NISTIR 8098

Calibration and Measurement 

Procedures for a High Magnification 

Thermal Camera 

Brandon Lane, Ph.D. 

Eric P. Whitenton 

http://dx.doi.org/10.6028/NIST.IR.8098
NISTIR 8098

Calibration and Measurement 

Procedures for a High Magnification 

Thermal Camera 

Brandon Lane 

Eric Whitenton 

Engineering Laboratory 

Intelligent Systems Division 

This publication is available free of charge 

from: http://dx.doi.org/10.6028/NIST.IR.8098

December 2015

U.S. Department of Commerce 

Penny Pritzker, Secretary 

National Institute of Standards and Technology 

Willie May, Acting Under Secretary of Commerce for Standards and Technology and Acting Director 
This page intentionally left blank
i

Calibration and Measurement Procedures for a 

High Magnification Thermal Camera

Brandon Lane

Eric P. Whitenton

National Institute of Standards and Technology, Gaithersburg, MD USA 20899-8223

ABSTRACT

Infrared thermography is a relatively mature measurement technology, however systematic 

evaluation of measurement uncertainty for multi-point measurement (an infrared focal plane 

array) is very limited, or, does not follow uncertainty evaluation guidelines outlined in the 

Bureau International des Poids et Mesures Guide to the Expression of Uncertainty in 

Measurement (BIPM-GUM). In order to approach uncertainty evaluation of thermographic 

measurements, a new thermographic calibration and measurement procedure was developed at 

the National Institute of Standards and Technology (NIST). This method takes from procedures 

used with single-point radiation pyrometers, which have a more developed background in 

uncertainty evaluation stemming from their use as calibration transfer instruments.

This report details a methodology for calibrating a thermal camera and obtaining temperature 

measurements, including measurement of emissivity and point spread function. This procedure 

is performed as a ‘post-processing’ technique which does not require the camera operating 

software to convert camera signal to temperature values. In-camera or manufacturer provided 

calibrations and temperature conversion algorithms may provide sufficiently low measurement 

uncertainty, however these results are often ‘hidden’ within the camera operating software and 

are not readily available for thorough uncertainty analysis, or, these algorithms are known but 

their form inhibits analytical means for uncertainty analyses.

The methodology described here is designed as a framework for future development of analytical 

derivations of measurement uncertainty. The detailed process also includes methods for 

measuring and reducing identified sources of uncertainty (e.g., point spread function) that are 

otherwise neglected in most thermographic measurement procedures.

Disclaimer: This document is an official contribution of the National Institute of Standards and 

Technology and is not subject to copyright in the United States. The National Institute of 

Standards and Technology does not recommend or endorse commercial equipment or materials, 

nor does it imply that the equipment or materials shown are necessarily the best available for the 

purpose.
ii

TABLE OF CONTENTS

Abstract i

Table of Contents ii

Table of Symbols iii

1 Introduction 1

2 Blackbody Calibration and Non-linearity Correction 3

2.1 Initial Imaging of the Calibration Blackbody 4

2.1.1 Calibration Blackbody Temperature Correction with Transfer Radiometer 5

2.1.2 Creating Blackbody Calibration Frames 5

2.1.3 Blackbody Imaging Results 6

2.2 Non-linearity 8

2.2.1 Nonlinearity Results 9

2.3 Temperature Calibration (Sakuma-Hattori) 11

2.3.1 Sakuma-Hattori Calibration Results 13

3 Point Spread Function Measurement 15

3.1 Measuring the ESF 17

3.1.1 Measurement and Projection of the KE image 17

3.2 Formation of the 3D PSF 23

3.3 PSF Deconvolution and Results 23

4 Emissivity Measurement 27

4.1 Calculating Emissivity 30

4.2 Emissivity Measurement Results 31

5 True Temperature Measurement 32

6 Discussion 34

7 Conclusions 35

References 37

Appendix A. Notation and Nomenclature for Image Statistics............................................. A-1

Appendix B. Spectral Responsivity and Sakuma-Hattori Equation Relationships............... B-1

Appendix C. Calibration Data Structure and Functions........................................................ C-1
iii

TABLE OF SYMBOLS

Symbol Name and units

Tbb True blackbody temperature [° C]

Tbb,TC Calibration blackbody thermocouple temperature [°C]

Tbb,rad Calibration blackbody radiant temperature [°C]

TTC Thermocouple temperature (during emissivity measurement) [°C]

Tapp Apparent or radiometric temperature [°C]

Sμ Signal (during emissivity measurement) [a.u.]

SmBB Average signal from micro-blackbody area [a.u.]

Smeas Signal image of a measured scene [a.u.]

η(S) Linearization function

ηbb Linearization function evaluated at blackbody calibration points

a,b,c,k Linearization function coefficients

F(S) Calibration equation (Sakuma-Hattori)

A,B,C Sakuma-Hattori equation coefficients 

c1L First radiation constant (1.1910439x10-16 [W⋅m

2

])

c2 Second radiation constant (0.014 338 [m⋅K])

λ Wavelength [μm]

τlens(λ),τfilter(λ). Relative spectral transmission of lens or filter [-]

R(λ) Relative responsivity of camera (sans optic components) [-]

wλ Relative responsivity of camera system (including filters and optics) [-]

μF( ) Frame mean; mean of every pixel value in one frame [various]

σF( ) Frame noise; standard deviation of every pixel value in one frame

μP( ) Pixel mean; mean of individual pixel value over multiple frames

μP( ) Pixel noise; standard deviation of individual pixel value over multiple frames

ε Emissivity [-]

Δε Difference in emissivity between two temperatures [-]

εmBB Micro-blackbody emissivity [-]

φbb Radiant flux from a blackbody source [W]

Lλ(T) Spectral radiance [W/sr/m3]
1

1 INTRODUCTION

The procedures outlined in this document are an extension and improvement on the authors’ 

previous work in uncertainty analysis for infrared thermography [1,2]. Targeted modifications to 

the equipment and calibration and measurement processes were made in order to reduce or 

quantify measurement uncertainty. Infrared thermography is a complicated measurement 

process, making it apt for Monte-Carlo simulations for uncertainty analysis [1,3]. As pointed out 

in [1], some components of uncertainty may be determined analytically, while more complicated 

factors likely require numerical simulation. Analytical solutions provide the benefit of yielding 

analytical equations which address the complex interrelationships between process variables and 

components of uncertainty. However, the complexities of some sources of uncertainty inhibit

analytical methods. Monte-Carlo methods allow all components of uncertainty to be evaluated

by iteratively reproducing the measurement process chain with input parameters specified by 

some probability density functions. While the Monte-Carlo method can calculate certain 

uncertainty component values which analytical methods cannot, they generally yield only 

statistical correlations which can miss details of input parameter and measurement uncertainty 

relationships. In addition to improvements to the physical thermographic measurement system, 

improvements to the calibration and measurement process were made to facilitate analytical 

methods for calculating measurement uncertainty and reduce measurement uncertainty overall. 

This report describes the new calibration and measurement process, and presents the foundation 

for a more thorough approach to the measurement uncertainty than in [1]. This report details 

only the calibration and measurement procedures, but does not show how the measurement 

uncertainty is calculated. These procedures are part of a newly developed measurement process 

chain shown in the schematic in Figure 1 and detailed throughout this paper. Commercial 

thermal camera software enables users to use factory calibrations and internal, sometimes 

proprietary calculations, making uncertainty evaluation difficult. To fully understand and 

analyze the process, each step in Figure 1 uses custom-developed pre- or post-processing 

software. The subsequent chapters in this document are organized according to this diagram. In 

developing this new calibration and measurement procedure, four new components are 

introduced:

1.) Nonlinearity measurement and correction using an analytical model applied on a pixelby-pixel basis

2.) Blackbody calibration using an analytical model applied on a pixel-by-pixel basis 

(Sakuma-Hattori equation)

3.) Measurement of the spatial resolution and the point spread function (PSF)

4.) Measurement and calculation of emissivity
2

Figure 1: Outline of the calibration and measurement procedure.

The example camera system and results in this document are for a modified commercial off-theshelf (MOTS) infrared camera shown in Figure 2. Modifications include water-cooled chill 

plates and insulating foam around the camera body which keep the camera body temperature at 

20.52 °C (± 0.28 °C, k=2.2 for 95 % confidence) measured 11 times via internal thermistor over 

4 hours, and 85 mm extension tube with 5x near infrared (NIR) microscope objective. Though 

not explicitly stated in [1], measurement uncertainty due to emissivity may be substantially 

decreased for NIR as opposed to the mid-wave infrared (MIR) systems for higher temperature (> 

500 °C) measurements since sensor signal values increase more rapidly with temperature as 

wavelength shortens. The objective also has a conical shroud made of brazed sheet brass, 

covered in fiberglass tape, and terminating in a bakelite standoff. This light cone blocks ambient 

sources and reflections from entering the objective lens while protecting it from stray hot chips 

commonly encountered in metal-cutting experiments. Multiple calibrations were performed for 

different filter and optics options and integration times, however only one is detailed here. All 

examples in this report are for the camera system shown in Figure 2. 

1.) Non-linearity

2.) Blackbody Calibration

Sbb

wl

S= h(S’)

Planck’s Law

Tbb

3.) PSF Measurement

S’KE

h

SKE PSF

4.) Emissivity Measurement

S’e

h

Se e

PSF

5.) Object Measurement

wl – spectral responsivity

Tbb – calibration blackbody temperature (oC)

S’, S – raw and linearized signal (DDL)

Sbb – signal at calibration points (DDL)

h – Linearization function

F(Tapp) – calibration equation (Sakuma-Hattori)

SKE – knife edge image (a.u.)

SESF – edge spread function (a.u.)

PSF – point spread function (a.u.)

Se – emissivity signal image (a.u.)

e – object emissivity value

Smbb – emissivity signal image (a.u.)

TTC – thermocouple indicated temperature (°C)

Tamb – reflected ambient temperature (°C)

Ttrue – true measured object temperature (°C)

Tapp – apparent or radiometric temperature (°C)

Curve fit

S’ Curve fit bb

h

Sbb

Tbb

S = F(Tbb)

Inverse Abel 

Transform 

3D rotation

SESF

Deconvolution Se

Emissivity 

Equations

TTC

Smbb

Ttrue

S’meas

h

PSF

Smeas Deconvolution

e

Tamb Measurement 

Equation

S’bb
3

Figure 2: Modified commercial off-the-shelf (MOTS) camera used in example calibration and 

measurements

Images are windowed to 256 pixels x 320 pixels. This set of optics gives the MOTS camera a 

field of view (FoV) of 1.10 mm by 1.38 mm, and a 3.365 μm/pixel instantaneous field of view 

(iFoV). This results in 4.46x magnification on the 15 μm/pixel focal plane array (FPA). 

2 BLACKBODY CALIBRATION AND NON-LINEARITY 

CORRECTION

For some calculations and results comparisons, the camera spectral responsivity is required. For 

the MOTS camera, spectral responsivity is calculated by taking relative spectral responsivity of 

the camera and relative spectral transmission curves for the microscope objective and filter from 

manufacturer’s specifications. These curves, shown in Figure 3, are copied and digitized to 10 

nm intervals, then multiplied together to form the system relative spectral responsivity, wλ . 

Use of the relative spectral responsivity does not require any particular scaling as wλ is typically 

scaled to unity in this report. 

 (l)  (l) (l) wl lens filter Rcamera  

(1)

Coolant manifold

& supply tubes

Camera body with 

insulating foam

85 mm 

extension tube

5x NIR objective 

with lightblocking cone
4

Figure 3: Combined spectral responsiviy of the MOTS camera determined from spectral 

responsivity/transmissivity of each component.

For this paper, the manufacturer’s specification for spectral responsivity is assumed accurate, 

though more accurate measurement methods are available. For example, several works provide 

measurement uncertainty for measuring spectral uncertainties [4,5]).

2.1 Initial Imaging of the Calibration Blackbody

Before any temperature measurements are made, the response of the camera FPA needs to be 

related to a thermodynamic temperature. This is done by imaging a commercial, variable 

temperature, spherical cavity calibration blackbody. The term calibration blackbody is used 

throughout this paper to distinguish the instrument used to calibrate radiometric devices from the 

physics definition of a blackbody (i.e., a perfect radiator). The calibration blackbody provides a 

nominally uniform emittance scene with spectral dependence closely resembling that of a perfect 

radiator. This essentially ties the thermal camera signal, through Planck’s radiation law, to a 

precisely known temperature. In reality, indicated temperature of the instrument does not

perfectly relate to the exitance from the blackbody cavity. This error may be partially corrected 

by cross-calibrating with a transfer radiometer, which in turn is calibrated based on temperature 

defined by the International Temperature Scale of 1990 [6].

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1500 1700 1900 2100 2300 2500

Relative Responsivity or Transmissivity (unitless)

Wavelength (nm)

MOTS Camera FPA Spectral Responsivity

Bandpass Filter Transmissivity

5x Microscope Objective Transmissivity

Combined MOTS Camera Spectral Responsivity

Relative Spectral Responsivity or Transmissivity
5

2.1.1 Calibration Blackbody Temperature Correction with Transfer Radiometer

The calibration blackbody was measured using the NIST Advanced Pyrometer Transfer 

Radiometer (APXR) traceable to the gold point (1064.18 °C) using the 650 nm narrowband 

filter. This wavelength is outside the sensitivity band of the thermal camera in Figure 3, which 

may incur error if the calibration blackbody emissivity is different at these different wavelength 

ranges. However, spectral dependence of the calibration blackbody’s non-unity emissivity is 

assumed to be negligible. The calibration blackbody setpoint temperature (measured via internal 

thermocouple) is compared to radiometric temperature value output by the APXR, and a 

regression curve was fit to the error at 5 blackbody set-points between 599 °C to 1000 °C. This 

regression resulted in a standard error of estimate (SEE) of 0.25 °C. This error formula is given 

as follows:

E  (Tb b,TC TAPXR )  4.384ln(Tb b,TC / C)  27.334 for Tb b,TC  510.3C

(2)

The error value is subtracted from the calibration blackbody thermocouple reading, Tbb,TC, to give 

radiant temperature of the blackbody Tbb,rad = Tbb,TC - E. The calibration blackbody has a known 

effective emissivity of εbb ≤ 0.995 , therefore the radiant temperature must be converted to a true 

thermodynamic temperature, Tbb. This is accomplished using the spectral measurement 

equation and numerical minimization routine detailed in [1] and [2] to solve for Tbb. 

     

l

l

e

l

e

l

l

l l

l

l

l

l

l

l

d

c T c T

w

d

c T

w

amb

b b

b b

b b

b b rad









































 

exp / 1

1

exp / 1 exp / 1

1

2 2

5

2 ,

5

2

1

2

1

(3)

Note that temperatures in (3) are in Kelvin. Bandwidth limits λ1 and λ2 are 1.5 μm and 2.5 μm, 

respectively, and ambient reflected temperature, Tamb, is assumed to be 20 °C. Bandwidth limits 

are not strictly chosen, but set such that the integration limits are well within the zero-value 

range of the camera relative spectral responsivity. Results for the calibrated conversion from 

setpoint to radiant to true blackbody temperature are given in Table 1.

2.1.2 Creating Blackbody Calibration Frames

When acquiring thermal images, researchers generally desire real-time images that are as clear as 

possible. All FPAs have inherent non-uniform sensitivity from pixel to pixel, which results in 

non-uniform or ‘stripy’ raw signal images. In addition, vignetting from the optics may cause 

pixels around the periphery of the FoV to appear darker. Thus, an in-camera non-uniformity 

correction (NUC) is typically used even if the images will be post-processed (as they are done in 

this report). A NUC assigns unique gain and offset values to each pixel so that the signal across 

the FPA’s pixels appears uniform when a truly uniform scene is imaged. A NUC may be applied 

to the pixel raw signal on the FPA chip (in-camera NUC), or through software after image data is 

acquired (software NUC), or both.

Before any calibration is performed, a single-point in-camera NUC is performed for each 

integration time used and stored on the hard drive. Whether calibrating or measuring a 
6

temperature, an integration time is first selected and the associated in-camera NUC is recalled 

from the hard drive. Since the same in-camera NUC is always used for that integration time, the 

calibration and subsequent temperature measurements will be valid.

If an unchanging scene is imaged at two different times, resulting camera signal may change due 

to systematic drift. By subtracting a standard uniform thermal scene from a target measurement 

scene, systematic drift is canceled (except for any drift that occurs during the time between the 

target measurement and standard scene measurement). A standard thermal scene is created by

imaging an emissivity plate (e-plate); an aluminum heat exchanger plate connected in-line with 

the camera liquid cooling system that is painted on one side with high emissivity black paint. 

The other side is a diffusely reflective, as-manufactured aluminum surface. The liquid cooling

system maintains the e-plate between 12.8 °C and 13.9 °C. The plate is held directly in front of 

the camera with the black side facing the camera and aluminum side facing the blackbody, and 

100 frames are collected immediately before or after a measurement. These 100 frames are then 

averaged and subtracted from the corresponding calibration frames, or frames taken during an 

actual measurement. 

Each time the calibration blackbody is imaged, the e-plate is imaged immediately before or after.

At each setpoint temperature, the calibration blackbody is allowed at least an hour to reach 

equilibrium, reaching the setpoint to within 0.1 °C (resolution of the blackbody digital readout),

and 100 frames are collected. The e-plate is measured and 100 frames collected either 

immediately before or after the blackbody measurement. The e-plate measurement in digital 

levels (DLs) are subtracted from the blackbody measurement in DLs, which are then expressed 

here as unit ΔDLs to indicated a subtraction has been made.

A bad pixel list is manually created after all blackbody setpoints are measured by visually

inspecting the calibration frames, and noting the location of unresponsive pixels. The 100 eplate frames at each setpoint are averaged and subtracted from the averaged 100 blackbody 

frames to create ΔDL data frames at each setpoint temperature. These data frames are plotted, 

and outlier pixels are manually selected and their pixel coordinates noted. Outliers are visually 

determined since their deviation from the frame mean is obvious and likely more than 10σ from 

the frame mean. Since calibration is conducted on a pixel-by-pixel basis, even pixels with 

responses significantly different than the mean response (e.g., more than 3σ from calibration 

frame means) may be utilized. 

2.1.3 Blackbody Imaging Results

In this example, 12 total calibration points were measured. Table 1 gives the calibration 

blackbody temperatures at each setpoint and the corrected values using Equations (2) and (3). 

Each calibration frame image is the mean of a 100-frame blackbody measurement (resulting in a 

single frame) minus the mean of a 100-frame e-plate measurement (also resulting in a single 

frame). Raw signal of either the blackbody or e-plate measurements is in units of digital levels 

(DL). After the e-plate measurement is subtracted, signal values are noted with units of ΔDL. 
7

Rather than show each of the 12 resulting calibration frames, results given in Table 1 give the 

frame mean (mean of all pixel values from 256 pixel x 320 pixel frame) of the pixel mean (pixelby-pixel mean of 100 frames). That is, 100 frames are averaged to form one frame, then all the 

pixels in that frame are averaged. The frame noise is also given, calculated as to standard 

deviation of a 256 pixel x 320 pixel frame. The frame noise gives a measure of non-uniformity, 

or pixel-to-pixel variation in a frame. The 12 calibration frames are stored as 256 pixel x 320 

pixel frames in the calibration data structure. Further explanation of frame mean, frame noise, 

pixel mean, pixel noise, and how calibration frames are created is given in Appendix A.

Table 1: Blackbody measurement results for the example calibration set. Note that bad pixels are 

excluded from these calculations.

Calibration Blackbody 

Temperature Calibration frame statistics

Set-point 

Tbb,TC

Radiant 

Tbb,rad

True

Tbb

Frame mean of 

the pixel mean

μF(μP)

Frame noise 

of pixel mean

σF(μP)

[°C] [°C] [°C] [ΔDL] [ΔDL]

300 300 299.8 25.26 0.66

350 350 349.7 68.50 0.89

400 400 399.7 158.57 1.76

450 450 449.6 326.49 3.08

500 500 499.6 619.42 5.64

550 549.7 549.2 1089.84 9.62

600 599.3 598.8 1797.35 15.13

650 648.9 648.4 2795.67 28.06

700 698.6 698.0 4155.31 35.11

750 748.3 747.6 5941.68 51.31

800 798.0 797.2 8269.20 68.31

840 837.8 837.0 10487.61 85.65

The frame noise at each calibration point was typically 0.8% of the frame mean. The frame 

mean of the pixel mean indicates that on average, a thermal image of a 300 °C blackbody will 

result in roughly 25 more DLs than the e-plate measurement. E-plate measurements ranged from 

frame means of 1687.80 DL to 1737.71 DL, with frame noise of pixel mean ranging from 0.92 

DL to 21.60 DL. Though it contributes negligibly to the calibration (assuming noise levels were 

symmetric about the mean), it was noticed that noise levels for some e-plate measurements were 

appreciably higher for some measurements. It was determined that this occurred when e-plate 

measurements were taken immediately after a blackbody measurement. It is possible that slight 

heating of the FPA may have occurred in the short time it was exposed to the blackbody, which 

may have influenced the baseline noise of the e-plate measurement. As a result, all future 

calibrations will ensure e-plate measurements are taken before the blackbody.
8

2.2 Non-linearity 

In most commercial thermal camera systems, raw camera signal is scaled by the in-camera NUC, 

then this signal is related to calibration blackbody temperature; most often through linear 

interpolation, though spline interpolation, polynomial, or other curve fitting may be used. This is 

relatively simple, sufficient for most measurement applications, often built into commercial 

camera software, and often calibrated by the camera manufacturer. For most commercial 

cameras, raw pixel signal is very linear with incident radiant flux, therefore it is presumed that 

measured signal is directly related to blackbody temperature through Planck’s law. However, we 

choose to separate the signal-to-flux and the signal-to-temperature relationships, which will later 

show how the MOTS camera response is slightly non-linear. By using multiple steps 

(linearization curve-fitting and then calibration-curve fitting), ‘linearized’ camera signal values 

are available that are directly related to flux without the assumption of linearity. This provides 

several benefits: 1) Measurement of the nonlinearity provides insight into the optimal 

measurement range of the camera (e.g., if the measured range is near the upper or lower limits of 

the camera’s dynamic range) and 2) Calculation of point spread function (PSF) or emissivity 

from measured signals may incur error if these signal values are in the non-linear ranges of the 

camera. 

For an incremental change in incident radiant flux on the FPA, the camera will give an 

incremental change in signal. Linearity describes how constant these incremental changes are 

over the measurement range of the camera. In other words, if incident flux is plotted vs. signal, 

it describes how close the data follows a straight line. This ties a camera signal to a radiant flux, 

which is related to the thermodynamic temperature of the target. Linearity can be measured 

using standard flux sources [7], or based on a calculated flux, which is done here by starting with 

Planck’s equation:

[exp /  1]

( )

2

5

1





c T

c

L T

L

l l

l

(4)

Integrated over the spectral sensitivity of the camera, the resultant temperature-dependent value 

is proportional to the radiance of the emitting surface. For a hypothetical perfectly linear 

camera, the measured signal S will be proportional to the incident radiant flux on the detector φ, 

and thus proportional to the emitted radiant flux of the blackbody. 

   

l l l   

S w L (T)d

(5)

Here, α is a constant of proportionality that depends on electronics amplification, pixel size, 

optics geometry and transmission [8], etc. and β is an offset representing the noise floor. Using 

this hypothetical linear signal calculated using equation (5) for each calibration point, a signal 

from a non-linear camera can be converted into values (in arbitrary units, or [a.u.]) that are 
9

linearly related to incident flux on the FPA by using a linearization function, η. The linearization 

function is calculated in equation (6. 1). Equation (6.1) says that the function η maps the ratio of 

calibration points (Sbb, or nonlinear signal in [ΔDL]) over the hypothetical linear signal (φbb) 

back to the calibration point. This mapping is realized by fitting a function, equation (6.2),

between ηbb (η calculated at the calibration points) and Sbb (signal measured while imaging the 

blackbody). Once the fit function is developed, equation (6.3) is used to convert any collected 

non-linear signal S’ [ΔDLs] into a linearized form S [a.u.]. 

bb

bb

bb S

S





h :

(6.1)

( ') [1 ( ( ') )] c h S  k  a  b S

(6.2)

S h(S') S'

(6.3)

The linearity fit equation in (6.2) is the same used in [7], although any functional form may be 

used, and will depend on how φbb is scaled. The real response of a thermal camera is typically Sshaped, with a linear response limited on the low end by electronic noise and saturated at the 

upper end [8]. If calibration points are chosen that reach far enough outside the nominally linear 

response, a different linearization function than (6.2) may be appropriate. The scaling parameter 

α in equation (5) was arbitrarily chosen such that φ was nominally equal to S’ at the highest 

calibration point (resulting in η≈1). This ensured (6.2) is an appropriate function to describe η.

If applied individually to each pixel in an image, linearizing acts as a non-uniformity correction. 

Generally, an in-camera NUC assumes linearity. That is, if a NUC is performed at one uniform 

temperature scene, capturing other uniform temperature scenes will also result in uniform 

images. In reality, the inherently different non-linearity of each pixel will result in a nonuniform image. By linearizing each pixel individually, this acts as non-uniformity correction 

beyond in-camera NUC taken before the blackbody measurement. This inherent NUC caused by 

linearization is limited by the uniformity of the original calibration frame and goodness of fit of 

the curve-fit for each pixel.

2.2.1 Nonlinearity Results

For each pixel, η was calculated from the 12 calibration points using equation (6.1) and a leastsquares fit used equation (6.2) to generate the coefficients of η(S). Example fit results are given 

for three pixels in Figure 4 (left). Four new 256 pixel x 320 pixel arrays/frames composed of the 

linearization coefficient values k, a, b, and c for each pixel are created and stored in the 

calibration data structure along with fit statistics. Using these linearization coefficient frames, 

the calibration frames (Sbb) were linearized and converted to [a.u.]. Figure 4 (right) gives the 

frame mean and frame noise of the linearized calibration frames. This shows the relative range 

of the values of η over the FPA.
10

Figure 4: Left: Example nonlinearity η [a.u. / ΔDL] and corresponding fit functions for 12 

calibration points and three example pixels. Right: Frame mean of η for each of the 12 linearized 

calibration frames. Error bars are one standard deviation (frame noise).

Table 2 gives the frame mean and noise for the four linearization coefficient frames, and two fit 

statistics frames. Each pixel has its own set of coefficients and fit statistics, and Table 2 gives 

mean values of all pixels. Fit statistics include the coefficient of determination (r

2

) and standard 

error of the estimate (SEE).

Table 2: Frame mean and frame noise of 256 pixel x 320 pixel frames consisting of each pixel’s 

linearization fit coefficients (a,b,c, and k) and fit statistics (r

2

and SEE)

Linearization 

Coefficient and Fit 

Statistics

Frame Mean

μF( )

Frame Noise

σF( )

a 0.8762 0.3762

b 0.2584 0.4281

c 0.2788 0.2494

k 0.4364 0.0366

r

2

0.860 0.1166

SEE 0.0078 0.0029

As mentioned, the linearization process acts as a post-processing NUC. Figure 5 and Figure 6

show how two example calibration frames become more uniform after linearization. Apart from 

the visibly diminished vignetting, frame noise and range diminish through linearization.

0 2000 4000 6000 8000 10000

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

Sbb [DDL]

h = bb / S

bb

hbb, Pixel 64,80

Fit curve, h(S')

RMSE = 0.0046

hbb, Pixel 128,160

Fit curve, h(S')

RMSE = 0.0096

hbb, Pixel 192,240

Fit curve, h(S;)

RMSE = 0.0086

0 2000 4000 6000 8000 10000

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

Calibration Frame Mean Signal, Sbb [DL]

Frame Mean of h(S

bb

)
11

Figure 5: Comparison of a Tbb = 837 °C raw blackbody image (minus e-plate measurement) 

(left) and the results after being linearized (right) showing reduced non-uniformity as a result of 

the linearization process.

Figure 6: Comparison of a Tbb = 349.7 °C raw blackbody image (minus e-plate measurement) 

(left) and the results after being linearized (right) showing reduced non-uniformity as a result of

the linearization process.

While linearization converts raw camera signal into values linearly proportional to the radiant 

flux on the FPA, a further radiometric calibration is required in order to convert linearized signal 

to a temperature.

2.3 Temperature Calibration (Sakuma-Hattori)

Equation (7) describes a nonlinear regression fit done for each pixel that relates a measured 

signal S to true blackbody temperature Tbb at each of the calibration points using the SakumaHattori (SH) equation [9]. Here, Sbb is the linearized signal after subtracting an e-plate 

Raw Calibration Frame (DLs) at Tbb = 837 oC

50 100 150 200 250 300

50

100

150

200

250

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06

1.07

1.08

x 104

mean = 10 488 [DDL]

s = 85.7 [DDL]

range = 1 913 [DDL]

Linearized Calibration Frame (a.u.) at Tbb = 837 oC

50 100 150 200 250 300

50

100

150

200

250

1.03

1.032

1.034

1.036

1.038

1.04

1.042

1.044

1.046

1.048

1.05

x 10 Original Calibration Frame [DDL] at T 4

bb = 837 oC

mean = 10 414 [a.u.]

s = 16.2 [a.u.]

range = 726.7 [a.u.]

Linearized Calibration Frame [DDL] at Tbb = 837 oC

Raw Calibration Frame (DLs) at Tbb = 299.8 oC

50 100 150 200 250 300

50

100

150

200

250

60

62

64

66

68

70

72

Linearized Calibration Frame (a.u.) at Tbb = 299.8 oC

50 100 150 200 250 300

50

100

150

200

250

62.5

63

63.5

64

64.5

65

65.5

66

66.5

67

67.5

349.73 oC 349.73 oC

mean = 68.5 [DDL]

s = 0.89 [DDL]

range = 14.0 [DDL]

Original Calibration Frame [DDL] at Tbb =349.73oC

mean = 64.8 [a.u.]

s = 0.69 [a.u.]

range = 5.9 [a.u.]

Linearized Calibration Frame [DDL] at Tbb = 349.73 oC
12

measurement for each pixel, and Tbb is the true blackbody values from Table 1. While 

linearization converts a measured signal into a value of arbitrary units, the Sakuma-Hattori 

equation converts radiometric or apparent temperature to signal. Equation (7.1) shows that the 

function F maps blackbody temperature Tbb (in Kelvin) to camera signal from a blackbody 

measurement Sbb, and that this mapping is easily invertible. Equation (7.2) gives the SakumaHattori equations used in the linear regression and calibration in the forward form and (7.3) gives 

the inverse form, which gives the apparent temperature Tapp. This is called apparent temperature 

because direct use of (7.3) to convert camera signal to temperature neglects the effects of 

emissivity or ambient temperature sources which are described in the measurement equation in 

Section 5.

 F Tbb  Sbb :

(7.1)

Forward 

Sakuma-Hattori 

Equation:

exp 1

( )

2

 















AT B

c

C

F T

 

(7.2)

Inverse SakumaHattori 

Equation:

  A

B

A C S

c

F S Tapp 



 



ln / 1

( )

1 2

(7.3)

Here, c2 = 14388 μm⋅K. Rather than fit the forward version of F in (7.2), the non-linear 

regression fits the inverse function using calibration frame signal Sbb as the independent variable 

and blackbody temperature Tbb as the dependent variable. This also make a more proper fit when 

using a standard least squares algorithm since it is assumed the relative uncertainty of the 

blackbody true temperature Tbb is less than the relative uncertainty of the calibration frame signal 

values. While Tbb is developed from a traceable standard, Sbb is subject to noise.

In fitting equation (7.3), the independent variable was weighted by 1/Sbb to force the regression 

to favor the lower temperature values. It was later learned that the weights should be chosen 

based on the variance of the measurement errors, i.e., each weight value should be wi = 1/σi

2

, 

though these errors were not calculated a priori. Table 1 showed the frame noise is 

approximately 0.8% of the mean frame signal, making the chosen weights 1/Sbb ∝ 1/σ, instead of

the preferred 1/σ2. In [10], they chose weights based on the 0.05% relative signal uncertainties, 

though it wasn’t stated if these were squared reciprocal, or just the reciprocal as done here. 

By fitting the Sakuma-Hattori equation to calibration data, converting temperature to signal or 

vice versa simply requires evaluation of Equation (7). Though not done here, this curve-fitting 

may also make use of regression methods such as ‘error-in-variables’ or ‘total least squares’, or 

methods for determining confidence or prediction intervals for each pixel’s calibration curve.

Though these are more effective methods for evaluating calibration uncertainty, the method 

using standard error of the estimate (SEE) used in [1] is used here. However, as a single value 
13

for each pixel, the SEE does not give information on how the goodness of the regression varies 

with signal or temperature. A more arduous and future evaluation will determine and report 

prediction intervals with each pixel’s curve-fit.

2.3.1 Sakuma-Hattori Calibration Results

A different 256 pixel x 320 pixel frame is stored for each SH coefficient and fit statistic similar 

to the linearization procedure outlined in Section 2.2. Table 3 gives the frame mean values for 

these arrays. Frame noise values are also given to demonstrate the variation in these values 

across the FPA. For example, the frame mean and frame noise of coefficient A means 68.2 % of 

all pixels resulted in an A value of 1.9331 ± 0.0606. Or, 68.2 % of all pixels resulted in an SEE 

of 0.0438±0.0149 °C.

Table 3: Sakuma-Hattori coefficients and fit statistics, and frame mean and frame noise of these 

values.

SH Coefficient 

or Fit Statistic

Frame Mean

μF()

Frame Noise

σF()

A 1.9331 0.0606

B 25.9214 21.871

C 7.8751E06 1.518E06

r

2

0.9999515 3.94E-05

SEE (

oC) 0.0438 0.0149

Though the inverse SH equation may result in temperatures below 300 °C, levels below the 

lowest calibration point of 25 ΔDL (300 °C) are not acknowledged. Though this example was 

given for one integration time (550 μs), the coefficient A should scale with the integration time. 

However, this relationship becomes nonlinear as integration time decreases, so it is not 

recommended to extend a calibration done at one integration time to another solely by scaling by 

the integration time without cross-checking several calibration points.

The regression in Table 3 proved much better than that for the linearization in Table 2 based on 

the r2 and relative SEE values. This is mostly due to the fact that the SH regression used 

linearized signal as the independent variable, and linearization had already acted as a 

nonuniformity correction and reduced the variance of all pixels in each frame. The average SEE 

of the SH curve-fit for all pixels is approximately 0.044 °C. Similar to the linearization, the SH 

curve-fitting acts as a per-pixel NUC. The example linearized calibration frames from Figure 5

and Figure 6 were converted to apparent temperature on a pixel-by-pixel basis using the inverse 

SH equation (Equation (7.3)) with results shown in Figure 7. 
14

Figure 7: Comparison of a Tbb = 837 °C calibration frame (left) and the linearized calibration 

(right) with frame mean, frame noise, and pixel value range.

For a perfect system, these apparent temperature frames should be uniform and equal to the true 

blackbody temperature Tbb. The calibration procedure error may be quantified by calculating the 

difference between the apparent temperature and Tbb. Figure 8 gives the frame mean of these 

values for each calibration point, and also shows the frame noise of each apparent temperature 

frame to give an idea of the relative non-uniformity of each frame. 

Figure 8: Temperature difference between calibration frames converted to apparent temperature 

(frame mean value) and the calibration blackbody temperature. Error bars indicate frame noise

(σF) of each apparent temperature frame.

In Figure 8, the lower calibration points have smaller errors between apparent and blackbody 

temperatures. Recall that the Sakuma-Hattori curve-fits were weighted towards the lower 

temperatures, resulting in better signal to temperature conversion at these values. Also note that 

the increasing temperature frame noise with higher temperatures is a result of higher signal frame 

Apparent Temperature Calibration Frame at Tbb = 837 oC

50 100 150 200 250 300

50

100

150

200

250

830

832

834

836

838

840

842

844

846

mean = 839.06 [oC]

s = 2.15 [oC]

range = 17.8 [oC]

Apparent Temperature [oC] of 

Calibration Frame at Tbb = 837 oC

50 100 150 200 250 300

50

100

150

200

250

348

349

350

351

352

353

Apparent Temperature [oC] of 

Calibration Frame at Tbb = 349.73 oC

mean = 350.02 [oC]

s = 0.66 [oC]

range = 5.8 [oC]

300 400 500 600 700 800

-1

0

1

2

3

4

Calibration Blackbody Temperature, Tbb (oC) Temperature Difference [

oC]

Frame Mean Apparent Temperature minus True Blackbody Temperature (& Frame Noise)

|mF(T

app

) - Tbb| (+/- sF(T

app

))
15

noise seen in Table 1. However, the relative magnitude of this noise is reduced when converting 

to temperature due to the nonlinearity of the inverse Sakuma-Hattori equation.

Saunders and White [11] gave formulae for estimating Sakuma-Hattori coefficients based on a 

detector’s relative spectral responsivity. The results of these formulae are applied to the frame 

mean values from Table 3 to compare to values based on the camera responsivity given in Figure 

3. These formulae for conversion are given in Appendix B.

Parameter

Based on SH 

coefficients

Based on 

Responsivity Units

Central Wavelength λ

0

1.975 1.956 μm

Variance, σ

2

0.012 0.023 μm

Bandwidth, Δλ 0.373 0.525 μm

Relative Bandwidth, σ/λ0 0.055 0.077

While the central bandwidth λ0 is well-determined by the calibration coefficients (to within 20 

nm), the bandwidth calculated based on the coefficients is approximately 30% narrower. For this 

camera, it would not be suggested to base calibration coefficients solely on the known 

responsivity, especially since the scaling (SH coefficient ‘C’ in Table 3) of each pixel varies 

across the FPA. However, when determining specifications of a potential camera system, these 

provide useful relationships between spectral bandwidth, system response, and their 

contributions to measurement uncertainty [1].

3 POINT SPREAD FUNCTION MEASUREMENT

Microscopic thermal imaging near the same scale as the detector wavelength (diffraction-limited 

scale) has the inherent problem of limited spatial resolution and blurry images. In thermography, 

where pixel temperature resolution as well as spatial resolution is important, blurring can cause 

significant error. High-temperature scenel areas may bias the apparent temperature of 

neighboring pixels higher, while in turn the high-temperature pixel is biased lower by the 

relatively cooler neighboring scenels. The scale of this effect depends on the local gradients of a 

particular measured scene and the intrinsic spatial resolution of the optical system defined by the 

point spread function (PSF). An ‘ideal’ image, or that without the optical blurring effects, may 

be extracted from a measured image by deconvolving a known or measured PSF from a collected 

image. 

Point-spread function is defined as the resulting image from measuring an infinitesimally narrow 

point source, i.e., it is the impulse response of the optical system. Creating an impulse source to 

measure PSF is difficult in practice since reducing the width of a source to near zero will in turn 

reduce the intensity, and thus approach the noise floor of the system. Rather, measuring the edge 

spread function (ESF) is simpler, more common in practice, and recommended in ISO 12233 for 
16

characterizing optical system performance [12,13]. Edge spread function (ESF) is the image 

obtained when viewing the straight-line boundary between high and low intensity sources. It is 

related to the PSF through the line spread function (LSF), which is the image obtained when 

viewing an infinitesimally narrow and infinitely long line source. The ESF, LSF, and PSF are 

related as follows:

 







esf x  lsf x  psf x y dy

dx

d

( ) ( ) ( , )

(8)

Another metric for characterizing optical system performance is the modulation transfer function 

(MTF, also called the optical transfer function). The MTF is the 2D Fourier transform of the 

PSF, therefore defined in the spatial frequency domain. Frequency-domain characterization is 

not done here, though much information may be found [8,12,14].

The ESF is measured by creating and imaging a knife edge (KE) scene, where a sharp, straight, 

opaque edge is placed at the focal plane between the camera and blackbody source. The KE 

image is used to create an esf(x) function by projecting all the KE image pixel values along the 

KE line into a super-sampled (i.e., data spacing is <1 pixel) digital ESF, then an appropriate 

esf(x) function is fit to this data. This process is described further in the next section.

Once an ESF is determined, it is differentiated to determine the LSF. A further step is required 

in order to obtain the PSF from the LSF. PSFs may be approximated as rotationally symmetric 

for systems with rotationally symmetric optics and defined over a radius (psf(r)) rather than 

Cartesian coordinates (psf(x,y)). Marchand [15] gave multiple methods for converting an LSF to 

a rotationally symmetric PSF and vice versa, one of which noted that the LSF is the Abel 

transform of the radial cross-section of the corresponding rotationally symmetric PSF:

Forward Abel 

transform:

dr

r x

r psf r

lsf x

x











2 2

( )

( ) 2

(9.1)

Inverse Abel 

transform:

 lsf x dx

x r

psf r

r

d x

d







  ( )

1 1

( )

 2 2

(9.2)

The rotationally-symmetric PSF function psf(r) may be calculated from the ESF measurement 

using Equations (8) and (9.2). The psf(r) function can then be digitally rotated about a center 

pixel to form the PSF image array.

Making use of certain properties of the Abel transform may allow more accurate evaluation of 

the ESF and transformation to the PSF. Abel transforms and their inverses (as well as the 

differential operator d/dx) are linear transformations [16]. Therefore, a linear combination of 

ESFs may be converted to a linear combination of LSFs and then PSFs. Tazannes and Mooney 

[14] mentioned the possibility of using a linear combination of functions to fit an ESF, however 
17

they did not detail how this was a mathematically sound approach. By fitting a linear 

combination of ESF functions, the number of fit parameters may be increased until a satisfactory 

fit is made. In addition, linear combinations of different potential ESF functions may be used. 

This property of the PSF is exploited here. 

Procedure for calculating the PSF:

1.) Create a knife edge scene in front of the blackbody source and collect a KE image in 

DLs. Collect a corresponding e-plate measurement.

2.) Subtract the e-plate measurement from the KE image, then linearize the KE image to 

convert from ΔDLs to a.u.

3.) Project the pixel values along the knife edge to form the supersampled ESF

4.) Fit various linear combinations of potential ESF functions to the supersampled ESF. 

Select the best fit as the esf(x).

5.) Convert the esf(x) into a psf(r) using the inverse Abel transform.

6.) Digitally rotate and resample the 2D psf(r) to form a PSF image array.

Once a PSF is determined, an image may be deblurred by deconvolving the PSF array from the 

measured image. Note that all of this is done with linearized signal (e.g., proportional to radiant 

intensity) before converting to temperature.

3.1 Measuring the ESF

3.1.1 Measurement and Projection of the KE image

The ESF as defined is perpendicular to the KE line. However, if the KE is aligned vertically or 

horizontally with the FPA, the location of the KE line will lie in the middle of a row of pixels, 

therefore cannot be determined to better than one pixel width. Tilting the KE diagonally across 

the scene is preferred. Multiple methods may be used to locate the edge in the image, mostly 

relying on some method of differentiating the image along some direction, and locating the 

maximum of this gradient, and fitting a line to the maxima locations. An example is given in 

Figure 9, which was made by placing a razor blade at the focal plane in front of a 700 °C 

blackbody source. The result of fitting a line to these points is the slope intercept form of the KE 

line, i.e., j = mi + b, where j and i are pixel indices, and m and b are slope and intercept. 

According to ISO 12233, a super-sampled ESF is generated from the KE image by projecting 

pixel values along the KE line onto a unified axis. This axis is defined by distance between a 

pixel and the KE line (therefore, making 0 the center of the knife edge and ESF). However, 

ISO 12233 also states that the KE line should only be tilted 5° from the vertical and the 

horizontal distance from pixel to KE line is used as the projection axis. Assuming a 5° tilt is 

equivalent to a vertical knife edge is computationally easier, though here we use the real normal 

(or minimum) distance to the KE line, as shown in Figure 9, and calculated in Equation (10).
18

Figure 9: Example knife edge image showing the KE line fit to image gradient maxima (denoted 

by a small white ‘+’ in the middle of a pixel). Calculation for determining each pixel’s normal 

distance the KE line is also shown.

1

2

,



 



m

j mi b

di j

(10)

Plotting pixel values (linearized signal, [a.u.]) vs. this distance results in the super-sampled ESF, 

shown in Figure 10 with three regions (a, b, and c) highlighted in subfigures. This was done for 

three blackbody temperature values. These values are normalized, as the vertical scale of the ESF 

is not important, and later PSF values are normalized as well.
19

Figure 10: Three super-sampled ESFs (normalized) at different blackbody temperatures.

The 900 °C supersampled ESF appeared similar to the others at the higher end, representing the 

optically clear region of the knife edge image, but different in the lower, opaque region. It is 

believed that the razor blade may have heated up while in front of the blackbody, thus having 

radiant emission levels higher than ambient condition.1 The 900 °C ESF is considered erroneous 

and not analyzed further. Due to the similarity between the 500 °C and 700 °C ESFs, only the 

700 °C ESF is further analyzed. 

At this point, ISO 12233 recommends the supersampled ESF be re-sampled into ‘bins’ where all 

points lying within 1⁄4 pixel increments are averaged together in order to reduce noise by 

averaging, but still provide 4x supersampled data to reduce aliasing. However, this document 

departs from ISO 12233 at this point, and fits a smooth, continuous curve to the super-sampled 

ESF data. One of the purposes of the ISO 12233 is to develop the MTF in order to characterize 

and compare camera systems. However, here we intend to develop a PSF to use as a 

deconvolution kernel to de-blur thermographic images. 

 

1

It should be noted that the lower (opaque) region of the ESF had camera signal values very near the lower 

boundary of the calibration (0 ΔDL to 20 ΔDL), while the higher (transparent) region varies in magnitude with the 

blackbody temperature. A more appropriate method may be to use a semi-transparent knife edge so that lower 

signal levels are in the more linear range and scale with the higher region and blackbody temperature.

-0.2 0 0.2

0.4

0.45

0.5

0.55

0.6

Projected X [pixels]

Normalized ESF [a.u.]

Normalized ESFs for 3 Temperatures

Distance to KE line, di j [pixels]

-150 -100 -50 0 50 100 150

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Projected X [pixels]

Normalized ESF [a.u.]

Normalized ESFs for 3 Temperatures

500 0C

700 0C

900 0C

Distance to KE line, di j [pixels]

Super-sampled ESFs at three Blackbody Temperatures

a

b

c

-140 -120 -100 -80 -60 -40 -20

-0.02

-0.01

0

0.01

0.02

0.03

0.04

Projected X [pixels]

Normalized ESF [a.u.]

Normalized ESFs for 3 Temperatures

Distance to KE line, di j [pixels]

a b

20 40 60 80 100 120 140

0.94

0.96

0.98

1

1.02

1.04

1.06

Projected X [pixels]

Normalized ESF [a.u.]

Normalized ESFs for 3 Temperatures

Distance to KE line, di j [pixels]

c
20

Curve-fitting the ESF:

Multiple authors have noted that common functional forms of the ESF resemble cumulative 

distribution functions, and claim the LSF or PSF are the associated probability distributions. 

Smith [17] attempted multiple single ESF functions (i.e., not a linear combination) that were fit 

to a measured ESF. These included error function (which generates a Gaussian distribution PSF) 

and arctangent (which generates a Cauchy distribution PSF). Tazannes and Mooney [14] fit a 

linear combination of Fermi distributions to a measured ESF. 

Various combinations of error function and arctangent function forms were fit to the supersampled ESF data. Expressions for lsf(x) were determined by simple differentiation, and 

conversion from lsf(x) to psf(r) through inverse Abel transform utilized tables of calculated 

transforms [16,18]. These functions are shown in Table 4. For single-function fitting of ESFs, 

any scaling coefficient is unimportant since the ESF and generated PSF may be eventually 

normalized. However, with linear combinations of ESF function, the proper Abel transform is 

important to maintain the relative scaling effect of each coefficient in the linear combination.

Table 4: ESF, LSF, and PSF relationships based on the assumption of Gaussian or Cauchy 

distribution shaped PSFs.

Gaussian PSF: Cauchy PSF:



















  

N

i i

i

b

x

esf x a erf

2 1

1

( )

(11.1)



















 

N

i i i

i

b

x

b

a

esf x

1

arctan

2

1

( )

(11.4)

















 

 

N

i i i

i

b

x

b

a

lsf x

1

2

2

exp

2

( )



(11.2)

 



N

i

i

i

x b

a

lsf x

1

2 2

( )

(11.5)

















 

 

N

i i i

i

b

r

b

a

psf r

1

2

2

2

exp

2

( )



(11.3)

 

 



N

i

i

i

b r

a

psf r

1

3/ 2

2 2 2

1

( )

(11.6)

Five linear combinations were attempted and are given in Table 5. These included from one to 

two Cauchy functions and one to three Gaussian functions. Mixed combinations of Gaussian 

and Cauchy PSFs were not attempted. Linear combinations of Gaussians provided superior fits. 

After the 3-Gauss function resulted in an RMS error of less than 0.2% full scale of the ESF, 

further attempts at fitting more complex ESF combinations were ceased. The multiple curve fits 

to the supersampled 700 °C ESF are shown in Figure 11, with lower, central, and upper region 

expanded for clarity. Resulting psf(r) functions, normalized to the area under the curve, are 

plotted in Figure 12.
21

Figure 11: Curve fits to the 700 °C ESF using various linear combinations of fit functions.

-40 -30 -20 -10 0 10 20 30 40

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Projected X [pixels] Normalized ESF [a.u.]

ESF Fit Functions

Raw ESF (700 oC)

1-Gauss

2-Gauss

3-Gauss

1-Cauchy

2-Cauchy

b

a

b

c

Distance to KE line, dij [pixels]

-18 -16 -14 -12 -10 -8 -6 -4 -2

0

0.02

0.04

0.06

0.08

0.1

Projected X [pixels]

Normalized ESF [a.u.]

ESF Fit Functions

a

Distance to KE line, dij [pixels]

-1 -0.5 0 0.5 1

0.3

0.4

0.5

0.6

0.7

Projected X [pixels]

Normalized ESF [a.u.]

ESF Fit Functions

Distance to KE line, dij [pixels]

0 2 4 6 8 10 12 14 16 18

0.8

0.85

0.9

0.95

1

Projected X [pixels]

Normalized ESF [a.u.]

ESF Fit Functions

c

Distance to KE line, dij [pixels]

-18 -16 -14 -12 -10 -8 -6 -4 -2

0

0.02

0.04

0.06

0.08

0.1

Projected X [pixels]

Normalized ESF [a.u.]

ESF Fit Functions

b
22

Table 5: Fit results for the five tested ESF functions.

Figure 12: Resulting PSFs, normalized based on the 2D area under the curve

Name ESF Function (fit) PSF functon Coefficients r

2

RMSE

1-Gauss a1 0.496 0.99949 0.0050

b1 2.475

2-Gauss a1 0.088 0.99989 0.0023

a2 0.412

b1 8.959

b2 1.871

3-Gauss a1 0.327 0.99994 0.0017

a2 0.019

a3 0.158

b1 1.577

b2 43.095

b3 4.692

1-Cauchy a1 0.295 0.99976 0.0035

b1 0.908

2-Cauchy a1 0.307 0.99980 0.0032

a2 -1337.412

b1 0.936

b2 3929.574

a1/b1*atan(x/b1) + 

a2/b2*atan(x/b2) + 0.5

a1/b1*atan(x/b1) + 0.5

a1*erf(x./b1)+0.5 2*a1/pi/b1^2*exp(-r.^2/b^2)

2*a1/pi/b1^2*exp(-r.^2/b1^2) + 

2*a2/pi/b2^2*exp(-r.^2/b2^2)

a1*erf(x./b1) + 

a2*erf(x./b2)+0.5

2*a1/pi/b1^2*exp(-r.^2/b1^2) + 

2*a2/pi/b2^2*exp(-r.^2/b2^2) + 

2*a3/pi/b3^2*exp(-r.^2/b3^2)

a1*erf(x./b1) + 

a2*erf(x./b2) + 

a3*erf(x./b3)+0.5

a1/2*(b1^2+r.^2).^-1.5

a1/2*(b1^2+r.^2).^-1.5 + 

a2/2*(b2^2+r.^2).^-1.5

-3 -2 -1 0 1 2 3 4

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

Radius [pixels]

Normalized PSF [a.u.]

Point Spread Functions

LSF

1-Gauss PSF

2-Gauss PSF

3-Gauss PSF

1-Cauchy PSF

2-Cauchy PSF
23

3.2 Formation of the 3D PSF

All calculations here, including integration, are done numerically. The smooth, algebraic 

function psf(r) given in Table 5 is oversampled to 10/pixel, and spline interpolation used to 

determine intermediary values. To ‘digitally rotate’ and re-sample psf(r) into an PSF image

array, the radius value is converted back into Cartesian coordinates and integrated over each 

pixel of coordinates (i,j):

   









 

0.5

0.5

0.5

0.5

2 2

,

j

j

i

i

PSFi j psf x y dxdy

(12)

After generating the PSF image array, it is normalized such that the volume under the PSF is 

unity. Figure 13 (left) demonstrates how Equation (12) develops the rotated and digitized PSF 

array. It shows the central 5 pixel x 5 pixel region, though the PSF array is defined over a 41

pixel x 41 pixel array. The majority of the PSF occurs over the central 7 pixel diameter, with 

values quickly tapering off outside this region as seen in the logarithmic scaled plot in Figure 13

(right). Despite their low magnitude, these exterior pixels are important due to the cumulative 

nature of 2D convolution and deconvolution. The 41 pixel x 41 pixel size of the PSF was chosen 

arbitrarily, but ensured that it encompassed all the PSF data to within four orders of magnitude. 

Also visible is the transition between the different contributing Gaussian functions that compose 

the linear combination PSF detailed in Table 5. 

Figure 13: Left: Demonstration of how central region of the smooth psf(r) function is averaged 

over each pixel to develop the PSF image array. Right: The full PSF image array plotted on a 

logarithmic scale for detail. 

3.3 PSF Deconvolution and Results

To test the PSF measurement, a calibration image was taken. This is made by placing a glass 

1951 US Air Force standard resolution chart (or Air Force target, AFT) in front of the calibration 

-2 -1 0

1

2

-2

0

2

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

X-distance [pixels]

Y-distance [pixels] Normalized PSF [a.u.]

Smooth psf(r) 

function

Averaged 

pixels 

-20

-10

0

10

20

-20

-10

0 

10 

20 

10-4

10-3

10-2

10-1

100

Y-distance [pixels]

X-distance [pixels] X-distance [pixels]

Y-distance [pixels] Normalized PSF [a.u.] Normalized PSF [a.u.]
24

blackbody. Because it was a glass target, the signal is attenuated, so signal values do not 

correspond to the blackbody temperature. Nonetheless, the target is still useful for determining 

the attenuation of small targets (triplet bars of multiple sizes on the AFT) due to blurring from 

the PSF. Table 6 gives the bar sizes for each element in each group of the AFT. 

Table 6: Bar width for group and element numbers in the 1951 AFT used to test the PSF

Bar width (μm)

Element Group 4 Group 5 Group 6 Group 7

1 31.3 15.6 7.8 3.9

2 27.8 13.9 7.0 3.5

3 24.8 12.4 6.2 3.1

4 22.1 11.0 5.5 2.8

5 19.7 9.8 4.9 2.5

6 17.5 8.8 4.4 2.2

Before the linearized signal AFT image is deconvolved with the PSF, it is symmetrically 

mirrored on each edge by the width of the PSF window (41 pixels) to reduce any edge effects of 

the deconvolution algorithm. The AFT image is then deblurred using a blind deconvolution

(maximum likelihood algorithm) with the 3-Gauss PSF array as an initial PSF. Figure 14 shows

the results of the AFT image with and without PSF correction, converted to apparent 

temperature. Two sub-regions are expanded to show the de-blurring effect of the PSF 

deconvolution. 
25

Figure 14: Example AFT image converted to apparent temperature without PSF deconvolution 

(left top image) and with PSF deconvolution (right top image), showing the deblurring effect on 

small features (bottom row).

Three traces are made of the AFT image to better observe the effect on temperature values vs. 

bar width. Figure 15 gives the results. The bar elements in the subfigures 1B and 2D in Figure 

14 are group 6, element 2 and 3, with 7.0 μm and 6.2 μm bar spacing, respectively. These are 

also shown in Trace #3 in Figure 15. 

Apparent Temperature, no PSF correction

140 150 160 170

155

160

165

170

175

180

600

620

640

660

680

700

720

Apparent Temperature, Deconvolved PSF

140 150 160 170

155

160

165

170

175

180

600

620

640

660

680

700

720

Apparent Temperature, no PSF correction

50 100 150 200 250 300

50

100

150

200

250

600

620

640

660

680

700

720

Apparent Temperature, Deconvolved PSF

0 50 100 150 200 250 300

0

50

100

150

200

250

600

620

640

660

680

700

720

Apparent Temperature, Deconvolved PSF

105 110 115 120 125

170

175

180

185

190

195

600

620

640

660

680

700

720

Apparent Temperature, no PSF correction

105 110 115 120 125

170

175

180

185

190

195

600

620

640

660

680

700

720

1A 1B

1A

1B

2C

2D

2C 2D

[

oC]

[

oC]

[

oC]

[

oC]

[

oC] [

oC]
26

Figure 15: Example traces of the AFT image (left) with and without PSF correction. Group and element numbers are given in each 

trace plot for select bar elements. 

Apparent Temperature, no PSF correction

50 100 150 200 250 300

50

100

150

200

250

600

620

640

660

680

700

720

Trace #2 Trace #1

Trace #3

Pixels

100 120 140 160 180 200 220

600

650

700

Trace #2

270 280 290 300 310 320 330 340

600

650

700

Trace #1

225 230 235 240 245 250 255 260 265 270

580

600

620

640

660

680

Trace #3

100 120 140 160 180 200 220

600

650

700

Trace #2

270 280 290 300 310 320 330 340

600

650

700

Trace #1

225 230 235 240 245 250 255 260 265 270

580

600

620

640

660

680

Trace #3

100 120 140 160 180 200 220

600

650

700

Trace #2

270 280 290 300 310 320 330 340

600

650

700

Trace #1

225 230 235 240 245 250 255 260 265 270

580

600

620

640

660

680

Trace #3

270 280 290 300 310 320 330 340

600 650 700

Trace #1

225 230 235 240 245 250 255 260 265 270

580

600

620

640

660

680

Trace #3

Deconvolved PSF No PSF Correction

G4 E4 G4 E3 G4 E2

G5 E1

G6 E2 G6 E6

G5 E3 G5 E6
27

Figure 14 and Figure 15 show that the temperature attenuation on the smaller bar groups caused 

by the PSF may be corrected. The limit may be seen at approximately group 5 element 3 (bar 

widths of 12.4 μm). For bars wider than this, the full 100 °C range of temperature is reproduced

without attenuation. Narrower than this width, the bars are attenuated even with PSF correction. 

However, for bar elements as narrow as 4.4 μm (group 6 element 6), PSF deconvolution 

decreases the attenuation effect.

In [1], it was found that for an example scene and PSF and measuring the maximum pixel 

temperature value, PSF correction provided an error of only -8 °C from the true maximum 

temperature of 973 °C, whereas without PSF correction, the error was -105 °C. The component 

of uncertainty due to error in the ‘measured’ PSF, as defined in [1], was 10.5 °C. In [1], the 

errors were based on a simulated image and a simulated PSF and PSF uncertainty. However, 

accomplishing the inverse problem of evaluating error from a measured image and measured 

PSF is still under investigation. It was shown, however, that PSF correction is necessary to 

reduce the biasing effect of optical blur on measurement error, and accurate measurement of the 

PSF is key to reducing the uncertainty due to this correction.

Motion blur was also investigated in [1]. This occurs when objects in the measured scene are 

moving or vibrating at a frequency such that blurring or persistence of motion occurs [2]. This 

blur increases with the velocity of the object, and decreases with the camera integration time. 

Though not mentioned here, it may be necessary to either compensate for motion blur by 

measuring its effect on the camera system, measuring the motion or vibration of the measured 

object and devising an appropriate Type-A uncertainty, or providing a Type-B uncertainty 

estimate [19]. 

4 EMISSIVITY MEASUREMENT

At this point, images may be collected, PSF-corrected, and converted to apparent temperature 

using the linearization and calibration scheme in Section 2. However, to convert to true 

temperature, the signal values need to be corrected for emissivity. It was noted in [1] that 

uncertainty in emissivity accounts for the largest source of measurement uncertainty. Therefore, 

more effort is taken here to provide better emissivity measurements. Madding [20] mentioned 

several methods for measuring emissivity; using a reference emitter, a reference temperature, 

measuring the target at two temperatures, or measuring reflectivity. In the authors’ previous 

work, only the reference temperature method was used by measuring the target temperatures 

with a thermocouple. Here, each method is used (except measuring reflectivity), although not 

always simultaneously. This depends if a reference emitter is available on the particular sample 

being measured. The reference emitter method is accomplished by electro-discharge machining 

(EDM) a small hole in the face of the tool shown in Figure 16. When the insert is heated, this 

hole acts as a micro-blackbody (mBB), with a reference emissivity value very near unity. 
28

The EDM process achieved mBB diameters of approximately 0.3 mm, with depths greater than 

3.5 mm measured by sticking a fresh, 0.2 mm EDM electrode into the finished mBB and 

measuring the displacement from the surface in a shadowgraph. It is well known that cylindrical 

cavities form blackbodies with effective emissivities greater than 0.990 if the depth to diameter 

ratio is greater than 10 and internal wall emissivity is greater than 0.5 [21]. 

Several steps are taken to prepare and heat the tool during emissivity measurement. After the 

mBB is made, the tools are ultrasonically cleaned in an acetone bath. From this point forward, 

any contact with the tool is made with cotton or nitrile gloves to prevent skin oil from 

contaminating the surface. The tool is wrapped tightly with copper foil, shown in Figure 16, 

with a type-K thermocouple held in close contact with the approximate center of the insert face. 

This is then wrapped in 18 gauge nichrome wire as the heat source. The copper foil better 

distributes heat from the nichrome around the insert. The nichrome-wrapped insert is held 

between two copper jaws in a vise (called the ‘thermal vise’), which are connected to a pulsed 

reflow soldering power controller (4.0 KV⋅A power). This controller allows timed, feed-back 

controlled (via thermocouple) heating up to 1000 °C by passing controlled current from the vise 

jaws through the nichrome wire causing it to heat. 

Figure 16: Example of a tungsten carbide tool in the thermal vise heating apparatus during 

emissivity measurement. Temperature and reference radiance are measured via thermocouple 

and/or micro-blackbody

While in the thermal vise, the tool is heated in cycles lasting 200 s each. These cycles increase in 

temperature from 100 °C to 550 °C for a total of 25 cycles. While developing this method, it 

was found that discoloration through oxidation, as well as emissivity values, ceased to change

after approximately 20 cycles. Images of the mBB are taken at nominally 550 °C and another 

cycle at 450 °C. If a mBB is not on a sample, then the bare face of the insert is measured at 

nominally 550 °C and 450 °C. After the emissivity images are taken, the raw signal image is 

Copper foil wrap

Type-K 

thermocouple

18 AWG 

nichrome wire

Microscope view (10x zoom) of 

micro-blackbody

Copper thermal 

vise jaws 

0.23 mm

0.27 mm ID

> 3 mm deep
29

linearized using the methods from Section 2, and PSF corrected using the PSF measured in 

Section 3. 

Figure 17 gives an example of two mBB images taken at two thermocouple temperatures (TTC). 

To get camera signal values representing the mBB radiance, a region of interest (ROI) is selected 

within the mBB, and mean signal pixel value, SmBB, is calculated (as well as standard deviation 

σ). An ROI outside the mBB is also made, with mean signal pixel value calculated for the tool 

region (Stool). These ROIs were separated to avoid the transition boundary between the tool 

region and micro-blackbody region, and a noticeable ring around the micro-blackbody. The 

source of the ring was undetermined, though it is hypothesized that this was residue from the 

EDM process.

Figure 17: Example micro-blackbody signal images (in [a.u.]) and user-selected regions of 

interest (ROIs) used to measure mBB signal and signal on the heated tool face.

Right away, it was noticed that the mBB and thermocouple gave differing temperature values for 

each insert measured. For example, in Figure 17, the mean signal measured in the mBB region 

(SmBB) converted to apparent temperature using the inverse calibration discussed in Section 2.3

give mBB temperatures of 435 °C and 525 °C; lower than the thermocouple readings of 446 °C 

and 545 °C, respectively. Two things may cause this; 1) the thermocouple is in fact at a higher 

temperature than the micro-blackbody or 2) the micro-blackbody has an emissivity less than the 

expected >0.990 value2.

Reason 1 is taken as true. The thermocouple lies under highly conductive copper foil which 

brings the thermocouple in closer thermal proximity to the nichrome heat source. The mBB

 

2

If the thermocouple temperatures are regarded as true, and the mBB is the same temperature, then this would result 

in a mBB emissivity of approximately 0.85. From previous work at NIST [22], tool emissivity was measured near 

0.85, whereas the mBB in Figure 17 is visibly emitting more than the surrounding tool face. This further gives 

reasoning for TTC being an inaccurate measure of the tool face temperature.

TTC = 446 oC

SmBB mean: 256.67 (s = 4.28) [a.u.]

Stool mean: 207.39 (s = 2.64) [a.u.]

F(TTC) = 301.2 a.u.

Tool 9k, TTC = 446 oC,

 Linearized+Deconvolved Signal [a.u.]

50 100 150 200 250 300

50

100

150

200

250 200

210

220

230

240

250

260

270

Tool 9k, TTC = 545 oC, 

Linearized+Deconvolved Signal [a.u.]

50 100 150 200 250 300

50

100

150

200

250

600

650

700

750

800

850

TTC = 545 oC

SmBB mean: 813.77 (s = 3.43) [a.u.]

Stool mean: 662.30 (s = 10.52) [a.u.]

F(TTC) = 1011.0 a.u.

Tool Region

mBB Region Ignored Region
30

occurs on an exposed face which is subject to radiative and convective cooling, while the 

thermocouple lies under the copper foil. In addition, the high depth to diameter ratio of the mBB

discussed previously ensures relatively high effective emissivity. Due to the potential difference 

between thermocouple and tool face temperatures, previously unconsidered in [1], it is noted that 

thermocouples should be placed very near the mBB on the exposed insert face; preferably fixed 

via resistance spot welding. Though this discrepancy is known here, methods for calculating 

emissivity based on the mBB, thermocouple, or both are given in the next section.

4.1 Calculating Emissivity

Values used to calculate emissivity are the micro-blackbody signal (SmBB), micro-blackbody 

signal converted to temperature (TmBB), or the thermocouple temperature (TTC). Multiple 

methods of calculating emissivity based on these measurements require certain assumptions of 

varying feasibility. Assuming negligible ambient reflection, measured signal from the heated 

tool face or the micro-blackbody is related to their respective temperatures, Ttool and TmBB:

( ) Stool toolF Ttool  e

(13.1)

( ) SmBB mBB F TmBB e

(13.2)

If it is assumed that the tool face temperature is well represented by the thermocouple 

temperature, then Ttool = TTC, and emissivity is:

( ) TC

tool

tool F T

S

e 

(14)

If a micro-blackbody is available, then one may assume that the tool surface and internal 

temperature are uniform and equal, or Ttool = TmBB. Then combining Equations (13.1) and (13.2) 

yields the following, noted in [20] as the ‘reference emitter’ technique:

mBB

tool

tool mBB S

S

e  e

(15)

The assumption can be made that εμBB = 1 in Equation (15). If it is assumed εmBB≠1, εmBB can be 

calculated from (13.2). However, TmBB is unknown. If you assume TμBB = TTC, then (13.2) gives 

the following: 

( )

TC

mBB

mBB F T

S

e 

(16)

Inserting Equation (16) into (15) yields (14), thereby requiring the same assumptions. If 

measurements are taken at more than one temperature, it may be assumed that the emissivity is 

the same at the two temperature (Δε = 0). The two temperature technique noted in [20] may be 

calculated by the following:
31

( ) ( ) ,1 ,2

,1 ,2

TC TC

tool tool

tool F T F T

S S





e 

(17)

And in a similar manner to the formation of Equation (15), where the temperature of the microblackbody is assumed equal to the temperature of the surface:

,1 ,2

,1 ,2

mBB mBB

tool tool

tool mBB S S

S S





e  e

(18)

These equations are summarized in Table 7 with the required assumptions for each technique. 

Table 7: Summary of emissivity calculation methods utilizing thermocouple temperature (TTC), 

camera signal in the tool ROI (Stool), and camera signal from the micro-blackbody ROI (SmBB)

Equation Equation # Required Assumptions

ε

tool = Stool / F(TTC) (14) Ttool = TTC

ε

tool = Stool / SmBB (15)

Ttool = TmBB

e

mBB=1

ε

mBB = SmBB / F(TTC)

ε

tool = εmBB * Stool / SmBB

(15), (16)

Ttool =TmBB= TTC

e mBB≠1

ε

tool = (Stool,1 - Stool,2) / (F(TTC,1) - F(TTC,2)) (17)

De

mBB= Detool = 0

Ttool = TTC

ε

tool = (Stool,1 - Stool,2) / (SmBB,1 - SmBB,2 ) (18)

De

mBB= Detool = 0

Ttool = TmBB, e

mBB=1

ε

tool = εmBB*

(Stool,1 - S

tool,2) / (SmBB,1 – SmBB,2 ) (16), (18)

De

mBB = Detool = 0

Ttool =TmBB= TTC , e

mBB ≠ 1

4.2 Emissivity Measurement Results

A total of five cutting inserts were measured, with resulting tool emissivity calculated based on 

the available data given in Table 8. Some of the insert numbers 9k and 8 included microblackbodies, whereas others did not, and insert numbers 9k, 1k, and 2k had measurements taken 

at two temperature values. Note that the row for Equation (16) gives the calculated microblackbody emissivity (εmBB) based on TTC, which is used to evaluate Equations (15) and (18) 

assuming εmBB ≠ 1. 
32

Table 8: Resulting emissivity values for five tungsten carbide cutting inserts

Insert Number : 9k 8 1k 2k 3

Thermocouple, TTC [°C]: 446 545 544 446 548 445 545 544

Equation (14) 0.689 0.655 0.741 0.799 0.771 0.850 0.800 0.773

Equation (15), εmBB = 1 0.808 0.814 0.850 * * * * *

Equation (16), (εmBB)

0.853 0.805 0.870 * * * * *

Equation (15) & (16), εmBB ≠ 1 0.689 0.655 0.739 * * * * *

Equation (17) 0.641 ** 0.760 0.782 **

Equation (18), εmBB = 1 0.817 ** * * **

Equation (16) & (18), εmBB ≠ 1 0.697 0.658 ** * * **

* No mBB available

** Only one temperature measured

As previously mentioned, it is assumed the micro-blackbody gives a more precise and accurate 

emissivity measurement than the thermocouple, and emissivity values calculated using Equations 

(15) and (18) are therefore more accurate since they rely on the micro-blackbody measurement. 

This is evidenced in Table 8. Emissivity values which used Equation (14) and relied on TTC

ranged from 0.655 to 0.850, whereas those that used Equation (18) and relied on the microblackbody ranged from 0.808 to 0.850. Though only one insert (9k) was measured at two 

temperatures and had a micro-blackbody, the assumptions required to use Equation (18) (the two 

temperature method) are assumed to be more accurate than Equation (15) since subtracting 

signal values may remove any erroneously incurred offset. From here forth in this document, an 

emissivity of 0.822 is assumed, being the mean of the four measurements made using Equations 

(15) and (18).

Though more accurate than the thermocouple, the micro-blackbodies are expensive and time 

consuming to fabricate. Placing the thermocouple nearer the micro-blackbody on the exposed 

face of the insert, and ensuring good thermal contact will ensure more comparable temperature 

values. Future efforts will test a new method of thermocouple placement alongside microblackbodies, and test correlation between emissivity measurements depending on their respective 

values. Given good correlation, it is hoped that better thermocouple placement may reduce the 

need for micro-blackbodies. 

5 TRUE TEMPERATURE MEASUREMENT 

Once the calibration, linearization, PSF, and emissivity of an object are determined, the true 

temperature of a measured object may be determined. The following steps are taken to obtain 

and prepare the measured signal, Smeas:
33

1.) Measure the object with the infrared camera using the same in-camera NUC used in all 

previous steps, same optics, integration time, etc. and collect camera signal in units [DL]. 

Obtain a 100 frame e-plate measurement immediately afterward.

2.) Take the frame average (μF) of the e-plate measurement and subtract it from each frame 

of the DL video pixel-by-pixel to obtain frames in units [ΔDL].

3.) Linearize each pixel of each frame using Equations (6.2)-(6.3) to obtain frames in units 

[a.u.].

4.) Deblur the image using the PSF by doing the following: 

a) Mirror the edge of each frame by 41 pixels (the width of the PSF, i.e., make the frame 

wider by 82 pixels) 

b) Deconvolve the PSF from the mirrored image using the blind deconvolution 

algorithm and the measured PSF as a seed value

c) Extract the un-windowed region from the deconvolved frame. This is the measured 

signal, Smeas.

Finally, the measurement equation is used to convert Smeas into true temperature, Ttrue:

( ) (1 ) ( ) Smeas F Ttrue F Tamb  e   e 

(19)

Here, F is the Sakuma-Hattori calibration function given in Equation (7.2). Tamb is temperature 

of the ambient surroundings reflected off the measured surface. It is typically taken as 20 °C, 

though for the calibration example in this document, and the relatively high emissivities, Tamb

accounts for less than 1 ΔDL signal and is therefore negligible for this example calibration. True 

temperature is evaluated by inverting Equation (19) on a pixel-by-pixel, frame by frame basis:











   





e

(1 e ) ( ) 1 meas amb

true

S F T

T F

(20)

Recall that F-1is calculated using Equation (7.3) for each pixel. Figure 18 shows the results of 

converting an example frame to true temperature using the measured emissivity value of 

ε = 0.822. The example is from an orthogonal cutting test using a tungsten-carbide grooving 

insert, oxidized in the thermal vise described in Section 4, and cutting a Ti-6Al-4V workpiece at 

90 m/min surface speed and 0.225 mm per revolution feed (or 0.225 mm depth of cut). Note the 

nonlinear conversion from signal to temperature due to the nonlinear calibration equation F.
34

Figure 18: Conversion of a linearized, PSF-corrected infrared frame (left) to a true temperature 

frame (right) taken during machining experiments with the MOTS camera. Field of view of the 

image is approximately 0.87 mm vertical by 1.09 mm horizontal.

6 DISCUSSION

At this point, all steps in the process to convert infrared measurements into true temperature 

measurements have been given. However, a video consisting of a series of true temperature 

images is not the final measurement. Once a true temperature frame or set of frames is created, 

then the real measurement result is based on however the measurand is defined. In addition, 

final calculation of a measurement uncertainty value would depend on how the measurand is 

defined.

For example, the measurand may be defined as the true, physical peak temperature on the 

imaged surface (e.g., the tool face in Figure 18). However, this value may not be directly 

measurable due to the size-of-source effects that include sub-scenel averaging and blurring due 

to PSF [1,2,23]. Sub-scenel averaging describes how a pixel value results from an integrated 

radiance over the scenel area, and any radiating feature smaller than the scenel is lost in this 

integration. The effect is greater for surface regions of high temperature or emissivity gradients, 

but may be neglected for very uniform surfaces. Figure 19 gives a simulated example of a hot 

spot whose temperature is reduced through sub-scenel averaging and PSF blurring. While the 

effect of PSF blur may be corrected using the method developed in Section 3, no method is 

known to correct for sub-scenel averaging.

Linearized Signal, Deconvolved PSF [a.u]

50 100 150 200 250 300

50

100

150

200

250 1000

2000

3000

4000

5000

6000 Smeas – linearized signal, PSF-corrected [a.u.] True Temperature (e = 0.822), [oC]

50 100 150 200 250 300

50

100

150

200

250 550

600

650

700

750

800 Ttrue – true temperature, e = 0.822 [oC]
35

Figure 19: Size of source effects on an example measurement where the source is smaller than a 

scenel. No method is known to correct for sub-scenel averaging.

Temperature error due to sub-scenel averaging increases with the local temperature and 

emissivity gradients (or more simply, the apparent temperature gradient). However, for 

apparently uniform regions, such as the peak temperatures on the tool face in Figure 18, it is 

unlikely that local hot or cold spots exist that are smaller than a scenel size. In addition, if 

apparent temperature is uniform, the effects of PSF blurring is minimized. However, it is still 

recommended to correct for PSF blur since a methodology is known. 

If the measurand is defined as the peak pixel value of the true temperature, no further steps are 

necessary beyond those mentioned in this document other than locating the peak pixel value. 

However, one might consider defining the measurand as the peak true temperature value of an 

average of N pixels, since uncertainty of an average is inversely proportional to the root of N. 

However, this also incurs greater error if the local apparent temperature near the peak is nonuniform. If the measurand is defined as the the peak pixel value of the true temperature on the 

object, and the object is moving, it may be necessary to include edge detection or object tracking 

algorithms to locate the object in each frame.

7 CONCLUSIONS

This document outlines the steps used to calibrate an infrared thermal camera and convert 

collected images into true temperature measurements. This process was developed based on 

previous work by the authors, and with the intent to reduce measurement uncertainty from 

known sources and more easily and accurately evaluate measurement uncertainty overall. The 

steps developed include: 1) linearization, 2) blackbody calibration, 3) PSF measurement,
36

4) emissivity measurement, and 5) object temperature measurement. While linearization, 

calibration and object temperature measurement processes (i.e., converting a measured signal to 

true temperature using an emissivity) are available from thermal camera manufacturers as 

internal operations of the camera or camera operating software, these algorithms are proprietary, 

therefore calculating a measurement uncertainty is difficult. Also, previous work by the authors 

determined that accurate emissivity measurement and PSF measurement were key to reducing 

overall measurement uncertainty. Therefore, new processes for PSF and emissivity 

measurement were developed, tested, and are disseminated in this document, with 

recommendations for future improvements.
37

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38

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A-1

APPENDIX A. NOTATION AND NOMENCLATURE FOR IMAGE 

STATISTICS

The measured signal data from the thermal camera is converted and stored as a 3-dimensional 

array. Here, the first indices (i, j) represent the pixels in each frame, and the third index k

represents a frame number. Statistics may be taken across a frame or along a pixel. The frame 

mean μF is the mean of the 256 pixels x 320 pixels in a single frame. During blackbody 

measurements, 100 frames are taken at a time. The pixel mean μP(i,j) is the average value of 

each pixel over the 100 frames, and results in a 256 pixel x 320 pixel array or frame. 

    

  

100

1

320

1

256

1

( , , )

100

1

( , , 1) ( , )

320

1

256

1

k

P

j i

F m S i j k m i j S i j k

(A1)

The frame mean of the pixel mean the gives a single value which is the approximate signal 

level at a calibration point:

 





320

1

256

1 256 320

( , )

( )

j i

P

F P

m i j

m m

(A2)

A measure of non-uniformity in a calibration frame is made by calculating the frame noise σF

which is the standard deviation of the 320x256 pixels in a single frame. A measure of the pixelto-pixel nonuniformity of a calibration frame is the frame noise of the pixel mean of a 

calibration frame:

   







320

1

256

1

2

256 320

( , ) ( ( , )) ( )

j i

P F P

F P

m i j m m i j

s m

(A3)

During a blackbody or e-plate measurement, each pixel experiences fluctuation from frame to 

frame over the 100-frame measurement. Taking the standard deviation of this fluctuation gives 

the pixel noise. 

 



 

100

1

2

( , , ) ( , )

100

1

( , )

k

P P s i j S i j k m i j

(A4)

Since each pixel has an associated pixel noise, a single-value measure of all the pixel noise is the 

frame mean of pixel noise

 





320

1

256

1 256 320

( , )

( )

j i

P

F P

s i j

m s

(A5)

A measure of how uniform the pixel noise is over the FPA is the frame noise of the pixel noise:
A-2

 







320

1

256

1

2

256 320

( ( , ) ( ( , )) ( )

j i

P F P

F p

s i j m s i j

s s

(A6)

Note that a calibration frame is the difference between a pixel mean of a blackbody measurement 

and pixel mean of an e-plate measurement:

( , ) ( , ) ( , )

, ,

S i j i j i j  m P bb  m P ep

(A7)

This is important to keep in mind for calculating the pixel-by-pixel calibration uncertainty which 

will require a sum of squares of the standard error of the means for the component of uncertainty 

due to electronic noise. However, in this document and in Table 1, statistics for the calibration 

frames are only meant as a representation of their mean values or a general measure of the FPA 

noise.
B-1

APPENDIX B. SPECTRAL RESPONSIVITY AND SAKUMA-HATTORI 

EQUATION RELATIONSHIPS

Saunders [11] developed a relationship between spectral responsivity of a detector and the 

expected coefficients A, B, and C of the form of the Sakuma-Hattori equation given in equation 

(7.2). Spectral responsivity of the detector is characterized by the central wavelength, λ0, and 

the variance, σ2of the spectral band. For a rectangular-shaped (or uniformly distributed) spectral 

responsivity, σ2= Δλ2/12, where Δλ is the bandwidth. Given a known spectral responsivity such 

as that of the MOTS camera in Figure 2, the central wavelength and variance are given as 

follows:







l

l l

l

l

l

w d

w d

0

(B1)









l

l l l

s

wld

( 0) d

2

(B2)

For sufficiently narrow bandwidths (σ/λ0 ≪ 1), these are related to the Sakuma-Hattori 

coefficients through the following:

c B

c A

2 12

2

0



l 

(B3)

2

2

2

2 2

( 12 )

2

c B

c A B



s 

(B4)

With these equations, either an equivalent spectral responsivity may be deduced from a 

calibration using the Sakuma-Hattori equation, or, a signal response to a blackbody may be 

deduced from a known responsivity:





























 

2

0 1 6

l

s

A l

(B3)

2

2

2















l

c s

B

(B4)





 

0

5

0

5

l

l

l

l a

d

w

C

(B5)

The coefficient a depends on the particular detector, electronics amplification, pixel size, optics 

geometry and transmission [8], similar to the coefficient α given in Equation (5) in Section 2.2.
C-1

APPENDIX C. CALIBRATION DATA STRUCTURE AND FUNCTIONS

All the linearization and calibration data and statistics results are stored in the ‘calibration 

structure’ in Matlab. This structure will be fed to other functions to calculate PSF, emissivity, or 

convert video files. For example, to access the ‘A’ Sakuma-Hattori coefficient for the (45,100) 

pixel, type: Cal_Mit_0550.SH.A(45,100).

Calibration Structure Description

Cal_Mit_0550 Calibration structure for the Mitutoyo lens, Spectragon 1640 filter, and 550 us integration time

.badxy [2x29] array of pad pixel pairs

.Tbb [12x1] vector of blackbody calibration temperatures (oC)

.e_bb Effective emissivity of the calibration blackbody

.Sbb_mean [12x1] vector of array averages of the blackbody image frames (DLs)

.frames [256x320x12] array containing the blackbody image frames (DLs with emissivity plate subtracted)

.responsvty [136x2] array of wavelength and repsonsivity pairs describing the camera normalized responsivity. 

.PSF [41x41] array containing the normalized PSF kernel

.Eta Linearization substructure

.a [256x320] coefficient array

.b [256x320] coefficient array

.c [256x320] coefficient array

.k [256x320] coefficient array

.r2 [256x320] array of fit statistic r2

.rmse [256x320] array of the root-mean-square error fit statistic

.function text string of the linearization function 

.frames_linearized [256x320x12] array containing the linearized blackbody image frames (arbitrary units)

.SH Sakuma-Hattori fit substructure

.A [256x320] coefficient array

.B [256x320] coefficient array

.C [256x320] coefficient array

.r2 [256x320] array of fit statistic r2

.SSE [256x320] array of the sum-square error fit statistic

.rmse [256x320] array of the root-mean-square error fit statistic

.function text string of the inverse Sakuma-Hattori function