Flatfield correction errors due tospectral mismatching
Nathan Hagen

Optical Engineering 53(12), 123107 (December 2014)

Flatfield correction errors due to spectral mismatching
Nathan Hagen*
University of Arizona, College of Optical Sciences, 1630 East University, Boulevard, Tucson, Arizona 85721, United States

Abstract. Flat field calibration of broadband imaging systems is widely used, and it has been said that users
should try to make the spectrum of the flatfield calibration light source as close as possible to that of the measurement object. However, a quantitative analysis of the error induced by a mismatch of calibration and object
spectra has been lacking. In order to develop this quantitative analysis, we provide a theoretical radiometricmodel for flatfield calibration and show how this spectral mismatching error arises. Simulations covering a variety
of measurement scenarios indicate that spectral mismatching can create quantitative errors of up to a factor of 5
in situations that are regularly encountered by researchers performing quantitative work. © 2014 Society of Photo-Optical
Instrumentation Engineers (SPIE) [DOI: 10.1117/1.OE.53.12.123107]

Keywords: detector arrays; radiometry; image quality; focal plane arrays; spectrometry.
Paper 141538 received Oct. 6, 2014; accepted for publication Nov. 4, 2014; published online Dec. 9, 2014.

1 Introduction
The goal of instrument calibration is to allow estimation of
the light entering the entrance pupil that is independent of theworkings of the instrument itself. Since detector arrays havesensitivities that vary from pixel to pixel and across different
wavelengths and optical elements and exhibit spatial andspectral variations in light collection and transmission, all
optical systems require calibration in order to achieve accurate measurements of light intensity. For broadband imaging
systems, the most common calibration corrections for intensity are “background subtraction” and “flat fielding.” Anexample of such a correction is shown in Fig. 1.
While flatfield calibration methods are widely used, little
has been shown about the theoretical basis for their use, theassumptions involved, and how the data are affected when
the assumptions are violated. A thorough look at how one
should do intensity calibration and how our assumptions
affect the accuracy of the results, is important to doing quantitative imaging work, such as in microscopy (such as formeasuring ion concentrations and membrane potentials),2
for astronomical photometry,3 hyperspectral imaging remotesensing,4 etc.
In the discussion below, we summarize the conventional
flatfielding correction and then present a detailed theoreticalmodel that exposes its underlying assumptions. From thetheoretical model, we derive several approaches for calibration and illustrate the errors that can arise when assumptions
inherent in the calibration model are not satisfied. In particular, we focus on the error produced when the spectrum of thecalibration light source differs from that of the object undermeasurement. At its root, this error arises from the fact thatwhile the detector may be linear in “at-sensor” photon flux, it
is not necessarily linear in “at-entrance-pupil” flux, and it is
the latter that we wish to estimate. For some wavelengths, thesystem sensivitity will be greater than at other wavelengths(due to differences in detector quantum efficiency, opticaltransmission, etc.), so that there is a nonlinear dependence
on the spectral distribution.
The analysis we present is limited to incoherent imaging,and so does not treat systems such as high-resolution spectroscopy, or those dealing with laser illumination, where
coherent imaging effects are important.5
2 Conventional Detector Array Calibration
Depending on a researcher’s background and the type of system being used, detector array calibration is often referred toas flat fielding,6–8 shading correction,9 gain/offset correction,10
nonuniformity correction,11,12 or intensity calibration.13 Eachof these refers to correction of the measured light intensity;
they do not calibrate for any blurring or multiplexing that
may occur.
In its most general form, a system measurement model isa mapping of an object’s light intensity distribution f (interms of the units of interest, such as radiance, irradiance,etc.) to an array of measurement data g, in digital counts,
given asg ¼ b þ Hf þ n;

where H is the “system matrix,” b is the background offset,
and n is the noise, with the noise defined as the difference
between g and its mean value: n ≡ g − ḡ. Here g, f, b, and n
are all vectorized representations of two-dimensional (2-D)
images. The overbar notation will be taken to mean a temporal mean (the average taken across a number of frames).
The system matrix maps elements of the object f n to pixels
in the image gm : H mn gives the sensitivity (or “responsivity”)
of pixel m to object element n. In general, the structure of
the system matrix H can be complicated, as it can include
effects such as blurring and scattered light, but a simple
model for an imaging system can take H to be diagonal(i.e., no blurring or scatter). In this case, it is clearer towrite it as a vector h, so that the model can be simplified to
g ¼ b þ hf þ n;

*Address all correspondence to: Nathan Hagen, E-mail: nhagen@optics.arizona.edu

Optical Engineering

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0091-3286/2014/$25.00 © 2014 SPIE

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December 2014 • Vol. 53(12)Hagen: Flatfield correction errors due to spectral mismatching

Fig. 1 An example correction of spatial variation in sensitivity across an image: (a) raw image, (b) corrected image. The images show a microscope image of a cell sample slide, taking by an IMS imaging
spectrometer,1 which uses flatfielding at each wavelength to remove sensitivity variations that produce
the line artifacts in the image shown at left. Note that the blurry dark spots are due to dust on the objective
lens or illumination lens and are calibrated out by the flatfielding process.

where it is now assumed that g and f have the same dimensions, and hf indicates an element-by-element multiplicationof h with f. The distinction between H and h is an important
one to keep in mind, because any spatial correlations amongneighboring pixels induced in the image (for example, due toblurring) will not be properly compensated for when using
the simpler model with h than the more general modelwith H.
Since all of the operations in Eq. (2) are element-byelement, we can re-express Eq. (2) in 2-D pixel coordinates
explicitly as

In textbooks and much of the existing literature, Eq. (3)may be found written in the equivalent form2,7–9,17–19
g−b X
ðg − bÞ;f^ ¼
gFF − b x;y FF(4)where gFF is an image of a “flatfield object,” and b is animage taken with no light source (e.g., with the shutter
closed). Comparing with Eq. (3), we see that the textbook
approach defines

gij ¼ bij þ hij f ij þ nij
for row i and column j in the image. In the literature, b isoften referred to as the “background,” “bias,” “offset,” or
DSNU (dark signal nonuniformity), while h is often referred
to as the “gain,” “responsivity,” fixed pattern noise (FPN),14
or photoresponse nonuniformity (PRNU).15,16 Taking themean of both sides of Eq. (2) gives ḡ ¼ b̄ þ hf̄. Thus, anestimate of the object is readily obtained by inverting thisequation:f¼

ḡ − b̄g−b→ f^ ¼;hh(3)where the hat over a variable indicates an estimate taken froma single realization, and the division is taken element-byelement. Thus, correcting a measurement image for detector
nonuniformity amounts to subtracting from it an offset image
and then dividing by a gain image.
Optical Engineeringg −b
h ¼ P FF:ðgFF − bÞx;yOther references that use a form equivalent to the above
include Ref. 6 (p. 82), Ref. 20 (p. 111), and Ref. 21 (p. 307
and 326).
3 Derivation of the Calibration Equations
In order to see the underlying assumptions involved in calibrating a sensor using the conventional flat-field correction,4
we derive the calibration equations from a general systemmodel. As in the previous section, the measurement imageat pixel ði; jÞ is represented as gij , and has units of digitalcounts (counts or ADUs). Each detected image is a sum ofbias, dark current, read noise, and a signal term:

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December 2014 • Vol. 53(12)Hagen: Flatfield correction errors due to spectral mismatching

gij ¼

Z

nrij þ ðt∕γÞdij þ bij|ﬄﬄﬄﬄ{zﬄﬄﬄﬄ} |{z}|{z}read noise

dark current

Λ̄ij ¼Ztηij ðλÞ τij ðλÞ Lij ðλÞ Aij ðλÞ Ωij ðλÞ dλ;þ
γ|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}(5)

signal term

where t is the integration time of the measurement, γ is thedetector translation constant (the detector “analog gain”
times the analog-to-digital conversion factor) in photoelectrons per count, b is the detector bias level, η is the detector’sinternal quantum efficiency (QE), and τ is the optical transmission of the system. Together, ητ can be considered thesystem’s “external quantum efficiency.” Informally, its units
can be described as detected photoelectrons per photon incident on the entrance pupil. Next, L is the spectral radiance (inunits of Wm−2 sr−1 μm−1 or photons s−1 m−2 sr−1 μm−1 ), A isthe area (units of m2 ), and Ω is the solid angle (units ofsteradians). Because their interpretation requires some care,
Appendix A provides a more complete discussion of L, A,
and Ω. The final measurement g is in digital counts andis proportional to the number photons incident on the
pixel during the integration period.
In Eq. (5), gij is what we “measure,” while the radiance
Lij is what we want to know about the scene. Writing themeasurement equation in terms of mean values givesZtt
ηij ðλÞ τij ðλÞ L̄ij ðλÞ Aij ðλÞ Ωij ðλÞ dλ:ḡij ¼ ½bij þ d̄ij  þγγ|ﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}
offset term

signal term

(6)
As earlier, the overbar indicates taking a temporal mean (theaverage taken across a number of frames).
While Eqs. (5) and (6) provide a general system model (or“forward model”) for an imaging system, it is common to
make a few assumptions in order to simplify the calibration
procedure. In many measurement situations, the absolute
radiometric value is not needed, so that what we want toknow is not the radiance Lij , but only the radiance to within
an unimportant scalar constant. This is simply the difference
between a “radiometric” and a “nonradiometric” calibration.
In order to handle both cases, we rewrite the radiance as
Lij ¼ sM ij I ij ðλÞ, where s is a radiometric scale factor carrying units of photon radiance (photons s−1 m−2 sr−1 ). The variable M is a unitless quantity giving the spatial distribution oflight across the scene, and I ij ðλÞ is the normalized spectral
intensity, carrying units of μm−1 , where the normalization
used is ∫ IðλÞdλ ¼ 1. Defining s this way, we can say more
precisely that s represents the peak spectral radiance value
within the brightest pixel in the image. Thus, for a radiometric measurement, the thing we seek to estimate is sM̄ij ,whereas for a nonradiometric measurement, we seek onlyM̄ij .
Substituting the components of Lij ðλÞ into Eq. (6), we
obtainttsḡij ¼ ½bij þ d̄ij  þ M̄ij Λ̄ijγγ
for
Optical Engineering

ηij ðλÞ τij ðλÞ Ī ij ðλÞ Aij ðλÞ Ωij ðλÞ dλ:

bias

(7)

Here, ½bij þ γt d̄ij  is an additive offset that varies from pixel
to pixel, and the spectral integral Λij is a pixel-dependent
gain factor—what is often called the “gain nonuniformity”
across the detector array.
Together with bij and d̄ij , Λ̄ij is one of the threepixel-dependent parameters we need to determine duringcalibration so that their influence can be removed fromthe measurement equation in order to determine M̄ij . We
use a subscript “cal” to mark quantities that vary with thecalibration setup; if the quantity is already subscripted,ðcalÞthen we use a superscript, as in ḡij . Likewise, for any quantities that depend on the setup for a measurement frame(sometimes called a “science” or “data” frame), we use subscript/superscript “meas.” Using Eq. (7), we can solve for the
spectral integral Λ̄ij during calibration in terms of the measured image and other parameters as
h
i
ðcalÞ
d̄
ḡij − bij þ γtcal
ij
cal
ðcalÞ
Λ̄ij ¼
:
(8)
ðcalÞ
tcal
s
M̄
cal
ij
γ cal
For a “measurement” frame, we havettðmeasÞ
ðmeasÞ ðmeasÞ
¼ bij þ meas d̄ij þ meas smeas M̄ ij Λ̄ij :ḡij
γ measγ meas(9)

The spectrum I ij ðλÞ at pixel ði; jÞ, and therefore the specðmeasÞ
tral integral Λ̄ijof the object under measurement is often
not something that we will know a priori. However, if we
can assume that the object spectrum is the same as that ofðmeasÞ
the calibration source, then we can replace Λij
with
ðcalÞ
Λij . Therefore, if we writeðmeasÞ

Λ̄ij

ðcalÞ

¼ Λ̄ij ϵ−1
ij ;then ϵ−1
ij defines the error factor that results from ourassumption of matched spectra:
ðcalÞ

ϵ−1
i;j ¼

Λ̄ij

R

¼R
ðmeasÞ

Λ̄ij

ðcalÞ

ηij ðλÞ τij ðλÞ Ī ij ðλÞ Aij ðλÞ Ωij ðλÞ dλðmeasÞ

ηij ðλÞ τij ðλÞ Ī ij

ðλÞ Aij ðλÞ Ωij ðλÞ dλ:The error factor will differ from 1 at pixel ði; jÞ whenðcalÞðmeasÞ
I ij ðλÞ ≠ I ij ðλÞ. Next we use the definition (8) for
the spectral integral, and solve Eq. (9) for the pixel radiance,
givingγtðmeasÞ
ðcalÞsmeas M̄ij
¼ scal M̄ij ϵij meas cal
γ cal tmeas0hi1ðmeasÞ
ḡij− bij þ γtmeas
d̄
ij
meas
B
h
i C
×@
A.
ðcalÞ
tcal
ḡij − bij þ γcal d̄ij
Finally, in order to represent a single calibrated measurement frame rather than a temporal mean, we drop the overbar
from M and g. Thus, the only parameters left that use theoverbar notation are the parameters obtained during calibration. Writing the result in vector form gives

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December 2014 • Vol. 53(12)Hagen: Flatfield correction errors due to spectral mismatchingγtsM ¼ scal M̄cal ϵ meas cal
γ cal tmeas



i1
h
d̄
gmeas − b þ γtmeas
meas
B
i C
h
×@
A:
tcal
ḡcal − b þ γcal d̄0(10)This is the most general expression providing the relationshipbetween the raw measurement image gmeas and the scene radiance distribution sM—a relation derived from the multiplication of four factors. The first factor corrects for spatial shape in
the calibration source. If the calibration uses a spatially uniform source, then M̄cal ¼ 1. The second factor is the error
introduced by any difference between the normalized spectrum of the calibration source and that of the measurement
object. The third factor scales the result to accommodate
differences in integration times and detector gain between
the calibration and measurement data. Finally, the fourth factor, consisting of the ratio of background-subtracted raw measurement data to that of raw calibration data, is the conventional
flat field correction [Eq. (4)].
While Eq. (10) provides a means of estimating the scene
radiance distribution, some users will prefer to estimate theradiant exitance, or possibly just the power, across each pixelin the image. For these quantities, users can simply estimatesM and then multiply the result by Ω to obtain the radiantexitance, or by AΩ to obtain the power. In addition, if theuser is not interested in an absolute radiometric calibration,then any radiometric scale factors s and scal in the equations
can be set to 1.
In deriving the calibration equations, we avoid describing
the result as “correcting” the raw measurement image, as it ismore accurate to describe the process as estimating the scene
light distribution—whether that is in terms of radiance, irradiance, or power. Also note that there is a difficulty in theestimation Eq. (10) when the measurement source has nearlyzero light, as it may, for example, in nonstar pixels ofðmeasÞ
will be aan astronomical image. In such a case, M ij
small number, therefore, the value of ϵij will be unstable.
ðmeasÞ
is small, it is probably best to take ϵ ¼ 1.
Thus, when M ij
4 Calibration Procedures
Equation (10) provides the most general form for estimating
the source spatial radiance distribution from a raw image.
For brevity, from this point on we will assume that any
unlabelled parameters correspond to measurement data andnot calibration data, except for those quantities such as
bij and dij that are unambiguously calibration parameters.
With this, Eq. (10) becomes10g − ½b þ γt d̄γtcal^ ¼ scal M̄cal ϵ
i A:
h
sM
×@
γ cal tḡ − b þ tcal d̄
cal

γ cal

(11)
The next step is to provide a procedure for estimating all of
the unknowns in this equation so that a corrected image canbe obtained from any raw measurement frame g.
Solving Eq. (11) requires knowing several parameters thatfor many users might not necessarily be of interest, so we
also show how to simplify the calibration procedure for thesecases. For example, in conventional flat-fielding, it is assumed
that the detector translation factor γ and the integration time t
Optical Engineering

used for the calibration are the same as those used for themeasurement. Of course, while these assumptions simplify
the calibration procedure, they also introduce potential
sources of error. Thus, after a discussion of the calibration
procedure, Secs. 5 and 7 below derive expressions for theerrors produced when various assumptions fail to hold.
4.1 Calibration with Fixed Camera Parameters
If the user fixes the integration time and detector gain so thatt and γ are always the same during calibration as they are
during a measurement frame, then the offset term ½b þ γt d̄can be collected into a single background offset image,which we can write as ō. And since they are fixed scalars inthis case, γ and t can be folded into the radiometric scale
factor s, so that the measurement Eq. (9) simplifies to
g ¼ ō þ sΛM þ n:(12)

Inserting the definition for ō into Eq. (11), we obtain theequation for estimating the scene radiance asg − ō
^sM ¼ scal M̄cal ϵ;(13)ḡcal − ō
When we allow s ¼ 1 (perform nonradiometric correction)and ϵ ¼ 1(calibration and measurement spectra identical),this is the equivalent to the conventional correction approach(Sec. 2).
So applying the correction to the raw measurement g
requires measuring offset ō and flatfield image ḡcal fromthe calibration, in addition to the radiometric scale factor
scal if a radiometric calibration is wanted. The widely used
procedure for obtaining these is as follows:
1. Estimate offset ō: As the first step in calibration, weclose the lens shutter or turn off the light source so thats ¼ 0. Inserting this into the measurement Eq. (12),
we are left with ḡ ¼ ō (ignoring noise). That is, if
we average the raw measurement image over many
frames, we have good SNR estimates for the background image.
2. Estimate ḡcal : Next, we turn on a flatfield source forcalibrating the nonuniformity factor, so that M̄cal ¼ 1.
Averaging the image obtained over many frames,
gives a good SNR estimate of ḡcal .
3. Estimate scal : If performing a radiometric calibration,
see Appendix C for a procedure to estimate scal .
Using the results from steps #1–3 above, we can then substitute M̄cal ¼ 1 (i.e., flatfield object) and estimates for theoffset ō, calibration image ḡcal , and radiometric scale factor
scal into Eq. (13) to obtain the estimated scene radiancedistribution.
4.2 Calibration with Camera Parameters Allowed
to Vary
The most common use case for imaging data correction
allows the camera parameters to vary—typically by varying
the integration time, and less often by adjusting the detector
gain. In this case, we cannot write the background offset
as a single variable, since the dark current makes the offset
vary with integration time. However, we can fold the fixed

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December 2014 • Vol. 53(12)Hagen: Flatfield correction errors due to spectral mismatchingdetector gain into the dark current by simply defining γ ¼ 1,
so that the dark current term is now in units of counts/s rather
than electrons/s, and the units of s also likewise change. So,
we will write out the most general form of the measurement
equation, and users that fix γ can insert γ ¼ 1 everywhere tosimplify:
g ¼ ½b þ ðt∕γÞd̄ þts
ΛM þ n;γ

slope of linear variation with integration time provides
a direct estimate of the gain term.
For this setup, we can use the following calibration
procedure:
1. Estimate offset and gain: Turn on the flatfield source
and collect images over a sequence of different t values. For each pixel, fit a line to the resulting data toestimate the offset and gain values:(14)

offset ¼ b;

and the corresponding equation for estimating the scene radiance from a given measurement g is given by Eq. (11).
A typical user will assume ϵ ¼ 1 and γ cal ¼ γ ¼ 1, so that
correcting the raw image requires estimating b, d̄, ḡcal ,and scal from the calibration (with the latter two parameters
located inside Λ), while also recording the values for t andtcal . The procedure for obtaining these is as follows:
1. Estimate offset: We close the lens shutter or turn off
the light source so that s ¼ 0. Inserting this into themeasurement Eq. (14), we are left with ḡ ¼ b þ γt d̄.
That is, if we measure the image obtained over a seriesof different values for t, we can fit a line to the resultand estimate the intercept value b and the slope value
1γ d̄ at each pixel.
2. Estimate gain: Next, we turn on a flatfield source forcalibrating the nonuniformity factor, so that M̄cal ¼ 1.
This leaves the measurement Eq. (14) as ḡcal ¼
b þ γt d̄ þ γt scal Λ̄. For a nonradiometric calibration,
we can set scal ¼ 1. Therefore, if we subtract the offset½b þ γt d̄, obtained from step #1, from each image, andaverage the result over many frames, we have a good
SNR estimate of the denominator in Eq. (11).
3. Estimate γ: If allowing γ to vary, or if performing aradiometric calibration, see Appendix B for methods
to estimate γ.
4. Estimate s: If performing a radiometric calibration,
see Appendix C for a procedure to estimate scal .
With the results from these steps, we now have all of theparameters needed to substitute into Eq. (11) and estimate
the scene radiance distribution sM. Note that this resultshows that while the user needs to collect a series of measurements at different integration times of the dark signal,
one need only take a single measurement with the lightsource on.
4.3 Calibrating Without a Dark Signature
An alternative approach to calibration is to measure the flat
field scene at various integration times and then fit a line to
the result at each pixel to get an offset and a gain. This canwork, but only if the dark current is small enough to neglect.
For this approach, we first invert the measurement Eq. (10) to
express the raw data in terms of M and the calibration parameters t, s, and γ:
g ¼ ½b þ ðt∕γÞd̄ þts
ΛM þ nγ≈bþtsΛM þ n:γ

Thus, if d̄ is small, the offset term is independent of t. And
if the calibration source is spatially flat (M̄ ¼ 1), then the
Optical Engineeringgain ¼ ðscal ∕γ cal ÞΛ̄cal

2. Estimate γ and s: Because these parameters areembedded inside the gain factor, it is only necessary
to estimate γ cal and scal separately if a radiometric calibration is desired. If so, see Appendices B and C for
estimation methods.
With the results from these steps, we now have all of theparameters needed to substitute into Eq. (11) and estimate
the scene radiance distribution sM.
5 Errors from Mismatched Spectra
Many discussions in the literature mention that the flat fieldcalibration image must have the same spectrum as the measurement object (e.g., Ref. 21, p. 326; Ref. 22), but fail tomention why and do not discuss the errors that result
when the assumption is violated. While Refs. 23 and 24 discuss how the difference in spectra can cause radiometricerror in the context of astronomical systems, they do not provide a quantitative model in the context of flat-fielding anddo not separate the different factors inside the system sensitivity that allows users to assess how much error they might
expect in a given measurement configuration, and how best
to mitigate the errors. In the calibration equations derived inSec. 3 above, we provide a way of quantitative modeling ofthese “spectral mismatching errors.”
The effects of any change in the spectrum at pixel ði; jÞbetween the calibration and measurement frame are contained within the spectral mismatch error term ϵij , which
we reproduce here:R
ϵij ¼ RðcalÞηij ðλÞ τij ðλÞ Ī ij ðλÞ Aij ðλÞ Ωij ðλÞ dλðmeasÞ

ηij ðλÞ τij ðλÞ Ī ij

ðλÞ Aij ðλÞ Ωij ðλÞ dλ

:

(15)

From this equation, we can see that the error ϵ can be considered as the ratio of two weighted averages, where the norðcalÞ
ðmeasÞ
malized source spectra Ī ij ðλÞ and Ī ij ðλÞ are theweights, and the quantity to be averaged is the overall systemsensitivity, given as the product of factors in the integral:
sensitivityðλÞ ¼ ηij ðλÞ τij ðλÞ Aij ðλÞ Ωij ðλÞ≈ AΩηij ðλÞ τij ðλÞ:

(16)

The approximation used in the last step here is a good one formost systems because ηðλÞ τðλÞ generally contains the majority of the spectral dependence and not the spectral variationin area AðλÞ or solid angle ΩðλÞ.
In order to see how nonunity values of ϵ are produced, we
can consider an example using simulated data, shown in
Fig. 2. Here the calibration spectrum uses a narrowband

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December 2014 • Vol. 53(12)Hagen: Flatfield correction errors due to spectral mismatchingsource (such as an LED) centered at 850 nm, while the measured object has a spectrum centered at 650 nm. The simulated detector quantum efficiency spectrum ηðλÞ and opticaltransmission spectrum τðλÞ are piecewise constant, so thatthe integral can be calculated as a simple sum. The entire
spectral range is from 400 to 900 nm, outside of whichτðλÞ ¼ 0. We also assume that A and Ω are wavelength-independent, so that they may be pulled out of the integral.
And since they occur in both the numerator and denominator,
they cancel out, so that ϵ becomes independent of them.
Using this approximation, and from Fig. 2 and Eq. (15),we therefore haveR
ϵ¼ RðmeasÞ

ηðλÞ τðλÞ I ijðλÞ dλðcalÞηðλÞ τðλÞ I ij ðλÞ dλ

:

(17)

are such as might be used in, for example, fluorescencemicroscopy). Figure 4 presents the quantum efficiency
curves for three different detector arrays: a Truesense CCD,
an sCMOS sensor, and a deep-depletion CCD. Figure 5
presents six different light source spectra as examples ofwhat may be used for calibration light sources. Figure 6
shows example spectra of six different object spectra: direct
sunlight, two different fluorescent dyes, green paint illuminated by sunlight, and two models of a quartz-tungstenhalogen lamp. Thus, selecting a curve from each of thefour figures provides each of the four inputs necessary forðcalÞðmeasÞ
calculating ϵ in (17): τðλÞ, ηðλÞ, I ij ðλÞ, and I ij ðλÞ.
Calculating the resulting ϵ for all of the 882 combinations
provided from these 23 different curves, we find that the largest errors occur for the following combinations:
1. Measuring the 3000 K graybody halogen lamp model
after calibrating with a white LED, metal halide, ora fluorescent light source produces ϵ ∼ 0.18.
2. Measuring either of the fluorescent dyes Alexa or
DAPI after calibrating with the 4500 K graybody
halogen lamp model produces ϵ ∼ 4.2.

We split the integral into five wavelength ranges: λ1 –λ2 ,λ2 –λ3 , λ3 –λ4 , λ4 –λ5 , and λ5 –λ6 . The integral over most of
the ranges will be zero due to the narrowness of the calibration and measurement spectra, so that the above equation
becomes0 þ 0 þ ð0.9Þð1.0Þð1.0Þð0.1Þ þ 0 þ 0
0 þ 0 þ 0 þ 0 þ ð0.5Þð0.25Þð1.0Þð0.1Þ
0.09¼ 7.20:¼
0.0125ϵ¼

So the large difference in the quantum efficiency and opticaltransmission in the spectral ranges of the calibration andmeasurement spectra translates into a large error in the flatfield correction: a factor of 7.2. Not only is this a radiometricerror, but if one region of the image has a different spectrum
than another, it will also induce a spatial uniformity error in^ because some pixels in an image will be incorrectlyMweighted relative to others.
In most cases, the spectral mismatch is not nearly as dramatic as that shown in the example, so we next provide somemore realistic simulations. Figures 3–6 show various inputs
to calculating the spectral mismatching error ϵ. Figure 3
presents six different optical transmission curves for broadband lenses and narrowband filters (the narrowband filters

Fig. 2 Simulated quantum efficiency and optical transmission curves,
together with example calibration and measurement spectra. Herethe calibration spectrum uses a narrowband source I cal centered at
850 nm, while the measured object has a spectrum I meas centeredat 650 nm. The simulated detector quantum efficiency spectrumηðλÞ and optical transmission spectrum τðλÞ. All four spectra are piecewise constant, so that their integral can be calculated as a simple sum.

Optical Engineering

These cases use the quantum efficiency curve of theTruesense KAI-16000 sensor and a broadband antireflection
coating as the optical transmission model. The reason is thatsince the KAI sensor has a much higher quantum efficiency
in the blue–green region of the spectrum than in the red–NIR
region, calibrating with a light source spectrum that weighs
one region much differently than the object spectrum does
produces substantial error. When using the sCMOS ordeep depletion CCD quantum efficiency curves, the spectral
mismatching error is generally half as large or less. For
example, for the case of measuring the 3000 K graybody halogen lamp model after calibrating with a white LED, metal
halide, or a fluorescent light source with these latter detector
arrays produce errors over a factor of 2 (i.e., ϵ ¼ 0.4 ∼ 0.5).
If we calculate ϵ for every permutation of the above
spectra (quantum efficiency, optical transmission, and thenormalized calibration and measurement spectra), then theFig. 3 Six example optical transmission spectra: Nikon represents
a measured transmission function for a Nikon 55 mm lens,25 BBAR
the transmission appropriate to a 4-layer broadband AR coating on
standard glass,26 BBP a broad bandpass filter transmission spectrum
designed to pass only the visible spectral range 400–700 nm,27 and
NBP4, NBP5, and NBP6 narrow bandpass filter transmission spectra
designed to pass 400–450 nm, 525–575 nm, and 635–685 nm,
respectively.28 The ideal filter (100% transmission everywhere acrossthe passband) is used in the simulations but not shown here.

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December 2014 • Vol. 53(12)Hagen: Flatfield correction errors due to spectral mismatching

Fig. 4 Three example detector quantum efficiency curves: KAI represents a Truesense Imaging KAI-16000 sensor,29 sCMOS a scientific CMOS sensor,30 and DeepCCD a deep-depletion CCD.31

mean difference from ϵ ¼ 1 among all 882 combinations is0.267, and the median difference is 0.073. Thus, we can saythat errors of ∼7% are typical, and those as high as ∼25% arecommon.
6 Color Cameras and Spectral Imagers
Color cameras and spectral imagers have an additional measurement dimension beyond that of broadband imagers—thecolor channel or the wavelength channel—which we
index by k. For these instruments, we incorporate the k’thchannel transmission passband ck ðλÞ into the measurement
model (6) astḡijk ¼ ½bijk þ d̄ijk γZtck ðλÞ ηij ðλÞ τij ðλÞ L̄ij ðλÞ Aij ðλÞ Ωij ðλÞ dλ: (18)þγ
so that the measurement is now indexed as gijk . Example
transmission curves for ck ðλÞ for a typical filtered RGB
color camera are shown in Fig. 7. For an imaging spectrometer, the ck ðλÞ take on the form of rectangle functions spanning the wavelength range of the channel. Thus, for example,
a spectral imager with eight spectral channels spanningthe 300–900 nm spectral range will have ck ðλÞ defined as

Fig. 5 Six example calibration source spectra: QTH shows the spectrum of a quartz-tungsten-halogen lamp,32 Metal halide the spectrumof a metal halide lamp,33 Deuterium the spectrum of a Deuterium
lamp,32 Fluorescent bulb the spectrum of a fluorescent bulb,34White LED the spectrum of a white-light LED35, and W4500 the spectrum of a graybody object at 4500K temperature.

Optical Engineering

Fig. 6 Six example measurement spectra: Sun shows the Sun as
seen through 1.5 atmospheric air masses,36 Alexa shows the emission spectrum of the fluorescent dye Alexa488,37 DAPI the emission
spectrum of the fluorescent dye DAPI,37 Green the spectrum of green
paint as illuminated with solar light [obtained from the ASTER spectral
library],38 QTH the spectrum of a quartz-tungsten-halogen lamp,32 andW3000 the spectrum of a graybody object at 3000K temperature.

rectangle functions 75 nm wide, none of which overlap
one another. This example set is shown in Fig. 8.
For a color camera, the filter transmission curves shown in
Fig. 7 show profiles that are strongly peaked, with extended
spectral regions in which the transmission is low comparedto the peak. This kind of profile exacerbates the system’s
sensitivity to spectral mismatch, and so we can conclude thatfiltered RGB cameras will be vulnerable to large radiometricerrors when the object spectra poorly match that of the flatfield calibration spectrum. This loss of precision is a tradeoff
for attempting to match the complex optical processing of
the human eye.
If, on the other hand, the spectral channels are relatively
narrow, such as in a spectral imager but not a color camera,then the parameters inside the integral will be approximately
constant over each channel passband. In this case, the factors
inside the integral can be pulled out, simplifying the model to
Z
h
i t
t
ḡijk ¼ bijk þ d̄ijk þ ηijk L̄ijk Aijk Ωijk
ck ðλÞ dλ
γ
γ
h
i t
t
(19)
≈ bijk þ d̄ijk þ ηijk L̄ijk Aijk Ωijk ðΔλÞk :γγ
If we develop a flatfielding approach for this simplified
measurement model, we find that the error factor ϵ drops

Fig. 7 Typical transmission curves for a filtered RGB camera, where
the curves R, G, and B represent c r ðλÞ, c g ðλÞ, and c b ðλÞ. [Thecurves are adapted from http://blog.teledynedalsa.com/2012/11/
spectral-cognisance-3-when-black-turns-red/.]123107-7

December 2014 • Vol. 53(12)Hagen: Flatfield correction errors due to spectral mismatching

c0

c6

c5

c4

c3

c2

c1

c7

Fig. 8 An example set of channel passband functions c k ðλÞ fora spectral imager having 8 spectral channels across the 300–
900 nm wavelength range. Small gaps have been inserted between
the individual channels for clarity.

Fig. 9 Mean signal versus variance measurement, showing experimental data (dots) and the fitted line. For the example here, a line
slope of 4.1 gives an estimate for the translation constant ofγ ¼ 4.1 photoelectrons/count. [Figure obtained from Ref. 46.]

out—the difference between the object and calibration spectra no longer has an effect on the radiometric accuracy. Thus,
as the individual channel passbands shrink, we can also
expect that the radiometric error due to spectral mismatching
will shrink with them.
7 Other Calibration Errors
So far the analysis has focused on calibration errors inducedby mismatched calibration and measurement spectra. We
briefly mention other sources of calibration error for users tokeep in mind. For example, Secs. 4.1–4.3 used the flatfieldassumption, i.e., that M̄cal ¼ 1. If, however, the calibration
source is not spatially uniform, then a glance at Eq. (11)shows that using an incorrect flatfield assumption will resultin the spatial pattern of the calibration source being^
imprinted onto the measurement scene estimate M.
For many interline CCDs, the “reciprocity law” (the rule
asserting that doubling the exposure time is equivalent todoubling the scene brightness) generally holds for integration times above a few microseconds.39 However, there aresome CCDs (such as Kodak’s KAI-16000 sensor) for which
the readout channels are insufficiently shielded from incident
light, so that at shorter integration times and higher lightintensities, the data are corrupted by light leaking into thereadout channels. Accurately correcting for this effect is not
Optical Engineering

trivial, as it requires what is effectively a scene-dependentbias signature.
For shutter-coupled full-frame CCDs, in which the integration time is not only governed by the CCD internal clock
but also by the speed of the shutter opening and closing, theintegration time is often unstable at a scale of about 20 ms.24
At these short time periods, the action of the shutter is notrepeatable enough to reliably achieve an accurate radiometriccalibration.
The detector dark current hdij i and quantum efficiency ηijboth vary with detector temperature, so that a calibration performed while the camera is cool will have significant error if
used after the camera has warmed up significantly.40 For
CCD detector arrays, this is especially true for NIR light,
where the quantum efficiency can vary by as much as 1%
per °C.24
The calibration equations provided in Sec. 3, and the subsequent analysis, all assume that after subtraction of the offset signal the detector measurement is linear in the source
radiance. This is known to be only an approximation.41,42
Fortunately, researchers have found that the primary sourcesof detector nonlinearity are wavelength independent.16 Thismeans that any nonlinearity correction factor can be placedoutside the spectral integral Λij used in the system modeling
equations, making it easier to correct as a postprocessing
step.
Another nonlinearity in the system is that, in general, pixels do not have uniform sensitivity across their measurement
region.43 As long as the spatial distribution of image intensity
varies slowly relative to the spatial variation in pixel sensitivity, these effects will be small and can be ignored.
8 Conclusion
Demonstrating the calibration errors induced by spectral
mismatching requires either a careful radiometric procedureor (if performing a nonradiometric calibration) a scene inwhich the correlations between pixel brightness and pixel
spectral differences are poor. For example, if a flatfield-calibrated was used to image an urban scene, then we may findthat the green trees are subject to a −10% spectral mismatching error, while the red brick buildings may be subject
to a þ30% error, and the blue sky a þ20% error. But in theimage, a user would probably notice nothing out of the ordinary because the trees would simply look slightly darker thanthey should, and the buildings and sky slightly brighter thanthey should. For most users, the tradeoff of this kind of error
for removing any systematic spatial sensitivity changes
across the image (such as shading due to cos 4 effects) is agood one. However, in the unusual case where the image isspatially somewhat uniform but covers a substantial change
in spectrum, such as a wide-angle view of a sunset catchingboth red and blue sky, the user may find shading patterns
induced as a result of spectral mismatching error.
Through simulations, however, we have demonstrated
that a poorly arranged measurement setup can induce spectral mismatching error factors in the flatfield calibration as
bad as a factor of 5× in over- or under-estimates, and commonly produce errors as large as 10 ∼ 25%. For precision
quantitative work, these are large errors, but the theoreticalmodel shows that the user can mitigate them through optimizing the choice of sensor, transmission of the optics, andchoice of calibration light sources if given knowledge of the123107-8

December 2014 • Vol. 53(12)Hagen: Flatfield correction errors due to spectral mismatchingtypical types of sample spectra that are expected to be seen
during measurements. Specifically, the mismatching error
can be minimized when each of the terms inside the error
integral—quantum efficiency ηðλÞ, optics transmission τðλÞ,
area AðλÞ, and solid angle ΩðλÞ—are all as close as possible
to uniform across the spectral range of detection. If thisalso produces errors that are too large, then working witha spectrally resolved instrument may be necessary.
Even for nonquantitative work, the spectral mismatchingcan produce substantial shading errors in images in which
the sample under measurement exhibits substantial spectralchanges across the field of view. This can produce artifacts
that may cause trouble for viewers and for algorithms used toanalyze images. If the highest precision (artifact-free imaging) is needed and the object spectrum is not known a priori,
then a carefully calibrated spectral imager instead of a broadband imager can be used to minimize the spectral mismatching error.
While this is the first time that an explicit model has been
provided for the spectral mismatching error, Lasarte et al.
have commented that in their calibration work, they havebeen able to produce a high-quality spatial nonuniformity
correction even when their calibration and measurement
spectra are substantially different.19 (This was done bychanging the current applied to their tungsten-halogen
light source, so that the hotter bulb would produce a bluer
spectrum.) However, the authors appear to have checked
their results using a scene in which the spectrum is uniform
for all pixels in the image, and since they were not performing a radiometric calibration, any resulting error in the estimated scene brightness would, therefore, not be noticeable.
The authors did not provide a test of their nonuniformity correction using a scene with a spatially varying spectrum, andso we can surmise that such a scene would have produceda shading error in their results.

Appendix A: Meaning of A and Ω
The precise definitions of the measurement model variables
A and Ω actually vary with the interpretation of the radiance
L. Since radiance is invariant to propagation, we can actually
use the measurement to estimate the radiance at any point
between the plane of the object and the image plane.
However, if the optical transmission τij ðλÞ is not everywhereequal to 1, then attempting to estimate the radiance at theimage plane amounts to ignoring the effects of the optics on
the amplitude of the measurement, resulting in radiometricerror. Thus, if we are to use a complete measurement modelthat includes the effects of τij ðλÞ, we must distinguishbetween radiance at the sensor plane and that at the object
plane or the system entrance pupil. And since it is only atplanes conjugate to the image and to the pupil where L is
easily evaluated, there are generally two ways of interpretingthe meaning of the parameters, as described below.
In the first case, L is taken to represent the radiance at theobject plane, so that Aij ðλÞ must take on the meaning of theeffective geometric area at the object plane that is imaged by
pixel ði; jÞ. This effective area may be spatially varying in thepresence of distortion, and may also have spectral variationin the presence of chromatic aberrations. The solid angle
Ωij ðλÞ takes on the meaning of the solid angle subtended by
the pupil of the optical system as seen by the point ði; jÞ on
Optical Engineering

the object plane. In this situation Ωij ðλÞ may vary withwavelength if the entrance pupil diameter or shape is wavelength-dependent, such as in the presence of strong pupilaberrations. For typical lenses these aberrations are small,so that the latter case Ωij can be taken out of the spectral
integral defining Λij . The solid angle often has significant
spatial variation due to the effect of the projected pupilarea decreasing with the field angle. Even in the presence ofideal optics, the projected pupil area varies with the cosine of
the field angle.
In the second case, we evaluate the radiance L at theentrance pupil. Here the area Aij ðλÞ takes on the meaning of
the area of the entrance pupil. This can vary with pixel location ði; jÞ and wavelength λ due to vignetting and pupilaberrations. The solid angle Ωij ðλÞ takes on the meaningof the range of angles whose light maps to pixel ði; jÞ.
Once again, Ω can vary with pixel location due to distortion,
and with wavelength due to chromatic aberrations.
In most treatments of the system measurement model inthe research literature, the area and solid angle are treated as
scalars, i.e., Aij ðλÞ ¼ A and Ωij ðλÞ ¼ Ω. While this is a reasonable approximation in many cases, assuming invariancewith location and wavelength can sometimes introduce
unwanted distortions into the model, particularly when imaging optics such as wide angle lenses or high-NA microscope objective lenses are being used. In addition, Ω andA can vary from frame to frame if the camera user changes
the detector binning or the focal length, such as with a zoom
lens.

Appendix B: Estimating the Detector
Translation Factor γ
The quantity γ is generally called either sensitivity20 orgain15,44 by different authors. This nomenclature can give
the wrong impression, however, since γ is simply a scale factor that has no relation to sensitivity (how well a system candetect small signals). Moreover, calling γ a “gain” makes it
easily confused with the detector analog gain and glosses
over the fact that while the inputs and outputs (photoelectrons and digital counts) may both be unitless, they do
not represent the same quantities. Thus, we name it the“translation factor,” as it translates units of photoelectrons
to units of digital counts.
Some users adjust the detector translation factor γ
between calibration and measurement, such as for camerasthat automatically adjust their internal analog gain value
in order to maximize dynamic range. For this case, and if
the user needs to perform a radiometric calibration, then it
is necessary to estimate γ cal . And if it cannot be obtainedby scaling γ cal , an independent estimate of γ meas may also
be necessary.
If the camera sensor and frame grabber are well-documented, then it may be possible to estimate γ simply bymultiplying the camera analog gain value (this is often visible in the software driving the camera settings) with a multiplier given in the documentation. If a documented multiplieris unavailable, then there are several ways of estimating γ
directly from data:

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1. One approach involves imaging a uniformly illuminated field with different illumination intensities.44
December 2014 • Vol. 53(12)Hagen: Flatfield correction errors due to spectral mismatching

At each illumination intensity level, perform the following steps:

factor s giving the brightness of the scene in absolute units.
As with estimating γ, there are a couple of ways to do this:(a) Acquire two sequential wide field images and crop
each of those images down to a 200 × 200 subimage, over which we are confident that shading
and other effects by the optics will have minimal
effect on the uniformity. The result is g1 and g2 .
(b) Calculate the mean intensity ȳ from the two
images: ȳ ¼ 12 meanðg1 þ g2 Þ.
(c) Calculate the variance across all of the pixels in the
difference image and divide the result by 2 toaccount for the increased noise due to the use of
two images: σ 2y ¼ 12 varðg1 − g2 Þ.

1. The easiest method for estimating scal is to image acalibrated light source whose spectral irradiance isalready known. This requires prior calibration (seestep #2 below), either by the user or by a service
that provides such calibrations. In the United States,
the latter are typically provided as NIST-traceable calibration.51 For a calibrated light source, either the
spectral exitance at the light source itself, or the spectral irradiance at a fixed location relative to the source
is provided, and so s is simply obtained as the integralof the provided curve over the spectral passband of thesystem: scal ¼ ∫ λλ21 Ecal ðλÞdλ, where EðλÞ is the spectral
irradiance or spectral exitance, depending on the quantity provided to characterize the light source. Thisvalue for scal can then be used directly in the calibration equations.
2. A second method for obtaining s involves using aradiometrically calibrated detector—often a photodiode. (For additional details, see Ref. 51.) In thiscase, the photodiode is placed at a convenient location
for measurement, such as an image plane, with a narrowband filter in order to block all light except in the
passband. Since the detector has been calibrated, theuser is provided with a responsivity curve showing theconversion of input optical watts to photodiode output
in volts, providing a direct link to absolute opticalunits. If the transmission spectra of the the narrowband
filter and the lens used to image the source onto thedetector are known, and the geometry of the measurement is well-characterized, then the user can remove
the effect of these elements by performing a division ateach spectral band. That is, the measurement modelfor this setup isZ(20)
Pk ¼ ΦðλÞ RðλÞ τk ðλÞ τlens ðλÞ dλ;Once the mean intensity ȳ and variance σ 2y values
have been calculated for a series of illumination
intensities, plot ȳ versus σ 2y and fit a line to the result,as shown in Fig. 9. The slope of the line gives theestimated γ value. If the dark current is negligible,
then it is possible to substitute a change in the integration time for a change in illumination (the “reciprocity law”),39 with the caveat that cameras using
shutters to change the integration time can only doso accurately when the integration time exceeds
about a second.45
2. Another approach is provided by Janesick, who gives
an equation for the translation constant γ as [Ref. 20,
p. 68; Ref. 47]γ¼ḡcal − ḡdark;
σ 2cal − σ 2dark

where ḡcal and σ 2cal are the average number of counts
and variance in the image while viewing a uniformly
illuminated scene, and ḡdark and σ 2dark are the average
number of counts while viewing without illumination.
The means and variances can be calculated either as instep #1 above (taking the variance across the pixels ina subimage) or by collecting a large number of images
and calculating the variances of each pixel individually
across the set of frames, and finally averaging the variances of the pixels in the image to give the result.
Note that, in both of the above procedures for estimating
γ, if the user calculates the signal variances using a spatial
ensemble of pixels rather than a temporal ensemble offrames, then the user is making an implicit assumption thatthe spatial variation in the image is much smaller than thenoise.

where Φ is the optical power at the photodiode, RðλÞ is the
wavelength-dependent responsivity of the detector (thedetector parameter that is known a priori) in units of volts
per optical watt, τk ðλÞ is the transmission spectrum of thefilter for spectral channel k, τlens ðλÞ is the transmission spectrum of the imaging lens, and P is the photodiode output involts. If the spectral channels used for the calibration are narrow enough, then we can approximate the integral with themean value for each of the functions in Eq. (20), multipliedby the width of the passband:Pk ≈ hΦik hRðλÞik hτk ðλÞik hτlens ðλÞik Δλk ;Appendix C: Estimating Radiometric Scale
Factor s
There are a number of imaging applications for which users
require a “radiometric” calibration, such as (a) when comparing the SNRs of different instruments,48,49 (b) comparing
objects measured at different times and/or under different
illuminations,4 (c) trying to estimate an object’s property
that is linked to the emitted radiance,50 etc. For these situations, it becomes necessary to estimate the radiometric scale
Optical Engineering

where h·ik indicates taking the mean value over the passbandof filter k, whose width is given by Δλk. This is now easily
inverted to givehΦik ≈Pk;
hRik hτk ik hτlens ik

so that now we have an estimate of the mean optical power Φ
in spectral channel k incident on the photodiode, assuming

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December 2014 • Vol. 53(12)Hagen: Flatfield correction errors due to spectral mismatchingthat we know hτk ik and hτlens ik . With this measure of opticalpower at the photodiode, one can then use the measurement
geometry to determine the corresponding exitance at the lightsource. For example, the camera equation can be used to estimate the radiance of a source given the first-order propertiesof the lens.52
Once this last result is obtained, the user can follow theprocedure in step #1 to estimate the radiometric scale factor
scal of the calibration source.
The large number of steps and additional equipment
needed to perform an estimation of s (especially for step
#2 above) is one of the reasons why radiometric calibration
is avoided unless necessary. One factor to keep in mind isthat sometimes the size of the calibrated photodiode is notenough to capture the signal from the light source. In thissituation, it is probably best to use a different setup suchthat the calibrated photodiode can be used to calibrate a different, larger detector, such as a large-format detector array.
One can then use the larger detector as a substitute for thephotodiode and continue the calibration as before, albeit one
more step removed from the original calibration reference.
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Biographies of the authors are not available.

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