CC BY
7
Visnyk N'l'UU KP1 Seriia Radiolekhnika tiadioaparatobuduummia, "2019, Iss. 78, pp. 74—78
УДК 681.78
Modulation Transfer Function of the Thermal
Imaging Monocular
Kolobrodov V. H.
National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute"
E-mail: Ikerrnо(Фикт. neI.
The modulation transfer function (MTF) of thermal imaging monocular (TIM) was investigated in this article. TIM consists of a lens, a microbolomet.ric matrix (MBM). an electronic system of video signal amplification and processing, a micro display and an eyepiece. The monocular is considered as a linear invariant incoherent system. It's MTF is equal to the product of the modulation transfer functions of the components. For the convenience of practical application, it is proposed that all MTFs are considered as a function of the angular spatial frequency in the space of objects. An example of TIM MTF calculation with given characteristics was considered. The study of the MTF showed that the spatial impact of the MBM. which is determined by matrix structure, has the greatest influence on the deterioration of this function.
Key words: thermal imaging monocular: modulation transfer function: angular spatial frequency
DOI: 10.20535/RADAP.2019.78.74-78
Introduction
Thermal imaging systems are widely used in various fields of human activity, including security systems, medical thermal diagnostics, aeronautical and space systems, remote sensing of the Earth's surface, military affairs, etc. fl 5]. In many cases, snch systems are small-dimensional thermal imaging monoculars (TO!) in which the heat-contrast image of an object is observed by the operator on the display screen with the help of an eyepiece. The main characteristics of TO! are the spatial and temperature resolution, the maximum detection and recognition distances. They depend on the modulation transfer function (MTF) of the monocular fC 8]. Significant amount of scientific papers [9 15] are devoted to investigation of the thermal imagers MTF. the main components of which are the lens and the radiation detector. At the same time, there is a lack scientific and technical information about the development of methods for determining the MTF of a TIM, which consists of objective lens, microbolometric matrix (MBM), electronic system, display and eyepiece. Therefore, the development of snch methods for determining the generalized MTF of thermal imaging monocular is an important task.
1 Problem formulation
The purpose of the article is to develop a method for determining the modulation transfer function of a thermal imaging monocular, which includes lens, mi-crobolometer matrix, electronic system, display and an
eyepiece that will optimize the characteristics of the monocular to solve a particular observation problem.
2 Physic-mathematical model of thermal imaging monocular
The functional scheme of the TO! is shown in Fig. 1. Infrared (IR) radiation from the object of observation is absorbed in the atmosphere and enters the entrance pupil of the lens. IR lens forms an image of the object and background on MBM sensitive surface. The electric video signal from the MBM is processed by the electronic system and enters the micro display, which forms the image of the object on the screen. The operator observes this image with an eyepiece.
Fig. 1. Functional scheme of TO!
The mathematical model of TO! will be considered in the frequency domain (spatial and temporal) considering that the monocular is a linear invariant system. It is supposed that he object and background emit incoherently and each element of TO! has its own MTF.
3 Modulation transfer function of thermal imaging monocular
Modulation transfer function of TO! is determined by the product of its separate components MTFs: lens. MBM, electronic unit, display and eyepiece. For a one-dimensional case we have
Ms(vx) = M0 (vx)MDa{vx)MDt (t)
■ Mei (t)Md(vx)Mep(vx), (1)
where Mo(vx), Mds(vx), Mdî(t), Mei(t), Md(vx) and Mep(vx) are the MTFs of the lens, MBM, electronic unit, display and eyepiece respectively.
High-quality lenses without central screening and with the entrance pupil diameter Dpo canbe considered as diffraction limited. Their MTF is determined by the function [7]
M0(vx)
■arccos x — x\J1 — x2, if 0 < x < 1;
if x > 1,
(2)
f
where x = Xvx.
To simplify mathematical transformations, we approximate the complex function (2) by linear fnncti-
Mo(Vx) =
'1 — 1, 218x, if 0 < x < 0, 821; 0, ifx> 0, 821.
(3)
Fig. 2 illustrates the MTF of diffraction-limited lens M_o (v_x )
1
0,8
0,6
0,4
0,2
0,2
0,4
0,6
0,8
Fig. 2. The MTF of diffraction-limited lens 1 and its linear approximation 2
One-dimensional spatial MTF of a MBM can be approximated by function [9]
MDs(vx) = sinc(WDvx) sinc(wDvx), (4)
where WD is a period of the matrix structure and wD is a size of sensitive pixel area.
The temporal MTF of the MBM is a spatial low pass filter, which is approximated by the function [7]
M
1
Dt
v/1+4^ f 2 '
(5)
where to is a constant time of the microbolometer.
The MTF of the electronic block Mei (t) is modeled by n-order Butterworth filters [ ]. Modern electronic blocks have Mei (f) « 1 [ ].
The ^ITF of the matrix display is approximated by a function similar to the MBM MTF, i.e.
Md(Vx) = sinc(Wd^x) sinc(wd^x).
(6)
The eyepiece MTF is approximated by a function similar to lens MTF. i.e.
Mep(C )
1 — 1, 218x, 0,
if 0 < x < 0, 821;
if x> 0, 821,
(7)
/e
where x = A d^ v.
Dpe
ep are the focal length
and the exit pupil diameter of the eyepiece, respectively.
As Fig. 1 shows, the MTFs of the lens and the MBM are defined in the lens back focal plane. The display and the eyepiece MTFs are defined in the display screen plane. It should also be noted that in most cases the spatial frequency vx is determined in objects space and is measured in mrad-1. The temporal MBM MTF and electronic unit MTF depend on the time frequency f.
The relationship between the spatial vx and temporal f frequencies is determined as [ ]
f = °D
J , ^x а,
to
Hz,
(8)
where af is angular pixel matrix size and ta is one pixel generation time.
The relationship between the angular spatial frequencies in the observation space vxa and the space of objects vxa can be established with use of fig. . Let the Foucault test (four-bar target) with a linear period Vtp be located in the plane of objects clt 3. distance R. Then the angular period and spatial frequency are determined as
atp
Vtp R
1 R
— = T^. (9)
atp
Vt
p
The lens forms an image of Foucault test with linear period Vtp and angular spatial frequency
Jo Vt
R
p
Vt
(10)
p
The ^IB^I forms on the screen Foucault's test image with the period Vtp. It is observed by the operator through an eyepiece with a focal length fep. The angular period of this image and the angular spatial frequency are determined as
Vt
p
tp fep 1 ytp
(ID
vf
p
i v
xa
0
0
1
V
V
xa
xa
f
u
V
x a
Screen
display EyePiece
Operator
Fig. 3. The relationship between the angular spatial frequencies in the observation space vxa and the space of
objects vxa
Taking into account (10) and (11). we get the relationship
fe
fe
fe p
-77 = -r^- = , ' I
VZ Vpßel f0ßel
tan a
p
Vtp B_
fep VtP
Y^P fo_ f' v'
e p p
foßel f'e
atp J ep v tp J ep • tp J ep
Therefore. (12) can be presented in the form
vx
!!
vx =
r /
Mo(vxa) =
'(1 - 1.218A, if 0 < Vxa < 0.821 ;
0,
if Vxaa > 0.821 .
(15)
MDs(Vxa) = Sine ( -j- Vxa
(VD \ . (VD \
17D aJ Sine I 70 a)
The temporal MTF of MBM is determined from (5) taking into account (8). So. we have
MD t (Vx a) =
The MTF into account (12)
£) '(I)2
-0,5
The MTF of the eyepiece can be expressed by (7) which, similarly to the lens MTF (15) and taking into account (12) will be
(12) Mep(Vx a) =
where ßei = Vtp/Vtp is an electronic magnification of p p
The angular magnification of the "TIM-operator" system is defined as (Fig. 3)
(1 - 1.218 A , if 0 <vxa < 0.821 ;
Dpc Af
If Vxa > 0.821 .
(19)
(13)
(14)
4 Analysis of the TIM MTF
Let put the MTFs of individual TO! components as functions that depend on the angular spatial frequency in the object space (Fig. 3).
The lens MTF is determined by ( ), where x = A Then
Let consider an example of TO! MTF calculation. TO! has following characteristics
• Lens: the focal length f'o = 70 mm, the entrance pupildiameter Dop = 70 mm;
• MBM: the pixel size is VD = 17 jm, the size of the sensitive areais wD = 14 jm, the size of the matrix is XD = 6.8 mm, the time constantis tD = 10 ms and the frame frequency is ff = 50 Hz;
• Display: the pixel size is Vd = 17 jm, the size of pixel color groupis vD = 15 jm, the screen size is Xd = 9.6 mm;
• Eyepiece: the focal length is fep = 25 mm, the entrance pupil diameter is Dep = 4 mm.
Fig. 4 shows the MTF of the TIM separate components and its resulting MTF (1).
The spatial MTF of MBM is defined from (4), which we represent in the form
(16)
1
0,8 0,6 0,4 0,2 0
M(v_xa)
.
0 0,4 0,8 1,2 1,6 2 2,4 2,8 3,2 3,6 4
v_xa, Mpag(-1)
Fig. 4. The modulation transfer functions. 1 MTF of , lens. 2 spatial MTF of MBM. 3 temporal MTF of 1 I 1 y 18 d0lm,",HHl )y (6) ta g MBM. 4 MTF of display. 5 MTF of eyepiece. 6
' ........." resulting TIM MTF
Md(vxa) = sine ^yj-VxO^ sine ^yjf^xa^j .
(18)
Analysis of (1), (15) (19) and their graphs showed
that
x a
x a
0
2
x a
1. The MTF of TO! components as a rule are defined in different locations. The lens and the MBM MTFs are defined in the lens focal plane: display and eyepiece MTFs are defined in the display screen plane. The temporal MTF depends on the time frequency. When determining the resulting TO! MTF. it is necessary that all the components MTFs are considered in a same location. For the convenience of practical application, it is proposed that all MTF sare considered clS ct function of the angular spatial frequency in the space of objects.
2. The greatest impact on the deterioration of the resulting TIM MTF Ms(vx) has spatial MTF of the MBM M-S(vxa), which is determined by the pixel size. The smallest influence on the TO! MTF have electronic system and the eyepiece.
3. At the Nyquist frequency vN = 2 mm-1 the contrast decreases duo to MBM to 50%. the diffraction limited lens to 71%. a display to 62%. an eyepiece to 71%. Under these conditions, the resulting MTF of TIM Ms(vN) = 0,095.
4. The resulting MTF of TO! is well approximated by a Gaussian function [6]
Ms,ap (Vx a) = exp(-2^rsVX), (20)
where rsa is an angular radius of the point source image which operator observes through the ocular on the display screen.
From the last expression it follows that the point spread function (PSF) of TO! has the form
hs(^x) =
/TO
M(S, ap)(Vxxa) exp a)d^xa =
-TO
=72k:exp (-1),(21)
where ux is a variable angle of field of view, mrad.
The radius rsa is determined by the angle between the center of the PSF hs(wx = 0) and its value hs(wx = rsa) = 0.606. In some cases, ( ) is used only within 0 < vxa < vN limits.
Conclusions
A method for determining the modulation transfer function of a thermal imaging monocular has been developed. The monocular consists of a lens, a mi-crobolometric matrix, an electronic system, a displayarid an eyepiece. It was proposed to consider the MTF of the monocular in the space of ''object TO!", which allows us to calculate the angular resolution of the contrast limited TO!. The obtained analytical
expressions for the MTFs of individual components allow one to optimize the characteristics of the monocular for solving a specific observation problem.
References
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M од у л я ц i й н а передавальна функщя теплов!зшного монокуляра
Колобродов В. Г.
В дашй статт! доонджуеться модуляцшпа передавальна фупкц!я (МПФ) теплов1зшпого монокуляра (ТПМ), до складу якого входять об'ектив, мшроболо-метрична матриця (МВМ), електронна система шдси-лення i обробки в!деосигналу, мшродисплей i окуляр. МПФ визначае просторове роздшення монокуляра, яке вплнвае на якють теплов1з1йпого зображення i макси-мальну дальшсть виявлення i розшзнавання об'еклв спостереження. Розроблена ф1зико-математичпа модель ТПМ, в якш монокуляр розглядаеться як лшшна ш-вар!антна некогерентна система, МПФ якого дор!вшое добутку модуляц!йних передавальних функцш окремих компоненте такого ТПМ. Запропоноваш апалиичш ви-рази для МПФ сучасних компонента ТПМ, а саме: об'ектива, матричного приймача випромшюванпя, еле-ктронно! системи, дисплея i окуляра. Для зручноста практичного застосування МПФ запропоновано розглядати МПФ ycix компонента в простор! "об'ект спостереження - ТПМ", що дозволяв розрахувати кутову роздшьну здатшсть контрастно обмеженого ТПМ. Встановлено зв'язок м!ж просторовими частотами в простор! спостереження i простор! предмета. Отримано формулу для розрахунку кутового зб!льшення системи " ТПК -оператор ". Розгляпуто приклад розрахунку МПФ ТПМ з заданими характеристиками компоненте монокуляра. Доондження МПФ такого монокуляра показало, що найбшыний вплив на попршення n,ie"i функци мае просторова МПФ МВМ, яка визначаеться i'l матричного структурою. Наприклад, на частот! Найкв1ста 2 мм-1 в!дбуваеться зниження контрасту за рахунок МВМ до
28%, дифракгцйно обмеженого об'ектива до 71%, дисплея до 62%, окуляра до 71%. За цих умов МПФ ТПК дор!вшое 0,094. Результуюча МПФ ТПК добре апрокси-муеться гаусовою функгцею.
Ключовг слова: теплов1зшний монокуляр; модуляцшна передавальна функгця; кутова просторова частота
Модуляционная передаточная функция тепловизионного монокуляра
Колобродов В. Г.
В данной статье исследуется модуляционная передаточная функция (МПФ) тепловизионного монокуляра (ТПМ), в состав которого входят объектив, микроболометрических матрица (МВМ), электронная система усиления и обработки видеосигнала, микродисплей и окуляр. Монокуляр рассматривается как линейная инвариантная некогерентная система, МПФ которого равна произведению модуляционных передаточных функций отдельных компонентов такого ТПМ. Для удобства практического применения МПФ предложено рассматривать МПФ всех компонентов в пространстве "объект наблюдения - ТПМ", что позволяет рассчитать угловое разрешение контрастно ограниченного ТПМ. Рассмотрен пример расчета МПФ ТПМ с заданными характеристиками. Исследование МПФ такого монокуляра показало, что наибольшее влияние на ухудшение этой функции имеет пространственная МПФ МВМ, которая определяется ее матричной структурой.
Ключевые слова: тепловизионный монокуляр; модуляционная передаточная функция; угловая пространственная частота
