ASNT NDT Handbook Volume 3 Infrared and Thermal Testing

and where NEP = noise equivalent power; D =
detectivity {reciprocal of NEP); D1.* =
1_ specific detectivity normalized from JJ;
and D.f =frequency band·width of infrared
<!>
thermographic system.
where 9 == diameter of objective lens Equations 12 and 13 lead to the
(millimeter), B =objective distance, f =
focal length (millimeter) and Fe = description in Eq. 14:
F-number of lens system. D.L'A due to !J.T~
can be derived from the minute change (14) (J_) I1
!J. \-111, in the spectral radiosity H\ N T, -410 ''•"'. -pZ-
determined from Planck's radiation law:
c

X -J.Af-t-.;;{;D •,. -IIIIW-T,1·

D.E1. can be rewritten as the following form Considering the \\'avelength
by substituting Eqs. 8 to 11 into Eq. 7: characteristics of the infrared

(12) 11£,, I IIW, 1 thermographic system, the factors, t 0, 1).
and D): and the minute change AW1:D.T;- 1
-410 '',"'. -A'-Ts· A0 -Fe2 in the spectral radiosity per unit
temperature change H'>,·Ts1 all depend on
Because the noise in the signal-to-noise ·wavelength. Consequently, these factors

ratio of the system depends on the noise affect or control the wavelength
characteristics of the infrared
of the infrared sensor, the signal-to-noise
thermographic system. By hypothesizing
ratio for the tmget surface maintained at the system to be in the range of )q to ),2 ,
the following expression can be derived
from Eq. 14:

Ts is expressed in Eq. 13:
(~ t. __(13) 1 Fv
l i E ). DIIEJ. 5 4 re' .[if
(15) ( ,)
NEP
l\ T,

v, J D;,J., -aa-wr·,,d),
.fAullf IIEJ.
X C]. tot),

"·

FIGURE 5. Relative specific detectivity with mercury-cadmium-tellurium sensor: (a) short wave
with thermoelectric cooler; (b) middle wave with stirling cooler; (c) long wave with stirling
cooler.

'\100 fr''\ fl" J 1\
..,90
.~•~•3Ij J r ) \I
80 1
l:S"' 70
(b)
rJ60 (c)
"':B (a)
* so
___J
"v 40
"·•iv'l' 30
·•S 20 I
W,..,J ~
m \

""' 10 \.

0

2 4 s 6 7 8 9 10 11 12 13 14

Wavelength), (J-lm)

136 Infrared and Thermal Testing

llecause the noise equivalent even to equipment where sensing and
temperature difference can be given by scanning systems differ.

the reciprocal of the signal-to-noise ratio
as mentioned in Eq. 6, it can be derived as
follows when e.1• =- 1.0:

(16) NETD

Spatial propagation coefficient 'to implies a
possible influence by vapor and carbon
dioxide in the measurement space, but
to = 1.0 can be assumed except for
measurements in special environments.
The integration part in Eq. 16 is set as XJ.'

On the other hand, the optical system
is composed of infrared transparent
materials with coating so that the
·wavelength characteristics are optimized
by cutting unwanted wavelengths. In
particular, the optical system is fabricated
to pass almost 100 percent infrared rays in
the wavelength bands indicating a peak
sensitivity. Figure 5 illustrates the relative
specific detectivities D>..1/ for three kinds
of infrared thermographic systems when
the maximum value of 11:D>..• for all the
optical system is treated as 100 percent.

There are three types of infrared
thermographic systems: short wavelength
(3.0 to 5.3 pm) system with for instance
mercury cadmium telluride or indium
antimonide sensor held at 213 K
(-60 "C ~ -76 "F) by thermoelectric
cooling; middle wavelength {5.5 to 8 pm)
system with for instance mercury
cadmium telluride sensor held at 77 K

H 96 "C ~ -321 "F) by a stirling cooler;

and long wavelength (8 to 13 pm) system
with for instance mercury cadmium
telluride sensor held at 77 K (-196
"C ~ -321 "F) by a stirling cooler.

Summary

Error impair the minimum detectable size
and the noise equivalent temperature
difference of the mechanical scanning
infrared thermographic system in both
theory and practice.

The alternative definition explained
above refers to evaluation of minimum
detectable size and noise equivalent
temperature difference and is applicable

Errors in Infrared Thermography 137

PART 2. Calculation and Evaluation of Errors

Following the above description of several 1.5, 5.0 and 10.0 mm (0.02, 0.04, 0.06,
sources of errors impairing infrared 0.20 and 0.40 in.) wide- are formed hy
thermography, the following section coloring with black matting material
describes the theoretical and empirical having an emissivity E approaching 1.0.
process of calculating and evaluating An electrical heater at the back uniformly
minimum detectable size and noise heats the specimen to a desired constant
equivalent temperature difference. temperature of about 311 K
(38 "C ~ 100 "F). The apparatus is in a
Apparatus for Error thermally insulated darkroom to avoid
Estimation interference due to ambient multiple
reflections. For this study, the surface
Minimum Detectable Size temperature distribution on the specimen
is measured ·with the infrared
Hgures 6 and 7 show a conceptual thermographic system adding a
drawing of the measurement apparatus. germanium single objective lens having a
The measuring specimen is a polished
aluminum plate, on \Vhich five slits focal length f = 30.0 mm and effective
numbered 1 to S- measuring 0.5, 1.0,
diameter $ = 24.0 mm. To evaluate the
FIGURE 6. Measurement configuration. Heater minimum detectable size M0 the specimen
(Fig. 7) is used in a series of measurements
Spedmen / with a change in slit width. The objective
distance B between the measuring
"x specimen surface and the infrared camera
is kept at 1.0 m (40 in.) in view of the
Infrared camera Controller specimen size and the system
performance. For measurements involving
FIGURE 7. Measuring specimen with measurements in the slit of 10.0 mm (0.40 in.) in width, an
millimeter (1 mm ~ 0.04 in.). iris of 10.0 mm (0.40 in.) in width with
the same temperature as the chamber
300 temperature is placed directly in front of
the black aluminum plate. The placement
't1 2 3 4 s 11100position is chosen to avoid a thermal
disturbance to the iris because of the
specimen.

Noise Equivalent Temperature
Difference

Figure 8 illustrates a conceptual
illustration of the measurement apparatus.
An ice block was used to approximate the
triple point of ice at about 273 K

FIGURE 8. Measurement configuration.

Blackbody

- -~ ~· ~ ~

10.0 5.0 1.5 1.0 0.5 Infrared camera Blackbody controller

138 Infrared and Thermal Testing

TABLE 1. Specifications of blackbody furnllces.

Aspect Furnace 1 Furnace 2

Core double conical cavity cylindrical cavity
Operating temperature 323 to 723 K(50 to 450 'C; 112 to 842 'F) adjustable, relative to ambient
Cavity aperture diameter ¢60
Radiator calibration platinum temperature detector ¢ 60
Other specifications temperature stability=± 0.15 K-1800 s-1 mercury thermometer
cavity effective emissivity= 0.993

TABLE 2. Optical designs with Figure 9 also shm\'S the radiance
mercury-cadmium-tellurium detecting temperature distribution on slits 1 to 5 in
element. Fig. 7. If 2x zoom is applied, focal length
is doubled and the scanning range is
Spectral Cooler compressed by SO percent. Although the
Wavelength Response (J.Jm) image size is doubled, the optical
resolution is not improved and the ratio
Short wave 3.0 to 5.3 thermoelectric between the slit width M and the
Middle wave 5.5 to 8.0 stirling minimum detectable size 1110 does not
long wave 8.0 to 13.0 stirling change. Thus, at B = 1.0 m (40 in.), by
substituting the practical values of a, f, bo
(-0 'C ~- -32 'F). Blackbody furnaces and 110 pertaining to the infrared
(Table 1) were used to cover all ranges of thermographic system into Eqs. 2 and 3,
blackbody temperatures. The infrared Mo becomes 1.86 mm (0.073 in.) where
thermographic systems have three kinds Mo ~ (Ar)0·5•
of detection wavelength bands as the
systems mentioned abovei their main To show only the parts required for
characteristics are summarized in Table 2. data, in Fig. 9, the thermal index
In a series of thermal index representing radiance temperature 7~~ is
measurements, radiance temperature Trs is composed of data for more than 100 dots.
measured with the infrared thermographic The distance between dots is 1.13 mm
systems under the ·optical axis where B is (0.044 in.) in this case. The solid line
the objective distance between the indicates Trs determined theoretically
blackbody furnace and the infrared ·when using an ideal distortion free lens
camera and B == 200 mm (8.0 in.). The without aberration. The dotted line
measuring target image is a rectangular depicts Trs determined theoretically when
area composed of 30 x 30 pixels in the using a real lens affected by lens
blackbody furnace where the temperature aberration. The dashed and dotted line
distribution is uniform. In the manner shnws 'l~s determined theoretically when
similar to the procedure mentioned using a real lens affected by both lens
above, the noise equivalent temperature aberration and signal amplifier frequency
difference can be determined by response. The fine dashed and dotted line
calculating the standard deviation indicates Trs determined theoretically
l(D.Trl>avJ0·5 of Trs measured with the when using a real lens affected by focus,
infrared thermographic systems. diffraction, lens aberration and the signal
amplifier frequency response. Diamond
Calculation and Evaluation shapes show the actual measured values.
The slits M designated by the slits 1 and 2
Minimum Detectable Size are wider than the slit 1.86 mm
(0.073 in.) wide whereas the slits 3, 4 and
'·Vhen a fixed objective distance B == 1.0 m 5 are smaller. Thus, Trs is influenced by
(40 in.), Fig. 9 indicates the radiance the peripheral, i.e., the aluminum plate
temperature distribution along the temperature. In these cases, the averaged
horizontal scanning line perpendicular to radiation energy, Ex (Tro), incident upon
a slit painted on the specimen surface. An the infrared sensor is indicated by Eq. 4
electrical heater at the back uniformly and the averaged radiance temperature
heats the specimen to a desired constant Trs.x!SLl• is expressed by the following
temperature of about 311.2 K expression with the aid of Eqs. 4 and 5:

(38.0 oc = 100 °F). The scanning line is ( ~T"
Mo rsl
crossing at the central portion of the slits.
+

Errors in Infrared Thermography 139

For instance, forB== 1.0 m (40 in.), 1~s M 0 == (AT)o.s. The solid lfne and the dotted
line indicate Trs determined theoretically
is determined theoretically for slit 3 when in the same manner as Fig. 9 when lens

measuring the specimen surface aberration is absent or present,
respectively. The dashed and dotted line
maintained at 311.3 K (38.1 oc = 100.6 °F)
represents Trs determined theoretically
and the distortion free lens is used. when using the real lens affected by lens

Figure 9 shows that the radiance aberration and signal amplifier frequency

temperature T151 of the adequately wide response. The fine dashed and dotted line
slit 1 colored with black matting material
indicates Trs determined theoretically
is 31 1.3 K (38.1 oc = 100.6 °1') as a when using the real lens affected by the
focus error in addition to lens aberration
thermal index and that the radiance
and the signal amp1ifier frequency
temperature Trs2 of the base plate made of response. The actual measured values of
aluminum is 303.4
K (30.2 oc = 86.4 oF) as Trs for each temperature are respectively

a thermal index. Then, substituting M == indicated by solid diamonds, solid circles

1.5 mm (0.06 in.) for slit 3 and M0 = and solid triangles. At a fixed slit width,
1.86 mm (0.073 in.) into Eq. 18 gives
l\.J == 10.0 mm (0.40 in.)j when B is
Tm(SL) = 309.8 K (36.6 °C =97.9 °F). changed, Afo < M when B S: 5.0 m (200 in.)
and M0 > M when B :o> 6.0 m (240 in.).
Afterwards, Trs can be theoretically
As a typical example for B = 7.0 m
determined in the same manner for all
(275 in.), Trs is determined theoretically
the slits- the solid line in Fig. 9. As can for slit 1 if the specimen surface is about
311.2 K (38.0 °C), Slit width M0 = (A-1)'1·5
be noted from Fig. 9, the effect of Tr52 on determined from Eqs. 2 and 3 becomes

Tm{SI.l increases with decreasing M and 11.86 mm (0.467 in.) in this case.

the thermal index on the slit declines Figure 10 implies that Trsl = 311.2 K

both theoretically and practically. (38.0 oc = 100.4 °F) as a thermal index for

Figure 10 illustrates the temperature

output when varying R for slit 1. After

substituting the practical values of the

above variables into Eqs. 2 and 3, then B

becomes 5.9 m (232 in.) for the slit where

FIGURE 9. Detected temperature of slit at objective distance B = 1.0 m (40 in.).

313 (40) 1104] r--~-~--~-~-~--~------~---,

311 (38) {100] , ..~----:'-~~{~.. • , • ·.. , : : , , ,' ' ~--~-~-- 1-~~~-"-'---~~~----~~----~~
: r•~- ·~'
'I I : f~ '\ '•, ' : :::
~ ----++ + f1! \~~'#.t: --:- +---+ -_;_309 (36) )97]
' '' ''• ' ' ''

- : I l : ' 1! ; '. : '•, ' : : ;
-• -~:~~- i: ---'1·1!t:.-...-.-'•-•:·i~:.~- :
-----::~---~• -----~: : :
'
'' ' '
.,: _____:, ___ :,-~-y:)93]

e
307 (34) '' .. •'\

.f~ '" m""' rirn:ftl:i\1~-- (ii·J~( -i(t;\'tf("·~·L(r,~~-:~+ • •'0
"1' ,i'Jc 303 {30) f86J ~#~.~-~~.•~·,:~ ~ ~ __ "''#_+~.~~~#.~~-#,.~~#!.' __ ~"' '•_,..~~+!',tf~~~+, +I~.,...,.,••.,,,'...~', •I!I,',..<~<,.."!'•'~ ~I~-~..~,'..+
'1 'I ' •' ':.to. t I\ • •+I '

<l t t <1I I t t

' ' t ! ! t <I ' I

'I ' 'I ' 't 'I 'I 'I '' '

!

' ' ' ' ' '' ' ' ' '
301 (28) [82] ~-~~-~-'-~-"-~--'~~--'--~~·-~__,_-~-'---'--~~-'-'

0 10 20 30 40 50 60 70 80 90 100

Quantity of pixels

legend

--"'theoretical line
.. -¢... =theoretical line influenced by aberration
·-·O·~· =theoretical line influenced by aberration and frequency response

- =theoretical l'lne influenced by aberration, frequency respome and defocus
·~·+·-·=experimental temperature

140 Infrared and Thermal Testing

B == 0.5 m (20 in.), where M0 is sufficiently quantitative evaluation of the causes is
smaller than M. Although not shown in discussed below.
Fig. 10, iris Trs2 == 295.2 K
Lens Aberration
(22.0 oc ~ 71.6 °F) can be substituted as
VVhere the objective distance H :::: 1.0 m
the thermal index base or the ambient (40 in.) the slit width M ~ 1.86 mm
temperature in Eq. 18, together with the (0.073 in.), corresponding to the
minimum detectable size M 0 determined
practical values of M, Jod0 and Trsl to derive theoretically, is the same ·width as the real
Trsx(SL)· Then the thermal index Tn :::: image of the infrared sensor on the slit in
308.9 K (3.5.7 °C ~ 96.3 F) theoretically absence of lens aberration. Therefore say
when B == 7.0 m (275 in.). Afterwards, the that the infrared thermographic system
detects the energy emitted from the slit
derivation of T1s determined theoretically 100 percent theoretically. But in reality,
with respect to each temperature and due to lens aberration, 100 percent of the
distance is indicated as the solid line. It is energy emitted from the slit is not
detected with the infrared sensor.
obvious from Fig. 10 that M0 increases Figure 11 illustrates the real image of the
with increasing B and that Tis as a thermal infrared sensor when M = 1.86 mm
index decreases at the same time both (0.073 in.) and B ~ 1.0 m (40 in.). The real
image is affected by the lens of a finite
theoretically and practically. The aperture. The figure indicates the light
reception ratio of energy detected with
surrounding temperature around the slit the infrared sensor to energy emitted from
plays a significant role on Tis. The thermal the slit. The slanted part represents the
energy actually detected with the infrared
indices for the respective slits actually
measured with the infrared thermographic
system shuw lower values than the

theoretically determined thermal indices

depicted by.the solid lines as shown in
Figs. 9 and 10. This reveals the extent to
which the resolution of the infrared

thermographic system is impaired. The

FIGURE 10. Objective distance 8 and detected temperature where slit width M == 10 mm
(0.4 in.).

328 (55) (1311 ,-,--,--r--,----,------,--,--------~

[i:' ---- i

G'-.. 318 (45) 11131 . '6.
''l;o..
"-'
+-·- t
1'

0

I

E

~

g 308 (35) [95J

ro

'6

~

298 (25) [77] :-~:"::-~-:':~-::-'=-~~~-:'-:-~-'-~~~..J_~~-_j

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
(40) (80) (120) (160) (200) (240) (280) (320) {360) (400)

Objective distance, m (in.)
Legend

- - " " theoretical line
· · -t:.· • "'theoretical line influenced by aberration
·-·1:5·-· =theoretical line influenced by aberration and frequency response
· "'theoretical line inflt1enced by aberration, frequency response and defocus

A. =experimental temperature= (325 K) 52 ~c
• =experimental temperature= {318 K) 45 "C
• ==experimental temperature"" {311 K) 38 T

Errors in Infrared Thermography 141

sensor. In this case, the infrared sensor (40 ih.). Siinilarly, the dotted line as
detects about 82 percent of all the energy. shown in Fig. 13 indicates the light
reception ratio for various R when M =
The real image of the infrared sensor, 10.0 mm (0.40 in.). These values
affected by lens aberration, is derived represented by the dotted lines are
from the light path through the lens. The affected by lens aberrations. \Vhen lens
dotted line as shown in Fig. 12 indicates aberrations are present, the light reception
the light reception ratio for various slits ratio corresponds to M·llift"il included in
with different dimensions ·when B = 1.0 m

FIGURE 11. Real image at sensor where slit width M = 1.86 mm (0.073 in.) and objective
distance B ~ 1.0 m (40 in.).

j..--. slit width M -~~

1.0

Portion of signal received: 82.1 percent

0.9 v./17: /), -
/./\
0.8

~
c
~ 0.7 v/ /

'0 y/ /

,g~ 0.6
.s 0.5
0 v/ 7
.sw
0.4 v./v/ / Semor
mro
0.3 I y/ / /-
«'iwii 0.2
7 1// \
0.1 !../ ' / v / '-.......

o.o -Mof2 0 M0/2

legend

~ = energy detected by sensor
M-=slitwidth
Mo -=minimum detectable size

FIGURE 12. Slit width and signal reception at objective distance B ~ 1.0 m (40 in.).

.........100 ~--~--.-.---r------r------r------~----~------.------.
....~----+---~·--··-~·:~"·~·-~-·~-- -1------+-----+------~
90 e'"~~=8=2=·'~=======f=======F====~~-~--;~_., c·.---~-----+-------:
80

70 ---- - - - + - - - - - - + - - - - - + " - - - - .. ..+---'''~'·1~---" -f----· ·---
(·
60 ---j----- - f----- --f-c- - ~f--'·-c~-;--- .. - -

.·~.

so 1--'----f----~----~-~-+-1---~---~-----
•,

40 ---+-------+--·-----i--l''-------1-__-_-_-_--_+ ··. '-··~~-~:.
30 - -·~1.86 ··~

20 --+-1- ---t---- ··---

10 f-----

0 5.0 3.0 2.5 2.0 1.S 1.0 O.S
10.0 (0.3) (0.12) (0.1) (0.08) (0.06) (0.04) (0.02)
(0.4)

Width of slit, mm (in.)

legend

-········ .. == abberatlon
-------- =defocus

142 Infrared and Thermal Testing

Eq. 4. The thermal index decline due to surface focused at the infrared sensor into
lens aberrations can be computed by an electrical signal. Thf influence of the
using M·Mt)1• Note that this technique is amplifier frequency response in
applied to derive the dotted Jines as processing the electrical signal is
shown in Figs. 9 and 10. considered. Note that the mechanicaHy
scanned infrared thermographic system
Amplifier Frequency Response detects the target by mirror scanning.
Fig. 14 shows the waveform of the
In particular1 the signal to determine electrical signal output after scanning; the
temperature is produced by converting real image of the infrared sensor
the infrared emission from the target originated on a slit corresponding to the

FIGURE 13. Objective distance and signal reception for slit width M = 10.0 mm (0.4 in.).

120 r----.-----r----~----,----.-----r----~--~,_---,

'@' 100 ~,-__ ... ·-c..::::;:: .. · - ·-c--·-·---t--+··--
~ ··.:.:·:· ..
.-3 ...80 ·----· - - ·
!---·- --- :·.:_· ·: -:_C~ - - - -·r--
--·

.g
e
60f---+-·-

40 1----+-

2o1---+- -+--+---+---+----r--1---+

0~---L--L--~--~-~--~--~--L-~

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
(40) (80) (120) (160) (200) (240) (280) (320) (360) (400)

Objective distance, m (in.)

legend

--..--- = abber<Jtion
-----·-·---=defocus

FIGURE 14. Output signal waveform where slit width M::::: 1.86 mm (0.073 in.) and objective
distance 8 ~ 1.0 m (40 in.).

1.0 J..0.821 .

0.9 I v\

0.8 I/ \ \

'[ 0. 7 .

5

0 0.6

"'c

·;O;;l 0.5

B~ 0.4

UJ 0. 3 /I 0 \
0.2
0. 1 /I \\
0.0
.....:: / '-..... ~

-M,I2 Mof2

legend

M =slit width (millimeter)
-=-real signal W<Jveform
---=signal waveform output by scanning

Errors in Infrared Thermography 143

minimum detectable size M0 determined the time 2T to scan an objective
theoretically. The solid line indicates the minimum detectable size is 16 ps and the
maximum amplifier frequency is about
real image of the infrared sensor when 100 kHz. The value of rom<t>- T/2 is therefon._·
using the lens affected by lens aberration derived as follows:

and slit diffraction. Because the dotted FIGURE 15. Triangular waveform.
line as shown in Fig. 14 indicates an
y
almost triangular waveform,
approximated by the purely triangular

one as shown in I:ig. 15. After fourier

transformation, the frequency
components of the waveform (Fig. 16) are

derived:

T T-t

(19) F(Ol) 2J cos(OJt) --dl
T

(I

r( s.m-O-JT-

T2
OJT
2

The amplitude attenuation ratio (see -T 0 T
Eq. 19 and Fig. 16) makes the triangular
waveform in Fig. 15 decline because of legend
amplifier frequency response. The T = time variable (second)
amplifier must have sufficient frequency I = time scale
response to analyze the minimum y ,_, waveform amplitude (relative units)
detectable size. Conversely, frequency
response greater than required is not
necessarily an advantage when
considering the noise equivalent
temperature difference. This is because the
noise is also amplified and the noise
equivalent temperature difference
increases. In the infrared thermographic
system used in the present measurement,

FiGURE 16. Fourier transform of triangular waveform.

F(to)
1.2 , . - - - - - - , - - - - - . . , - - - - - - - . - - - - - ,

1.0

E o.s -----+-·--
COrm< T/2 = 2.51
c I

~

~
·.;:; 0.6

gro

~ 0.4

3

0.

~ 0.2

0.0 1--==-~'1

-0.2 .___ _ _ __ j_ _'-...._ _ __.c_ _ _ _ ___L_ _ _ _ __.

-10 0 5 lO

legend
T = time {second)
to>"" angular frequency

r = frequency

144 Infrared and Thermal Testing

T Noise Equivalent Temperature
O)max Difference
2(20) 1I fmax 'I'
Before calculating the actual value of the
2.51 noise equivalent temperature difference
based on the measured results, the noise
\'\7ith respect to Fig. 16, the amplitude equivalent temperature difference is
attenuation ratio of the waveform due to estimated theoretically for a long
amplifier frequency response corresponds wavelength thermographic system -
to the ratio of the area representing the taking the third type in Fig. 5c as an
slanted part enclosed by (OmaxT/2 == ±2.51 example, with a ble~ckhody furnace at
to all the area in the range of--= to oo, temperature T, ~ 303 K (30 oc ~ 86 ol'). At
This value comes out to be 91 percent. In first, the integration X is calculated as
other words, the temperature output is shown in Eq. 17 by integrating the
attenuated to 91 percent because of the product of the relative specific detectivity
amplifier frequency response when D)Jt and the minute change in spectral
measuring a target with dimension M0• radiosity per unit temperature 61V1:6.T;-1
The dashed and dotted lines in Figs. 9 with wavelength)~ in the range of 1~ 1
and 10 denote the radiance temperatures (7.7 pm) to ),2 (13.5 pm). Change AW,,in
7~s determined theoretkally when using spectral radiosity is given by Planck's
the lens affected by combining amplifier
frequency response with lens aberration. radiation law. The distribution of v;.R

Defocus versus A is refined every 0.2 pm in the
integration process. To obtain the absolute
The measured radiance temperature 'f~s as value of X, the integration is further
the thermal index of the slit - multiplied by the specific detectivity
represented by closed diamonds, closed 1Jj1b = 1.7 x 1010 (in cm·Hz0·\t,V-1) and the
circles and closed triangles - is slightly practical coefficient = 3.5 for this case).
lower than theoretically determined The practical coefficient is needed to
values shown by dashed and dotted lines adjust the absolute value of temperature
in Figs. 9 and 10. The difference is due to detected as an electric output to the
a cause other than lens aberration and blackbody temperature. The noise
amplifier frequency response. equivalent temperature difference (Kelvin)
can be determined by assigning the
Such attenuation is due to defocus. For following factors into Eq. 16: F number of
example, it may be present ·when the optical system Fe 1.36; sensor area An =
objective lens is focused digitally. The 25 x 10-6 (square centimeter); and
light reception ratio due to defocus is frequency bandwidth J).{ = I05 (hertz):
indicated in Figs. 12 and 13 and is related
to lens aberration. The fine dashed and (21) NETD ~ 4 x 1.36' ~1\11010-5 6
dotted lines in these figures indicate the
light reception ratio when using the lens X 9.28 X l{)-X
with the aberration affecting only the
lowest bits of information, i.e., signal 4.2 x w-z
attenuation. \.Yith this as a reference, the
fine dashed and dotted lines in I;igs. 9 and According to the above process, when
10 denote Tr\ determined theoretically T, ~ 303 K (30 oc ~ 18 °F) for the
when using the lens affected by lens thermographic systems, then the Fig. Sa
aberration, amplifier frequency response NETD ~ 0.23 K (0.23 °C ~ 0.41 °F) and the
and defocus. These lines are well Fig. Sb NETD ~ 0.14 K (0.14 oc ~ 0.25 °F).
correlated with the empirical data.
As an exmnple of the measured results,
Additional Error Causes Table 3 shmvs part of the data with
respect to the radiance temperature T,~,
Other causes of discrepancies between the every pixel in a rectangular area being
measured values and the theoretically composed of 30 x 30 dots when
determined values are attributed to such measuring a standard blackbody furnan·
sources as aperture iris refraction, by using the 1ong wave1cngth
diffraction, internal multiple reflections in thermographic system shown in Fig. Sc.
equipment and nonuniformity of In this case, noise equivalent temperature
temperature distribution on a target difference can be calculated with the
surface. But the detrimental effects from standard deviation [(/).T,_hwJ0.S ofT,~ for
these sources are negligible compared to each pixel in the rectangular area.
the sources mentioned earlier.
Figure 17 shows noise equivalent
temperature difference versus blackbody
temperature T~ when using thermographic
systems configurations shown in Figs. Sa,
Sb and 5c. The measured and theorE'tical

Errors in Infrared Thermography 145

values are indicated by the closed and Because the values of noise equivalent
open markings, respectively. It is obvious temperature difference determined
from this figure that, in near ambient theoretically are well coincident with the
conditions, the third system type (Fig. Sc) measured values as shown in Fig. 17, it is
has the smallest noise equivalent considered that the present theory and a
temperature difference with a sensitivity series of measured results are adequate for
in longer wavelength bands. On the other evaluating noise equivalent temperature
hand, it is seen that the noise equivalent difference. The unevenness in noise
temperature difference of the first system equivalent temperature difference of the
(Fig. Sa) with a sensitivity in shorter measured results in Fig. 17, as compared
wavelength bands decreases suddenly to the smoothness of the theoretical
·with increasing 1~. The first system values, is due to the variation in a
(Fig. Sa) is superior to the third system standard deviation because of the slight
inequality of the temperature distribution
(Fig. Sc) above about 423 K in the measuring range of the standard
blackbody furnace. The rise of noise
(ISO "C = 302 "F) for the theoretical value equivalent temperature difference
measured with the third system (Fig. Sc)
ofT, and above 403 K (130 "C = 266 "F) in the high temperature range is due to
for the actual measured value of Ts. In the larger inequality of the temperature
other words, in high temperature ranges, distribution of the standard blackbody
tlle first system (Fig. Sa) is more useful furnace maintained at a high temperature.
than the third system (Fig. Sc) in terms of Figure 18 shows the histogram of Tr~
noise equivalent temperature difference. based on pixels composed of 30 x 30 dots
measured within the rectangular area with
These facts can be proven in a standard blackbody furnace at 294 K
accordance with Planck's radiation law.
Figure 5 shm\'s that DJ.R• for the first (21 "C = 70 "F). The closed diamonds in
system (Fig. Sa) indicates a peak value in Fig. 18 represent the measured results. lf
the range of 3 to 5.3 pm; for type 3, of 8 the curve is based on the averaged value
to 13 pm. On the other hand, according of the 900 dots, T, = 294.08 K
to Planck's law, the radiosity peaks at
around 10 pm near ambient conditions, (21.08 "C = 69.94 F) and [(LIT,l).,.]o.s =
where T, = 300 K (27 "C = 80 "F), and the
0.0606 K (0.0606 "C = 0.!091 "F),
position indicating the peak shifts to a
shorter wavelength band with increasing represented by the dotted line, then the
temperature Ts.

TABLE 3. Example of thermal index. Arrayed pixels are individually assigned hue and
intensity on basis of thermogram box data, which represent radiant temperature
measurements from long wave (-8 to -13 J.Jm) mercury-cadmium-tellurium detector
with stirling cooler. Tm., = 473 K (200 "C); Tmrn = 223 K (-50 "C). Standard

deviation= 0.060601. Zoom= 5x on both axes. Absolute temperature: "C = K- 273.15 =

CF - 32)/1.8.

Y Axis X Axis Data ("C)
--- --

Data 133 134 135 136 137 138 139 140

53 20.87 20.80 21.01 20.94 20.87 20.94 20.80 20.87

54 20.94 21.01 20.94 20.94 20.94 20.87 20.87 20.94

55 20.87 20.80 20.94 20.94 20.80 20.94 20.94 20.94

56 20.94 20.94 20.94 20.94 20.94 20.94 20.87 20.94

57 20.94 20.87 20.87 20.94 20.87 20.94 20.94 20.87

58 20.94 20.87 20.87 20.94 20.94 20.94 20.94 20.87

59 20.87 20.94 20.94 20.94 21.01 20.94 20.94 20.94

60 21.01 21.01 20.87 20.87 20.94 20.94 20.94 20.87

61 20.94 20.87 20.94 20.87 20.94 20.94 20.87 20.94

62 20.87 20.94 20.80 20.94 21.01 20.73 20.87 20.94

63 21.01 20.887 20.94 21.01 20.94 20.94 20.87 20.73

64 20.87 20.87 21.01 20.87 20.94 21.01 20.87 21.01

65 20.87 20.87 20.87 20.87 20.94 20.94 20.87 20.94

66 20.94 20.94 20.80 20.94 20.94 20.87 20.87 20.87

67 21.01 21.01 21.01 20.94 20.94 20.94 20.87 20.94

68 21.01 20.87 20.87 20.94 20.87 20.87 20.94 21.01

69 20.94 20.94 21.01 21.01 20.94 20.94 20.87 20.94

70 20.94 20.94 20.87 20.87 20.94 21.01 21.01 20.94

71 21.01 20.94 20.87 20.94 20.94 21.01 21.01 20.94

72 20.87 20.94 20.94 20.94 20.94 20.94 21.01 20.94

146 Infrared and Thermal Testing

FIGURE 17. Noise equivalent temperature difference, based on standard deviation of radiance
temperature.

~ 0.6 -·-r- ···- ----1--+---
i~c
0.5 f~'_\.\A. ,_.-.\..-t-~---~---+t---_-_--t+---_-_: _------+r---_---~-+t---_-_:_--_-_-- 1 ---
04
- ·-

! 1\0.3 '•, '•, ... - - -1--+--- -·-+-- -1----t--

i 0.2 ~
·--: ....i......... .................. ...~.~2-

$!
0.1 f---j- ·-.•.>-....: ~~. •• - --- ---1---t--·-
.........~
-.-.-:.-:~~--::n:~~· ,.::;:·t::: ::::·:::: ---------~·

0.0
273 293 313 333 353 373 393 413 433 453 473
(0) (20) (40) (60) (80) (100) (120) (140) (160) (180) (200)

(32] (68] (1 04] (140] (176] (212] (248] (284] (320] (356] (392]

Blackbody temperature, K ("C) [°F}

legend

Theoretical

-e-- =short wave ;:o 3 to 5.3 pm
......()-- = middle wave"" 5.5 to 8 /Jffi
---e- = long wave= 8 to 13 !Jm

Experimental
·-·•·· "'" short wave= 3 to 5.3 pm
- ··•·- = middle wave= 5.5 to 8 !Jm
···•·· = longwave=8to13~m

fiGURE 18. Histogram of radiance temperature (standard deviation = 0.0606).

1.2r----------.-----------.-----------,-----------.----------·

(413)
1.0 1-------+------+-.·.·~r<>,.,-'.-'-----t--·----l---------l

.· ' ...

~ o.8r-------r------.~-~-----~-~-------t---------l

c
.~
e'0 (265)
---1--------
~ 0.6 f------t----_,_--j--f-----....+-

-~

~ 0.4 --------+- ----:-- .... (154)

0.2j-------cct--T---·--t---' ·. (~', (2)
:>_
~~r..(4) ........
O.OL-~~~~--L-J-~~--~~~~~--~~~~~~~~~~4-~~~

293.9 294.0 294.1 294.2 294.3 294.4

(20.7) (20.8) (20.9) (21.0) {21. 1) (21.2)

(69.3] (694] (69.6] (69.8) (70.0) (70.2)

Blackbody temperature, K (0 C) [°F]

legend

= average temperature= 294.08 K(20.93 oq

-~~-~--~ = gaussian distribution
(x) = quantity x of pixels

Errors in Infrared Thermography 147

shape of the curve becomes very similar
to that of gaussian distribution. The
distribution implies that, to determine the
noise equivalent temperature difference
objectively and quantitatively, it is useful
to regard a standard deviation i(ATrl)av1°·s
of the thermal index representing Tr~ as
the noise of the infrared thermographic
system.

Summary

Lens aberration, lens defocus and
amplifier frequency response are likely to
be the major factors impairing the
minimum detectable size of the
mechanical scanning type of the infrared
thermographic systems. The standard
deviation f(8Trhwl 0·5 of thermal index
representing radiance temperature 7~~ can
also be defined as the noise equivalent
temperature difference of the mechanical
scanning type of the infrared
thermographic systems. The thermal
index and its deviation can be treated
statistically and quantitatively by using
the technique proposed. Qualitative
differences in reading data do not occm
when evaluating the noise equivalent
temperature difference. The present
techniques for analyzing the error causes
or sources affecting the minimum
detectable size and the noise equivalent
temperature difference are also applicable
to other types of infrared thermographic
systems even if the causes of noise differ.

148 Infrared and Thermal Testing

Part 3. Statistical Processing of Errors

Infrared thermography can be made Apparatus for Error
rompletely quantitative and useful only Estimation
when applied to a measuring target
greater than the minimum detectable size Before considering the statistical
and with temperature variations greater processing 'of errors, it is helpful first to
than the noise equivalent temperature understand the measurement apparatus
difference. In addition to the fact, the for evaluating a measurement uncertainty
quantitative visual measurement when using infrared thermography.
Figure 19 shows a schematic illustration
technique possesses many error factors of the measurement apparatus. The
with respect to the statistical treatment measurement field of 1900 mm (75 in.}
for the data measured, even though the long x 1500 mm (60 in.) wide x 1500 mm
measurement is performed for larger than {60 in.) high is covered with pseudo
the minimum detectable size and noise blackbody surfaces of black velvety
equivalent temperature difference. It is material. The boundary walls are filled
normal that the measured data will always with adiabatic materials made of glass
include an error in a way similar to other ·wool. The internal surfaces are
techniques such as those that use maintained at a temperature as constant
thermocouples. The following discussion as possible are also covered with the
shows how to estimate the confident pseudo blackbody surfaces made of black
levels or the uncertainty levels by material \Yith a velvet texture to eliminate
analyzing respective error strata multiple reflections between surfaces. The
pertaining to the measurement process, ambient temperature 7~ is monitored with
classifying the error factor with several a precalibrated K-type thermocouple of
error strata. 300 pm (0.012 in.) diameter and is
controlled uniformly by using a suction
blower and a natural cooling heat

FIGURE 19. Schematic illustration of experimental apparatus for evaluating measurement
infrared thermography.

1.9 m (75 in.)

r AHeater
oTb"Hjenctg

700 mm (27.6 in.)

processing
unit

Adiabatic blackbody wall
ray tube

"!::===::3:~- C('ntral processing unit

Errors in Infrared Thermography 149

exchanger installed in the measurement with an ultrasonic cleaner after polishing
and the roughness before painting Is
field at the same time. estimated to be about 0.1 pm (4 x 10--6 in.)
The preliminary measurement for by using an electronic micrometer. In
some cases, the surface is painted with
calibrating the thermocouple is performed black matting material to simulate a
in a water tank maintained at a uniform blackbody surface.

temperature to obtain a calibration curve, The surface is heated to a desired
·which represents the relationship between temperature with a silicon heater attached
to the back of the measuring target. The
the two temperatures obtained with a amount of heating is controlled ·with a
constant alternating current power supply
thermocouple and with a standard by adjusting the calorific power. The
thermometer. The thermocouple surface temperature ']~as a true value is
temperature T15 can be calibrated quantitatively determined by applying
previously throughout this process. Fourier's law under thermal equilibrium
Therefore, the true temperature T~ can be after measuring temperature by using a
precalibrated K-type thermocouple of
determined quantitatively by applying the 100 pm diameter mounted 1 mm
calibration curve linking T5 and 71~· (0.04 in.) below the central surface of the
Measurement of T15 needs to be calibrated measuring target. Figure 20 shows the
under the same conditions as an detailed view of the measuring target with
application measurement. That is, for the heater and the thermocouple
applying the calibration cmve to attached. This temperature is used for a
determine the true temperature, it is reference value when checking
quantitatively a temperature indicated
desirable that both the calibration process with the infrared thermographic system.
All the targets in the measurement space
and the application measurement are are inclined at 15 degrees from a
maintained under the same conditions as perpendicular line to attain an ideal
observation angle for using the infrared
much as possible. The rate of flow by the camera. The defocus error is negligible
blower is so small that the flowing air when the target is very large relative to
slit width.
does not affect the environment inslde,
other than maintaining uniform The infrared thermographic system can
generate a two-dimensional thermogram
temperature conditions. with a thermal index representing
The measuring target and the optical radiance temperature Trs on a color
monitor. The apparatus is composed of a
device are installed in the enclosure at the mercury cadmium telluride sensor with a
same time to attain an ideal measurement selective wavelength band of about 8 to

field. The objective distance B between
the measuring target surface and the
infrared camera is 700 mm (28 in.). The

measuring targets~ which

measure 100 mm (4 in.) long, 100 mm
wide (4 in.) and 10 mm (0.4 in.) thick-

are made of pure copper (99.99 percent)1
carbon steel (S3SC), stainless steel

(15Cr-10Ni-6Mn-1Mo) and acrylic resin,

respectively. All these surfaces are cleaned

FIGURE 20. Detailed view of measuring target. Thennocouple
Thermocouple

10 mm (0.4 io.) 40mm 40mm

Copper base Test piece
sides patch

wall

ISO Infrared and Thermal Testing

13 pm, an optical device (camera), a then calibrated quantitatively to coincide
cooler, a color monitor, and signal with the true value after obtaining an
conditioning hardware. The infrared appropriate calibration curve linking T\
sensor installed in the ap'paratus is
simultaneously cooled with liquid with Tr,.
nitrogen. The true value of J~ has already been

Note that T,s is a thermal index determined by the preliminary
indicated with the infrared thermographic measurement mentioned above. The
system. In general, the resolution for multiple reflections between the
detecting a picture element depends on surroundings may affect the
an objective distance B between the measurement. Therefore, the above
measuring target surface and the infrared calibration process between T~ and Tr,
camera. For instance, the horizontal under the same conditions should be
resolution and the vertical resolution of required for quantitative infrared
the present infrared thermographic thetmogwphy.
system become 1.13 mm (0.044 in.) and
1.32 mm (0.052 in.) when B = 1000 mm A series of measurements are started
(10.0 in.). They are 0.6 mm (0.024 in.) under thermal equilibrium when T~ or Tr,
and 0.7 mm (0.028 in.) respectively when becomes constant or steady state. Their
B = 500 mm (20.0 in.). One frame of the distributions become uniform on their
thermograph on the monitor consists of surfaces at the same time. Under a
207 horizontal scanning lines having time constant temperature, individual
response of 1/207 s individually. Each line experiments are repeated eight times and
consists of 255 picture elements. 30 data sets are obtained individually. The
Therefore, the time to make one temperature data are then ensemble
thermogram is 1.0 frame per second. averaged after eliminating unusual values
by using a modified thompson T
Each scanning line composed of technique in a way similar to
255 picture elements has already been ASME PTC 19-1-1985, PerfOrmance Test
calibrated to adjust radiance temperature Codes. 1 (The thompson t technique is
to blackbody temperature ·when observing used to eliminate unusual values one hy
a standard blackbody furnace with the one during the calculation process by
present infrared thermographic system. In using a standard deviation and a
general, 1~s is not necessarily consistent difference between an instantaneous value
with T~, because emissivity E is not usually and an averaged value.ll) All these data
equal to unity. Although I~~ can be are recorded with an optical magnetic disk
treated as -Ts on surfaces after painting through a data sccmner. Table 4 describes
with black matting for the quantitative the performance of the analyzed infrared
measurement, it is necessary to calibrate thermographic system.

Trs precisely throughout an appropriate The measurement uncertainty for
calibration procedure even if the surface is infrared thermography h<1s never been
painted with black matting material, investigated and evaluated systematically
because emissivity E is about 0.97 on the and quantitatively. To establish the precise
surface. At first when using quantitative temperature measurement and its
infrared thermography, the absolute value applicability, the uncertainty levels should
of 'f~s• which is an indicated vall!e, should be estimated by using a procedure or
be verified in order to adjust it to the standard issued by an appropri<~tc
blackbody temperature preliminarily by international association. In this
measuring a standard blackbody furnace. discussion, the measurement uncertainty
Next, a thermal image of a central square is addressed with the aid of the
area of 10 mm x 10 mm (0.4 x 0.4 in.) in uncertainty analysis based on
the measuring target surface is collected AS"tvfE PTC 19-1-1985, Performance Test
by the infrared thermographic system. Its Codes. I According to the procedure, at
area averaged value is defined as Tw Trs is first, the error f<~ctor is classified with
three error strata: c<~libration stratum, data

TABlE 4, Common performance of infrared thermographic system with
mercury-cadmium-tellurium sensor.

Aspect Detail

Detection wavelength bands 8 to 13 pm
Effective temperature range 223 to 2273 K (.-50 to +2000 '·c; -58 to +3632 "f-)
Focus range 150 mm (6 in.)- oo mm
Frame time 1.0s
Horizontal scanning per frame 207 lines
Measurement precision (full range) 0.5 percent
Minimum temperature resolution 0.10 K (0.10 "C "0.18 "F)

Errors in Infrared Thermography 1S1

acquisition stratum and data reduction as negligibly small or previously
stratum. The targets of the analysis arc all predictable.
physical quantities pertaining to infrnred
thermography. According to the uncertainty analysis
based on ASME PTC 19-1-1985,
Statistical Treatment of Pe1{umumce Test Codes, 1 the measurement
Errors uncertainty can be analyzed by classifying
the error factor with three strata:
Jn general, the standard deviation SD can calibration stratum, data acquisition
be used to analyze a statistical stratum and data reduction stratum. In
characteristic of various phenomena by this case, the bias limit R can be defined
collecting physical quantities Xi composed as B; and the precision index as S; (j = 1, 2,
of population data N: 3 in order). However, they should be
rewritten as B;; and S;; when there arc k
elemental factors respectively in the three
strata. Therefore, B and S can finally be
expressed as follmvs by summarizing Bi;
and S;; for the three error strata:

(22) so (28) B

(23) X N (29) s [ ··--
K
~> L.si;z

hd \ J=l i=l

N

When using the averaged value of N The measurement uncertainty,
confidence level or uncertainty level, can
data instead of using only one measured
be given by calculating ui\DD and URSSt
value, the precision index S for the uncerwinty levels on the b<lsis of the 99
averaged value Xav of Xi can be defined: percent coverage and the 95 percent
coverage. They arc defined as follows \Vith
(24) s SD the aid of a student value t:

.JN

If individual measurements are (30) UAilD B + IS

repeated M times under the same
---- ---condition and N data are collected
(31) Ultss
respectively, the standard deviation SD
can be described as the following: A statistical quantity, t represents a
r6M N ~(X;; -X/) 2 value at a point indicating a 95 percent
area of a symmetrical t distribution. \'\7hen
(25) so 1 M(N-1) the symmetrical distribution of the error
is applied to the uncertainty range, in
In this case, S for the grand averaged contrast to Eq. 31, it can be expressed as
folluws:
value (Xav)av of Xp can be rewritten as
follows: (32) X ± UAllll
or
(26) s SD
X± Unss
~,/~N;
Furthermore, In ASME I'TC 19-1-1985,
\ '=1 Pnjlnmance Test Code.~, 1 a degree of
freedom v represents the number of data
IH N used for calculating Band Sand can be
_Lx;; expressed in the following form:

(27) X i=l i=l
hJN

The bias limit B is usually constant (33) v
throughout all the measurements. It can
be treated as negligible by practicing an
appropriate calibration process, treating R

152 Infrared and Thermal Testing

Note that v1i represents the degrees of The standard error of t'stimate can then
freedom for the error factors in the three be calculated:
error strata. It is important to cdnsitler
I(l'; -Y/)N 2 N 2
and keep a necessary data number for an -a-bX;)
appropriate statistical treatment. In (37) l:(li
general, v decreases by one with each
statistical quantity calculated on the basis h=1 i=l
of the population. A resultant R =
Note that a and b are as follows:
f(Pl,P2, ... ,P;), where {=function linking R
and parameter P1• For example, emissivity (38) a Y - bX
E may be affected by several parameters or
independent variables, such as radiation (39) b N
energy E, surface temperature '/~, radiance
temperature Tr~ and so on. A ratio 81 is I(x,- x)'
sometimes useful to evaluate the
uncertainty level of a resultant; 8; i=l

represents the ratio of the error of the

resultant to that of the parameters and
then can be defined as Eq. 34 for relative

index and Eq. 35 for absolute index:

(34) 8; aR
aP,

aR (40) X N
_R_
(35) e,' aP,

pi INY;

The preliminary, fundamental and (41) y i=l
statistical treatment for analyzing a
measurement uncertainty according to N
ASME PTC 19-1-1985, Performance 1l'Sf
Codes has been described above.1 The Finally, the standard error of estimate
practical process and another important can be obtained as the following form:
aspects in determining the uncertainty
levels pertaining to infrared (42) SEE
thermography will now be explained.
However, the uncertainty level of the data TABLE 5. Stratum of calibration error for thermocouple.
measured with the thermocouple should
be discussed before infrared Error elements Bias Precision Degree of
thermography, because the thermocouple UmitB Index S Freedom v
is the standard procedure throughout all
the measurements. '!~1bles 5 and 6 Scattering of thermocouple 0.1 0.399 >30
illustrate the list of representative absolute Standard error of estimate 0.1 0.399 >30
bias limit Band absolute precision index S Total >30
for both the calibration and the data
acquisition strata pertaining to the TABLE 6. Stratum of data acquisition error for
measurement with the thermocouple after thermocouple.
the appropriate statistical treatment. The
data reduction stratum is negligible Error elements Bias Precision Degree of
because of the high calculation accuracy limit 8 Index 5 Freedom v
\Yhen using a high specification computer.
In these tables, the elemental error linked Time variation of data 0.1 0.00348 >30
to the standard error of estimate (SEE) Time change of sensor and 0.1 >30
represents a scattering around a least 0.1
squares regression curve in the calibration error of data reading 0.14 0.108 >30
error stratum. The standard error of Error of zero compensation >30
estimate becomes the largest among all Total
the errors listed in these tables.

Here is the procedure for evaluating
the standard error of estimate. At first, a
least squares regression curve is
established:

(36) l/

Errors in Infrared Thermography 153

The constant C represents the number Within the· uncertainty levels, the
of constants to make the least squares calibration curves can be used to
regression curve as shown in Eq. 36 and determine the true temperature from a
temperature indicated by the
the value is 2.0 in this case. N stands for thermocouple.
the data number for the calculation.
Next, according to·the process
V\1hen the degrees of freedom v is over 30 mentioned above, the uncertainty levels
for all the data, the student value t are evaluated for data Trs measured with
becomes 2.0 at 95 percent confidence the infrared thermographic system.
Tables 7 and 8 illustrate the Jist of
leveL Therefore, for example, the present representative B and S as absolute values
final relative uncertainty levels when Ts = for both the calibration and the data
acquisition strata after appropriate
328 K (55 "C = 131 "F) are as follows with statistical treatment. The data reduction
the aid of the results listed in Tables 5 and stratum is also negligible for the same
6: B·T;;1 = 0.313 percent, S·T;;1 = 0.751 reason mentioned above. The item of the
spatial nonuniformity when diagnosing a
percent, v > 30, t =2, UAno·Ts' = 1.815 two-dimensional objective area is ne·wly
added to Table 8. This item can be
percent and URSSITs = 1.535 percent. evaluated by rotating the scanning line in
five degree increments on the target
Temperature T, =328 K (SS "C = 131 "F) surface. The B and S for the spatial
nonuniformity can be calculated as
implies the intermediate temperature follows:

when heating from 293 K (20 "C = 68 "F)

to 353 K (80 "C = 176 "F) on the surface of

the measuring targets. In a series of

measurements when using the
thermocouple, the calibration curve is

used to calibrate thermocouple
temperature T1s with the more exact
value. Figure 21 illustrates two typical
cases using the least squares technique.

FIGURE 21. Calibration curve for thermocouple.

373 (100) [212) r---.---.---.---......,---,---......,--...---.--.----,

353 (80) [176) ''

;Y

G:' (60) [140)
(40) [104)
~

E

"' 333

,__>

~

i3

~
o~_

E 313

~

~
~

""

293 (20) [68)

273 (0) [32) '---'--_J'-----'---......l---'---......l--...L-__.l._ _.l__ _J

273 293 313 333 353 373

(0) (20) (40) (60) (80) (100)

[32) [68) [104) [140) [176) [212)

Thermocouple temperature Tg, 1< \C) [°FJ

Legend

0 == th~rmocouple 1
o == thermocouple 2

154 Infrared and Thermal Testing

2 using the least squares technique. From
the calibration curves (Fig. 22), the true
- x) temperature from Trs indicated with the
infrared thermographic s.ystem can be
(43) s~. determined within the uncertainty levels.

(44) X L After considering the few examples
above for evaluating the uncertainty
(45) BL IS levels of P1 representing the
temperatures measured with the
5. thermocouple and with the infrared
thennographic system, the estimation of
Note that L represents the number of an uncertainty level of a resultant U
influenced by P; (P; being composed of N
rotating direction and L = 34 in the data) can now be considered. The
resultant is a final target to be estimated
present measurement. In Eqs. 43 to 45, by ASME PTC 19-1-1985, Per(omumcc Test
the subscript L differentiates SL and Br Codes. 1 VVhen obtaining the uncertainty
from other quantities. From these tables, level, R needs to be expressed by using
the elemental error linking to the individually averaged values of Pi as
follows:
scattering of the infrared thermographic
system in the calibration stratum becomes (46) II

the largest among all the errors listed in (47) P;
these tables. Because v > 30 for all the
data1 t becomes 2.0. Therefore, for Note that I represents the number of
instance, the present final relative parameters that are functions to the
resultant R. UADD and URSS for R can be
uncertainty levels when Ts = 328.15 K obtained as follows in the manner similar
(SS.O °C = 131 °F) are as follows with the to Eqs. 30 and 31:

aid of the results listed in Tables 7 and 8; (48) UJ(mn
B· T;:1 = 0. 791 percent, S· T~ 1 = 1.065
percent, v > 30, t = 2.0, UAnn·T;:1 = 2.922
percent and URSs' T;: 1 = 2.273 percent. In a
series of measurements when using the
infrared thermographic system, the

calibration curve is used to calibrate Trs
with the true value and is shown in

Fig. 22, which shows four typical cases

TABLE 7. Stratum of calibration error for Infrared The uncertainty range of U can be
thermography. defined:

Error elements Bias Precision Degree of BR and SR in Eqs. 48 and 49 appear in
Umit8 Index S Freedom v the general expressions for ahmlute index
(Eqs. 51 and 53) and relative index
Scattering of infrared camera 0.4 0.3 >30 (Eqs. 52 and 54):
Standard error of estimate 0.289 >30
Thermocouple calibration 0.172 0.413 >30
Total 0.435 0.586 >30

TABLE 8. Stratum of data acquisition error for infrared
thermography.

Error elements Bias Precision Degree of (52)
Limit 8 Index 5 Freedom v (53) SR

Scattering of thermocouple 0.0075 0.0532 >30
Standard error of estimate 0.0125 >30
Thermocouple calibration 0.0125 0.0546 >30
Total 0.0146 >30

Errors in Infrared Thermography 155

(54) /±i(~'~)2 Example Applicable to
\ i=l I, Radiative Quantity

where in a manner similar to Eq. 26 the It may be useful to consider an. example
following form gives the respective of how to evaluate the uncertainty levels
precision indices for the averaged values of radiosity coefficient as and emissivity c
of Pi: in the way similar to the process
mentioned above. For instance, according
(55) Sf>; to studies,6,7 a., and £can be obtained
from an energy conservation equation
Quantities 9; and 9/ and the bias limits applicable around an infrared sensor with
the aid of ambient temperature Tm true
BO'i)a\' and precision indices B(PiJa\' have surface temperature T~ and radiance
already been determined through a series temperature Trs> which have already been
of processes as shown above. In Eqs. 48 measured by using some procedures as
to 54, the subscript R is set to a series of mentioned earlier. They can be defined as
statistical quantities tentatively to follows:
discriminate them from others.
(56) a, - ( -T" )"

T,

FIGURE 22. Calibration curve for infrared thermography.

373 (100) [212) , - - - . - - . - - - , - - , , - - . . , - - - . - - - , - - - , - - , - - - ,

353 (80) [176)

G:'
'-..

G
"--
333 (60) [140)
"J2'· (40) [104)

~
il
r~u
<>
E
J'i 313

ru
~

"'

293 (20) [68)

273 (0) [32) ' - - - - ' - - - ' - - - - ' - - - ' - - - ' - - - ' - - - ' - - - ' - - - - ' - - - '

273 293 313 333 353 373

(0) (20) (40) (60) (80) (100)

[32) [68] [104) [140) [176) [212)

Radiance temperature Tr11 K ("C) {°F)

legend

D=lestrun 1
o"" test run 2
tJ. =test run 3
<> = test run 4

156 Infrared and Thermal Testing

a, - (T,,J" (60) B,

Ia~, ~;·J . JE
(57) E mr'[~ il£
~ Rr,
J.O-
+ i)'f5 T;-

a, T,

Note that the power index 11 is defined JI~ ,. rtil£
with 4.31 within an uncertainty level by [aT"+_-£__R_r._, +
the studies.6.7 Therefore, it is natural that
the uncertainty levels of the individual '1;, au 11
'1~, II
essential physical quantities as a
parameter included in the right hand side [ J I. J(61) s,d£
t
of these equations necessarily propagate -t T:E Sa - Sr

into the radiative properties as a resultant (lcl5 + iJJ,:,
when determining a.\ and E. In general,
tis
the measurement uncertainty levels of 'f:u
T_\ and Trs linking to a5 and E transfer to [ l [": ~]r,+
the radiative properties by the following
Taylor series in a way similar to
ASME PTC 19-1-1985, Per{im11ance Test
Cudes: 1

-d£ sr;, + au 11

€

(58) iJT, T;;

T, II

2 Note that these equations arc presented
as the relative indices. Figures 23 and 24
i;·+~d::a,s, ) indicate the measured results of as and f
versus T5 along with UADD and URss, which
[ are averaged among all data of individual
uncertainty levels throughout all
aa, 2 Yz measurement conditions. It is revealed
[-a.na~ from these figures that all the data of a,
) and e are ·within the uncertainty levels
and that the present measurement by
+ !!E._ using the infrared thermographic system
may have an appropriate reliability.
11 Furthermore, the sin1ple forms of Eqs. 56
to S7 are adequate for practical
II measurement of radiative properties.

2 Summary

a;(59) aa, Sr, ) The calculation of errors and estimation
of measurement uncertainty pertaining to
[ ~sT.\ infrared thermography are explained
above by applying ASME l'TC 19-1-1985,
[ ~a,_ )2 Yz Performance Test Codes. 1 The uncertainty
level and the cause of errors are addressed
an+ -ll·s -51-1 by analyzing individual error strata
11 pertaining to the measurement process
II after classifying the error factor with
several error strata.

Because the infrared thermographic
system can diagnose a t\vo-dimensional
temperature field instantaneously,
simultaneously and nondestructively even
though the objective target has a
complicated shape, there is a possibility
that the nondestructive test method will
be useful in various engineering
disciplines that demand high
measurement accuracy.

Errors in Infrared Thermography 157

FIGURE 23. Radiosity coefficient and its measurement uncertainty. I I
III I III
-
1.2 i-

o" 1.1 i- -

c~
·~o
'0: q,-_t_tt--~ - -
• -u0 f- - '9 -- ~~- ~: - -
- - -
•uc -I() -

'~•0 0.9 r- -

0.8 I- II IIII II -
360
280 300 320 340 (80) I
(7) (27) 380
[45] (47) (67) [176] (1 07)
[81] [225]
1117] [153]

Legend

0 = test run 1
0 = test run 2
t::.. = test run 3

- - =uncertainty level UADo for 99 percent coverage (averaged= 0.0232)

-----·- = uncertainty level UR~~ for 95 percent <overage (averaged= 0.0184)

FIGURE 24. Emissivity E (relative to blackbody) and its measurement uncertainty.

I II III I I I

1.2 '- -

1.1 i- -

w - - - -crn-o u-- ·a- -

0 -- - - -- -~ - ~t::.<>t::.<> - -<> - --=
- - ---
:~ - -0 -

.:E;;

w

0.9 - -

0.8 - g II II II -

280 I 01 320 340 360 I
(7) (47) (67) (80)
[45] 300 1117] 1153] 1176] 380
(27) (107)
181] 1225]

legend

0 = testnm 1

0 = test run 2
t::.. = test run 3

<> = test run 4
- - = uncertainty level UADo for 99 percent coverage (averaged= 0.0345)
---·-·- = uncertainty level URss for 95 percent coverage (averaged== 0.0292)

158 Infrared and Thermal Testing

References

1. ASME PTC 19-1-1985, Performance Test 9. Kurokawa, K., T. Inagaki, M. Agu and
Y. Okamoto. "Analyses of Factors
Codes, Supplement on Instruction a/UI Having Effects on Detection

Apparatus: Part 1, Measurement Resolution of Mechanical Scanning
Uncertainty. New York, NY: American Thermograph with Single Infrared
Society of Mechanical Engineers Detecting Element.11 Journal ofthe
\fisualization Society ofJapan. Tokyo,
(1990). japan: Visualization Society of japan
2. ASTM E 1543, Test Met/iod for Noise (to be published).

Equivalent Temperature Difference of 10. Kurokawa, K., IvL Agu, ·c lnagaki and
Thermal Imaging Systems. ·west
Y. Okamoto. A11 Study of Noise
Conshohocken, PA: American Society Equivalent Temperature Difference of
for Testing and Materials (1994). Thermograph." Transactions of the
3. ASTM E 1862, 1est Metlwds for Japan Society of Mecllanical Engineering,
Measuring and Compensating for Series C. Tokyo, japan: Japan Society
Reflected Temperature Using Infrared of Mechanical Engineers (to be
Imaging Radiometers. \Vest
Conshohocken, PA: American Society published).
11. Thompson, VV.R. "On a Criterion for
for Testing and Materials.
4. ASTM E 1897, 1est Methods for the Rejection of Observations and the
Distribution of the Hatio of the
Measuring aud Compensating for
Tra1151nittance of an Attenuating Medi11m Deviations to Sample Standard
Using Infrared Imaging Radiometers. Deviation."
\'Vest Conshohocken, PA: American Annals ofMathematical Statistics.
Vol. 6. Hay\'·.'ard, CA: Institute of
Society for Testing and Materials. Mathematical Studies (1935):
5. ASTM E 1933, Test Met/iod (or p 214-219.

Measuri11g and Compe1tsati11g for
Emissivity Using Infrared Imaging
Radiometers. Vlest Conshohocken, PA:

American Society for Testing and
Materials.
6. Inagaki, 'J: and Y. Okamoto. 11Surface

Temperature Measurement Using

Infrared Radiometer Applying
Pseudo-Gray-Body Approximation:

Estimation of Radiative Property for

Metal Surface." Journal of Heat Transfer.
Vol. 118, No. 1. Ne·w York, NY:

American Society of Mechanical
Engineers (February 1996): p 73-78.
7. Inagaki, T. and Y. Okamoto. 11Surface
Temperature Measurement near

Ambient Conditions Using Infrared
Radiometers ,Nith Different Detection

Wavelength Bands by Applying a

Grey-Body Appr9xirnation: Estimation
of Radiative Property for Non-Metal
Surfaces." Nondestructive Testing and
Evalrtatiou Intemalional. Vol. 29, No. 6.
Oxford, United Kingdom: Elsevier

Science Limited (December 1996):

p 363-369.
8.JIS T 1141-1986, Medicalln{Tared

Thermogmplls. Tokyo, japan: japanese
Industrial Standards (1986).

Errors in Infrared Thermography 159



CHAPTER

Parameters in Infrared
Thermography

Arnold Daniels, Coherent, Incorporated, Auburn,
California (Parts 1 and 2)
Xavier P. V. Maldague, University Laval, Quebec,
Quebec, Canada (Part 3)

PART 1. Performance Parameters for Optical
Detectors

This chapter presents some commonly square output voltage within this same
used descriptors of detectors. Detector harmonic component. In general, the
performance can be described in terms of responsivity 9t(f) of a detector decreases
various quantities of merit such as as the modulation frequency firicreases.
responsivif)~ noise equivalent power and By changing the angular speed of the
detectivily. These parameters enable the chopper of Fig. 1, the responsivity can be
user to compare relative performance obtained as a function of frequency. A
among detectors. typical curve of responsivity versus
frequency is plotted in Fig. 2.
Responsivity
The response time of a detector is
Responsivity gives the detector response characterized by its response time constant,
magnitude and provides information on the time that it takes for the detector
gain, linearity, dynamic range, and output to reach 63 percent (1 ~ e-!) of its
saturation level. The rcsponsivity is a final value after a sudden change in the
measure of the transfer function between irradiance. For most sensitive devices, the
the input signal photon power or flux and response to a change in irradiance follows
the detector electrical signal output: a simple exponential law. 1:or example, if
a delta function pulse of radiation QoO(t)
(1) Output signal is incident on the detector, an output
Input flux voltage signal (that is, the impulse
response) of the form:
where the output signal can be in volt or
ampere and where the input can be in (3) \'(l) = \'0 exp-t-
watt or photons per second. The
nomenclature is 9ti for current '
responsivity and 9~\' for voltage
responsivity. A common technique ln is produced, where tis the time constant
detection is to modulate the radiation to of the detector and t :2': 0. Transforming
be detected and to measure the
modulated component of the electrical FIGURE 1. Detection of temporally modulated radiation.
output of the detector, as shown in Fig. 1.
This technique provides some (a)
discrimination against electrical noise,
because the signal is contained only in Chopper
the fourier, or harmonic, compOnent of
the electrical signal at the modulation ' Detector
frequency whereas the electrical noise is
often broadband. Furthermore, it avoids Radiation beam
baseline drifts that affect electronic
amplifiers because of alternating current. Blackbody
The output voltage would vary from peak
to valley as shown in Fig. 1. (b)

An important characteristic of a Time (relative unit)
detector is how fast it can respond to a
pulse of optical radiation. The voltage

rrcsponsivity to radiation modulated at

frequency is defined as:

(2) l'sJgnal(f}
$5ignal(f)

where $signal(() is the root mean square
value of the signal flux contained within
the harmonic, or fourier, component at

frequency r; and 1'signa1(() is the root mean

162 Infrared and Thermal Testing

this time dependent equation into the contains all wavelengths of radiation,
frequency domain by using the independent of the spectral response
corresponding fourier transform yields: curve of the detector. Two standard
blackbody temperatures are used to
(4) l'(f) ~ ''ot evaluate detectors: (1) 500 K
(227 oc ~ 440 "F) for infrared
I + j2n{T measurements and (2) 2850 K
(2577 oc ~ 4670 ol') for visible and near
which can be extended to responsivity as: infrared measurements. Given a
blackbody that produces a spectral flux cV.J.
(5) 9lo in \.\/·tlm-1, the output voltage l'out 1ut
I + j2n{T (volt) of the detector is calculated from
the following overlap integral:
The modulus of Eq. 5 can be \\'ritten
as: f),

where 9\0 = v0t·Qo-1 is the detector (8) ~'output ~ $, 9lv(A.)d!,

responsivity. The cutoff frequency fcutoff 0
(Fig. 2) is defined as the modulation
Equation 8 determines the
frequency at which 19(, (frutorr )12 falls to contribution to the detector output in
half its maximum value and is related to those regions where the spectral flux and
the voltage spectral responsivity overlap.
response time as:
Although the responsivity is a useful
(7) fcutoff I measurement to foresee a signal level for a
2nT given irradiance, it gives no indication of
the minimum radiant flux that can be
In general, the responsivity depends on detected. In other words, it does not
the ·wavelength of the incident radiation consider the amount of noise at the
output of the detector that will ultimately
beam, and thus the spectral response of a quantify the signal-to-noise ratio {SNR).
detector can be specified in terms of its
Noise Equivalent Power
responsivity as a function of wavelength
The ability to detect small amounts of
<ft.(l..l). Spectral responsivity is the output radiant energy is inhibited by noise in the
detection process. Because noise produces
signal response (voltage or current) to a random fluctuation in the output of a
monochromatic radiation, of wavelength radiation detector, it can mask tlle output
produced by a weak optical signal. Noise
/,, incident on the detector, modulated at thus sets limits on the minimum input
spectral flux that can be detected under
a frequency f Correspondingly, the given conditions. One convenient
description of this minimum detectable
blackbody responsivity nomenclature is signal is the noise equivalent power (NEP),
9l(T,{). It is defined as the output defined as the radiant flux necessary to
produced in response to 1.0 V\' input give an output signal equal to the detector
noise. In other words, the noise
optical radiation from a blackbody at equivalent power is the radiant power
incident on the detector that yields a
temperature T modulated at frequency f signal-to-noise ratio of 1 and can be
expressed as the root mean square noise
\.Yhen measuring blackbody responsivity, divided by the responsivity of the
the radiant power on the detector detector:

FIGURE l. General shape of responsivity of detector as (9) NEP
function of frequency.
where l'11 denotes the root inean square
~ 1.000 voltage produced by a radiation detection
system and where NEP is measured in
c watt. Similarly, the noise equivalent
power in terms of current responsivity is:
[ 0.707

:f

c
a0 .

£

Frequency f (relative unit) (10) NEP ~

Parameters in Infrared Thermography 163

where i11 denotes the root mean square blackbody D*(T,f) is the signal-to-noise
current produced by a radiation detection output when 1 W of blackbody radiant
power (modulated at frequency f) is
system. incident on a 1 cm 2 detector area, within
If the responsivity is spectral in Eqs. 9 a noise equivalent bandwidth of 1 Hz.

and I0, the noise equivalent power is also A typical D*(i,,f) curve is plotted in
spectral jNEP(A,f)j and is defined as the Fig. 3. The peak value of the D'(i,,{) is
defined as the peak spectral D*p~;~k(/~,f ),
monochromatic radiant flux necessary to and corresponds to the largest potential
produce a signal-to-noise ratio of one at signal-to-noise ratio. In addition, any
optical radiation incident on the detector
the modulation frequency t: On the other at a wavelength shorter than the cutoff
wavelength, Acutoff• will have a D*(l,f)
hand, if it is the blackbody responsivity, reduced from the peak D*J!l-·akU,{) in
proportion to the ratio A·{A cuton)-1• This
the noise equivalent power nomenclature relationship is linear as seen in J:ig. 3.
is NEP(T,f), the blackbody radiant flux
Noise in Optical Detection-
necessary to produce a signal-to-noise Photon Noise Limited
Performance
ratio of one at the modulation frequency
The ability to detect small amounts of
f. Substituting Eq. 2 into Eq. 9 yields: radiant energy is inhibited by the
presence of noise in the detection process.
(11) NEP ~ <l>.<>ignal' ~·n o/slgnal The ultimate performance on optical
detectors in general is reached when there
v ...ignal SNR is no amplifier noise, there is no noise
generated within the detector itself and
It can be seen from Eq. 11 that the there is no radiating background against
sensitivity of the detector improves when which the signal must be detected. Under
the noise equivalent power is small. these conditions the only events produced
within the detector are due to signal
The disadvantage of using the noise photons. A detection system, in which all
equivalent power to describe detector other sources of noise contributions are
performance is that the noise equivalent small compared to the photon noise, can
power does not allow a direct comparison be considered in a sense the best possible
of the sensitivity of different detector condition. Such a detection system is
mechanisms or materials. This is because called pllotonnoise limited. In determining
of its dependence on both the square root
of the detector area and on the square FIGURE 3. Spectral D* as function of wavelength.
root of the elcctronlc bandwidth. A
descriptor that circumvents this problem Wavelength /,(relative unit)
is called D* (pronounced dee star), which
normalizes the inverse of the noise legend
equivalent power to a 1 cm2 detector area
and 1 Hz noise band\vidth. This =o• specific or normalized detectivity
normalized descriptor is presented below.
f = frequency
Specific Detectivity ). = wavelength
.; = spatial frequency
The parameter D*, called the specific or
110n1Wiized rletectivil)~ is measured in
'(Adet'llfl·\",H and is defined by:

{~{

(12) D* ~
NEP

D* is independent of area of the detector
and the electronic bandwidth because the
noise equivalent power is also directly
proportional to the square root of these
parameters as well. From Eqs. 9 through
12, it can be seen that D* is directly
proportional to the signal-to-noise ratio as
well as to 9\.

Unlike the noise equivalent power, this
descriptor increases with the sensitivity of
the detector. Depending on whether the
noise equivalent power is spectral or
blackbody, the D* correspondingly can be
either spectral or blackbody. D'(A,f) is the

detector's signal-to-noise ratio when 1 '"'
of monochromatic radiant flux
(modulated at frequency f) is incident on
a 1 cm 2 detector area, within a noise
equivalent bandwidth of 1 Hz. The

164 Infrared and Thermal Testing

the photon noise limited performance of (1 7) SNR l1lJ$q,nm
detectors, it is important to distinguish
between noise in the signal to be detected ~2 $ qil f
and noise in.the radiation background 11<J
present in the absence of the signal. This
distinction is particularly important at H the incident flux on the detector is
infrared wavelengths, where the thermal
emission from the atmosphere as well as assumed to be discrete quanta, Eq. 14
from the optical components arc likely to
create a radiation background. results in $q,rm~ = <l\J· The noise

equivalent photon f uxor NEPq required
for SNR ::::: 1 is then obtained by squaring

and simplifying Eq. 17 yielding:

Signal Dependent Limited (18) NEI~1
Detection
To find noise equivalent power in
The output signal~to-noise ratio for the energy derived units (subscript e) from the
case of signal radiation alone, without any noise equivalent photon Oux, each
extraneous radiation present, should be photon is multiplied by the eneq,'y of a
considered. It is applicable when the photon ilc-),~ 1 :
detector is 'Nell filtered and is looking at a
narrow spectral source or when the (19) NEP,
background radiation or scatter is
negligible. This ideal case occurs when the where 11 is Planck's constant, c is the speed
signal photon noise is the dominant noise of light in free space and ), is the working
contribution. wavelength.

Consider a photodiode viewing an Equation 19 can be interpreted as the
infrared source. Its photo generated signal noise equivalent power ultimate limit for
current is: the case where the photon flux is the
dominant noise. The corresponding
(13) islgnal ::::: 11q$q,rms normalized detectivity D~ is obtained
using Eq. 12 and is given by:
where 11 is the quantum efficiency, q is the

electronic charge of an electron in
coulomb, and /{>q,rms is the root mean
square value of the fluctuation and is
given by:

(14) ~q,nm t

_!_ J¢~(t)dt where subscript pnl indicates photon
noise limit.
\ t0
Background Noise Limited
where $q is the photon flux in photons Detection
per second incident on the detector. The
subscript q denotes photon or quanta The detector in the presence of signal as
derived units. Alternatively in Eq. 14, well as background radiation is now
detector time constant t can be replaced considered. In this case the background
with a large time value. photon flux is the dominant source of
noise as is commonly the case in infrared
Supposing that all the internal noise is SGmning systems as well as focal plane
negligible and that the background arrays. Under this condition, the detector
radiation falling on the detector is is a background limited infrared plwtodelector
negligible, then the detector output (BLIP). Background limited performance
circuit noise current will be simply the for a photon detector depends on the
shot noise: spectral distribution of the background,
the spectrC~l response of the detector, the
where D. {is the measurement bandwidth temperature of the detector, the mode of
and iav is the average current given by: operation of the detector and the field o(
view (H)\') within which the detector
The signal~to~noise ratio at the detector receives bC~ckground radiation.
output is the just the ratio of the signal
and shot noise cmrents: To determine the spectral D*(),, {)
under background limited infrared
photodetector conditions for a
photovoltaic detector, the root mean
square fluctuations in the rate of arrival of
detectable background photons must be
equated to the average rate of arrival of

Parameters in Infrared Thermography 165

signal photons. The phOtogenerated D** (Dee Double Star)
average current in this case is expressed
as: The background irraditmce falling on the
detector is controlled by the detector
where Eq,bk· and E ~,asiJckagrerotuhnedpahnodtosnignal geometry and its cold shield as shown in
irradiance for the Fig. 4

respectively. Assuming that under In this case the background irradiance

background limited infrared depends on the planar half angle ewith

photodetector operating conditions Eq,bl..s which the detector views the background
through the cold shield. Prom this
>> Eq,'iig and substituting Eq. 21 into geometry, the background irradiance can
be expressed as:
Eq. 15, the shot noise in this case is given

by:

(27)Eq,bkg nLq,bkg sin 2 8

Therefore, the signal-to-noise ratio can be -where sine is the numerical aperture (NA)
written as: and F1# is the F-number of the infrared
imaging system. Substituting Eq_ 27 into
(23) SNR Eq. 26 the specific detectivity can be
expressed as:
The signal-to-noise ratio is set equal to
one and the signal power corresponds to ,,.- --(28) D'BI.ii' (/,,f) ~ " ), - -21]-
the signal photon irradiance: Rhe q,bkg

(24) ~e,<ig Hence, the D* expression for the

Thus the spectral noise equivalent power background limited case is also a function
is derived from .Eq. 23:
of gthleeeF.1A~; annedwcpoarrreasmpeotnedriDng*l*y of the half
an

(pronounced dee double star) is introduced

to remove the need to specify the field of

view when listing D*. This allows a

comparison of detectors normalized to

hemispherical background. D**(/..,f) is

defined for background limited infrared

photodetector conditions as:

where lq,hkg is background radiance in johnson Noise limited
photon units. Performance

Substituting Eq. 25 into Eq. 12 yields The thermal agitation of electrons
the specific detectivity under background contained in a resistor gives rise to a
limited infrared photodetector conditions: fluctuating voltage across the leads of the
resistor. These fluctuations are known as
(26) Dnur(1,,f) ~ -1. ~~2EZq,E~kDg. johnson noise. 1•2 In a sense, johnson
noise is less fundamental than the photon
1lC noise, because it is not an inherent aspect
of the detection process. However, in
fiGURE 4. Background irradiance falling on infrared detector. cases where the photon flux reaching the
detector is low, this thermal noise
Background Cold shield becomes dominant.
photon noise
Background Recalling Nyquist's formula: 1·2
(Lq,,l..___. irradiance
(Eq_b •.g) ( 3 0) iJohnson
Signa! e------------- 0
the signal-to-noise ratio can be calculated
Detector fens in terms of energy derived units as:

. 166 Infrared and Thermal Testing

(31) <Psignill
SNR
lltJ ---,1(-

),

where k is Boltzmann's constant, b.{is the
electronic bandwidth, Tue~ is the detector
temperature and Rdet is the detector
resistance.

Forcing Eq. 31 to be equal to one, the
spectral NEP(A,f) can be written as:

~(32) NEP(i.,f} = 14kTdetl1f
qq), ~ Roe~

Substituting Eq. 32 into Eq. 12, results
in the spectral D'JOuO~,f) for a detector
device that is johnson noise limited
expressed as:

(33) Djou (i.,f) 1Jqi.)lloetAdet

21Jc~kTdet

The problem of maximizing the

detectivity of a detector is thus equivalent

to that of maximizing the RdetAdet product

of the p-n junction or schottky barrier, as

well as maximizing the quantum

efficiency 11 of the device. The RuetAdet
product is an inherent characteristic of
the detector material and fabrication
process.

Parameters in Infrared Thermography 167

PART 2. System Performance Parameters

A thermal imaging system collects, sources, each with a strength proportional
spectrally filters and focuses the infrared to the brightness of their original object at
scene radiation onto a multielement that location. The final image g(x,y)
detector array. The detectors convert the obtained, is the superposition of the
optical signals into analog signals, which individual weighted impulse responses.
are then amplified, digitized and This is equivalent to the convolution of
processed for display on a video monitor. the object with the impulse responsc:3
The main function of the imaging system
is to produce a picture or map of (34) s(x,r) ~ t(x,r)'*il(x,r)
temperature differences across an
extended source target. Therefore, the where the double asterisk denotes a
imaging system performance depends on two-dimensional convolution.
both the spatial resolution and the
thermal sensitivity. Both attributes are The validity of Eq. 34 requires shift
necessary to produce good thermal invariance and linearity- a condition
imagery. Spatial resolution is related to called isoplanatism. These assumptions are
how small an object can be resolved by often violated in practice but, to preserve
the thermal systemi thermal sensitivity is the convenience of a transfer function
concerned with the minimum analysis, the variable that causes
temperature difference that can be nonisoplanatism is allowed to assume a
discerned above noise leveL set of discrete values. Each set has its own
separate impulse response and transfer
Modulation Transfer function. Although h(x,y) is a complete
Function specification of image quality, additional
insight is gained with the transfer
The modulation transfer function (MTF) is function. A transfer function analysis
the parameter that describes both the considers the imaging of sinusoidal
spatial resolution and image quality of an objects, rather than point objects. It is
imaging system in terms of spatial more convenient than the impulse
frequency response. The interpretation of response analysis because the combined
image quality in the frequency domain effect of two or more subsystems can be
makes the entire range of linear systems calculated by a point·by·point
analysis techniques available, ·which multiplication of the transfer functions,
facilitates insight, particularly when rather than convolving the individual
several subsystems are combined. impulse responses.

Modulation Transfer Function Using the convolution theorem of
Definitions fourier transforms, 3 Eq. 34 is rewritten as
the product of the corresponding spectra:
The image quality of an optical or
electrooptical system can be characterized where F(~.~) is the object spectrum, G(~,ll)
by either the system's impulse response or is the image spectrum and H(~,11) is the
its fourier transform, the transfer transfer function 1 which is the fourier
function. The impulse response h(x,y) is transform of the impulse response. The
the two-dimensional image form in variables~ and 11 <lie spatial frequencies in
response to an impulse or delta function
object. Because of the limitations imposed the x andy directions respectively. Spatial
by diffraction and aberrations, the image frequency is defined as the rt>ciprocal of
quality produced depends on the the crest-to-crest distance (the spatial
wavelength distribution of the source, on period) of a sinusoidal waveform used as a
the F-number (F1#) at which the system basic function in the fourier analysis of an
operates, on the field angle at which the object or image. The concept of spatial
point is located and on the choice of frequency is schematically shown in
focus position.
Fig. 5.
A continuous object f(x,y) can be Spatial frequency is typically specified
decomposed using the shifting property of
delta functions, into a set of point in Simg (cycle per millimeter) in the image
plane and in angular spatial frequency
S~ng,o!JJ (cycle per milliradian) in object

168 Infrared and Thermal Testing

space. For an object located at infinity, transfer function and the phase transfer
these two representations are related function alter the image as it passes
through the focal length f(in millimeter) through the system. For linear phase shift
of the image forming optical system as: invariant systems, the phase transfer
function is of no special interest because
1(36) ~ong,obj ~ ~img X ~3 it will only indicate a spatial shift with
respect to an arbitrary selected origin. An
The transfer function H(~.n) in Eq. 35 image in which the modulation transfer
function is drastically altered is still
is usually normalized to have a unit value recognizable whereas large nonlinearities
in the phase transfer function can destroy
at zero spatial frequency. This recognizability. Generally, phase transfer
normalization is appropriate for optical function nonlinearity increases at high
systems, because the tra1isfer function of spatial frequencies. Because the
an incoherent optical system is modulation transfer function is small at
high spatial frequencies, the linear phase
proportional to the two-dimensional shift effect is diminished.
autocorrelation of the exit pupi!,4 and the
autocorrelation is necessarily maximum at The modulation transfer function is
the origin. In its normalized form, the then the magnitude response of the
transfer function J-J(l~,ll) is·refcrred to as imaging system to sinusoids of different
the optical transfer fimction (OTF). The spatial frequencies. The response can also
optical transfer function that plays a key be defined as the attenuation factor in
role in the theoretical evaluation and modulation depth:
optimization of an optical system is a
complex function having both a (38) M ~ Amax - Amin
magnitude and a phase portion: Amax + Amin

(37) OTF(~, n) H(~.n)

IH(~. n)l ·exp[;e(~. n)J where Amax and Amin refer to the
maximum and minimum values of the
The absolute value or magnitude of the waveform that describe the object or
image in VV·cm-2 versus position as shown
optical transfer function is called the
modulation transfer fimctioo (MTF) whereas in Fig. 6a. The modulation depth is
actually a measure of visibility or contrast.
the phase portion of the optical transfer The effect of the finite size impulse
function is referred to as the phase transfa response (that is, not a delta function) of
function (PTF). The system modulation
the optical system is to decrease the
FIGURE 5. Definition of spatial frequency S,. modulation depth of the image relative to

that in the object distribution. This
attenuation in modulation depth is a

function of position in the image plane as
seen in Fig. 6b. The modulation transfer
function is the ratio of image-to-object
modulation depth as a function of spatial

frequency:

(39) MTF (~, 11) ~ M;mg (~,11)
Mobi(~.n)

Position x within target A classical modulation transfer
function curve is shm\'n in Fig. 6c. The
Observation point area under the modulation transfer
legend function curves measures huw well a
system will faithfully reproduce a scene.
0 "'" angle= T,R--1 (rad'tan) The highest frequency that can be
R "' distance between observation point and target faithful1y reproduced is defined as the
Tx "' spatial period cutoff frequency of the system. However,
x "' position within target the modulation transfer function may be
; "' spatial frequency= 103·o,-1(cycfe per 1 mrad-1) incomplete when presented a single
curve. That is, the modulation transfer
function differs for different portions of
the field of view and for different
orientations. In general, the vertical and
horizontal modulation transfer functions
are different.

Because of its frequency dependence,
the modulation transfer function is more
descriptive of system performance than a

Parameters in Infrared Thermography 169

single value suCh as limiting resolution. that spatial frequency when the
Resolution can be defined when the modulation transfer function curve drops
modulation transfer function curve equals to two to five percent of its maximum
zero. However, for most systems1 the value. Two specific circumstances for two
modulation transfer function curve does different systems are shown in Fig. 7. In
not abruptly reach zero bt1t approaches it Fig. 7a, two systems named A and B have
asymptotically. Therefore, in most cases, identical resolution but different
the limiting resolution can be defined as performance at lower frequencies; in
Fig. 7b, the system H ·with the best
FIGURE 6. Modulation transfer function defined as attenuation resolution performance is worse at
factor in modulation depth as function of spatial frequency ~- midband frequencies than is system A1
with poorer resolution. In other words,
(a) the resolution specification alone can give
a misleading picture of system
Am~' performance.

~ Modulation Transfer Function
Calculations
v
The overall transfer function of an
~ electrooptical system can be calculated by
multiplying the individual transfer
"'·c~ functions of its individual subsystems.
The majority of thermal imaging systems
2 operates with broad spectral band passes
and detects noncoherent radiation.
E Therefore classical diffraction theory is
adequate for analyzing the optics of
Arr.·r>
Position in object plane (relative unit)
(b)
FiGURE 7. Difference between modulation transfer function
r;' and resolution.

E (a)
c /A
v :2 \

~ cv
2\
"'·c~ \ B
~ \
2

E

Position in image plane (relative unit) c '''''' '' ' limiting resolution
(c)
g /

.,c

.0

"•3
"0

2

Spatial frequency (relative unit) '

(b)

A

t3/ .c \

c 0

2\

Spatial frequency S(relative unit) -!: \ Resolution A
gc
legend '' ' ' ' '
An~~= maximum value of waveform .,c
Am;,.= minimum value of waveform / / Re>olutioo B
A =wavelength (tJm) .0
'
S= spatial frequency "•3

"0
2

Spatial frequency (relative unit)

170 Infrared and Thermal Testing

incoherent electrooptical systems. The curve as the upper envelope. Aberrations
broaden the impulse response ll(x,y),
optical transfer function of a diffraction resulting in a narrower and lower
limited optics depends on the radiation modulation transfer function, with less
integrated area. The area under the
wavelength and the shape of the entrance modulation transfer function curve relates
pupil. Specifically, the optical transfer to a figure of merit called the strehl
function is the autocorrelation4 of the intemity ratio or simply strehl ratioJ• The
strehl ratioS quantifies the degradation of
entrance pupil function with entrance image quality and is defined as the
irradiance at the center of the actual
pupil coordinates x and y replaced by impulse response divided by that at the
center of a diffraction limited impulse
spatial frequency coordinates ~ and 11 response. It shows that small aberrations
respectively. The change of variable for reduce the intensity at the principal
the coordinate x is: maximum of the diffraction pattern, that
is, at the diffraction focus, and that the
where xis the autocorrelation shift in the removed light is distributed to the outer
pupil, J.. is the working wavelength and di parts of the pattern. Using the central
is the distance from the exit pupil to the ordinate theorem for the fourier
image plane. From Eq. 40 a system with transforms,3 the strehl ratioS can be
an exit pupil diameter D has an image written as the ratio of the area under the
space cutoff frequency: actual modulation transfer function curve
to that under the diffraction limited
(41) Scutoff ~ 1 modulation transfer function curve:
-F)
(44) s
' /#
The strehl ratio is a number that can
which is when the autocorrelation reaches range between 0 and 1 but its useful range
zero. The same analytical procedure can is approximately 0.8 to 1. (This ratio was
conceived for highly corrected optical
be performed for they coordinate. systems.)
A system without ·wave distortion
Geometrical aberration optical transfer
aberrations but accepting image faults due functions can be calculated from ray trace
to diffraction is called diffraction limited. data, by fourier transforming the spot
The optical transfer function for such a density distribution, without regard for
near perfect system is purely real and diffraction effects. The optical transfer
function thus obtained is accurate if the
nonnegative (that is, modulation transfer aberration effects dominate the impulse
function) and represents the best response size. The optical transfer
function of a uniform blur spot can be
performance that the system can achieve, '''ritten as:
for a given F111 and ·wavelength )...
(45) OTF(i,)
The modulation transfer functions are
now considered that correspond to where IJ() indicates the first order bessel
function of first kind and B is the
diffraction limited systems with square diameter of the blur spot. The overall
(width 1) and circular (diameter D) exit optics portion modulation transfer
pupils.3AThe square aperture has a linear function of an infrared system can be
modulation transfer function along the determined by multiplying the ray trace
spatial frequency ~ given by: data modulation transfer function by the
diffraction limited modulation transfer
(42) MTF(i,) ~ I - _i,- function of the proper F1# and /...

~cutoff T\\'O integral parts of modern infrared
imaging systems are (1) the electronic
and when the exit pupil of the system is subsystems, which handle signal
processing and image processing and
circular1 the modulation transfer function (2) the sensor(s) of the imaging system,
is circularly symmetrical. The ~ profile is Characterization of the electronic circuitry
MTF(i,) ~ 0 for i, > Scutoff and is: and components is well established by the
temporal frequency in hertz, To cascade
(43) MTF(s) {cos-'(2 the eledronic and optical subsystems; the
temporal frequencies must be converted
rr ~cuit,off )

.y

Sn~nff [] - (Sc~oJ]} 2

for S :-::; ~utoH·

The modulation transfer function curve

for a system with appreciable geometric
aberrations is bounded by the diffraction
limited modulation transfer function

Parameters in Infrared Thermography 171

to spatial frequencies. This is achieved by sensor systems like focal plane arrays
dividing the temporal frequencies by the (FPAs) and line scanners to have a
scan velocity of the imaging device. In particular kind of shift variance (that is,
contrast to the optical transfer function, spatial phase effects), in which case, they
the electronic modulation transfer depend on the position of the target
function is not necessarily maximum at relative to the sampling grid to measure
the origin and can either amplify or the modulation transfer function of the
attenuate the system modulation transfer
function curve at certain spatial system~· 12
frequencies.
Different measurement techniques can
The detector modulation transfer be used to assess the modulation transfer
function can be expressed as: function of an infrared imaging system.
These include the measurement of
where dh and dv are the photosensitive different types of responses such <lS:
detector sizes in the horizontal and (1) point spread function, (2) line spread
vertical directions respectively. Although function, (3) edge spread function,
the detector modulation transfer function (4) sine target response, (5) square target
is valid for all spatial frequencies, it is response and (6) random target response.
typically plotted up to its cutoff A generic modulation transfer function
test configuration is shown in Fig. 8. All
frequencies (i; = l·dh-J and fJ = l·dt:-1). targets except the random ones should be
placed in a micropositioning mount
The nyquist frequency of the detector containing three degrees of freedom
array must be taken into consideration to (x,y,e) to account for phasing effects.
prevent aliasing effects.7
AJJ optical and electrooptical
It is the combination of the optical and components comprising the infrared
electronic responses that produce the imaging system should be placed on a
overall system modulation transfer vibration isolated optical table. The
function. aperture of the collimator should be large
enough to overfill the aperture of the
Modulation Transfer Function system under test. The optical axis of the
Measurements infrared camera has to be parallel to and
centered on the optical axis of the
In an infrared system, the infrared flux is collimator and the camera's entrance
focused on one or more detectors, pupil must be perpendicular to the
converted to a voltage, amplified, collimator optical axis. The display gain
sampled, processed and then displayed on and brightness should be optimized
a computer monitor to present a visual before the modulation transfer function
image. Optical systems tend to be measurements to ensure that the display
isoplanatic, but when they are sampled setting is not limiting the performance of
(digitized), new frequencies are created the detector array.
that were not originally present in the
analog signal. Digitization alters and The imaging of a point source O(x,y) of
distorts the signals. Sampling causes an optical system has an energy
distribution called the point spread flmction
(PSF). The two-dimensional fourier

FIGURE 8. Modulation transfer function measurement test configuration.

Target"" Collimating Detector
optics lens
~

Blackbody
source

Focal distance distance to r----'----,
to collimator
sensor
Controller Frame grabber

c__o_'_'P_''_Y_~I~•~--~-c-o_m_r_"_''_'_~

172 Infrared and Thermal Testing

transform of the point spread function the data, which can corrupt the resulting
yields the complete two-dimensional modulation transfer function. It is
OTF(~.~) of the system in a single
measurement. The absolute value of the important to ensur.e that th.e edge" js
optical transfer function gives the straight with no raggedness. To increase
modulation transfer function of the the signal-to-noise ratio for both the line
system. The impulse response technique
can be practically implemented by placing and edge spread techniques, the
a pinhole, which should be as small as one-dimensional fourier transform should
possible, at the focal point of the
collimator (see Fig.· 8). If the flux passing be taken and averaged over all the rows of
through the pinhole produces a the image. In addition, the system gain
signalwtownoise ratio that is below a usable
value, a slit target can be placed at the should be reduced to reduce noise and the
focal plane of the collimating optics. In target signal should be increased if
this case, the output is referred to as the
line spread function (LSF). The cross section possible.
of the line spread function is obtained by
integrating the point spread function The modulation transfer function of a
parallel to the direction of the line source, system can also be obtained by measuring
because the line image is simply the
summation of an infinite number of the system's response to a series of sine
points along its length. The line spread wave targets, where the image modulation
function yields information only about a depth is measured as a function of spatial
single profile of the twowdimensional frequency. Sinusoidal targets can be
optical transfer function. Therefore, the
absolute value of the fourier transform of fabricated on photographic films or
the line spread function yields the transparencies for the visible spectrum;
onewdimensional modulation transfer
function of the system. however, they are not easy to fabricate for
the testing of infrared systems because of
To obtain other profiles of the
modulation transfer function, the line materials limitations. They require
target can be reoriented as desired. The interferometric13 or halftone
slit angular subtense must be smaller than
the installtaneous field of view (IFOV) with transparencies14 techniques. A less
a value of 0.1 x IFOV recommended. The
phasing effects are investigated for the expensive and therefore more convenient
system under test, by scanning the line target is the bar target, which is a pattern
target relative to the sampling sensor grid
until maximum and minimum signals are of alternate bright and dark bars of equal
obtained at the sensor output. The width, The square wave response is called
measurements are performed and
recorded at different target positions. contrast transfer function (ern and is a
Averaging the system output over all these
locations yields an average modulation function of the hmdamental spatial
transfer function. Huwever, this average
modulation transfer function is measured frequency Sr of the specific bar target
using a finite slit aperture. This
undesirable component is removed by under test. The contrast transfer function
dividing out the fourier transform of the is measured on the peak~to~valley
finite slit yielding a more accurate
modulation transfer function result. variation of the image irradiance and is
defined as:
The modulation transfer function can
also be obtained from an edge spread (47) err(~,) 1\'1 square rcspom('(Sr)
T!'sponse (ESF), which is the response of
the system under test to an illuminated Sr)1\.finput square wav{'(
knife edge target. Jt is also called the edge
response, knife edge response or step The contrast transfer function is
response. There are two advantages in
using the knife edge target over the line typically higher than the modulation
target. A knife edge target is simpler to transfer function at all spatial frequencies,
build than a narrow slit and there is no because of the contribution of the odd
modulation transfer function correction, harmonics of the infinite square wave test
as the slit requires. The edge is
differentiated to obtain the line-spread pattern to the modulation depth in the
function and then fourier transformed. image. Following Cottman's derivation,I5

However, the derivative operation the contrast transfer function is expressed
accentuates the system noise present in
as an infinite series of modulation transfer
functions. A square wave can he expressed

as a fourier cosine series. The output
amplitude of the square wave at

frequency Sr is an infinite sum of the

input cosine amplitudes modified by the
system1s modulation transfer function:

(48) CTF(~r) ~ ~ [~rrF(~r) J

*MTF{:l~l)

+ !5MrF(s~1)

- y~rrr(7s1) +

Parameters in Infrared Thermography 173

conversely, the modulation transfer imaging system tend to average out the
function can be expressed as an infinite phase effects. Random targets of known
sum of contrast transfer functions as: spatial frequency content allow
measurement of the shift invariant
(49) MTF(sr) ~[cTl(sr) modulation transfer function because the
information of the test target has random
+ }crF(3sr) position with respect to the sampling sites
of the digital imaging systems. A typical
- ~CTF(ssr) band limited white noise random target is
shown in Fig. 9. Its output power spectral
+ ym(7sr) + ·) density PSD0u1 (~) is e_stimated by imaging
the target through the optical system onto
Optical systems are typically the detector focal plane array (see Fig. 8).
characterized with three-bar and four-bar
targets and not by an infinite number of The output image data are then
square wave cycles. Therefore, for these captured by the frame grabber and
practical cases, the contrast transfer processed to yield the out put power
function might be slightly higher than spectral density as the squared absolute
the contrast transfer function curve for an value of the fourier transform of the
infinite square wave. For bar targets output imaged data, averaged over the
whose spatial frequency is above one rows of the image. The input and output
third of the cutoff frequency (that is, the power spectral densities are related in the
spatial frequency where the modulation following manner:
transfer function approaches to zero), the
modulation transfer function is equal to FIGURE 9. Band limited white noise random
rr/4 times the measured contrast transfer target.
function. These modulation transfer
function and contrast transfer function
measurements, although they appear
straightforward, may be difficult to
perform because of electronic
nonlinearity, digitization effects and
sampled scene phase effects.16

All the above techniques use
deterministic targets as system inputs.
Imager systems that contain a detector
focal plane array are nonisoplanatic and
their responses depend on the location of
the deterministic targets relative to the
sampling grid, thus introducing problems
at nearly all spatial frequencies. Random
target techniques!7,JH for measuring the
modulation transfer hmction of a digital

FIGURE 10. Experimental setup for testing optics without including detector modulation
transfer function.

Blackbody

source

!.em under IE' It

174 Infrared and Thermal Testing

(50) PSDoutput(l,) ~ IMT1'(1,)12 i'SDtuput(S) high enough numerical aperlure to
capture the entire ima~e forming cone
for which the modulation transfer angle.
function can be calculated by simplifying
Eq. 50. Using an infrared setup similar to that
in Fig. 10, a long wave infrared zoom
Common path interferometers,19 have projector lens was characterized using the
been used for measuring the transfer line spread function technique. This
functions of optical systems. An gimbaling projector shown in Fig. 11
interferogram of the wave front exiting makes possible the testing of a variety of
the system is reduced to find the phase on-board sensor systems. The on-axis and
map. The distribution of amplitude and off-axis modulation-transfer function
phase <lCIO~s the exit pupil contains the curves are shown in Fig. 12.
necessary information for calculating the
optical transfer function by pupil Noise Equivalent Temperature
autocorrelation. Difference

The modulation transfer function of The parameter that characterizes the
optical components can be measured thermal sensitivity is called the noise
without including the detector equimlenl temperature difference (NETD)
mo~ulation transfer function, by placing and is defined as the target-to-background
a miCroscope objective in front of the temperature difference that produces a
detector focal plane array as shown in ratio of peak signal to root mean square
Fig. 10. The microscope objective is used noise (that is, the signal-to-noise ratio) of
as a relay lens to reimage the system's one given by:zo
response formed by the optics under test
onto the focal plane array with the ir j~(51) NETD
appropriate magnification. Jn this case, !·Uti
the detector is no longer the limiting rr D' (A d/,
component of the imaging system, \ del dT
because its modulation transfer function
response becomes appreciably higher than where F111 is the F-number of the system,
the optical modulation transfer function b.fis the electronic bandwidth, D* is the
curve. The microscope objective must be detectivity of the detector, Adrt is the
of high quality to reduce degradation of
the measured response function and have

FIGURE 11. long wave infrared zoom FIGURE 12. Modulation transfer function results of zoom
projector.
projector.
1.0

0.8
c

·0e

c

.2

,g: 0.6

gc

.,.c0 0.4

cro;

"0

:0;;

0.2

0 20 40 60 80 100

Spatial frequency~ (cydes·mm-1)

Legend
- - "'vertical on axis
- --- =horizontal on axi>
- - - "'vertical off axis
·- "'horizontal off axis

Parameters in Infrared Thermography 175

effective area of the detector and the background flux. Also Fq. 52 has a linear
partial derivative of the radiance ()L with
respect to temperature OTis the radiosity dependence on F1*' rather than a square
contrast (exitance contrast). It is dependence as in Eq. 51.
important to notice that E.q. 51 applies
strictly to a situation not limited by The noise £guivalent temperature
background, as with a background limited difference measurement is usually Gtrricd
infrared photodeteftor. out using <1 square target. The size of the
square must be several times the detector
A smaller noise equivalent temperature angular substance (that is, several
difference indicates better thermal instantaneous fields of view) of the
sensitivity. For the best noise equivalent extended source, to ensure that the spatial
temperature difference, D* slwuld be response of the system does not affect the
peaked near the wavelength of maximum measurement. This target is usually placed
radiosity contrast of the source. According in front of an extended area blackbody
to Eq. 51, lower noise equivalent source, so that U1e temperature difference
temperature differences result from lower between the square target and the
F;u. A smaller F;u collects more flux, background is several times the expected
yielding a more accmate estimate noise noise equivalent temperature difference,
equivalent temperature difference. A to ensure a response clearly above the
smaller electronic bandwidth yields a system noise. The noise equivalent
temperature difference test imaged pattern
larger d·well time, obtaining a smaller <Is well as the resulting temperature profile
is shown in Fig. 13. The peak signal and
noise voltage and thus lowering the noise root mean square noise data arc obtained
equivalent temperature difference. A by capturing, averaging and taking the
larger detector area gives a larger standard deviation of sel;eral images. The
instantaneous field of view, thus noise equivalent temperature difference is
collecting more flux, resulting in a better then calculated from the experimental
noise equivalent temperature difference. data as follows:
Basically, sensor performance depends on
both thermal sensitivity and spatial (53) NETD />,T
resolution. Therefore, the fundamental SNR
drawback of noise equivalent temperature
difference as a system level performance fiGURE 13. Noise equivalent temperature difference: (a) test
descriptor is that whereas the thermal pattern waveform; (b) resulting image.
sensitivity improves for larger detectors,
the image resolution deteriorates for larger (a)
detectors. One is obtained at the expense
of the other. Thus, although the noise -------- Peak signill
equivalent temperature difference is a
sufficient operational test, it cannot be l~f----Tuget profife
applied as a design criterion.
t Root
Equation S1 must be modified when t~----------------~..,. ..s-.r.,~,~,.,_'""""~;I~,..,.~J'I
the system operates under background .,....., ~ fr'I""''-V''- Y
limited infrared photodetector conditions. \··}~~-----mean
As seen in Eq. 28, D*mw is proportional to
the F1#. Substituting this equation into -----square
Eq. Sl, the equation for noise equivalent noise
temperature difference under background
limited infrared photodetector conditions Spatial coordinates {arbitrary unit)
is given by:
(b)
(52)

X

where }, is wavelength, 11 is Planck's
constant, cis the velocity of light in
vacuum, L(J,hkg is the background radiance
and 11 is the quantum efficiency of the
detector. In Eq. 52 the noise equivalent
temperature difference is inversely
proportional to the square root of the
quantum efficiency and proportional to
the square root of the in-band

176 Infrared and Thermal Testing

where !'J.T = T1argrt- Tbkg and SNR is the because it accounts for both spatial
signal-to-noise ratio of the thermal resolution and noise level. Therefore, the
system. minimum resolvable temperature
difference is a more useful overall design
Care must be taken to ensure that th~ criterion.
system is operating linearly and that no
noise sources are included. Because of the Minimum resolvable temperature
dependence of noise on bandwidth, the difference is a measure of the ability to
nois~ equivalent temperature difference resolve detail imagery and is directly
must be measured with the system proportional to the noise equivalent
running at its full operational scan rate, to temperature difference and inversely
obtain the proper dwell time and proportional to the modulation transfer
bandwidth. function. This proportionality is shown
by Eq. 54, which includes the main
Minimum Resolvable Temperature variables of interest:
Difference
(54) MRTD oc NETD ~~ [lllio\'- V-IFOV
The minimum resolvable temperature
difference (MRTD) is a subjective MTF(~t )~teye -F
measurement that depends on the
infrared imaging system's spatial where ~1 is the fundamental spatial
resolution and thermal sensitivity. Allow frequency of the target being observed,
spatial frequencies the thermal sensitivity tc-ye is the integration time of the human
is more important whereas at high spatial eye, F is the frame time, MTF is the
frequencies; the spatial resolution is the overall transfer function of the system at
dominant factor. The advantage of the that particular target frequency and
minimum resolvable temperature HIFOV and VlFOV are the horizontal and
difference is that it combines both the vertical instantaneous fields of viev.• of the
thermal sensitivity and the spatial system respectively. The derivation of an
resolution in a single measurement. The exact analytical expression for minimum
minimum resolvable temperature resolvable temperature differences is
difference is not an absolute value but is a complex because of the number of
perceivable temperature differential variables that contribute to the
relative to a given background. The term calculation. It can be calculated using
difference is sometimes omitted because it computer aided performance models such
is understood that it is a differential as the NVL modeF0 Substituting Eq. 5 J
measurement. into Eq. 54 yields:

Conceptually, the minimum resolvable (55) MRTD ~1 ~HIFOV- Vll'OV
temperature difference is the temperature
difference required between the bars and MTF(~1 )[r;- F
spaces of a four-bar test target having a
F1~ ~"i,j
fundamental spatial frequency S1 so that
-~vfinimum resolvable temperature
the bars are just discemable by a trained difference depends on the same variable~
observer with unlimited viewing time. as NETD (F;H, tJ.f~ D* and radiance
The minimum resolvable temperature contrast). However, it is not possible to
difference is a measure of the observed increase the thermal performance of the
thermal sensitivity of a system as a system by increasing the area of the
function of spatial frequency. These tests detector or instantaneous field of view.
depend on decisions made by the The modulation transfer function
observer. The results vary with training, decreases at higher frequencies. Therefore,
motivation and visual capacity, as '''eli as the amount of !'J.T required for a target to
the environmental setting. Because of the be discernable increases for smaller bars.
considerable variability between one The minimum resolvable temperature
observer and another, and between difference increases when modulation
observations by the same observer, several transfer function decreases hut the
observers are required. The underlying minimum resolvable temperature
distribution of observer responses must be difference increases faster because of the
known, so that the individual responses
can be appropriately averaged together. extra factor St in the numerator of Eq. 55.

Minimum resolvable temperature A typical minimum resolvable
difference is a better system performance temperature difference curve is shown in
descriptor than the modulation transfer Fig. 14. The effect of the observer is
function alone because the modulation included in the factor t1Cr1CF. Increasing the
transfer function measures the frame rate gives more observations within
attenuation in modulation depth, without the temporal integration time of the
regard for a noise level. 1vfinimum
resolvable temperature difference is also a
more complete measurement than the
noise equivalent temperature difference

Parameters in Infrared Thermography 177

human eye and then the eye brain system Some typical problems associated with
will tend to average out some of the the minimum resolvable temperature
noise, leading to a lower minimum difference measurements include the
resolvable temperature difference. related distance between the display
screen and the observer, background
In the generic test minimum resolvable brightness and strain. In general, the
temperature difference configuration, the contrast sensitivity increases with
four bar targets located in front of the background radiance; however, during the
blackbody source are placed at the focal minimum resolvable temperature
plane of the collimator, so the radiation difference tests, the observer can
from each point of the surface of the continually adjust the system's gain and
target is coHimated. Because high spatial level and monitor the brightness and
frequency is of interest, it is necessary to contrast to optimize the image for the
mount the minimum resolvable detection criterion. This is expected to
temperature difference setup on a cause considerable inconsistencies
vibration isolated optical table. Because between the results obtained by different
the minimum resolvable temperature observers. Also, each of the observers
difference is a detection criterion for noisy expends great effort during these tests.
imagery, the gain of the infrared imaging Consequently, over a long period of time,
system must be sufficiently high so that the human eye brain sensitivity decreases,
the image is noisy. A typical minimum causing unreliability.
resolvable temperature difference test
pattern with a set of various sized four~bar The minimum resolvable temperature
targets is shown in Vig. 15. Infrared difference is also somewhat limited
imaging systems are subject to sampling because all the field scenes are spectrally
effects.lO The minimum resolvable selective (that is, the emissivity is a
temperature difference does not have a function of wavelength) whereas most
unique value for each spatial frequency minimum resolvable temperature
but have a range of values depending on difference tests are performed ·with
the location of the target with respect to extended area blackbodies.
the detector array. Therefore, the targets
must be adjusted to achieve the best It is of practical interest to measure the
visibility. It is important that the observer minimum resolvable temperature
count the number of bars to ensure that difference without the need of a human
the required number is present. The observer. Automatic tests or objective tests
targets should range from low spatial are desirable because of insufficient
frequencies to just past the system cutoff. number of trained personnel and because
Targets must span the entire spatial the subjective test is time consuming. In
frequency response. this context Eq. 54 can be written as:

fiGURE ·14. Typical shape for minimum resolvable temperature r(56) MRTD ~ K(l; ) NETD
difference curve. MTF(sr)

where the constant of proportiona1ity and
any spatial frequency dependent terms -
including the effect of the observer~ are
taken up into the function K(~1). To
characterize the average effects of the
observer, for a given display and vie-wing
geometry, a minimum resolvable
temperature difference curve is measured
for a representative sample of the system
under test. Along with the minimum
resolvable temperature difference data, the

FIGURE 15. Minimum resolvable temperature
difference image test patterns, backlit by
extended source blackbody and imaged
through zoom projector lens of Fig. 11.

Spatial frequency S(relative unit)

178 Infrared and Thermal Testing

noise equivalent temperature difference FtGURE 17. Forward looking infrared tester
and modulation transfer function are apparatus.
measured and recorded for the system.
From these data, the function K(~1) can be
determined and subsequent tests of
similar systems can be performed without
the observer.

Figure 16 shows a comprehensive
automatic laboratory test station, which
provides the means to measure the
performance of an infrared imaging
system (that is, the modulation transfer
function, noise equivalent temperature
difference and minimum resolvable
temperature difference). A field test
system measuring the forward looking
infrared parameters of a military
helicopter is shown in Fig. 17.

fiGURE 16. Automatic laboratory test station.

Parameters in Infrared Thermography 179

PART 3. Effects of Atmosphere

Atmosphere is a complex mix of various wavelength and meteorological
gases, particles and aerosols in different conditions.
concentrations. Transmittance of
electromagnetic energy through this mix In order of importance, the main gases
is complicated. Belo\v is a brief description that absorb the transmitted radiation are
of this involved processes; interested
readers can consult published sources21,22 first water vapor (H20), then carbon
for more information. dioxide (C02 ) and finally ozone (0.~). In
combination with the other gases (such as
The atmosphere has a few
characteristics annoying for infrared carbon monoxide, oxygen etc.) they make
thermograpllic nondestructive testing. up for the following transmission windows:
The main problem concerns the 0.4 to 1 pm (including the visible
transmittance t of electromagnetic energy, spectrum); 1.2 to 1.3 pm; l.S to 1.8 pm;
which is less than 100 percent.
Self-absorption by gas (water, carbon 2.1 to 2.5 pm; 3 to 5 pm; and 8 to 13 pm
dioxide, ozone etc.) in the atmosphere (Fig. 18). Of these, the two main bands of
and diffusion due to particles such as
aerosol propellants and other molecules interest for infrared thermographic
are responsible for variations and
interference in transmission of nondestructive testing are discussed
electromagnetic radiation. !vforeover, below.
various factors such as the presence of
thermal gradients and of turbulence make In accordance with Planck's Jaw, the
the index of refraction heterogeneous and longwave band (8 to 13 pm) is of
fmther contribute to the degradation of
infrared measurement through the particular interest in measuring radiation
atmosphere. Finally the atmosphere itself from objects near room temperature (an
emits infrared energy, which is captured
by the infrared sensor. Most of these example is for the detection of intruders
effects arc not easily taken into account by law enforcement agencies). The
because they depend on the distance,
shortwave band (3 to 5 pm) is best suited
for warmer objects (an example is any

process releasing carbon dioxide, such as

combustion engines, in applications such
as military tracking of targets). Absorption

by gases is a complex phenomenon and,
because of the scale, J1ig. 18 does not
reveal the intricate pattern of individual
absorption lines.

FIGURE 18. Atmospheric transmittance for the 1976 US standard atmosphere, 288 K (15 oc ~59 oF), 5.9 mm of precipitable
water, 46 percent relative humidity, 101 kPa (1 013 millibar) atmospheric pressure at sea level for a horizontal path of 1 km
(0,6 mi). Also shown are the main gases causing absorption in the infrared bands.

Water Carbon Water Carbon Ozon£> Carbon Water
dioxide dioxide
dioxid£>

100 r1rU ' lI"' 11"'1 • ~,.,, 'T

c 80 .l

wc ~ -UI
w~
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10 11 12 13 14 15 17 19 22
8 60

gcwv

E 40

c~

0

F

20

0

o..s 1.0

Wavelength (1-Jm)

180 Infrared and Thermal Testing

It is nevertheless possible to describe of the atmosphere following Gladstone\
mathematically the absorption. One way law:
is through the exponential absorption
Jaw, which applies to the case of a given (60) Hair
monochromatic wavelength. It can also
be used in the case of spectral hands if where k11 ls a constant and P01 ir is air mass
absorption does not depend on the density. Normally PJit·:::::: 1. It is difficult ~o
\Vavelength: take into account turbulences because au
mass density is affected by local
(57) F temperature variations but also by ~vine!~,
humidity etc. In general, computallon of
where kr is the medium absorption atmospheric transmittance assumes
coefficient that depends on the turbulences are homogeneous and
wavelength ), and where x is the distance. isotropic. In the presence of turbulences,
At null distance (useful for calibration radiation is transmitted but the
purposes)1 x:::: 0 and thus F:::::: F0 • The propagation path does not foilm'l' a
coefficient t of transmittance is then straight line anymore. Various
expressed by: phenomena are observed. Because of thl'
stratification of air layers, radiation rays
(58) I curve in low atmosphere (this explains
mirages). Displacement of the points from
Sometimes, the medium is their normal position on the image plane
characterized in term of optical density D: causes images to fluctuate. This
fluctuation is due to the rapid
(59) D log 10 I displacement of large zones (with respect
to transmitted rays) with heterogeneous
- 11 . values. If these zones are small, the
l im"'age sciHti/lates - spots appear. 1···ma IIy,
defocalization of the images and
log10 [exp(-krx)] degradation of spatial cohesion are other
observed phenomena.
0.43 krx.
It is not an easy task to take into
As mentioned before, diffusion of account all these effects because involved
radiation by particles is another concern. parameters are to be measured first.
This process changes the spatial Parameters may include the wind speed,
distribution of the transmitted energy the temperatures along the transmitted
whose intensity is also affected. Large path of interest and also the
particles (with respect. to wa~•el~1~gth. of concentration of the various gases (mainly
interest) do not contnbute s1gmflcatJvely water, carbon dioxide and ozone).
to the diffusion process whereas small I\.foreover, these parameters are not
ones having a size similar to the constant in time due to meteorological
wavelength of interest affect the diffusion conditions and solar heating through the
following the theory of diffraction. day. In fact1 a measurement is likely to be
Interestingly, the diffusion process of the obsolete as soon as it becomes available.
solar radiation by particles explains why For these reasons, models of the
the sky is blue at noon (sun at the zenith) atmosphere have been developed.
and turns red at dusk (sun at the horizon).
A variety of radiative transfer models
In the presence of humidity, particles have been developed. The discrepancies
suspended in the air (of size around among models vary up to about
O.S pm) agglomerate water molecules. 10 percent, depending for example on
This creates mist that turns to fog as water cloud optical thickness. Among common
continues to agglomerate, making models are a band model and a
droplets or icc crystals (in this case line-by-line radiative transfer model.
diameter rises to 3 or 4 pm (about Commercial and freeware versions of
1.S x IQ-4 in.). Finally when droplet size these programs are available fr<?m various
reaches about 0.25 mm (0.01 in.) in vendors. In the infrared part ol the
diameter, the droplets become too heavy electromagnetic spectrum, the situation is
and it rains. ·water vapor absorption more complex than in the shortwave
occurs particularly at 2.6 pm, in the 5.5 to range because of assumptions made on
7.5 pm range and over 20 pm. features such as line sh<1pr and line cutoff.

As the sun heats up atmosphere layers, On a practical point of view, the global
turbulences are created by air convection transmittance t1. can be expressed by the
because of gas densities inversely following formula (for a given wavelength
proportional to gas temperat~ne..These J, and distance x):
turbulences affect the refractJOil mdex H<lir
(61) 1),

Parameters in Infrared Thermography 181

where t1•.Hzo is the transmittance due to ambient operation temperature silicon
water vapor in the atmosphere, tJ,,c02 is detectors.
the transmittance due to carbon dioxide
in the atmosphere, t>..d is the Another important point to consider is
transmittance due to particle diffusion the detectivity JJ* of the detector used. A
and train takes into account the effect of
rain, if any. Parameters in Eq. 61 can be 77 K (-196 •c ~ -320 "I') cooled indium
obtained from tables2i,Z3 for various
scenarios of air temperature, relative antimonide detector operating in the 3 to
humidity, distance of visibility, length of 5 pm has seven times higher detectivity
path of interest. Atmospheric
transmittance computations for standards than a 77 K (-196 •c ~ -320 •F) cooled
atmospheres are also available
interactively on the web. mercury cadmium tellmium detector
operating in the 8 to 12 pm range. That
Selection of Atmospheric means that, even if the emitted radiation
(temperature of interest, spectral
Band emissivity) is higher in the 8 to 12 pm for
a speciflc application, the contrast
Because the atmosphere has not perfectly obtained may be stronger in the 3 to S
flat transmission prOperties (Fig. 18), the pm range because of the superior D* of
selection of the operating wavelength the indium antimonide detector.
band will be conditioned by the final
application. For the majority of Detailed studies have concluded that,
nondestructive testing applications, the for temperatures from 263 to 40~~ K
useful portion of the infrared spectrum
lies in the 0.8 to 20 pm range. Beyond (-10 to +130 •c 1+14 to +266 "F]),
20 pm, applications are more exotic such
as high performance fourier transform measurements can be done without much
spectrometers which operates in the difference in both bands (3 to 5 pm and 8
25 pm. The choice of an operating to 12 pm). For some special applications
wavelength band dictates the selection of (for example, for the military), bispectral
the detector type. Among the important cameras operating simultaneously in both
criteria for band selection are operating bands have been developed to
distance, indoor/outdoor operation, characterize target thermal signatures
temperature and emissivity of the bodies more accurately.
of interest.

Planck's law stipulates high
temperature bodies emit more in short
wavelengths; long wavelengths are of
more interest to observe objects near
room temperature. Emitted radiation from
ordinary objects at ambient temperature

(300 K 127 •c ~ 70 "F]) peaks in this long

wavelength range. Long wavelengths are
also preferred for outdoor operation
where signals are less affected by radiation
from the sun. For operating distances
restricted to a few meters (about 10 ft) in
absence of fog or water droplets, the
atmosphere absorption has little effect.

Spectral emissivity is also of great
importance because it conditions the
emitted radiation. Polished metals with
emissivity smaller than 0.2 can not be
obsen•ed directly because they reflect
more than they emit. A high emissivity
coating (such as black paint) or a
reflective cavity must be used.

Although no specific rule can be
formulated, generally the most useful
bands are 3 to 5 pm and 8 to 12 pm
because they match the atmospheric
transmission bands. Most of the infrared
commercial products fall in these
categories while near infrared (0.8 to
1.1 pm) is easily covered by standard

182 Infrared and Thermal Testing

References

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Parameters in Infrared Thermography 183

'"'·R.20. Ratches1 ]., Lawson/ L.P. Obert1

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184 Infrared and Thermal Testing

,, . -',:'

CHAPTER

Noncontact Sensors for
Infrared and Thermal
Testing

Xavier P.V. Maldague, University Laval, Quebec,
Quebec, Canada (Parts 1 and 2)
jussi Varis, University of Helsinki, Helsinki, Finland
(Part 3)
Yoshizo Okamoto, East Asia University, Shemonoseki,
japan (Part 4)

Portions of Parts 1 and 2 adapted with permission from Infrared Methodology and Technology,
© 1994, Gordon and Breach Science Publishers, Langhorne, Pennsylvania.