ASNT NDT Handbook Volume 3 Infrared and Thermal Testing

19. Boccara, A. C., D. Fournier, 'vV. jackson 27. Coufal, H. and P. Hefferle. "Thermal
Diffusivity Measurements of Thin
and N.M. Amer. "Sensitive Films with a Pyroelectric
Photothermal Deflection Technique Calorimeter." Applied Physics A.
Vol. 38. Berlin, Germany: Springer
for Measuring Absorption in Optically Verlag (1985): p 213-219.
Thin Media." Optics Letters. Vol. S,
No.9. Washington, DC: Optical 28. Mandelis, A. "Theory of

Society of America (1980): p 377-379. Photothermal-\'\'ave Diffraction and
20. Bertolotti, M., G.L. Liakhou, Interference in Condensed Media."
Journal of the Optical Society ofAmerica,
R. Li Voti, S. Paoloni and C. Sibilia. series A. Vol. 6, No. 2. \oVashington,
DC: Optical Society of America
"Analysis of the Photothermal
Deflection Technique in the Surface (1989): p 298-308.

Reflection Scheme: Theory and
Experiment." journal ofApplied
Physics. Vol. 83, No. 2. College Park,
MD: American Institute of Physics
(1998): p 966-982.

21. Bertolotti, M., G.L. Liakhou,
R. Li Voti, S. Paoloni, C. Sibilia and
N. Sparvieri. A11 Cryostatic Set-Up for

the Low Temperature Measurements
ofThermal Diffusivlty with the
Photothermal Method.'' Review of
Scientific Instruments. Vol. 66, No. 12.
College Park, MD: American Institute
of Physics (1995): p 5598-5602.

22. Bertolotti, M., G.L. Liakhou, R. Li
Voti, Ruo Peng Wang, C. Sibilia,
A.V. Syrbu and V.P. Yakovlev: 11An
Experimental and Theoretical

Analysis of the Temperature Profile in
Semiconductor Laser Diodes Using

the Photodeflection Method."

Measurement Science and Tecllnolagy.

Vol. 6. London, United Kingdom:
Institute of Physics (1995):

p 1278-1290.
23. Touloukian, Y.S., R.W. Powell, C.Y. Ho

and M.C. Nicolaou. Thermophysical
Properties ofMatter. Vol. 10. New York,

NY: Plenum (1973).

24. Reyes, C.B., ]. jaarinen, L.D. Favro,
P.K. Kuo and R.L. Thomas. Review of

Progress in Quantitative Nondestructire

Evaluatiou~ ed. D.O. Thompson and
D.E. Chimenti. Vol. 6. Ne\\' York, NY:

Plenum (1987): p 271.
25. Bertolotti, M., R. LiVoti, G. Liakhou

and C. Sibilia. "On the

Photodeflection Method Applied to
Low Thermal Diffusivity

Measurements." Rel'iew ofScientific
Instruments. Vol. 64, No. 6. College
Park, MD: American Institute of

Physics (1993): p 1576.

26. Bertolotti, M., V. Dorogan, G.
Liakhou, R. LiVoti, S. Paoloni and
C. Sibilia. "New Photothermal
Deflection Method for Thermal

Diffusivity Measurement of
Semiconductor \Vafers." Rel'iew of
Scientific Instruments. Vol. 68, No. 3.
College Park, MD: American Institute

of Physics (1997): p 1521-1526.

86 Infrared and Thermal Testing

CHAPTER

Fundamentals of
Infrared Radiometry

Stephan Offermann, Universite de Reims
Champagne-Ardennes, Reims, France
jean Louis Beaudoin, Universite de Reims
Champagne-Ardennes, Reims, France
Christian Bissieux1 Universite de Reims
Champagne-Ardennes, Reims, France

PART 1. Fundamental laws

Electromagnetic Spectrum unchanged in the medium, whereas the
and Thermal Radiation wavelength /, becomes:

The intimate nature of radiation ·was first (4)
established by Maxwell as electromagnetic
waves, ranging from cosmic rays to radio and the wave velocity F becomes:
waves and including gamma rays, X-rays,
ultraviolet radiation, visible light, infrared (5) {/ = !___
radiation and microwaves. The behavior
of the electromagnetic waves is ·well II
described by a set· of mathematical
relations, named ~vfaxwell's eguations. 1•2 The frequent use by spectroscopists of
In a vacuum, the most simple solution is the Wtll'£' llllmber a instead of the
the monochromatic plane waw: wavelength should also be mentioned.
The wave number, a= /,- 1, is commonly
(1) E expressed in cm-1.

The expression of a plane
monochromatic wave, propagating along
the Z axis in the medium, becomes:

(6) E E0 exp(-2nk_!/,_o_)

·where £0 is amplitude, v is frequency, z is x exp [J.2 rr ( vt - ··1),1z )]

distance from the origin of the coordinate 0
axis and /..0 is wawleugt/1 in a vacuum. At
any instant, the wave amplitude is The first exponential term represents
spatially periodic and the wavelength the progressive attenuation of the
represents the spatial period. The amplitude, as long as the wave travels
wavelengths corresponding to the deeper in the medium, according to the
radiation visible to the human eye range value of the extinction index kl,!)\' The
from about 0.4 to 0.8 pm ·whereas the ab5orpliou cocfficimt K for energy is
infrared domain extends beyond 100 pm. proportional to the square of amplitude
The wavelength appears related to the and is related to k:
velocity c of the electromagnetic wave in
a vacuum: (7) K

c Various processes can be responsible for

v the emission of radiation: collision with

\'\'hen the ·wave has to propagate in <l particles in electric discharges, excitation
medium, things are much more involved. by light beams or heating of matter.
On a microscopic scale, it is clear that the
electric and magnetic fields of the wave Among these processes, only the latter
do interact with the electric charges of relates to thermal radiation.
matter. Because the present discussion is
concerned only with the macroscopic A medium is said to emit thermal
scope of things, Maxwell's theory can still radiation when the interactions between
be used provided that a new macroscopic
parameter is introduced, the complex its constitutive particles are strong enough
refractive index of the medium: to keep their enert,'T distribution in a
statistical t'quilibrium. This equilibrium i~
(3) ll ~ II - ik mathematically described by a function of

where n is the refractive index and k is the the local temperature- generally the
extinction index. The frequency v remains !vlaxwcll-Boltzmann's distribution. Then
the medium is at local tlu:rmutlynmuic
equilihrium. For example, in an electric

wire heated by the joule effect, the
unceasing collisions between electron.<.

88 Infrared and Thermal Testing

and atoms permit establishment of this This quantity wil1 appear to be
statistical equilibrium. This is not the Ci}se fundamental, because any imaging
system, such human eyes or the detector
in the rarefied atmosphere of a discharge of infrared cameras, responds
lamp, where the electric arc is able to proportionaBy to L. Because no varieties
force a particular energy distribution of radiance are usually introduced, it must
be specified whether a radiance is emitted
mainly related to its strength. or received or reflected or even
transmitted at a surface. In the absence of
At local thermodynamic equilibrium, a specification, it is assumed that the
the mathematical expression for the radiance of a surface is derived from the
radiosity.
spectral distribution of emitted radiation
is the product of a universal expression, Of less utility is the quantity radiant
depending only on temperature (Planck~~ i11tensity I, mainly useful to characterize
point sources:
law)1 by a quantity specific to the
material: its emissivity. In this case, where (10) I ~ eM>
radiation has a thermal nature1 and in the
usual conditions of temperature, the dQ
emission 'i\'avelengths ·will appear mainly
in the infrared range. However, the word iutensit)' to describ<:'
radiance must he used carefully in many
Radiometric Quantities contexts.

Some adequate quantities must now be Blackbody Radiation
defined to quantify the radiation leaving
or reaching a surface. At first, two The basic laws of thermal radiation
categories of terms are to be considered: describe the emission of an ideal emitting
material, usually known as a blackhod)'.:P
1. Total quantities deal with energy This emission, for which physical laws are
summed over the whole spectrum; thoroughly settled, actually exists inside
an isothermal closed cavil}~ whatever the
2. Spectral or monochromatic quantities nature of its walls. The emission of this
characterize the ener&'J' contained in a completely closed isothermal cavity is
narrm\' interval of "\vavelength; these impossible to observe. Indeed, a small
will be written with a subscript],. aperture must be made through a cavity
wall. The resulting device is called a
In the same way as for conduction and laboratory blackbody. Geometrical
convection, the radiant flux or radiant calculations show that its actual emission
power <P, measured in watt, is the time rate is very close to that of the ideal
of radiant energy. The radiant flux per blackbody, which represents both the
unit area, over all directions of space, is perfect absorber and the best emitter at
thus measured in watt per square meter local thermal equilibrium.
(W·nr2):
The total emissive power /o.J0 (\·V·m-2) of
(8) tp d<t> an ideal blackbody, summed over the
dS whole spectral range, is given by the
Stefan-lloltzmal'ln law:
If the energy is emitted by the surface,
then this quantity is called emissil'e (11) d<t>
dS
power M.
with:
If the energy is received by the surface,
(12) (J
then it is called irmdiance E.
Because the total radiance of thl' ideal
If the energy is leaving the surface, blackbody 1.0 (\·V·m-2·sr-1) is isotropic, the
whatever the physical cause of which
blackbody is said to be lambertian and its
(emitted plus reflected plus eventually radiance is simply related to its emissive
transmitted), then it is called radiosit}' f.
power:
So, 111, E and I appear as varieties of the
same physical species <fl1 introduced to
describe more precisely the radiation
exchanges at a surface element.

To quantify now the tlux passing
through a small surface of area dS within

a small solid angle dO., around a direction
making an angle 0 with the normal to the
surface, the radiance L (or lwuiuance), is

defined and measured in watt per square
meter per steradian (\·V·nr2·sr 1):

(9) L d2cJ>
dSdQcose
(13) L" n
dtp
d!lcosO

Fundamentals of Infrared Radiometry 89

Planck's Jaw, already evoked, describes shorter wavelengths. Indeed, in an
the spectral distribution of the flux obscured room, the blackbody emission
becomes visible (dark red) for a
emitted by the blackbody in a vacuum tempemture higher than 823 K (550 "C =
(very close to that emitted in air, where), 1020 °F).
is close to ),0 ):
It must be pointed out that more than
(14) dM 0 95 percent of blackbody radiation lies in a
d), spectral range between 0.5 J. m and 5 )'111 -
that is, one order of magnitude in
1)),5 (exp_llc_ - l\'avelength. For a blackbody at room
ki.T temperature, Am is about 10 pm
nC11.-5 (2898/293) and the essential of the

expC--z- - emitted energy lies in the infrared

),T between Sand 50 run. At 823 K (550 oc =

with the first and second radiation 1020 oF), ),m is about 3.5 pm (2898/823)

constants defined respectively as C1 = and the main emission still lies in the
2!JcZ ~ 1.191 I0-16 W·ll1 2 = 3,742 !QH infrared, but the human eye is a very
good detector and sees the tiny emission
W·nr2·p111'4 and C2 = IIck-1 = 1.4388 104 at the limit between red and infrared.

run-K. The sun radiates nearly like a

M01, is the spectral {or monochromatic) blackbody at 5800 K
emissive power of the blackbody. The
(5500 oc = 10000 °F). The solar surface
associated spectral radiance L). is
exhibits its emission peak around 0.5 pm
isotropic: and essentially emits between 0.25 t-Jlll
and 2.5 pm. Aside from this shift toward
(15) L~ shorter ·wavelengths, it should not be
forgotten that a hotter blackbody always
emits more than a colder one, at any
wavelength.

(16) L~ dL0
die

dSdQ cosSri),

(17) 21!c2
ics ( e x pli-e-
kAT

At a given blackbody temperature, M 0,
A1f, L0 and Lf are thus weB known and
easily computable.

Mf presents a maximum value at a
\<Vavelength Am, simply related to the

absolute temperature by \•Vien's law:

(18) ),,T = 2898 run-K = C1

The spectral emissive power M-2m at /,m
follmvs the Jaw:

(19) Mf,

with c4 = 1.287 X w-s (\·V·nr3.K-s).

Objects at low temperatures are not
actually perceptible to the eye, because
their thermal emission lies mainly in the
infrared. When the temperature of the
blackbody increases, its thermal emission
will contain more and more energy at

90 Infrared and Thermal Testing

PART 2. Radiative Properties of Materials

Directional and Spectral which describes the real material's
Emissivities emissive ability versus wavelength and
emissive direction. (See Fig. 1.) Note that
Because the blackbody is an ideal radiator, the knowledge of E't. or €1.. can be restricted
it emits as much radiant energy as to the 0.5 ]..111 to S A111 interval because L0>.
possible at local thermodynamic is negligible out of this interval. From the
equilibrium. The emissive power of a real knowledge of e'1_, the other three
material will then be defined by emissivities can he derived:
comparison with that of a blackbody at
the same temperature, the _ratio being (25) € ~
called the emissivity of the material.
JEt.;4 L}_ d),
To characterize the angular and spectral
dependence of the emissive behavior of (J u
the real material, four parameters are
usually defined:3-7

(20) € ~

for total (hemispherical) emissivity; (26) E 1
- -JLcosBdn
(21) <' ~ .!-_ nL0

Lo n
for directional (total) emissivity;
- 1-JE'L0 cosedn
(22) ''· rrL0

n

J~ E'cos8dQ

n

for spectral (hemispherical) emissivity; (27) E'
and
(28) €), ~ .r!r .Jr',•cos9dn
(23) <\
n
for spectral directional emissivity. The
adjectives between brackets arc usually A frequently used simplification supposes
not mentioned. The radiances under the surface to he gm}', meaning that its
consideration here are strictly the emitted spectral radiative properties do not
ones. depend on wavelength:

According to these definitions, the four '·(29) €'·
ratios are obviously equal to unity for a
reference blackbody and range between and
zero and unity for real materials. The
emissivities are linked by the spatial (30) E), ~ E
integral relation between emissive power
and radiance: The second usual simplification
supposes the surface to be diffltse,
(24) M JM,,dk JLcos8tfQ meaning that its directional radiative

0 n

The spectral directional emissivity E'J.
appears as the fundamental parameter,

Fundamentals of Infrared Radiometry 91

properties do not depend on the emission For the most -common materials,
angle: emissivity tables can be found in the
literature. Generally, these tables present
and total emissivities, rarely spectral
emissivities and almost never directional
(32) e' ~ e emissivities. The tables often give a good
idea of the approximate emissivity values
Moreover, the emissivity effectively but in actual temperature measurement
involved in a radiometric measurement is conditions the onl>' knowledge of the
actually an averaged value over the total emissivity may not be sufficient.
spectral sensitivity range of the
radiometer. Its spectral response 5(/..) is Absorptivities,
different from zero only over a spectral Reflectivities and
sensitive interval t'1A of the detection Transmittivities
device. Then, the effective emissil•ity of the
material over the spectral range of the Once the emissive behavior of the
apparatus is defined as: material has been characterized, the
response of the material must be·
Je'1.L~S(i.)di. considered when an external radiation
impinges on its surface. The total
/,), absorptivity a of a surface element of area
dS is simply defmed as the ratio between
and would be computed, provided that the absorbed flux d<I>a and the incident
e')_, L0 >. a11d S(l~) are available. Obviously, if flux dct>i:
the material can be considered as a
graybody over D./.., the effective emissivity (34) a.
E'& remains equal to the spectral
emissivity e').· For a blackbody, which is perfectly
absorbing, a is equal to unity. For a real

fiGURE 1. Emissive power versus wavelength.

Blackbody

/"T'/Mf=nif

I: \
I : :'"'\
)<I :' \ '\
Graybody
I --.i .
/ ;·\ ,· '- '{ '-...., '-....., Real surface
/// \.),/ '' ~--.' "/?<.... __ _. .
I '/ , / ·----, -
I..' '.· ••••...• .•..••··........-··
I
//
/....--··'

),

Wavelength

legend

Lf = spectral radiance associated with blackbody

Mf = spectra! (or monochromatic) emissive power of blackbody

M:.. =spectral emissive poWN{W·m-l·pm-1)
}. = wavelength

Am = maximum wavelength

92 Infrared and Thermal Testing

medium, at least a small part of the (37) p

incident radiation is reflected. Then, the As in the case of absorption, the
absorptivity remains strictly less than reflectivity is generally dependent on the
unity. Absorptivity a generally depends on angular and spectral distributions of the
the angular and spectral distributions of incident radiation. However, the angular
distribution of the reflected radiation
the incident radiation. Analogously to the must also be taken into account, leading
emissivities, spectral and directional to defihition of more parameters: total or
spectral, hemispherical, directional
absorptivities must be defined1 hemispherical, hemispherical directional
introducing four parameters: a, a', a1. and and bidirectional reflecth•ities.
a'1,~ the first three being computable from
the latter- that is1 the directional The fundamental quantity, specific to
spectral absorptivity: the sample, from which every parameter
can be computed is the bidirectional
Ia',,L1.i cos8dQ reflectana' distribution function (BRDF) (,1.:

(35) U.), In (38) {,),
Lj.jCOS8dQ
where dhr is the reflected radiance in a
n direction 0,(6r,lflr) and d£1. is the irradiance
due to the incident radiance L1.1 in an
(36) u.' Ja'J. L1.i d), elementary solid angle dQi around the
direction Qi(ej,{}lj).s (See Fig. 2.)
),

'lb describe now the reflection of the

incident flux by the material, its total
reflectivity is first defined:

FIGURE 2. Reflection of rad'1aflon at opaque surface can be described by defining total or
spectral, hemispherical and directional reflectivities.

dL;_,

legend reflected radiance in direction 0,(9, <.p,)
surface area
dL).r elementary solid angle (steradian) around direction n,(O,tp,}
dS
elementary solid angle (steradian) around direction n,(G,,<.p,)
dO; incident radiance (W·m-2·sr·'·J.lm-l)
angle (degree~) relative to normal of direction D.(fl,<.p,)
do, angle (degrees) relative to normal of direction H,{fl,<.p,)
azimuth angle of direction n,(O~q:>,)
LJJ azimuth angle of direclion n,(O,~.p,)

e,
e,

••••

Fundamentals of Infrared Radiometry 93

Finally, in a similar way, the (44)
transmission of radiation by a nonopaque
material can he described by defining its Another important relation between
total or spectral, hemispherical or radiation properties is Kirchhoff's law if
directional transmittivities, all deriving the material remains at local
from the bidirectional transmittance thermodynamic equilibrium:
distribution function (BTDF) (0..
The ability of a medium to emit
Every parameter definition has not
been given here but the actually useful radiation at specified ·wavelength and
quantities will only be specified later.
Moreover, the reader should notice rat11er direction is thus related to its ability to
disturbing confusions in the literature, absorb the same radiation. This
where authors use different and
sometimes inconsistent definitions. A fundamental property extends well
reliable reference is the monograph from
the National Bureau of Standards and its beyond the thermal radiation domain,
considerations and nomenclature have taking its origin in quantum physics.
been closely followed here.8
Emissivity r't. and its integrals r,.. r' and
The radiant energy conservation is care intrinsic properties, depending only
obviously verified at any surface element: on the sample temperature. On the
contrary, a'1• is the only intrinsic
Thus, according to the definitions of the absorptivity, because its integrals are
total radiation parameters:
functions of the angular and spectral
(40) (J. + p + T ~ J distributions of the incident radiation.

Neglecting any phenomenon inducing Consequently, the total absmptil'ity is
a change in frequency, like raman
scattering or nonlinear optics, the energy generally different from tile total emissivity.
balance can be restricted to an elementary As an example, in solar collectors, a
spectral interval: maximum value of a toward solar
irradiation and a minimum value of£ arc
Searching for a relation between
directional spectral quantities, the simultaneously required. In this case, the
scattering of radiation by the sample must latter is total emissivity integrated over
be taken into account. Even in the case of the spectral range of a blackbody at the
a unidirectional incident radiation, the collector temperature whereas its total
reflected and transmitted radiation are
mostly scattered over their respective half absorptivity is integrated over the solar
spaces. So, directional hemispherical irradiancc spectrum.
reflectivity and transmissivity must be
used, which are defined respectively as The radiative balance associated with
the ratios of the reflected and transmitted Kirchhoff's Jaw brings an indirect path
fluxes over the whole half spaces to the
directional incident flux. They can be toward the directional spectral emissivity:
expressed as spatial integrals of the
bidirectional distribution functions: PT"" a't, = 1 ~ ~

(42) PT which becomes fairly simple for an
opaque material:
and
- p').'
J(43) •T ~ fi>. case, dll1
Physical Fundamentals of
n Radiative Properties

The radiative energy balance links the As said above, according to the
directional spectral properties: electromagnetic theory, a medium is
conveniently characterized hy its complex
index:

The two indexes 11 and kex arc often
called the optical constants of the

medium, despite the fact that they are
actually strongly variable with
wavelength.

94 Infrared and Thermal Testing

Fresnel's formulae permit the electromagnetic waves, which cannot
propagate in such a medium.
calculation of the reflectivity as a function
Via the theoretical expression of p'J.(9),
of the indexes n and kf.'x for a Fresnel's formulae allow the knowledge of
nonscattering medium with a perfectly t';.. versus the emission angle. For clean
smooth surface. Reflection on such a polished metals; c';..(9) remains constant
medium is called spewlar and leads to till 40 degrees, then strongly increases
simplified notations: with a maximum beyond 80 degrees
before diving back to zero at grazing
(49) PT p'), angles. Thus the hemispherical E is higher
than E'11 because it integrates the higher
and values of r' at large angles. Typical values
for polished metals in the infrared are:
(5o) tT
(54) E;, < 0.1
The emissivity can then be derived from
the radiation balance: and

(51) (55) 1.2

where Fresnel's formulae give the specular However, these values strongly depend
reflectivity under normal incidence: on surface conditions: roughness, oxidation
and even pollution. Roughness leads to
(52) P\ P't.n + k"2 more diffuse surfaces and greater apparent
emissivities, increasing the actual radiance
If this sample is sufficiently thick or its by cavity effects. Oxidation can even
absorption coefficient high enough, there make metals behave like dielectrics and
is no transmitted radiation: the emissivity of the oxidized metal
generally depends on the oxide thickness.
(53) £),, 1 - Pin Finally, the emissivity of a metal gener(llly
411 decreases toward long wavelengths and
increases rather slightly versus

If the sample surface is rough, the FIGURE 3. Typical directional emissivities. 80 degrees
theoretical study is much more difficult. 1.0
10 degrees
Many works have dealt with this problem,
leading to complex mathematical 0.25 0.5 0.75

developments but not to very good F)'.
agreement with experiment. However, the
legend
particular but fundamental case of the a =- polished metal emissivity curve
b == lambertian surface emissivity curve
specular reflection allows the prediction c == opaque dielectric material emissivity curve
of ty'pical radiation behaviors, mainly d = blackbody emissivity curve
c).=- spectr<Jl directional emissivity
distinguishing two groups of media:
metals and dielectrics.

For metals, the extinction index is

high, namely in the infrared where kex is
most often higher than 11. They are

opaque for a thickness far lower than
1 run (4 x JQ-5 in.). If k,,, is much higher
than n, p\.~1 approaches unity and r.'}.J/
approaches zero.

Thus, in a wavelength range where a
medium absorbs very strongly (the

absorption coefficient is equal to
4nkf.'x·l.-1) it becomes almost totally
reflecting and is able neither to emit nor
to absorb. There is only an apparent
inconsistency: the absorptivity and
absorption coefficients must not be

mistaken for each other. The small
non reflected part of the incident wave is

indeed very strongly attenuated. A perfect
mirror would totally reflect the

Fundamentals of Infrared Radiometry 95

temperature, apart from oxidation effect. multiple reflections induced by the
microstructure. Dielectrics can then
(See Fig. 3.) appear opaque (essentially because of
For dielectrics, the extinction index k is scattering) over their macroscopic
thickness whereas their grains remain
nearly always negligible, when compared transparent. Such a material can present a
to 11. Then, the specular reflectivity only strong directional hemispherical
depends on the refractive index: reflectivity and consequently a weak
emissivity: this medium may he described
(56) p',., (n - 1)2 as a white body by analogy with materials
(n + 1)2 appearing white in the visibie·range.

For example in the case of glass, 11 takes The radiation behavior of gases should
a value around 1.5 in the visible leading also be discussed because of their
difference in their behavior from the
to a reflectivity value p}.11 = 0.04. For behavior of condensed matter. The
water, the index 11 is as low as 1.33 and a interaction between the gas molecules is
reflectivity of 0.02 is calculated. very weak and their energy levels remain
spectrally sharp. Consequently, the
Though k remains most often spectral transmissivity of a gas present\ <l
negligible, the sample thickness can be great number of quasimonochromatic
spectrum lines, with spectral hanch•.:idths
sufficient to ensure sample opacity. Then: of about J0-2 cm-1.

411 All these spectrum lines correspond to
rotation and vibration energy levels of
(n + 1)2 molecules, to which countless translation
levels are superposed. The study of gases
Nevertheless, it is often advisable to is thus very intricate because spectral
check the sample for opacity. correlation phenornena appear: the
transmissivity, integrated over a large
For opaque dielectrics, the normal spectral bandwidth much more wider
directional emissivity can only be less than the Jines, does not exponentially
than O.S if 11 is greater than 6, ·which is decrease as a function of the distance.
not realistic. Emissivity E'(S) remains Such an exponential behavior is only
constant till 60 degrees before decreasing verified in the strictly monochromatic
to zero at grazing angles and there is no case.
intermediate maximum as in the case of
Unfortunately, the transmissivity of gas
metals. Hence, E is smaller than £;1 and is an essential factor in many industrial
applications (for example, combustion in
typically: furnaces or motors) and radiometric
measurement are nearly always carried
(58) £ 0.95 out through the atmosphere. Because
modeling the transmissivity of gas is very
The influence of surface condition is difficult, empirical formulae are
far less critical for dielectrics than for commonly used for its computation (for
metals, essentially because the -emissivity example, l.owtran's formula for the
values are already close to unity. atmospheric transmissivity)Y

The emissivity of dielectrics is often Semitransparent Media
said to increase from near to far infrared
but dielectrics actually present more or Radiation always travels inside a real
less numerous reflection bands. So, medium, at least over a very short path.
dielectrics cannot be considered as gray, Following the same idea, a sample of any
except over limited spectral intervals. material transmits radiation if it is thin
Close to their strongest absorption bands, enough. Thus, semitransparency appears
dielectrics often behave like metals and as the general behavior of real media.
present typical reflection peaks, which
locally decrease their emissivity. This is Two different mechanisms cause the
notably the case of glass at a wavelength attenuation of radiation inside the
of about 10 ~tm, where p)J1 "" 0.3. medium:

Outside these absorption regions and 1. Absmpliuu corresponds to the encrb"')'
for common dielectrics, whose refractive transfer from the incident wdiation to
indexes are close to I.S, the normal the electrons, atoms or molecules,
emissivity values fall around 0.95 but only yiPlding heat conduction in the
if the sample remains opaque. Indeed, at material.
·wavelengths far enough from the
absorption bands, dielectrics become 2. Scatterins corresponds to random
semitransparent and their emissivity is changes in the propagation direction
tlms lower. because of multiple reflection and
refraction by small heterogeneities.
Moreover, scattering inside the material
can increase its reflectivity because of the

96 Infrared and Thermal Testing

These random changes produce an The product" ~;:I is called the optical
attenuation of the radiation propagating · tllic:kness of the material and permits
in the incident direction: fog scatters the
rays of automobile lights and can even quantification of the notion of
become opaque at large distances semitransparency (for material thickness I
although water itself is transparent to and for radiation at a wavelength /,):
visible light. The same differences are
observable at solid state between snow 1. For opaque material, P;.:l > S; then

and ice. 1;, ~ cxp(-P>:/) < 0,01.
Because the elementary processes of
2. For transparent material, P1:1 < 0.01;
scattering by the microstructure are very
complex, they are taken into account here then 1,, > 0.99.
only from the macroscopic point of view. 3. For semitransparent materials,
Matter can then be considered as
homogenous again and characterized by 0.01 < p,.f < 5; then 0.01 < 1;, < 0.99.
its complex index, its absorption and
scattering coefficients and its phase The general problem of radiation transfer
function. across ari emitting, absorbing and
scattering medium is rather intricate as
The radiance losses along an can be seen from the mathematical form
of the transfer equation (Eq. 63).3.4,10,11
elementary path ds in a direction n is
The crudest approximation for
expressed in terms of the absorption K>. one-dimensional geometry assumes that
and scattering Ot spectral coefficients: the radiance distribution consists in two
isotropic components: this is known as
At local thermodynamic equilibrium, Schuster-Schwarzschild approximation
the emission gain is proportional to and leads to rather simple and easily
Planck's law at the local temperature: interpretable results. For example, suppose
a standard value of 10 crn-1 for an
(61) dL>.,e ~ K,,L0 dT(s)jds isotropic scattering coefficient. If the
following set of values are considered for
Besides, scattering along the path the absorption coefficient- 0.01, 0.1, 1,
induces a radiation gain in the direction 10 and 100 cm-1 -then the values of the
Q because of scattering from all the other bulk reflectivities are respectively: 0.94,
0.82, 0.54, 0.1 7 and 0.02. The medium
directions n'. This incoming scattering is thus appears as white (highly reflective)
when its absorption coefficient is much
described and quantified by introducing a lmver than its scattering coefficient and
biangular phase fimclion p1.(s,Q',Q): black in the opposite situation.

(62) rlL,,ct ~ds The radiance transmitted through a
macroscopic thickness l within the
4n medium, as given by Bouguer's law,
appears as the solution of the equation of
x Jp,,(s,Q',n)L,,(s,n')dll' transfer when the gain terms are omitted:

4n Within a nonscattering medium, the
equation of radiation transfer is greatly
simplified:

Summing these four contributions to (65) dL,,(s) + K,,L,,(s)
achieve a radiation energy balance along ds

the path tis, the classical form of the Integrating the equation of transfer
tmmfer equation is derived: along the thickness I gives the radiance
expression at the boundary of the
(63) rlL\ (s,n) ~ -(K,, + a,)L,,(s,n) material for the direction normal to the
surf<Ke:
'"
(66) 1.,, (1) L,, (o) exp(-K,,I)
+ K}. L0},[T(s)] + -4"n'-
I
x JP;(s,ll',n)L,,(s,n')drl'
+ K,,J LnT(s)jexp(-K,s)ds
4n
0
The two Joss terms can he gathered by
letting ~A == K,. + O). be the extinction
coe{f1cienl.

Fundamentals of Infrared Radiometry 97

L0 }JT(s)j is the spectral radiance of the typically semitransparent, apart from
blackbody at the local temperature T(s). If thick black paints. Effective emissivities
the material is isothermal, then IY1.!T(s)] = have been measured from )._ = 3 to S.S pm
L0,_(T0) and: to 0.5 or 0.6 pm for painted sheet steels.

(67) L1.(1) L1•(o) exp(-K1J)

fI

+ K1. L? (To) exp(-K).S}ds

0

L1•(o) exp(-K,l)

+ L? (q [1- exp(-K,I)j

The first term, exogen, originates
outside of the medium and is transmitted
through it whereas the second term,
endogen, is emitted within the medium.

Besides, according to the definitions of
E}. and 1").:

yielding by identification:

and

- exp(-K;J

These results are obtained for an
isothermal medium and without
considering the reflections at the
interfaces. This can be used to evaluate
the spectral radiation properties of the
atmosphere layer between a radiometer
and the surface of the radiating object.

The semitransparent behavior of a
material has tv-.ro main practical
consequences for radiometry:

1. The emissivity depends on the
thickness of the material if the
material is isothermal. The emissivity
loses its physical meaning if there is a
temperature gradient along the
thickness.

2. A part of the radiance coming out of
the material arises from sources
behind the material.

In the short wavelength atmospheric
l1and (3 to S.S pm), many common
dielectrics such as glass and plastics
exhibit a semitransparent behavior for
material thickness up to 1 mm (4 x lQ-2).
This phenomenon occurs less frequently
in the long wavelength atmospheric band
(8 to 12 tJm), where the absorption
coefficients are stronger, but this problem
must still be solved especially for thin
submil1imeter layers. Paint coatings are

98 Infrared and Thermal Testing

PART 3. Temperature Measurements

The principles of radiometric The spectral radiance leaving the
measurement of temperature are well surface of an opaque material at the
covered in the literature.t2,B temperature 1'u is the sum of the endogen
and the exogen contributions:

Equations of Radiometry L (T)E}',00
},

As mentioned above, real materials J-t fr). L}J cosa' dQ'
present emissivities lower than unity.
Thus, a radiometer in front of an object n
detects not only the emitted (endogen)
radiance but also a part of (exogen) Generally, a radiometric measurement
radiance due to the reflection of the is carried out in air, a medium that has a
ambient fluxes by the object surface. Both refractive index equal to unity (that is, the
contributions are then attenuated by refractive index of air is one) but that
absorption through the atmosphere bilt absorbs a part of the infrared radiation.
simultaneously reinforced by the emission Consequently, air can be actually
of the atmosphere. It is necessary to take considered as a semitransparent medium
into account these phenomena for every and the calculation of the radiance needs
measurement purpose, especially for the solution of the equation of radiation
objects with low temperature or weak transfer. To simplify, air is taken here as
emissivity. homogenous and nonscattering. The
equation of radiation transfer becomes:
An infrared camera measures the flux
of the incoming radiation. The Computing the radiance L).(s) would
radiometric signal H' measured by the need the knowledge of the temperature
infrared camera is then given by triple profile along the optical path between the
integration of the monochromatic sample and the camera. Because this
radiance from the object: temperature profile is rarely known in
actual test conditions, the radical
III(71) w ~ R(J.,n,s)LJ.(1,,n,s)dS simplification of an isothermal
atmosphere is widely used:
1. n s

x cose dn dl.

where R(A,Q,S) is the response of the The complete solution of this
detection device, including the detector differential equation is:
sensitivity, the transmissivity of the
optical device and the amplification by (76) LJ.(s) L1.o exp(- KJ.s)
electronics.
rr+ ('I~tm)
In most radiometric measurements, the
detection solid angle and the target Jt must satisfy the boundary condition
surface area remain small and can be for s ::: 0, that is, at the surface of the
validly considered as elementary. This is sample:
not the case for the spectral interval f..A,
·which is most often taken large enough to (77) L,.(o) E'). L~(To)
increase the signal to noise ratio of the
device. The calculation of the signal then
contains only an integration over the
spectral sensitivity domain to..A of the
camera. Defining S(A) as the relative
spectral sensitivity, the measured signal
can be written as an effective radi<mce L:

(72) L JS(1,)LJ. di- J+ fr~. L).i cosa' dQ'

'" n

Fundamentals of Infrared Radiometry 99

Arriving at the camera, ·where s == I, the If the incidmt nulianceL1,1 is supposed to
radianre is L>, (/): be isotropic, the reciprocity and

(78) L1,(1) [L1,(0)- L~('l~tm)] Inormation properties of frl. yield:
x exp(-K1J) (83) L1,, LA\ t;i, cos8'dl2'

+ L}.(Tatm) Then, applying the radiation balanr('
for an opaque material:
L1,(o) exp(-K1J)
+ L~,(I:,.m)[l - exp(-K,J)] The isotropic condition on the incident
radiance is more especially realized in an
from tl1is equation, the atmosphere's isothermal enclosure. If a room can
transmissivity and its emissivity, which reasonably be considered as isotherm£~! at
can be introduced because of the the temperature 1~ then the radiance
isothermal hypothesis, can he identified: incident on any object inside the room is

and isotropic and equal to the radiance of a
blackbody at the wall temperature. In this
exp( - K1J)
case, the reflected radiance is simply
evaluated by:

Finally, the radiance arriving at the
camera is the sum of three terms:

(81) L1, t> r), Lf('l~) This situation, leading to the simplest
result, is commonly used in infrared
t}.J+ fri.Lt.icosS'dO.' radiometry. The isotropic condition on
the incident radiance is thus included in
Only the first term appears to be the standard comlWuns o(rndiomelr)'.
endogen and related to the sample Jf the incident flux cannot be supposed to
temperature. The two latter terms are be isotropic, some assumptions should be
exogen because they represent the made about the object surface. For <m
reflection of the ambient radiation on the opaque object with a diffusely reflecting
sample surface and the proper emission of surface, the distribution function is
the atmosphere. Because the atmosphere independent of incidence and reflection
has been considered as isothermal, angles:
homogenous and nonscattering, the last
term has been written here under a very Then:
simple form.
JP~. L~.i cosO' dO.'
Above, the radiance reflected by the
sample towards the radiometer has strictly ·where E Is irradiance at surface, Lis
been expressed in terms of the reflected radiance, d!l is solid angle (sr), t;,
is spectral emissivity, 8 is angle (degrees)
r;,,bidirectional reflectance distribution relative to normal and p is reflectivity.

function (BRDF) and of the spatial The irradiance E\ could be theoretically
distribution of the radiance L;.i incident
on the sample surface: evaluated from the experimental
conditions, both geometrical
J{,(82) L,_, ~ 1,L1_; cos a' elf!' (configuration factors) and radiometric·
(radiances of the different surfaces
n surrounding the object under
investigation). Practically, the quantity
Generally, neither frA nor LH are knmvn Ei..·n- 1 can be evaluated by measuring the

and the reflection term must be simplified

with respect to practical applications.
Simplifying assumptions must be made,

concerning either the incident radiation
or the reflection properties of the surface.

100 Infrared and Thermal Testing

radiance reflected by a rather good difftlse (91) L
surface (such as a crinkled aluminum foil).
I+ *h. (1- e',)L~('l;,)dA
r;,,Ideally, for a perfect diffuser: E). = 0, = I"'·

1·111 and L!.r = E>:rr---1• + s(1.)(1 - ,·,. )Lnr.,,m) r11,
For a perfectly specular surface, the
,\1,
distribution function is zero except for the
direction symmetrical to the measurement
direction and can be expressed by means
of Dirac's delta function. Then, the
integration reduces to:

where Lts is the incident radiance in the This intt'gral relation is the actual
spewlar direction. This radiance can he (tmdamental equation ofradiometry.
simply calculated from the sole part of the llowever, strong theoretical difficulties
surroundings situated in the symmetrical appear if it is attempted to keep a rather
direction relative to the normal to the simple expression for the detected
specular olJject. Practically, the radiance radiance. Effective radiation properties
L1,\ can be measUred ·with a quasi perfect over the device sensitivity range must be
mirror (gold, silver, aluminum) for which used. For example:
t'1, = 0 and then LAr = L>.s· Ideally, in the
general case, a perfectly reflecting replica (93) '"'· I s(<)r',.E}J~(To)di,
of the real surface should be used. I""·

It is useful to return to the main s(l.) L?, (To) di,
problem of the radiometric measurement
as a whole. Through a nonscattering "'·
isothermal atmosphere at the temperature
Tatm with transmissivity t'1• in the I s(l,)t',.L?,(1~)di,
measurement direction, the radiometer IA).
actually receives:
S(I,)L?.(To)dA
(89) L,, <), [r)J,f(To)
A).
+ [r,,. r,.; cos8'dn']
These effective properties depend on
+ (1 - t).)Lf(I:,.m)
the temperature and are consequently not
If the surroundings can be considered as
isothermal at the temperature '/~, the the same at Tm T..1 or 1~tm· Furthermore,

spectral radiance takes its standard form: e;.neither S(k), nor the Planck's intensity,

(90) L,, "'>.e\L~ ('/~) nor and particularly t}. are constant
over the interval/1/,.
+ 1',(1 - e',.)L~('I~)
Let£ be the effective emissivity of the

material under investigation and the
effective transmissivity t of the

atmosphere over the sensitivity range of

the device. Usually, the material is
supposed to be gray over IJ.),,

Transmissivity tis practically evaluated
with semiempirical formulae or tables 12

and is taken to be unity at the calibration

distance. \'\'ith these assumptions, the
standard equation is finally obtained:

+ (1 - t' >J L~ ('J~tm) (94) L Tf.L0 (T0 ) + t(1-E)L0 ('J~)

As mentioned above, the spectral + . (1-t)Lo(Tatm)
interval/1/.. is most often taken large
enough to increase the signal~to~noise This relation, sometimes called the
ratio of the device and cannot be validly fundamental equation ofradiometty, is
considered as elementary. According to commonly used to derive the temperature
this, the expression of Li, written above T0 of the material from the metlsurement
must be integrated over the spectral of the radiance L. (See Fig. 4.)
sensitivity range of the device (essentially
that of detector and optics): Many assumptions have been necessary
to derive this quite simple exprt>ssion, so

that the accuracy of the temperature
measurements depends on the validity of

these assumptions in the actual
measurement situation.

Fundamentals of Infrared Radiometry 101

Practical Remarks small ·windows are available. Sapphire,
fluoride or silicon \Vindows are used in
Nonoxidized metals can reasonably be the short wave domain, whereas for the
considered as gray and opaque. On the long wave domain windows are made in
contrary, for many dieft>ctrics the gray zinc selenide or germanium. Less
assumption is more especially doubtful in expensive windows can be realized using
the short wavelength range, ·where they thin plastic foils (e.g., most polymers for
are semitransparent, transmissivity 1), short wavelengthsi polyethylene for long
being neither zero nor constant in this wavelengths).
range.
Because windows are never full
Variations ofF-'>. due to selective transmitters, at least because of reflection,
reflection (variation of p}, with I~) often the attenuation of the infrared flux
occur in the long wavelength range for through these windows must be taken
dielectrics (e.g. the reflection peak of into account. At calibration distance,
ordinary glass around 10 pm). In this without window, the infrared camera
case, restricting the wavelength sensitivity receives the radiance L from the object at
range with a filter can help select a temperature ~):
limited domain of constant (and if
possible higher) emissivity. However, (95) L
using a filter \VOU!d reduce the received
infrared flux and is only of interest when where f is the object effective emissivity
hot material makes the infrared emission within the ·wavelength domain of the
strong enough. camera.

Measurement through Through a \Vindow of transmissivity tw,
Windows this radiance is attenuated by a factor tw:
Additionally, the infrared camera also
A current problem is to measure the receives both the radiance reflected from
temperature of a surface through an the ambient by the window and the
infrared window, e.g. samples under proper emission of the window. Finally,
controlled atmosphere or in a vacuum. the camera measures the radiance I:
Otherwise, it can be necessary to protect
the camera from an aggressive (96) L Tw£L0 (To)
environment- for example, hot gas
flows. + Tw(l-e)L"(J;,)

Unfortunately, almost perfectly + Ewl0 ('~.,,) + PwC'(Ta)
transparent windows within the
wavelength domains of infrared cameras It must be pointed out that the
(short wave 3 to 5.5 pm, long wave 8 to window properties tw, fw, Pw are generally
14 pm) are very expensive and only quite

FIGURE 4. Standard radiometric measurement configuration. Temperature 70 of materia! is
derived from measurement of radiance L where effective atmospheric transmissivity t of over

sensitivity range of device is unity at calibration distance.

Isothermal environment <(1-<)!,(TJ ~}

~<)L'(T,)

Isothermal atmosphere tEL0 (T0) ~

(I - t) L'(T,~) ~

" Object surface at T0

legend
L0 = blackbody radiance (W·m-2·sr-1)
T~ = temperature of isothermal surroundings (K)

Tatrn =atmospheric temperature (K)
=T0 object temperature (K)

r: = effective object emissivity
t = effective atmospheric transmissivity

102 Infrared and Thermal Testing

not constant over the sensitivity On a blackbody at a temperature T
bandwidth of the infrared camera but close to Tw, the transmissivity for the
depend on the wavelength. Consequently, ambient radiation is simply given by:
the transmissivity of the ·window depends
on the spectral distribution of the (103) 'w L - L0 (1~\')
considered infrared radiation coming L0 (T) - L" ('1;,)
from the sample. A window which is
more transparent at shorter -wavelength Hereafter, the transmissivity for the
than at longer wavelength within the radiation coming from a bhlCkbody at a
camera bandwidth would be globally temperature close to ~. can be
more transparent for radiation from a determined:
higher temperature source (compare
Planck's law): (1 04) Two L- (1- Tw)L"(T,,)

(97) Two *- lw L" (To)

because:

J*h,wL~.(T0 )di, Radiometer Calibration

(98) Two f~~- It is very difficult to determine the
s(ic)L~ (I;,)dic spectral sensitivity of an infrared
radiometer such as an infrared camera.
"'· Consequently, the link between the
effective blackbody radiance and the
whereas: blackbody temperature cannot be
established by integration. Furthermore,
f S(ich.wL~ (T,)di, the sensitivity varies with time and a
calibration is necessary, at least once a
"'·f(99) Tw s{lc)L~ (T,)dic year or when the detector or an electronic
component is replaced. During such a
"'· calibration procedure, the correlation
between the temperatures and the
h\-\1 en the sample is at a notably radiances L0 (1) is experimentally
different temperature from the ambient, established using a laboratory blackbody,
the window transmissivity Two for the situated at the calibration distance /0 from
sample radiation differs from its the camera. The measured radiance Lis
transmissivity tw for the ambient then:
radiation:

(1 00) L Two EL" (7~) f(105) L

+ Tw (1- e)L"(I;,) S(i,)T\L):(T)dic

+ (1- Tw - Pw)L"(Tw) f"'·+ s(!c)(l - T',,)L~ (T,m)di,

+ PwL0 (1;) "·

If the \Vindow temperature T... is near

ambient temperature T3 then this
equation can be simplified:

(101) L

The values of the window transmittivities
Two and Tw can be determined by
measuring through the window the
radiance coming from a blackbody (c = 1)
at the corresponding temperatures:

(102) L

Fundamentals of Infrared Radiometry 103

f(106) L to S(A)L~ (T)di,

f&

+ (1 - t,.,) s(i,)L~JT,.,)dA

A),

Effective atmospheric transmittivities at
the calibration distance I ::: 10 are taken to
be equal to unity and the correlation
between temperature and radiance of the
blackbody is then simply given:

f(107) L ~ L0 (T) ~ S(I.)LnTJdA

"'·

To minimize calibration differences due
to atmosphere variations, the calibration
distance is taken as small as possible,
generally /0 < 1 m. Moreover, the aperture
of the blackbody must be seen under an
angle sufficiently large to avoid
underestimation of the measured
radiance, because of diffraction
phenomena.

In practice, the radiometer stands in
front of a laboratory blackbody and the
radiance I is expressed in arbitrary
customer units. It is measured for each
optical configuration as a function of the
blackbody temperature, then a calibration
curve is determined, fitting the
measurement points by a function of
three variables (say, A, B and C). Because
the physical response of the radiometer
·would be Planck's Jaw in the
monochromatic case, the fit is generally
taken of an analogous form:

(108) I ~ A

Even when the radiometer is calibrated
in arbitrary units, radiance I is
nevertheless proportional to the radiance.
The standard equation of radiometry
remains verified in arbitrary units:

(109) I tEio + t(l - £)I,

+ (1- t)I,1m

Closing

The relationships described above govern
the 'Nave behavior of infrared radiation
and can be used for the scientific
description of radiometry.

104 Infrared and Thermal Testing

References

I. Born, M. and E. Wolf. Priociples uf
Optics. London, United Kingdom:
Pergamon Press (1959).

2. Stone, j.M. Radiation and Optics. New
York, NY: MacGraw Hill (1963).

3. Siegel, R. and J,R. Howell. Thermal
Radiation Heat Transfer. New York, NY:
MacGraw Hill (1972).

4. Ozisik, M.N. Radiative Transfer and
Interactions with Conduction and
Convection. New York1 NY: john Wiley
(1973).

5. Hottel, H.C. and A. F. Sarofim. Radiative
Transfer. New York, NY: MacGraw Hill
{1967).

6. Sparrow, E.M. and R.D. Cess, Radiation
Heat Transfer. \.Yashington, DC:
Hemisphere Publishing Corporation
(1978).

7. lncropera/ P.I. and D.P. De Witt.
Fundamentals ofHeat and Mass Transfer.
New York, NY: john \\Iiley and Sons
{1981).

8. Nicodemus, F.E., ).C. Richmond,].].
Hsia, I.\'\1• Ginsberg and T. Limper is.
Gem1wtrical Considerations mul
Nomenclature for Reflectance.
Monograph 160. Gaithersburg, MD:
National Institute of Standards and
Technology [\·Vashington, DC:
National Bureau of Standards] (1977).

9. Airborne Visible/Infrared Imaging
Spectrometer (AVIRIS): Airborne
Geoscience H'orksl.wp Proceedings.
Pasadena, CA: California Institute of
Technology, Jet Propulsion Laboratory

(2000).

10. Chandrasekhar, S. Radiative Trans{e1~
New York, NY: Dover Publications

(1960).

11. Kortiim, G. Re{lectallCe Spectroscopy.
Berlin, Germany: Springer-Verlag
(1969).

12. Hudson, R.D. Infrared System
Engineering. New York, NY: john Wiley
and Sons (1969).

13. De\'\'itt, D.P. and G.D. Nutter. Tllemy
and Practice ofRadiation Thermometry.
New York, NY: VViley-lnterscience
Publications (1988).

Fundamentals of Infrared Radiometry 105



.' .

.I

CHAPTER

Noise in Infrared
Thermography

Nik Rajic, Defence Science and Technology
Organisation, Melbourne, Australia

PART 1. Definitio111, !Effects and Measurernent

The term noise refers to any spurious or where k is Boltzmann's constant {\V·s·K~ 1 ),
unwanted signal in a system.' Because all
measured signals are affected by noise, it Tis absolute temperature (K), R is
is essential to have some understanding of
the origins and properties of noise. This electrical resistance (n) and d{is
chapter describes the various types of
noise that arise in infrared thermography bandwidth. It .will be noted that johnson
and the techniques for quantifying the
noise increases with respect to both
noise.
In the broadest sense, noise falls into temperature and bandwidth.

one of two categories: random or fixed Generation recombination noise occurs
pattern. Noise can also be classified
according to whether it impacts on a in semiconductor materials (involving
signal in either an additive or a
multiplicative sense. In general, random most photon detectors) and is caused by
noise is usually additive, meaning that the
magnitude of the random fluctuation is fluctuations in the rate of generation of
independent of the signal intensity, ·whilst
fixed pattern noise is most commonly free charge carriers1 produced by the
multiplicative, as in the specific case of
detector sensitivity variation. incident photon stream and of the

Random Noise recombination of oppositely charged

Random uoise, known also as stochastic carriers. It tends to have a flat pmvcr
noise or uncorrelated noise, is characterized
by a signal whose value at any particular spectrum up to a frequency that
position or time is independent of values
that precede or follmv. By definition corresponds to the free carrier lifetime
therefore, it has no deterministic
description but can be described by \Vay and declines rapidly ·with further increase
of certain statistical properties. Random
noise can arise from a variety of sources in frequency.
that mostly relate to the detection system.
In tl1eory, the radiating object and its The mechanism that generates {-1, or
surroundings can also contribute random
signals through fluctuations in the excess noise, is somewhat more
emitted photon flux but these variations
tend to be very small compared to those mysterious with its underlying
that occur in the detector system. \'\7ithin
the detector, the process of transforming mechanism not well understood. Possible
the incident photon flux to an electrical
signal can involve at least three theories have been proposed2 and
fundamental mechanisms of noise
production: johnson noise, generation evidence suggests a dependence on
recombination noise and £-1 noise.
semiconductor processing.3 The noise
Johnson or thermal noise (also known
as nyquist noise) occurs in all conducting exhibits a power spectral density that
materials and is a consequence of the
chaotic or random motion of free varies inversely with frequency, hence its
electrons. The root mean square voltage
produced by this noise is: 2 name, and is significant only at low

(1) ~'nns frequencies. Figure 1 provides an

illustration of the noise spectrum for

torr-'these detector based sources. As shown, it
is usual noise to dominate at the

lower end of the spectrum,

generation/recombination noise in the

midrange and johnson noise at higher

frequencies.

The three noise types discussed thus far

relate to the process of transforming

incident photon flux into an electrical

signal. 1v1odern infrared imagers usually

convert this electrical signal into a digital

FIGURE 1. Spectrum of primary semiconductor noise sources.

J-1 nolse dominant
Generation and recombination

johnson noise dominant

-- • • . . - - - - --- ~c,---

Frequency (log scale)

108 Infrared and Thermal Testing

equivalent through a process known as instances of each possible value, the
resulting curve is bell s}laped ~ typical of
digitization. This leads to another source a gaussian or normal ct\stti_bttll?D .•
of noise called quantizatio11 noise, which (Fig. 2b). Note that the distribution is
arises whenever an analogue value is discrete because the intensity at each
converted to a discrete level. The amount pixd is represented by an eight-bit
number and thus has only 256 possible
of quantization error fntroduced depends values (and the subtraction result had
on the quantization level. Given a true been rescaled around 128). A gaussian
value v that is converted to a digitized or distribution is commonly assumed when
quantized value of vq, the variance describing the properties of noise in
introduced by quantJzation error is: 4 infrared systems. As Fig. 2h suggests, the
assumption is often very good but the
+.il2l: possibility that some noise processes may
- J (v -1 I' 2 produce nongaussian noise should be kept
(2) '1 in mind. For a continuous random
~1' v, ) d1' variable x the gaussian distribution has
v _Al_
1 FIGURE 2. Image containing random noise: (a) difference
image; (b) histogram showing underlying normal
"' distribution.
-1 (M)2
12 (a)

where Llv is the quantization increment.
This shows that the quantization error
falls as the quantization increments
decrease, as might be expected. In the
limit of a continuous representation
(lw = 0), the error vanishes.

Fixed Pattern Noise (b)

Fixed pattem noise or correlated noise refers 15
to noise having a distinct pattern. In an
imaging context, such noise may appear
in the form of an object distortion (barrel
or pincushion), although this is
uncommon in modern optical systems.
Intensity variations are a far more
common type of fixed pattern noise.
These can be caused by mechanical or
optical vignetting, noticeable by a reduced
irradiance toward the periphery of the
image, or by sensitivity variations
between detectors in a focal plane array. It
is interesting to note that because humans
are quite adept at perceiving patterns,
fixed pattern noise tends to be more
noticeable to the observer than an
equivalent amount of random noise.

Measurement of Noise

Noise is characterized by its probability

density function, which in essence defines 0 10 f--------
how often a particular value of the

random variant is observed. The

probability density function can be

estimated by forming a histogram of the

noise population. For example the image

shown in Fig. 2a ·was formed by

subtracting two thermal images of a static

scene, acquired using a 512 x 512 focal

plane array infrared camera. It will be 0 L___ll .,, I,
noted that the image is not completely II-
featureless but contains :-:.light intensity 0 100
variations produced by the random noise 150 200 256
sources discussed earlier. If a histogram of

the image is formed by counting the Gray level

Noise in Infrared Thermography 109

the following prClbability density cameras is the noise equivalent temperature
function: diffi'rt'IIC£'. This parameter represents a
measure of the signal-to·noise ratio
(3) p(x) evaluated in relation to the temperature
of the viewed object. 1n effect it defines
where p is the mean and cr2 is the the temperature change that a viewed
variance of x. It will be noted that these ohject must undergo for the change in
two parameters completely describe tl1e signal power to exceed the power of the
distribution. The gaussian distribution has system noise.
a number of use.ful properties ~ for
example, averaging N separate
measurements of a signal containing
gaussian noise leads to a reduction in
noise level by a factor \IN. (This effect is
not restricted to gaussian distributions.)
Other useful properties are discussed
elsewhere.:l

Signal-to-Noise Ratio

The signal-to-noise ratio (SNR) is defined
as the ratio of the signal power to the
noise power. By providing a measure of
the system noise relative to the proper
signal it yields a useful indication of the
true significance of the noise. Maldague5
describes a procedure for evaluating the
signal-to-noise ratio for a thermal
inspection system. Given a static scene,
two images are captured successively in
time. Designating these as A and B, t11e
signal-to-noise ratio is simply:

(4) SNR _!_~'Mal_
where: 2

and

Here, i and j index respectively the x
and }' directions in an image of N x A,f
pixels and J-1 is the mean of the noise
distribution E. \Vith this definition the
signal-to-nOise ratio depends on the scale
used for A and B. The result varies
according to whether A is expressed in
units of Kelvin or degrees celsius.

~vfanufacturers often define the
sensitivity of an instrument based on its
intrinsic signal-to-noise ratio. A related
measure specific to infrared imaging

110 Infrared and Thermal Testing

PART 2. Noise Reduction thrrough Image
Processing

All image acquisition systems are system configuration. Given a scene with
susceptible to noise processes. By uniform brightness and ~1ssmning a linear
implication it follows that all images will detector response, the process is to
contain some noise. Even those that acquire an image at each of two knmvn
appear entirely satisfactory when observed intensity levels and to then write Eq. 7 for
casua1Iy will invariably betray perceptible each case. Because only two unknowns
levels of noise when exposed to more are involved these equations me easily
exacting scrutiny. The levels at which solved:
image noise is considered significant is a
subjective matter that depends entirely on (8) Ai,i li/2 -Iijl
the usc to which the information is Sz -Sr
applied. Whereas the image quality
furnished by modern infrared detection (9)
apparati would satisfy most
thermographers, such data may appear Any scene is then recoverable by
exceptionally noisy when exposed to means of the following operation:
machine interpretation where sensitivity
to noise is vastly more acute than that s(lO) 1,; I;,; - B;,;
possessed by the human observer. The Ai,J
trend toward automation in concert with
the growing imperative for quantitative The most notable disadvantage of this
interpretation should ensure the simple approach is a high sensitivity to
continued relevance of image processing noise. This arises from the division in
as an important means of noise Eq. 10 by a distribution A that tends to be
suppression for thermographic rather noisy if evaluated as suggested.
applications. Processing techniques Note in particular the escalation of noise
effective in this regard are described implicit in Eq. 8, which is caused by the
below. The treatment given here is subtraction of two uncorrelated noise
necessarily brief, so in the event that distributions. A simple means of
more detail is required, the reader is improving robustness is to form the
directed to texts on the subject.6 images 11 and /2 based on a temporal

To facilitate discussion of image noise,
it is helpful to adopt the following simple
model for the image formation process
illustrated diagrammatically in l'ig. 3:

FIGURE 3. Image composition process.

£(x,y)

·where the subscripts i and j respectively 8S(x,y) t t(x,y)
index the horizontal and vertical position
of pixels in the image. For a given scene legend E8
radiance S, degradation of the image I A -= gain factor
occurs through the random noise term £ 8 -= offset factor I
and through distortion (nonuniform 1 =image
distribution) of gain factor A and offset B. S = scene radiance B{x,y)
x = first dimension
Fixed Pattern Noise y =second dimension
£ = noise term
Its inherent stability makes the removal of
fixed pattern noise a relatively
straightfonvard exercise. Equation 7
implies that if the distributions of A and B
can be characterized then the sceneS is
recoverable hy mathematical substitution.
A simple experiment can be devised to
determine both parameters for a particular

Noise in Infrared Thermography I I 1

average (recall that for N frames the noise Here, the constants Care estimated by
variance diminishes by a factor N if the least squares. Figure 4 shows a fixed
noise is gaussian). It may however be pattern gain aberration pertaining to a
impractical to perform the average over a 512 x 512 focal plane array camera,
period sufficiently long to yield a evaluated by least squares.
satisfactory noise reduction. In such cases
an alternative approach is to prescribe A Another alternative is to apply linear
and R functional forms and to fit these to regression independently to each pixel.
the noisy distributions. A biquadratic Because Eq. 7 describes a straight line the
function usually suffices, particularly if task of estimating A and B is equivalent to
the pattern derives from an optical estimating the slope and intercept of a
distortion: linear regression model. To best exploit
the inherent noise rejection advantages of
(11) A(x,y) Ctx2 + C 2 j12 + C3X)' the least squares approach, the pixel
response needs to be sampled many times
+ C4x + CsJ' + C6 for a given excursion of the scene
radiance. To this end, a peltier cell is quite

FIGURE 4. Gain aberration for 512 x 512 pixel focal plane array infrared camera: (a) from
unaveraged pair of frames; (b) from averaged pair (30 frames); (c) biquadratic fit shown as
contour plot with raw data included for comparison.

(a) (b)

(c)

200

" 150

~

e"'

8
0

·~

c

21

c 50
0

112 Infrared and Thermal Testing

useful in providing a scene with the Instead, if the scene variation is
sufficiently repeatable, as is often the case
requisite spatial uniformity and variable in _pulse thermography, averaging may he
applied on a synchronous basis. This
radiant intensity. ! involves capturing many image sequences
of the same dynamic scene, which are
It should be noted that contemporary then averaged to produce a single cleaner
sequence. This approach entails additional
infrared systems apply corrections for experimental burden, requiring either the
ability to perform real time processing or
pixel sensitivity variations as part of a the means to store and later to process
copious data.
standard internal calibration procedure.
More often than not, however, a
These corrections rely on internal dynamic scene is effectively m•ersampled,
that is, acquired at a rate higher than the
reference flags to help determine pixel minimum necessary to preserve important
temporal information, which permits
sensitivities for a particular configuration. frame averaging to be applied within a
sequence. The process is called box car
However, these corrections apply only to twerasins and the kernel is defined:

effects inherent to the detector and

cannot therefore account for fixed pattern

noise of optical or mechanical origin.

Some focal plane array cameras correct for

nonuniformity of AiJ and B1;·

Random Noise (12) jk 2v + 1

Random noise is principally dealt with by where k is frame index. The width 2v + 1
using smoothing or filtering. Each of the averaging window dictates the
involves a process whereby a degree of smoothing. The selection of an
measurement is moderated according to appropriate value for v involves a
some prescribed, usually linear, operation compromise between the degree of noise
on neighboring values. Such operations rejection required and the amount of
can be applied in either the temporal or smearing that is tolerable. The same
spatial domain, however a preference will consideration applies in the practical
usually emerge based on an assessment of context of setting the infrared detector
the required outcome and the nature of integration time, equivalent in effect to
the data. Because most filtering the window length of a moving temporal
approaches cannot distinguish between average. Indeed, an understanding of hm\'
noise and true signal, the need to preserve the moving average works is found hy
signal integrity in a particular domain considering the frequency response
often establishes this choice. If for function of a simple integrator:
instance temporal fidelity cannot be
compromised, filtering is best applied <BJ r J'
with respect to the spatial domain and
vice versa. exp(-i wt) dt

Temporal Averaging t=O

Temporal averaging was briefly discussed - 1-[l- exp(-ion)]
as an effective means to improve the
robustness of gain and offset distortion iCJH
characterizations. Of course, the principle
applies equally well to the routine Here, ro is the circular frequency (rad·s-1}.
treatment of image noise. The principle is Note that the integration time 1 (second)
restated thus: for random gaussian noise is equivalent to filter window length. The
with unit variance, averaging N frames of magnitude of this complex function gives
an image sequence reduces the noise the attenuation response:
variance lJy a factor N. It is quite common
in thermal imaging for the scene to be This expression suggests that the
dynamic, that is, to change as a function
of time. The rate at which these changes attenuation of an integn-1tor increases as
occur relative to the rate at which a frame both a function of (1) and T, with complete
sequence is acquired is important in
deciding 'iVhat type of averaging to apply. rejection of frequencies satisfying
A static scene is the simplest case to treat (1) = 2n:n·r1 and 11 = 1,2,1 .... This latter
as frames in a time sequence differ only in characteristic is undesirable from the
the distribution of random noise. As such,
all frames can be averaged to form a viewpoint of hath detector integration
single image with noise variance reduced
proportionally to the sequence length.

\-\7here a dyn<lmic scene is involved,
each frame is representative of a unique
scene state and thus an ad hoc averaging
of <Ill frames is obviously inappropriate.

Noise in Infrared Thermography 113

and filtering, particularly if the null attenuated in a fashion analogous to low
response coincides ·with an important pass filtering.
signal frequency. Although unavoidable in
the COTlft'":t r_lf detector integration, this The smoothing effect of a low pass
filter is characterized by its frequency
behavior can be averted in a moving response function, which defines the
variation in signal attenuation as a
average approach by applying appropriate function of spatial frequency. Below are
weightings to the elements in the kernel. described the more common variants of
More will be said of this in discussions on this type of filter.
spatial filtering.
If the emissivity field has a high
Cross correlation provides another frequency content, spatial filtering is
means for achieving noise reduction in impractical~ for example, in
the temporal domain, an approach that is thermography of electronic cards.
often referred to as phase locked
thermography when applied to the Box Filter (Neighborhood Averaging). Box
nondestntctive evaluation context. This filtering, also known as neighborhood
technique is described in more detail awraging, is the spatial equivalent of box
elsewhere· in this book. It essentially car averaging in the temporal domain.
involves comparing the measured pixel Smoothing is achieved by replacing each
response to a time varying harmonlc pixel value with the arithmetic average of
thermal excitation applied to the subject its neighbors. The average for pixel i,j is
under examination. The response at the given by:
excitation frequency is extracted by
performing a cross correlation between Figure Sa shows the computational
the excitation signal (which may be molecule for a 3 x 3 box filter. For the
available as a lamp voltage, for instance) most part, such averaging is effective and
and the detector response at each pixel. simple but there are situations where the
The calculation can be applied in real
time or off line, though storage FIGURE 5. Example of 3 x 3 box filter: (a) convolution mask;
requirements for the latter may become (b) frequency response function.
prohibitive for a long sequence. Assuming
zero phase difference between the (a)
reference and detector signals, the
amplitude of the pixel response at the
excitation frequency can be written thus:

(15)

'ivhere L is the reference signal. 0.111 0.111 0.111
The longer the correlation proceeds U1e 0.111 0.111 0.111
0.111 0.111 0.111
greater the extent of noise rejection. As
with conventional averaging, noise (b)
variance diminishes in proportion to
increasing N. '0
-~ 0.8
Spatial Filtering
'E"
Smoothing is a process of low pass
filtering in which features of high spatial .0s
frequency are attenuated or rejected
entirely \\'hercas those of lower frequency .,~
are passed largely unaltered. Fundamental
to its success is the presumption that "0
discarded frequencies pertain principally
to the image noise and are thus irrelevant il
to the scene. This, of course, can only be
the case if the scene and emissivity field "';;~o
are band limited, a situation not
uncommon when a thermal scene Frequency fy -1.0 -1.0 1.0
involves a process of heat diffusion. It (normalized)
should be recalled that temperature Frequency f~
gradients are an impetus for heat (normalized)
diffusion, which acts to suppress these
gradients. The consequence of this is that
high frequency contributions are

114 Infrared and Thermal Testing

approach can lead to unsatisfactory where the bottom half of the image has
results. This shortcoming is apparent been filtered using neighborhood
when the frequency response function of averaging. A comparison between the top
the filter is examined closely (Fig. Sb). and bottoril halves illustrates the potential
Rather than a progressive increase in for spurious amplitude and phase
attenuation with frequency, the box filter behavior.
exhibits a harmonically modulated
response that implies irregular attenuation The practical implication of these
across the frequency spectrum. The effects are probably greatest in the context
behavior can be examined in more detail of the inspection of composites. Figure 8
by ·writing the one-dimensional frequency shows a thermogram derived from a flash
response directly from Eq. 14: inspection of a woven composite
laminate. The image is noticeably
(17) A(J..,a) ~ _2_n!a:_ vrL;rl-'(1-- cos A2 degraded by noise and is a good candidate
"") for smoothing. Hy deliberately choosing a
poor filter length, equal to the weave

where "A is signal wavelength and a is filter FIGURE 8. Woven composite subject

length. inspected by flash thermography: (a) raw
image; (b) 15 x 15 box filter image.
'"'hen the function is plotted against
the ratio of these two parameters (Fig. 6) (a)

the filter is observed to have a null
response at window lengths equal to
integer multiples of the signal wavelength
{a= n),, 11 = 1,2,3 ...). Furthermore, the
phase is also periodic and wraps from -n
ton for a= O.S·(2n + l)A, 11 = 0,1,2,3 ....
This behavior can lead to spurious effects
if the filter length is not chosen carefully.
Figure 7 shows an image with a linear
frequency swept intensity variation,

FIGURE 6. Frequency response function of one-dimensional
box filter.

f ·--~~- ----~~-- 2.0
16
10 -Magnitude 12 (b)
0.9
0.8
"o;- o.8j Phase
"'.s0.4
!;(
~ 071 0 ::\
ro
·E 0.6 r
-0.4 ~
-" I ~
.1; 0.5
-0.8
~ oA Ii -1.2

c"0 -1.6

-~ -2.0
4.0
"':r:;o; 031
0.2

l_,0.1

0
0 0.5 10 1.5 2.0 2.5 3.0 3.5

Normalized frequency o.).-1

FIGURE 7. Box filter of length a applied to image with variation in linear frequency A.

0.5 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
Normalized filter length a-),-1

Noise in Infrared Thermography 115

pattern wavelength in this case, some where the standard deviation a of the
scene information has been lost. However, distribution controls in concert with the
because the woven pattern has been mask size the extent of filtering that
removed, visibility of the four occurs.
discontinuities is enhanced. A
discontinuity having the size of one It is to be noted that the frequency
weave pattern would practically disappear response function (Fig. 9) is generally
- the price of detecting faint and huge monotonic. However, spurious behavior
discontinuities. akin to that of the box filter can arise if
the value of standard deviation a is small
It should be noted that a less expensive relative to the mask size. In the case of
camera with few pixels can obviate spatial the woven composite example, the
filtering. Spatial filtering should be performance of the gaussian filter is
considered an occasional remedy. shown (Fig. ] 0) to better preserve the
weave pattern (in case it is needed).
Gaussian Filter. Ry weighting each pixel in
the mask equally, neighborhood averaging Butterworth Filter. It should be noted that
ascribes as much importance to the pixel the filters discussed thus far have been
value being replaced as to any other designed by prescribing the convolution
member in the mask. Experience suggests mask. In fact, a more sensible approach is
however that the relationship between to prescribe the transfer function first,
pixel values diminishes with increasing bec<mse it determines the filters'
separation. The gaussian filter performance, and to then seek a
accommodates this by weighting pixel convolution mask with the necessary
contributions according to a gaussian characteristics. The butterworth filter
distribution centered about the pixel i == 0, (Fig. 11) is created using this approach. Its
j = 0: transfer function is:

(19) H(fx,fj.) ~ 1

(18) il(i,j) ·2 ·2)N 1+(r(f,,!J)2"

rr r,,
M p ( -~

ex 2cr

j,] jo;J where:

FIGURE 9. Example of 3 x 3 gaussian filter: (a) convolution (20) r(f, f)·)
mask; (b) frequency response function.
where r is tile distance from the origin
(a)
and f0 denotes the wavelength (cutoff
0.057 0.124 0.057

0.124 0.272 0.124 FIGURE lO.Thermal image of woven composite filtered with
15 x 15 gaussian kernel (cr = 3).

0.057 0.124 0.057

(b)

'@ 1.0

.':!

" 0.8

E

s0 0.6
w
u 0.4
·icl
0.2
eron
:;;
0

1.0

Frequency /r -1 -1 1.0
(normalized)
Frequency fx
(normalized)

116 Infrared and Thermal Testing

value) where the attenuation of the filter cases ·where an image is interspersed ·with
has fallen by 50 percent. large amounts of binary or shot noise,
characterized by either a null or saturated
The order of the filter 11 controls how pixel value. Because such values carry no
steeply the filter transitions through the
cutoff value. As 11 gets very large the FIGURE 12. Test image: (a) unfiltered; (b) ideal low pass
transition approaches a discontinuous filtering; (c) second order butterworth filtering.
step and the filter begins to behave like an (a)
ideal low pass fHter by passing largely
unattenuated all frequencies below the 1.0
transition and rejecting entirely all
frequencies above. The term ideal refers to 0 X
the shape of the transfer function and not
to its practical functiona1ity. To the y
contrary, the step transition is to be (b)
avoided because it can lead to a
phenomenon known as ringing, one of the 1.4
principal reasons why the ideal low pass 1.2
filter is rarely used. The effect is shown in 1.0
Fig. 12. 0.8
0.6
Median Filter. The box, gaussian and 0.4
butterworth filters rely on a convolution 0.2
process and are thus classed as linear 0
filters. In contrast the median filter relies
on a nonlinear operation. Instead of
calculating a weighted smn of values
within a prescribed neighborhood, the
median filter replaces the central value
·with the median of the population. By
depending on a ranking operation, the
median filter is particularly useful for

FIGURE 11. Second order butterworth filter: (a) convolution
mask; (b) frequency response function.

(a)

0.063 0.125 0.063

0.125 0.250 0.125

y X
X
0.063 0.125 0.063 (c)

(b) 1.0
0.8
'0' 1.0 0.6
r.o~ o.s 0.4
0.2
E0 0.6 0

.s ~1 -1 1.0

(]) 0.4 Frequency t~
(normalized)
-o

·E0 o.z
;"'!' 0

1.0

Frequency fy
(normalized)

Noise in Infrared Thermography 117

FIGURE 13. Thermogram showing composite useful information the best approach i~ to
delamination: (a) image contaminated with remove them from the image, which is
binary noise; (b) 3 x 3 median filtering; what median filtering is designed to
(c) 3 x 3 box filtering. achieve. Figure 13 illustrates comp"r,llive
(a) effects of median filtering and
neighborhood averaging for an artificially
(b) degraded thermal image of an impact
damaged composite plate. Whereas
averaging merely smears t11e shot noise
over the filter length, median filtering
discards the offending values and also
better preserves sharp detail in the image.
(Note in particular the fiber indications
that run diagonally across the image.)

Harmonic Filter. As stated earlier, the
efficacy of low pass filtering depends
largely on the assumption that scene
information is biased toward the low end
of the image wave number spectrum.
However, given specific knowledge of the
underlying phenomenological process
from which the measurements have
evolved, it is possible to design a filter
that is more selective regarding the
frequency components that are attenuated
in uniform emissivity and absorptivity
fields. An example is the harmonic fi1ter 7
designed to filter measurements known to
arise from a harmonic process1 that is,
from a process governed by Laplace1S
equation:

(c) Equation 21 describes heat diffusion
under steady state conditions and
-- ..- consequently the harmonic filter is
potentially useful in thermographic
applications. The filter \Vorks on the

a,principle of constrained optimization.

Given a noisy function a function II j~
sought that minimizes the following
relationship over a prescribed domain:

JJ ;~ ) ~~2 2

(22) ( + ( ) dx d)'

subject to the following constraint:

(23) NMs2 NM

2.; 2.; [u(x,,n)

h=O j-""0

''(x,,)'jlf

·where s is a smoothing tolerance and N
and M define the grid size.

118 Infrared and Thermal Testing

PART 3. Techniques to Increase Emissivity

Emissivity E is defined as the fraction of Surface Coating
power emitted at a given temperature
(and in a given spectral range) relative to The energy emitted by an object radiates
that of a blackbody at the same from a surface layer only 3 or 4 pm (about
temperature. It is expressed as: 1.5 x lQ--4 in.) thick.1 In dielectric
materials the distance can be a thousand
(24) E). times greater. Consequently, it can be
asserted that emissivity is principally a
where E). is the result of an integration surface property. \¥here access to the
target object is unrestricted, an effective
over 2rr steradians and where h ~- is means of enhancing its emissivity is to
coat the surface with a high emissivity
radiant emittance of a blackbody, defined paint. This practice is commonplace in
by Planck's Jaw: thermographic nondestructive testing and
in thermoelastic stress analysis. As well as
(25) ensuring high emissivity such coatings
have the added benefit of producing a
exphe--- 1 uniform finish, a factor often important
HT in nondestructive testing where emissivity
variations complicate the task of
and where Ais ·wavelength (meter), Tis discovering concealed structural damage.
temperature (K), cis speed of light (m·s-1),
fi is Planck's constant and k3 is It is worth noting that a paint may
Boltzmann's constant (\oV·s·K-1). need to satisfy more than just the sole
criterion of high emissivity. If the paint is
Emissivity ranges in value from unity likely to impact adversely on the
for a perfect emitter (a blackbody) to zero functionality of an object and cannot
for a nonemitter. In relation to thermal remain in place after inspection,
imaging, high emissivity is critically important to restoring the object to its
important for two key reasons. Hrstly, it original state is that the paint be easily
promotes enhanced radiant emittance and removed. To this end, water based carbon
so assists the remote infrared detection suspensions are finding widespread usage,
process by improving the signal-to-noise in preference lo solvent based paints,
ratio. Secondly, it implies low reflectance particularly in aerospace applications
(p = 1 - e) thus reducing scope for signal where existing surface finishes have a
corruption by reflected emittance from specific functional purpose and need to be
extraneous sources. Spurious reflections preserved. Such paints are also Jess likely
are a serious hindrance to the proper to react chemically with most materials.
interpretation of thermographic data. For As a general rule, the compatibility of the
example, reflectioilS causing erroneous chemistries of the paint and object should
temperature measurements could, in a be verified.
condition monitoring exercise, initiate
costly supplemental inspection or trigger It is important not to draw inferences
unnecessary preventive maintenance. about the infrared emissivity of a paint
based purely on its color as perceived by
Applications like nondestructive testing the human eye. Although most paints
are also susceptible. For instance, it is promoted on the basis of efficient heat
entirely possible to mistake a localized radiation are black, it is erroneous to
reflection for an indication of structural assume that only and all black paints
damage. Broader reflections are equaJiy have high emissivity. V\1hite titanium
disruptive by potentially obscuring actual dioxide paint, for example, has an
discontinuities that can be imaged under emissivity of 0.94, surpassing that of
more favorable conditions. This all serves many black paints on the market. A
to underscore the critical importance of reliable source of data or advice should
ensuring that object emissivity is high. always be consulted before judging the
The following discussion provides suitability of a particular paint.
guidance on how this may be achieved
and points out relevant factors to consider Paint is by no means the only
when choosing an approach. compound capable of producing a high
emissivity finish. Puwder is a possible
alternative, although adhesion and spatial

Noise in Infrared Thermography 119

uniformity are likely to be inferior to 'that where x = L; '' [r., - T,)
of paint. Powder in practice is
uncommon. Liquid latex is reportedK to -k iJTP
have good emissivity and has the (30) P dx
attraction of easy removal once cured.
Another option is to apply adhesive tape. where x =a;
It should be noted however that tape is
likely to have a more significant impact (31) k uT k aT,
on the measured thermal response than s Ox
paint and may thus demand more p _axI '
diligence in the interpretation of the
measurements. where x = n;

Response Modification Due to (32) Tp(.t,O) 0

Surface Coating where l = 0;
(33) 7~(x,O) ~ 0
Surface coatings change more than just
the emissivity of an object. They also where t = 0. ln Egs. 26 to :u, Tis
inevitably modify the manner in which a
structure responds to a thermal stimulus temperature (K), Q(x,t) is pawer density,
and may thus have an important bearing k is thermal conductivity, o: is thermal
on the interpretation of the thermal diffusivity, 11 is the interfacial thermal
response measurements. The extent to
wl1ich such an effect needs to be conductance, subscript Ji refers to the
considered in practice depends on factors
such as the thermal properties and paint layer and subscripts refers to the
relative thickness of the object and its substrate.
coating, as well as on the use to which
the response measurements are applied. The temperature response to a pulsed
excitation of duration t0 and intensity QH
In general the effect of a coating ranges absorbed at the surface x = 0 is given by:
from insignificant in the case of relatively
thick objects made of low diffusivity FIGURE 14. Two-layer slab of indeterminant width.
material, to quite substantial for an article
comprised of thin aluminum. For the
most part, the effect is far more
pronounced in the transient regime than
under quiescent thermal conditions and is
therefore most relevant to active
thermography of m.etallic objects.
Consequently further discussion of the
issue is confined to pulsed thermography.
The prospective effect of a coating can be
examined by studying the heat diffusion
equation for a two-layered slab (Fig. 14) as
defined by the equations:

o2T.

,
.i(26) + I Q· p(x,t)

-k
uX p

where 0 < x = a, and:

(27)

where a< x < L, with the following
boundary, interface and initial conditions:

(28) -()Tp ~0 legend

dx a "'paint thickness (arbitrary unit)
L =substrate thickness {arbitrary unit)
where .x = 0; x =dimension normal to part surface

(29) ()Ts 0

dX

120 Infrared and Thermal Testing

(34) T(x,t) objective is merely to fmd a discontinuity;
(2) quantitative, where a discontinuity is
where: detected and then.characterizcd. From the
effect iiJustrated in Fig. 15, a coating is
(35) l for 0 < t :5: t0 unlikely to reduce the prospect of merely
fort0 <f<= detecting a hidden structural
discontinuity- except where the
and diffusion time across the discontinuity
diameter {in substrate) is shorter than the
(36) Q = {~0 for 0 < t :5: fo diffusion time through the coating. On
fort0 <f<oo the contrary, emissivity equalization
rendered by a properly applied coating
and where the quantity p11 satisfies the may assist detection by excluding
reflection as a tangible cause of an
transcendental equation: indication. The scenario is quite different
in quantitative applications, primarily
(37) 8tan(p,a) + tan[yp,(L-a)] = 0 because of the delay effect mentioned
above.
and where the quantity 3 is expressed as
follows: It is instructive in discussing this
matter further to focus on the contrast
(38) E zplla +sin2Pn X) response, that is, the deviation in
response at a suspect site compared to
4P, that from a sound reference. Such
contrast curves form the basis for most
Before considering practical case quantitative approaches to thermographic
nondestructive evaluation. Figure 16
studies, it is important to note that a illustrates such a response for a case study
coating of about 10 to 30 pm (0.4 x J0-3 involving a coated 1.6 mm (0.06 in.) plate
to 1.2 x lQ-3 in.) in thickness is necessary with a 0.8 mm (0.03 in.) deep void
discontinuity. Conventional practice
to obscure a polished metal surface with would see the discontinuity depth
paint applied by aerosol. Under normal estimated on the basis of the time taken
field conditions, the urge to avoid missing for a characteristic event to occur in the
a spot may lead an inexperienced contrast curve. The time to peak contrast
technician to the deposition of even and the time to contrast inflection
heavier coatings. Figure 15 shows the (stationary point of the first time
derivative) are good examples. Figure 16
response preclictecl by Eq. 34 to flash shows the delay induced in the contrast
heating of a 10 mm (0.4 in.) aluminum response by t11e presence of a coating and
from this the potential for a biased depth
plate (1) when bare and (2) when coated
with acrylic based paint. The curves FIGURE 15. Surface response as function of layer thickness for
derived for the coated structure suggest a acrylic paint as predicted by Eq. 34 for 10 mm (0.4 in.) thick
three-stage response comprising aluminum plate.
successively: (1) an lnitial regime
dominated by the coating; {2) a transition Aluminum 10' 10'
reference Time (ms)
phase where the response involves
heterogeneous behavior; and (3) a steady 1o-'
period of substrate dominated lJehavior. 10- 1
The delay of the substrate response

induced by the coating is the key effect in
terms of impairing quantitative analyses.

Note that this delay, even for relatively
thick coatings, is quite short in any
practical sense, which implies that only

objects that exhibit relatively fast
transients are likely to be affectt>d,
aluminum being a good example.

Active thermography is applied in two
distinct forms: (1) qualitative, where the

Noise in Infrared Thermography 121

estimate if the delay is not considered. are made with respect to the visible
Indeed, a coating may render any spectrum, the issue of emissivity becomes
estimate of discontinuity depth virtually irrelevant. Absorptivity, however, remains
impossible it, as shown for the 55 p.m ari issue in tests using optical energy.
coating, the curve is shifted beyond values
that can be explained by the presence of a There are, as with all techniques,
discontinuity alone. This underscores the limitations and disadvantages. Paints and
importance of ensuring that coatings are films have an inferior temperature
made only as thick as required to produce resolution and limited dynamic range
a high emissivity finish. compared to modern infrared photon
detectors. And, as with all coatings, they
Thermochromatic Paint or disturb heat flow through the article and
Film consequently yield a biased temperature
response, as discussed above. Films in
A thermochromatic substance changes particular have rather slow response times
color in response to a change in and are thus unlikely to be suited to roles
temperature. Someone unfamiliar with where thermal event times are very short.
the effect need only visit the local toy
store where toys can be found that Surface Modification
undergo changes in color when clasped in
the hands or dropped into a bucket of Surface modification is another means of
achieving a useful increase in emissivity.
cold or hot '''ater. The past decade has Table 1 lists the emissivities of several
seen the advent of thermochromatic common metals for a variety of different
paints and thin films that offer surface preparations. A polished finish
temperature sensitivities suitable for consistently yields the lowest emissivity.
application to a wide range of thermal Indeed, polished metal can serve as an
inspection problems. By translating a excellent infrared mirror and there are
change in temperature to a change in many examples of such usage. An abraded
color these materials offer a very low cost surface has higher emissivity, due in most
alternative to conventional thermal part to the cavity effect described below.
inspection techniques that require Nevertheless, the emissivity of an abraded
expensive thermal imaging apparatus. surface remains quite low, to the extent
Importantly, because the measurements that abrasion has relatively little practical
benefit; at an emissivity of 0.2 a surface
FIGURE 16. Effect of painting on contrast evolution calculated reflects four times as efficiently as it emits
for 1.6 mm (0.06 in.) aluminum skin with 50 percent and is thus vulnerable to stray reflection.
material loss. Oxidation and anodizing are chemical
processes that result in the deposition of a
1.0 ... - .... ...f-- -· / l.--;:· compound that typically has higher
r- .~·· intrinsic emissivity than its parent metal.
~ 0.9 .- If permitted on functional grounds,
. r-·- ~- · - ·: - promotion of oxidation is quite an
c 0.8 i . +-i- c-;~ - r-~-~- effective way of raising emissivity to
- 1- useful levels.
~

"'.-.ee~ 0.7 .I r-- - ···-
0.6
-.~ --r ·---; f-- -· r--- 1- --
c~ 0.5 -· Table 1. Emissivlties for common metals.
-'-... I_ -
Material Surface Finish Emissivity
·.,R-0

~ · - ~. - - -· - 1--
11<ii 0.4 1-- · · - - · - ··-

"~ 0.3 ..L :-.;~ - 1-- - f---1--1- 1-- Aluminum polished 0.04
sandblasted 0.21
.~ I Brass anodized 0.55
Copper oxidized 0.11
§ 0.2 - '1---;- ~- f"- '-- 1- Steel polished O.Q3
abraded 0.20
!/ r.-z0 0.1 --;· - . - -- · -f - f--- 1---1-- oxidized 0.61
polished 0.02
/ heavily oxidized 0.78
polished 0.02
0 oxidized 0.69 to 0.95

5 10 15 20 25 30 35 40 45 50 55 60

Time (ms)

legend

• = 27 111n coating)

" =55 prn coating

-- =uncoated
----=uncoated, 3 percent materia! loss

122 Infrared and Thermal Testing

Reflective Cavity fiGURE 17. Conical shaped reflector.
Pyrometry

\.Yhcn the radiant emission from a small
opening in an isothermal enclosure is
examined, the spectral response is found
to closely approximate that of a
blackbody. This Cat'il)' effect applies to all
materials and enclosure geometries as
long as the opening is sufficiently small.
Reflective cavity pyrometry is a means of
exploiting this effect to overcome low
emissivity problems. Figure 17 sho'\\'S a
conical reflective cavity positioned above

a low emissivity object. The observed or
apparent emissivity as inferred from the
radiant flux through the fiber bundle may
be substantially increased by judicious
selection of the emissivity and geometry
of the reflector, as ·well as the standoff
distance to the object. In a noncontacting
approach, some clearance needs to be
maintained between the reflector and
object. The clearance limits the extent to
which blackbody conditions can be
approximated. Nevertheless, given
prudent reflector design high emissivities
arc achievable. Various aspects of reflector
design have been examined in the context
of online temperature measurement
during sheet metal fabrication.9 The
approach is fundamentally limited to spot
temperature measurement and cannot be
applied in any conventional way to
thermal imaging.

Noise in Infrared Thermography 123

PART 4. Techniques to Overcome low Emissivity

Emissivity is arguably one of the most (40) R).,).,
important properties in determining the
viability of a thermographic inspection. exp--h-e --I
An unprepared metallic surface usually ~ --"-),"oz:.:_k_:_T__
can be dismissed out of hand because it /q exp--l-ie --1
reflects far more efficiently than it emits
and any data obtained are thus likely to A1kT
be heavlly contaminated by
environmental emittance. As discussed Figure 18 illustrates the principle. It
above, however, low emissivity is easily should be noted that Eq. 39 relates only
remedied by means of an appropriate to radiant emittance and as such the
surface modification, most conveniently treatment cannot account for reflected
by application of a thin layer of paint. emittance arising from low emissivity.
Although widely practiced, cases do arise Recall that the emittance from a
where this simple treatment is not viable. nonblackbody comprises not only its
Possible reasons are the need to preserve radiant emittance but also the reflected
an existing surface finish for some emittance, the strength of which wiJl
functional purpose or a simple case of depend on the object reflectance and the
object inaccessibility. In either situation, surrounding environment (transmittance
alternative treatments to the low is negligible for most engineering
emissivity problem wi1l need to be materials). The dual band approach is
considered. The present discussion used mainly for hot surfaces (several
outlines inspection techniques as well as hundred degrees celsius). Therefore, where
postprocessing techniques that offer a dual band approach is prescribed it is
useful solutions where direct surface important that the potential for reflection
modification is not feasible. !Je minimized as far as practical. Note also
that a ratio approach in general implies
Dual Band Thermography increased sensitivity to image noise as
discussed above.
Dual band thermography is a useful
technique by which to account for spatial FIGURE 18. Photon emittance at T ~ 298 K (25 °C ~ 77 of) for
variations in object emissivity. The
principle is simple. The spectral photon blackbody and graybodies with emissivity E :::o 0.8, 0.5 and
emittance of an object with emissivity £ is
given by: 0.2. Ratio of emittance at wavelengths /q and ),2 is identical
for each curve.
2nc 1
(39) Q). £;, }4 '0
llc
' exp ------ I w
AkT .~ 1.0 £ = 1.0

"E' f = 0.2
.0s 0.8
The emittance is a function of both cwv · - ji
temperature and emissivity. For a ~ 0.6
graybody, emissivity is constant with Ew 8 10 12 14 16 18 20
respect to wavelength. This constancy
suggests that tl1e ratio of emittance c 0.4
measured at two independent
wavelengths (hence the term dual baud) ~
should he independent of object -C
emissivity and consequently a function Q_
only of temperature. To illustrate the
principle, consider a pair of 'c•6 0.2
monochromatic measurements taken at g~
the wavelengths 1~1 and 1..2• Graybody 0 '··
behavior implies that Eu :::o LJ.z, so the ratio w
of the emittances is: 46
Q_
~

2

Wavelength (IJm)

124 Infrared and Thermal Testing

Thermal Transfer Imaging state. The function is especially useful
,...,hen inspecting scenes cluttered with
Thermal transfer imaging is an irrelevant features (objects in the
experimental methodolot,ry that aims to background) and where only a relatively
provide a reliable means by which to small region is likely to respond to the
inspect low emissivity objects. The stimulation. Recall that the radiant power
principle is to transfer the temperature of an object as measured by a detector
distribution of the target object to a host comprises two components:
structure of superior emissivity thus
providing an improved platform for (41) J ~ £10 + (! -c) I,
thermal imaging. Figure 19 illustrates a
possible experimental configuration. A The radiant emittance 10 is caused by
low conductivity transfer material is thermal stimulation; the environmental
usually recommended to promote radiation Je is reflected by the object.
persistence of a transferred thermal
image.s Such materials reduce the loss of Pulsed stimulation will cause those
signal intensity that occurs in the delay segments of the object with high
between completion of a transfer and emissivity to emit a higher signal than
image capture. The need to establish those segments that are highly reflective,
physical contact limits the practical thus generally improving the contrast
application of transfer imaging to cases within those features. To illustrate the
where this is both permitted and feasible. potentia) for improved contrast and to
Accordingly, production line settings highlight important limitations, a test
where objects undergo motion along a case was developed using a rectangular
prescribed path are well suited to this titanium specimen with a narrow channel
approach. milled into the back face. The surface was
prepared with a uniform coating of matte
Reference Image black paint followed by the application of
Subtraction three thin strips of gold paint known to
be reflective in the infrared band. The
Useful reductions of the appearance of article was inspected by means of pulsed
emissivity artifacts as ·well as other thermography in the arrangement
background effects may be achieved by indicated by Fig. 20. Thermograms
subtraction of an image taken of the
object in a quiescent state, sometimes FIGURE 20. Experimental arrangement at mome~t of flash
called a reference image. Such an operation discharge. Specimen size is exaggerated for clanty.
serves a similar role to the calibration
button found on many infrared imagers.
Its role is to affect relative radiance
measurements with respect to a reference

FIGURE 19. Illustration of transfer imaging arrangement.

Heat slnk line heater
Transfer medium

/

Object

Noise in Infrared Thermography 125

obtained before and after excitation arc inspection process. Specifically, the time
shown in Fig. 21. Subtraction of the history at each pixel location is
reference from the response (Fig. 22a) transformed to a complex fou~j_er
shows noticeably improved contrast for spectrum by:
the material loss indication but also for
the reflective strips. Other approaches where the phase at each frequency u is
involving normalization are possible.5 In given by a relationship between the
general the reference will need to have imaginary and real parts of F11:
both offset and contrast adjusted to
account for the difference in average (43) $, Imaginary IF;,l
temperature compared to that of the
excited response (Fig. 22b). atan Real lr.,l

Pulsed Phase The insensitivity of a phase map to
Thermography emissivity variations is partly explained
by the fact that a surface feature,
An object with an irregular emissivity providing it is thermally thin, has no
distribution will give rise to a thermal means by which to affect the time history
image containing corresponding intensity and consequently the phase distribution
variations. These can mask underlying of a thermal event in the substrate. A
discontinuities and generally serve to phase map is actually sensitive to
unnecessarily comp1icate interpretation. absorptivity variations, which are often
Pulsed phase thermography10 is an correlated ·with emissivity variations. A
effective means of removing such mathematical interpretation is that
artifacts, as well as other background
effects. The technique is based on a
temporal fourier transform applied to an
image sequence acquired during an active

FIGURE 21. Thermograms (average of 40 frames each): (a) quiescent; (b) response to pulsed
illumination.

(a) (b)

126 Infrared and Thermal Testing

emissivity behaVes likC a multiplicative Computation of
constant and is effectively cancelled Reflections
through division of the in-phase and
quadrature components in Eq. 43. \-\'here the measured radiant power of a
Figure 23 shO\\'S hmv a phase map can low emissivity object comprises a large
reduce surface emissivity artifacts and component of reflected radiation, it may
thus clarify the presence of underlying be possible to account for and eliminate
surface features, in this case a thickness this component provided the emissivity
variation. distribution of the object is measurable
and the environmental radiation field is
Selection of Detector known. Jn relation to a
Spectral Band microthermographic application, a
procedure has been used to measure the
Objects are sometimes encountered that reflectivity of an object by using a time
exhibit a wavelength dependent modulated infrared source and an infrared
emissivity. These are termed colored or detector tuned to the excitation
selective emitters. For such objects, the frequency. 11 The apparatus is scanned
radiant spectral emittance curve differs in over the surface to form a
shape to that of a blackbody in that local two-dimensional map of the reflectivity
maxima may appear at wavelengths where distribution. }!or an opaque object,
the emissivity is high. VVhere a local Kirchoff's Jaw permits emissivity to be
maximum is found to occur within one of inferred from reflectivity, which provides
the common detector operating bands a basis for decomposing the apparent
(3 to 5 pm or 8 to 12 pm) common sense radiant emittance into a part stemming
suggests that the detector be selected on from self-emittance and a part arising
the grounds of peak spectral emittance. from reflection. Radiant emittance Ic must
be known.

FIGURE 22. Improved discontinuity contrast through reference subtraction: (a) subtraction of
image in Fig. 21 b from image in Fig. 21 a; (b) as in Fig. 22 but with prior adjustment of
contrast and offset of reference image.

(a) (b)

Noise in Infrared Thermography 127

fiGURE 23. Artifact reduction by use of phase mapping: (a) as in Fig. 21 b; (b) phase image
calculated from time sequence of 40 frames.
(a) (b)

128 Infrared and Thermal Testing

References

1. Hudson, R.D. Infrared System 11. Chen, Z.H., ·c Uchida and S. 1vfinami.
Engineuing. New York, NY:
"Emissivity Correction in Infrared
Wilcy-lntcrscience (1969). Microthermography." Measurement.
2.jamieson, ].A., R.H. McFee, G.N. Plass, Vol. 11, No.1. London, United

R.H. Grube and R.G. Richards. Infrared Kingdom: Elsevier Science, for the
Physics and Engineering. New York, NY: International fvfeasurement
McGraw-Hill (1963). Confederation (March 1993): p 55-64.
3. Burke, M. YV. Handbook of.Machine
Vision Engineering: Vol. 1, Image
Acquisition. London, United Kingdom:
Chapman & Hall.
4.]ahne, B. Digital Image Processing.
Berlin, Germany: Springer-Verlag
(1991).
S. Maldague, X.P.V. Nondestructive
Evaluation o(Materials by Infrared
Thermosraphy. London, United
Kingdom: Springer-Verlag (1993).
6. Lim, J. S. Two Dimensional Signal ami
I1nage Processit/g. Englev·wod Cliffs, NJ:
Prentice-Hall Signal Processing Series
(1990).

7. Grundy, I. SMOOFF- A Fortran

Smoothing Program.

ARL-STRUC-TM-490. Melbourne,
Australia: Department of Defence,

Defence Science and Technology
Organisation, Aeronautical and

1v1aritime Research Laboratories (1988).

8. Kallis, J. M., A.H. Samuels and

R.P. Stout. "True Temperature
Measurement of Electronic

Components Using Infra-Red
Thermography." Circuit l·Vorld. Vol. 12,
No. 2. Bradford, \o\'est Yorkshire,

United Kingdom: MCB University
Press for Institute of Circuit

Technology and Printed Circuit &
Interconnection Federation (1986):

p 25-30.

9. Krapez, ].C., P. Cielo and
M. Lamontagne. Reflecting-Cavity IR

Temperature Sensors: An Analysis of
Spherical, Conical and Double-\Vedge
Geometries." Proceedings SPJE Infrared
Technology aud Applications. Vol. 1320.
Bellingham, \o\1A: International Society
for Optical Engineering (formerly the

Society for Photo-Optical
Instrumentation Engineers) (1990):
p 186-201.
10. ?vfaldague, X.P.V. and S. ~vfarinetti.

"Pulse Phase Thermography." journal
of Applied Physics. VoL 79, No. 5.

College Park, J'vJD: American Institute
of Physics (March 1996): p 2694-269H.

Noise in Infrared Thermography 129



CHAPTER

Errors in Infrared
Thermography

Terumi lnagaki, Department of Mechanical
Engineering, lbaraki University, lbaraki, Japan
Katashi Kurokawa, NEC San-ei, Kodaira, Tokyo, Japan

PART 1. Sources of Errors

Introduction ANSI/ASivJE standard, discussed below. 1
The source of errors are considered and
Infrared thermography is a convenient calculated. The confidence levels or
nondestructive test method that can help uncertainty levels are addressed by
visualize and estimate various analyzing respective error strata
two·dimensional temperature fields pertaining to the measurement process,
simultaneously by sensing classifying the error factor with several
electromagnetic energy emitted from error strata.
target surfaces. Increased recognition of
the technique's app1icability has The American Society for Testing and
prompted development of remote sensing Materials has issued several standards for
diagnostics for various engineering dealing with c;-rrors.2-5
applications. Infrared thermography has
been applicable across many engineering There are many types of infrared
disciplines such as discontinuity thermographic systems possessing
detection, condition monitoring and heat different types of sensing and scanning.
transfer measurement. The technique can The present chapter involves a typical
also be applied to critical diagnostics such system using a single sensor element and
as those in nuclear facilities ·with high mechanical scanning. A similar approach
can be used for elements with multiple
radioactivity. sensors.
Although infrared thermography has
Definitions
t11ese convenient features, it is normal for
Below, a minimum detectable size and a
the measured data to include an error in a noise equil'alent temperature difference are
referred to affecting the resolution of the
way similar to other techniques, such as infrared thermographic system. Each
thermocouple measurements. However, cause affecting the resolution is evaluated
the quantitative error of the measurement quantitatively. Previous discussions of
has been difficult to evaluate precisely qualitative measurements and
and systematically- not only in high evaluations6,7 did not fully quantify the
temperature conditions but also in causes. Not only the noise equivalent
conditions near ambient. It is necessary to temperature difference (NETO),
develop quantitative infrared representing minimum detectable
thennography further for various temperature difference, but also the
engineering applications by analyzing the minimum detectable size (MDS) are
confidence level of data through an important performance indicators of the
appropriate evaluation technique. In infrared thermographic system and
particular, the accuracy might decrease express the smallest resolution that can be
when applying the technique to near detected. r:or instance, when the surface
ambient conditions, because the radiation temperature of a target below the
energy detected at an infrared sensor, minimum detectable size is measured, the
which is usually used for determining shape cannot be discerned whereas the
temperature, will always include nois!' for surrounding energy also enters the
instance from the reflected energy detector as a noise signal, giving rise to
incident in the surroundings. For various error in judging a thermal index measured
engineering disciplines, a quantitative with the infrared thermographic system.
basis of infrared thermography needs to
be established that is (I) applicable to The thermal index is composed of
ambient conditions and (2) accurate and pixels representing radiance temperature.
free of significant error. The minimum detectable size for the
infrared thermographic system used for a
This chapter analyzes the sources of medical infrared imaging is rated with
errors in infrared thermography and then respect to a japanese standard.H According
discusses the calculation and evaluation to the rating, the size threshold is
of errors. Factors of minimum detectable calculated from the separately
size and noise equivalent temperature distinguishable horizontal resolution.
difference are fundamental in Thus, the rating is adequate for the signal
maintaining thermographic measurement output to be recognizable. This definition
confidence. Furthermore, measurement may be suitable for applications where it
uncertainty can be evaluated by using is necessary merely to judge the shape of
uncertainty analysis based on an

132 Infrared and Thermal Testing

the image hut for visual gaging and value indicates better temperature
quantitative temperature measurement
the definition needs to clarify the resolution of the system. \Vith respect to
relationship between the detectable target
size and the thermal index. It is also the noise equivalent tempera!ure .
important to clarify the rC'Iationship
between the detectable temperature difference, a japanese industnal standardH
difference and the thermal index. The
sources of the error pertaining to the has already been published for medical
infrared thermographic system will be
explained below with the help of an infrared thermographic systems. The
alternative definition.
standards define that the signal to
Factors Impairing
Performance determine temperature is obtained from a

In general, any type of measurement signal ·waveform as displayed by
includes errors due to a number of causes.
The infrared thermographic system is no deflection modulation of temperature
exception, and quantitative evaluations
from multiple aspects are required to signal and that the noise is obtained from
clarify the measurement accuracy. There
are many factors impairing the a peak-to-peak value of noise voltage NJlP'
performance indices of typical
mechanically scanned infrared The noise for this case is defined to
thermographic equipment. Two factors
used to describe and compensate for be the peak-to-peak value of
errors are the minimum detectable size
(,MDS) and the noise equivalent J"'pp X l/(2·2"·5) ~ NJlP/(2·2°·5).
temperature difference (NETD). However, because the noise depends on

The minimum detectable slze is a frequency components of the signal
primary index of the infrared
thermographic system's spatial waveform, the procedure, in which the
resolution.9 The smaller value indicates
better spatial resolution of the system in peak-to-peak value of tile signal W(lveform
measuring the temperature of a target
\urface on the display screen. VVhen is multiplied by 0.2 to 0.1 1 is sometimes
measuring temperature quantitatively, applied to calculate noise ~qu!v~lent
identifying minimum detectable size is
important. Also needed is darificatio!1 of temperature difference. It IS difficult for
the relationship between the target SIZe
and its temperature reading. This point is this technique to determine noise
significant for quantitative discussion of
lens aberration, diffraction effects, equivalent temperature difference ..
amplifier frequency response and defocus
as major factors impairing the quantitatively, bec(luse personal readmg
performance indices of the typicat
mechanically scanned infrared errors and the frequency components of
thermographic system. The detrimental
effects of other causes of errors are the signal waveform affect it. There is a
negligible: aperture iris refraction,
nonuniformity of temperature Jack in objectivity of the data measured-
distribution on a target surface and (in
some cameras) internal reflections. the measurements are simult.:meously

The noise equivalent temperature affected by the sensor's sensitivity and by
difference is also the primary index of the
infrared thermographic system. 10 This its time response. Noise equivalent
index indicates the temperature resolution
and shuws the minimum detectable temperature difference is therefore defined
temperature difference appearing on a
target surface whose emissivity E is nearly by calculating a standard deviation of
equal to 1.0. In other words, it is defined
as a temperature change !!Ts such that the thermal index distribution when a
signal~to-noise ratio of the infrared
thermographic system becomes 1.0 when standard blackbody furnace maintained at
measuring the target surface. A S111aller
a constant temperature is observed; noise

equivalent temperature difference refers to

the standard deviation [(dTr..,l)a\·1°·5 of

radiance temperature Th measured with

the infrared thermographic system. This

technique confums the objectiveness of

data with no personal error when reading

the scale.

Theoretical Background

Minimum Detectable Size

Planck's radiation law is known to govern
the spectral radioslty H\ of a blackbody. A
simple expression of the relationship
between blackbody temperature T~ and
radiance L is the extended power of
radiosity. In this rase, when the radiation
from a bhKkbody having a core
temperature T~ and a temperature change
!!T, (< 7~) is n.wasured with th~ infrared
thermographic sy~tem possessmg a
detection wavelength band Q having a
certain amplitude, then radiance L can be
approximated by the following formul<l
using Li.:H

Errors in Infrared Thermography 133

(1) L 111~ 11 distance h + /Jo from the lens to the target
surface being linked:

(2) - + 1 1

In the above, A is a constant and t 1• is the a b + 1>0 r

spectral radiance depending on ~he .following formula is applicable
wavelength/.... The exponent u can be
derived by integrating the product of !mkmg the square sensor area, An to the
Plan~k'~ radiation _law and the detectivity
pertammg to the mfrared thermographic ~ rmstantaneous measurement area AT:
system with wavelength /,,6,7 A detected (3) Ay ( b : bo AD

energy spectrum is obtained {Fig. I) and, Note that the distance from the lens to
the window is b0 •
by using the least squares technique,
11 is then derived from a nearly Because ll minimum detectable size M 0
determined theoretically corresponds to
straight line to become 4.31 for the the real image of the infrared sensor

mercury cadmium telluride infrared projected on the target surface to be
sensor, having a 8 to 13 pm detection
measured, by inserting the practical values
wavelength band. This value of 11 s!10uld pertaining to the infrared thennographic
system into Eqs. 2 and 3, Mo = (kr)o.s can
vary as a function of n and of the
be derived for every objective distance
temperature range because of the
B = b + b0 . When the size of a target to be
transmission function. The temperature measured is larger than lv/0 , the radiance
range considered is 300.2 to 373.2 K temperature 1~5 of the target surface itself
can be determined. But if the size of the
(27.0 to 100.0 "C; 80.6 to 212.0 "F). target is smaller than M0, even when
Figure 2 is a schematic of the infrared using an ideal device with a distortion
free lens, the data include not only energy
thermographic system as vie'ived from the
emitted from the target surface but also
aspect of a temperature gage (the
scanning system is omitted here). The ;,nergy emitte.d from the surroundings.
I he surroundmgs affect the signal and
following formula links (a) the lens focal hence the quantitative determination of
length{, (b) the separation a between the
infrared sensor and the lens and (c) the temperature.

FIGURE 1. Energy detected with mercury-cadmium-tellurium . Ordinarily, in a thermal image from an
sensor having 8 to 13 ~m wavelength band): In (y) ~ mfrared thermographic system, the
Ao +A, In (x) where A0 ~ 31.334 and A1 ~ 4.3069.
thermal index Trs indicated with the
4 infrared thermographic system is

"::-:: Ec=·' -~t:~~ A1 ln {x) represented as an area averaged Tr~ within
Mo composing the image. As indicated in
- In (y)
= = ='I
-- Ao=-31.334 -
E

.s t= •~
3 1- ~ - A1 "'4.3069 -~

1- - -

-I-I---- --
r= =CJ ,-
!~ r--- 1:= -~ -;:,'·I -
=~ 2 - - - c- ~,b -~
I- - FIGURE 2. Optical model.
-~ - ~ --
b--~-bo-t-- o-~
-- -
cID - ·- - - -- - ~
-
ID -~ ~'¢
~-
,.".'{ju - - -
--
-- " -~
-

* + l;v1- ·-I-I--- ~ - -

0 1- 1\~

473 S73 673 - .. ~-1;[]
(200) (300) (400)
[392] [S72] [7S2] Measurement area J~"'

Blackbody temperature r.,, K (0 C) {°F) ~

legend \"'indow

- ==instrumented measurements Objec!ive
Ao =area lens
A1 ==area
Ot, =- blackbody emis5ion legend
T = temperature
An = sensor area
r=x =variable
variable A, =measurement area
a= standard deviation a=- distance from sensor to !en~
b =distance from target surface to observation window
bo =distance from lem to ob5ervation window

134 Infrared and Thermal Testing

Fig. 3, when measuring a target surface is defined as a temperature change such
having radiation energy Ez(Trsz), that the ratio of signal S to noise N of the
representing Tn2, consider the case that infrared thermographic system becomes
the component (slit width) M having the 1.0 around T~:
radiation energy E 1 (7~~ 1 ), representing Tr~l
and being higher than E2(Tr52), exists as (6) NETD
an adequately long vertical shape (slit)
within M0 . The averaged radiation energy A simplified schematic of the infrared
E,;,(Tr~_,) received at the infrared sensor is thermographic system from which the
determined in proportion to the area scanning unit is removed is shown in
ratio M·M0- 1: Fig. 4. \.Yhen the minute temperature
change /1T5 occurs on the target surface,
~MoE 1(Trsl ) the corresponding minute change LJ.E1•
occurring in the energy E>. incident on the
+ M0M- M E (T ) infrared sensor can be expressed as Eq. 7
2 rs2 with transmission coefficient r0 of air,
0 optical system transmission coefficient 1).,
spectral emissivity£>._, measurement area
Trsl, Trs2 and Trsx are the thermal A'f; effective solid angle nT(dfJ and minute
indices of the infrared thermographic change M>. in spectral radiance LJ,:
system as indicated below:
(7)
(5) E, "cr1~s 1 '
The physical quantities used in Eq. 7
Ez aTrs211 ' are linking to each other in accordance

"Ex cr7~s.\ =(:with the following relations when B >> f

where cr = Stefan-lloltzmann's constant and n
and 11 = sensor intrinsic constant
(mentioned above).

Noise Equivalent Temperature (8) Ar B2 Jl2
Difference
-a2A v -[ 2A n
The noise equivalent temperature
difference (NETD) for a target surface l~r
maintained at a constant temperature T~
n -B-z -
(9) QT{eff)

fiGURE 3. Average energy.

FIGURE 4. Optical schematic.

: 1-,------- ' t I· .~8--
f1{T,1) <o
I 1
I f,(T,,)
/\~~
Legend \ r;l] Ao
[ = radiation energy ih(ffl) \I--------~
E,. =averaged radiation energy A, ,,
M =slit width
legend
M0 == minimum detectable 5ize
A0 =sensor area
A1 = mea5urement area
o"' distance from 5ensor to !ens (millimeter)
B =objective distance from targel5urface to !em
E = emhsivity
4> =lens diameter (millimeter)
'~=special propagation coefficient
t)_ =propagation coefficient of optical system
H1 <•-II) =effective solid angle

Errors in Infrared Thermography 135