimum resolution temperature difference.

34

V. Kolobrodov

Fig. 9 Test object for determining NETD (a) and the appearance of the signal at the output of the electronic system (b)

Signal transfer function. To obtain the functional dependence u s (L t ), let’s consider Fig. 1. Suppose that the object of observation has uniform spectral brightness L t (λ), over its area, and its angular dimensions ξt x × ξt y are significantly larger than the instantaneous field of view of the TI located at a distance R from the object of observation. Let’s assume that the surface of the object radiates according to Lambert’s law. Then, the spectral brightness of the object’s surface can be determined by the following formulas: L t (λ) =

1 εt (λ)Mλ (λ, Tt ), π

(96)

where εt (λ) is the spectral emissivity of the surface of the object; Mλ (λ, Tt ) is the Planck’s function. If the surface of the object is placed perpendicular to the axis of observation, then the spectral flux of radiation enters the entrance pupil of the lens of the thermal imaging system. Φλ (λ) = τ A (λ)L λt (λ)AΩo ,

(97)

where τ A (λ) is the spectral transmission coefficient of the atmosphere; At is the area of the object within the instantaneous field of view of the TI; Ωo = A p /R 2 is the solid angle within which the radiation from the object reaches the entrance pupil of the lens with an area of A p . The signal at the detector output with spectral sensitivity R D (λ) will be equal to

Mathematical Models for Calculating the Spatial and Energy Resolution …

{λ2 us = λ1

Ap Φλ (λ)τo (λ)R D (λ)dλ = At 2 R

35

{λ2 τ A (λ)L λt (λ)τo (λ)R D (λ)dλ.

(98)

λ1

During measuring the STF function, it is assumed that the test object is located at a short distance from the thermal imager τ A (λ) ≈ 1 and the spectral transmittance of the optical system within the operating spectral range has an average value τo . Then, the STF function (98) of the thermal imager, taking into account the gain factor of the electronic unit C El and the ratios A p = π Do2 /4 and At /R 2 = A D / f o,2 , that follow from Fig. 1, will have the form ( )2 {λ2 Do π τo L t (λ)R D (λ)dλ. u s (L t ) = C El A D 4 f o,

(99)

λ1

The function u s (L t ) complex appearance primarily depends on the operating spectral range and spectral sensitivity of the detector, making it difficult to measure the true brightness of the object. Formula (101) does not consider the spectral composition of the electrical signal, which is determined by the scanning system. The signal magnitude u s is influenced by the gain C El and system noise. Noise-equivalent temperature difference. To derive the calculation formulas N E T D, we will utilize the assumptions and results presented in Sect. 3.2.1. Additionally, we will make several assumptions [11]. The test object is placed at a slight distance from the thermal imager. Then it can be assumed that the radiation is little absorbed while passing through the atmosphere, i.e. in the operating spectral range τ A (λ) ≈ 1. The effective noise band of the electronic system is determined by the formula 1 Δf = N E Pmax

{∞

| |2 N E P( f )| H f,t p ( f )| d f,

(100)

0

where N E P( f ) is the spectral density of the noise power; H f,t p ( f ) is the transfer function of the reference filter, which is determined by Formula (95). A large test object is placed on a uniform background and has a temperature contrast ΔT. The test object and the background emit as black body. Therefore, when there is a temperature contrast between the object and the background, a useful (informative) signal is generated, allowing for detection and analysis. u s = u st − u sb ,

(101)

where ust and usb are signals formed by the object and the background. If the object and the background radiate according to Lambert’s law, then Formula (101) taking into account (98) can be written in the form

36

V. Kolobrodov

1 Ap u s = At 2 τo π R =

{λ2 R D (λ)[Mλ (λ, T + ΔT ) − Mλ (λ, T )]dλ = λ1

1 Ap At τo · ΔT π R2

(102)

{λ2 R D (λ) λ1

∂ Mλ (λ, T ) dλ, ∂T

where τ o is the average transmittance coefficient of the OS. By substituting Formula (33) into (102), we find the signal/noise ratio at the output of the reference filter Ao ΔT 1 us = At τo 2 √ SN R = un π d ADΔ f

{λ2 D ∗ (λ) λ1

∂ Mλ (λ, T ) dλ. ∂T

(103)

The formula for calculating N E T D is found from (103), assuming that S N R = 1. Then √ πR 2 A D Δ f . (104) N E T D = ΔT = {λ ) dλ At τo A p λ12 D ∗ (λ) ∂ Mλ∂(λ,T T Let’s express this equation in another form by using Fig. 10. It is evident that A p = πDo2 /4 and At /R 2 = A D / f o,2 . Then N ET D =

Fig. 10 Test object (Foucault measure) to determine MRTD(νx ), where spatial frequency is νx = 1/2Vt p

τo

{ λ2 λ1

4ke2f f D∗

) (λ) ∂ Mλ∂(λ,T dλ T

/

Δf . AD

(105)

Mathematical Models for Calculating the Spatial and Energy Resolution …

37

Using relation (105), we can indicate the following ways of reducing N E T D. Using high-power lenses with a small aperture number ke f f = f o, /Do and a high transmittance coefficient τo is the most efficient way, because N E T D is approximately proportional to the inverse square of the aperture diameter. Using detectors with a large specific detection capacity D*(λ). Reducing the effective noise bandwidth Δf of the reference filter. However, to obtain high spatial resolution, this bandwidth needs to be increased. Therefore, Δf is chosen as a compromise between spatial and temperature resolution. It also follows from Formula (105) that N E T D ~ A−0.5 D , decreases with an increase N E T D in the area of the sensitive area of the detector. However, the spatial resolution of TI is approximately equal to δωs ~ V D ~ A0.5 D . Therefore, the area of the detector pixel, which mainly determines the spatial resolution, is chosen to be minimal. As a rule, it is limited by OS aberrations, and for the spectral region 8…14 μm, it is about 17 × 17 μm2 [18]. In thermal imaging systems with detector, in addition to the noise of individual pixels, the geometric noise of the matrix must be taken into account. In this regard, the following components are considered for the definition of NETD: 1. N E T Dth caused by thermal Johnson noise and 1/f noise (proper noises); 2. N E T D ph caused by radiation (photon) noise and determines the theoretical minimum value of NETD; 3. N E T Dg caused by geometric noise. If the dominate self-noises of the detector determined by the detection ability D* specified in the passport, Formulas (104) or (105) are used to calculate NETD. It’s important to note that the NETD parameter fully characterizes the energy sensitivity of the TI, but it also has some disadvantages [11]: 1. Formula (104) is only valid for large objects; 2. The NETD parameter doesn’t consider the characteristics of the electronic video signal processing system and the display; 3. The operator who perceives the image of the test object from the display screen wasn’t taken into account when deriving the NETD formula. Minimum resolvable temperature difference. The most important characteristics of a thermal imaging system are spatial and temperature resolution. Therefore, it is desirable to establish a functional connection between them. Such a connection is established with the help of the MRTD(νx ), function, which is the temperature difference between the object and the background, at which it is possible to separate the bar measures of a certain spatial frequency by eye. Bar measures of different spatial frequencies constitute the Fourier decomposition of the structure of the recognition object. The thermal test measure (Foucault measure) is shown in Fig. 10. The test object is a four-bar measure, in which the bar and the intervals have the same width them and a ratio of height to width of 7:1 ensures a certain periodicity in the x direction. Minimum Resolvable Temperature Difference MRTD is the minimum effective temperature difference ΔT between the stripes (strokes) of the test object with a

38

V. Kolobrodov

given spatial frequency νx and the background, which allows you to visually separate the strokes of the object on the display screen. The test object is a four-bar measure, in which the bars and gaps have the same width Vt p , and their height is equal 7Vt p (Fig. 10). The bars and gaps emit as black body with different but constant temperatures. Spatial frequency can be expressed both in linear and angular values (most often in mm−1 or mrad−1 ). To describe the test object, we use the MTF of the entire TI M s (νx ), which is denoted as M s (νx ) However, is defined for a sinusoidal measure, and the test object for determining the MRTD is a rectangular measure, we need to introduce a correction to the Fourier distribution of a rectangular measure. From example 1.3 of the textbook [8], we know that the amplitude of the 1st harmonic of a rectangular signal is 4/ π times greater than the amplitude of a sinusoidal signal of the same period and amplitude. Moreover, we need to take into account that the eye registers the average brightness of a rectangular stroke on the background. Specifically, the average value for the half-period of the 1st harmonic of a rectangular signal is 2/π times smaller than the amplitude. Therefore, the MTF of the TI, taking into account the perception of the test object by the eye, can be expressed as follows: Mt p,E =

8 Ms . π2

(106)

The process of image perception by the eye is considered by analyzing the signalto-noise ratio at the output of the reference filter, where the proportionality holds true SN R f ≈

Mt p,E ΔT , N ET D

(107)

where ΔT is the temperature contrast. NETD determines the noise at the output of the reference filter with an effective noise band Δf , and Mt p,E ΔT is the signal taking into account the resolution of the TI. To calculate the signal-to-noise ratio, it is necessary to consider the difference between the effective noise band of the TI, which takes into account the power spectrum of the detector N P S( f ) and MTF of the electronic path and display M El ( f ), from the effective noise band of the reference filter Δf . Then the signal-to-noise ratio on the display screen: ⎤0.5

⎡ S N RS =

Mt p,E ΔT ⎢ ⎢ ⎢ N E T D ⎣ {∞

Δf 2 N P S( f ) · M El ( f ) · M S2 ( f )d f

⎥ ⎥ ⎥ . ⎦

(108)

0

We take into account the integrating properties of the eye by changing the signal/ noise ratio. At the same time, the number SNRr , is entered, which guarantees with a given probability the recognition of the test measure against a noisy background. Let us take into account the laws of visual perception [8, 11]:

Mathematical Models for Calculating the Spatial and Energy Resolution …

39

1. The eye performs time integration, increasing the signal-to-noise ratio over time, where f f is the frame rate and t E is the eye constant of 0.2 s. 2. The eye carries out spatial integration along two coordinates:

/

f f tE

2.1. During integration along the receiving axis y, the signal-to-noise ratio ( )0.5 ξ times. increases in βtDp 2.2. During integration along the x coordinate, the perceived signal-to-noise ratio increases by a factor equal to the square root of the ratio between the effective noise bands of the target, present in the TI, and the TI itself taking into account the MTF of the eye M E ( f ), in ⎡ ⎢ ⎢ ⎢∞ ⎣{

{∞

⎤0.5 2 N P S( f ) · M El ( f ) · M S2 ( f )d f

0 2 N P S( f ) · M El ( f ) · M S2 ( f ) · M E2 ( f )d f

⎥ ⎥ ⎥ ⎦

times.

0

Thing into account all the above factors, the signal-to-noise ratio perceived by the operator would be actual ]0.5 [ Mt p,E ΔT Δf {∞ S N RE = 2 2 2 N ET D 0 N P S( f ) · M El ( f ) · M S ( f ) · M E ( f )d f ( ) ( )0.5 ξt p 0.5 . (109) f f tE βD The probability of detecting the test object on the display screen depends on the signal/noise ratio S N R E = S N Rr perceived by the operator. Solving Eq. (109) relative to ΔT, we get the general formula for calculating MRTD: 8 ΔT S N R E = S N Rr = Ms (νx ) N E T D π2

/

f f t E 7ωt p . kΔ f β D

(110)

where {∞ kΔ f =

0

2 N P S( f ) · M El ( f ) · M S2 ( f ) · M E2 ( f )d f Δf

(111)

the bandwidth coefficient, which shows the difference between the real bandwidth of the TI, taking into account visual perception, from the effective noise bandwidth of the reference filter. In a simplified form, this coefficient is determined by the formula [8]:

40

V. Kolobrodov

kΔ f =

Γs ν x α D , to Δ f

(112)

where Γs is the angular magnification of TI. The required number of SNRr for the recognition (detection) of the bar measure is known [8]. For example, if the probability of recognition is 0.9, it is necessary that SNRr = 4.5. If we substitute this numerical value in (110), then ΔT will be the desired MRTD function of the spatial frequency ν x . From relation (110) we get the formula for calculating the MRTD: Γs ν x ΔT = M RT D(νx ) = 0, 66 · S N Rr · N E T D Ms (νx )

/

αD βD . Δ f to f f t E

(113)

Note that the angular spatial frequency νx in Formula (113) is defined in the “display-observer” space. Formula (113) has some limitations: 1. The formula for the MTF of the eye is valid for the case when the size of the image of the measuring line exceeds ω,, t p the angular resolution of the eye. 2. Spatial integration of the eye stops if 7ω,, t p > 40, . Many researches have shown that Formula (113) produces good results that agree with experimental data for TI. However, MRTD has some drawbacks as a generalized characteristic, since it does not consider the real MTF of the eye and does not account for real objects and backgrounds. Nevertheless, Formula (113) enables us to analyze the effects of the main TI parameters on spatial and temperature resolution. The importance of the MRTD (νx ) as a generalized characteristic of the TI is that it determines the temperature sensitivity at low spatial frequency (for large objects) and the spatial resolution at high frequencies. MRTD (νx ) allows you to effectively compare thermal imagers, because it takes into account the entire system as a whole, its noises and the perception of the image by the operator. Since MRTD is a function of spatial frequency and TI parameters, as well as NETD, using the MRTD (νx ) dependence, you can find the spatial resolution of the system νr es The MRTD (νx ) function, as well as the NETD parameter, are determined by the background temperature Tb = 300 K. However, MRTD does not take into account some important factors that affect its value [19]: 1. MRTD determines the dependence of the temperature contrast of the Foucault measure on its spatial frequency (Fig. 11). But TI perceives not the temperature contrast ΔT, but the brightness contrast of the measure ΔL = L t −L b . According to the Stefan-Boltzmann law for a Lambertian radiation source, we have that the brightness contrast of the measure is equal to

Mathematical Models for Calculating the Spatial and Energy Resolution …

41

Fig. 11 Alignment of the MTF of the lens Mo (νx ) with the MTF of the detector M Ds (νx )

] σ{ ] σ{ 4 Tt − Tb4 = (Tb + ΔT )4 − Tb4 π π ] σ{ 3 4Tb ΔT + 6Tb2 ΔT 2 + 4Tb ΔT 3 + ΔT 4 , = π

ΔL =

(114)

where σ = 5.67023 × 10−8 is the Stefan-Boltzmann constant W × m−2 × K−4 . It follows from this expression that the background temperature T b affects the determination of MRTD. For a small temperature contrast ΔT