Camera Spectral Sensitivity Estimation from a Single Image under Unknown
Illumination by using Fluorescence
Shuai Han1∗ Yasuyuki Matsushita2 Imari Sato3 Takahiro Okabe1
1
The University of Tokyo, Japan
2
Microsoft Research Asia, China
3
National Institute of Informatics, Japan
Abstract
Camera spectral sensitivity plays an important role forvarious color-based computer vision tasks. Although several methods have been proposed to estimate it, their applicability is severely restricted by the requirement for aknown illumination spectrum. In this work, we present asingle-image estimation method using ﬂuorescence with norequirement for a known illumination spectrum. Under different illuminations, the spectral distributions of ﬂuorescence emitted from the same material remain unchanged up
to a certain scale. Thus, a camera’s response to the ﬂuorescence would have the same chromaticity. Making use ofthis chromaticity invariance, the camera spectral sensitivity can be estimated under an arbitrary illumination whose
spectrum is unknown. Through extensive experiments, we
proved that our method is accurate under different illuminations. Moreover, we show how to recover the spectra ofdaylight from the estimated results. Finally, we use the estimated camera spectral sensitivities and daylight spectra tosolve color correction problems.

1. Introduction
Light spans a wide range of wavelengths. With a digitalcolor camera, the scene radiance is recorded as RGB values
via color ﬁlters that specify light in different wavelength
ranges to be observed. Therefore, the recorded RGB values
are dependent on the spectral sensitivity of color ﬁlters (orsensors), i.e., different sensors yield different RGB outputs
for the same scene. Camera spectral sensitivity plays animportant role for a lot of computer vision tasks that use
color information, such as spectral reﬂectance recovery [17,10], color correction [16] and color constancy [8, 9]. Thatmeans estimation of camera spectral sensitivity is necessary
to guarantee various color-based methods work well.
∗ This work was done while Shuai Han was an intern at Microsoft Research Asia.

Yoichi Sato1

A standard technique for estimating camera spectral sensitivity is to take pictures of monochromatic light whose
bandwidth is narrow [2]. The spectral sensitivity of thecamera can be reliably estimated from recorded observations and known spectral distributions of monochromatic
light. Although it gives accurate estimates, the method requires expensive hardware to generate and measure a series of monochromatic light, thus its use has been limited
to well-equipped laboratories only. In addition, the wholeprocedure is laborious because of the need for multiple observations of different light.
To simplify the procedure, methods using calibration targets whose reﬂectance is known have been proposed [7, 21,15]. These methods estimate the camera spectral sensitivity by recording a calibration target, e.g., IT8 Target, underillumination with a known spectral distribution. These approaches reduce the effort needed for measurement. However, their requirement for a known illumination spectrum
limits their practicability due to the need for a spectrometeror a speciﬁc light source with known illumination.
In this paper, we propose a new method for estimating
the camera spectral sensitivity from a single image captured
under an unknown illumination. The key idea is to use ﬂuorescence: a physical phenomenon whereby the substanceemits speciﬁc wavelengths of light by absorbing radiationof different wavelength. Its physical property of a ﬁxed emission spectrum is particularly useful for the camera spectral sensitivity estimations because the spectral proﬁle of theﬂuorescence remains unchanged up to a certain scale underarbitrary illumination. To make a single image estimate, we
use a chart of ﬂuorescent materials whose emission spectra are pre-determined. Then, from the emission spectra ofthese ﬂuorescent materials, we derive an analytic solution
for determining the camera spectral sensitivity.
There are three key properties about the proposed
method. First, it does not require the illumination spectrum
to be known. The estimate can be made under unknown
lighting conditions, e.g., outdoors under sunlight, under askylight, or indoors under a ﬂuorescent lamp, without hav-

ing to measure any spectrum. Second, the estimation requires only a single shot of a scene with the calibration
target. This reduces the cost of data acquisition. Third,
the ﬂuorescent chart used in this work is made from ﬂuorescent paint, which is inexpensive and readily available atstationery stores. These properties will make the camera
spectral sensitivity estimation more widely available.

2. Related Works
Camera spectral sensitivity can be estimated by establishing a relationship between incident narrow-band lightwith different wavelengths across the visible wavelength
range and the camera’s outputs. To achieve this goal, thestandard technique uses a monochromator or narrow-band
ﬁlters to generate a series of monochromatic light. Eachmonochromatic light is cast on a standard white board, andthe reﬂected light is observed by the target camera. At thesame time, the spectral distributions of the monochromatic
light are measured by a spectrometer. The camera spectral
sensitivity can then be calculated using the captured images
and measured spectra [2]. Because measurements are independent of each other, this method is accurate. However,it requires expensive optical equipments and a dark room.
Moreover, capturing dozens of images and spectral distributions makes the whole procedure time-consuming.
To make measurement more practical, methods using
calibration targets are proposed. These calibration targets,e.g., the IT8 target or Macbeth ColorChecker, contain several patches whose spectral reﬂectance is known. Under illumination with known spectrum, spectra of reﬂected lightfrom these patches can be computed. The camera spectral
sensitivity can also be estimated from the RGB values ofthese patches in captured images [7, 4, 6, 21]. However,these methods require controlled lighting conditions, whichmeans the light should be spatially uniform and its spectral
distribution is known. These requirements can be satisﬁed
only in laboratories. To extend the scope of these methods
out of the lab, Rump et al. [15] devised an imaging modelaccounting for specularity and spatially varying illumination. By using this model, measurements can be conducted in an environment where specularity or shadows exist.
However, it still needs a known illumination spectrum.
To avoid the requirement for a known illumination spectrum, we make use of ﬂuorescence. Fluorescence has been
receiving more and more attention recently. In [13, 12],methods describing how to model and render ﬂuorescentmaterials are presented. Modiﬁed color constancy methods
to deal with ﬂuorescent surfaces are shown in [3]. In [11],ﬂuorescence is used to sample the geometry of transparent
objects that cannot be sampled with traditional methods. In
this work, we explore the spectral properties of ﬂuorescenceand use them to estimate the camera spectral sensitivity.

3. Fluorescence
Fluorescence is a common phenomenon that has beenused for a variety of purposes, such as in lamps, stationery,safety vests, etc. Fluorescence is the emission of light bya substance that has absorbed light of different (generally
shorter) wavelengths. In this absorption and re-emission
process, two terms about ﬂuorescence are involved, i.e., absorption (or excitation) spectrum and emission spectrum.
The absorption spectrum represents how strongly the ﬂuorescent material absorbs ﬂuorescence-exciting light as afunction of its wavelength. The emission spectrum describes the spectral proﬁle of ﬂuorescence emitted from theﬂuorescent material. Both of the absorption spectrum andemission spectrum are determined by the properties of theﬂuorescent material itself.
The emitted ﬂuorescence from the p-th ﬂuorescent material, fp (λ) (p = 1, 2, 3, ..., P ), is described asap (λ )l(λ )dλ ep (λ),(1)
fp (λ) =where λ and λ are the wavelengths of the incoming lightand the outgoing ﬂuorescence, ap (λ ) and ep (λ) are the absorption and emission spectra of the p-th ﬂuorescent material, l(λ ) is the spectral distribution of the incoming light.
We can see that the integral part in Eq. (1) is determinedby the absorption spectrum and the spectrum of the incoming light. Therefore, it is independent from the spectrum of
the outgoing ﬂuorescence. Replacing that part by a scale
factor kp , we can rewrite Eq. (1) as
fp (λ) = kp ep (λ).

(2)

Eq. (2) means that the emitted ﬂuorescence from a certain
ﬂuorescent material under different illuminations remains
unchanged up to a certain scale.
To verify this unique property of ﬂuorescence, we measured the ﬂuorescence emitted from a patch smeared withﬂuorescent paint under different monochromatic light. The
measured spectral distributions are shown in Fig. 1. As expected from Eq. (2), all these distributions are different onlyin scales.
When the emitted ﬂuorescence from the p-th ﬂuorescentmaterial is observed with an RGB camera, the relationshipbetween the pixel intensity (Rpf , Gfp , Bpf )T and the emitted
ﬂuorescence is represented by1

Rpf = fp (λ)cR (λ)dλ.
(3)Here, cR (λ) (cG (λ) and cB (λ)) is the spectral sensitivityof the red (green and blue) channel. Gfp and Bpf can be
1 We assume the camera has a linear intensity response function. Without loss of generality, geometric factors is omitted here.

1

Intensity

320nm
340nm
360nm
380nm
400nm
420nm
440nm
460nm
480nm
500nm

0
520

Wavelength

740

Figure 1. Spectral distributions of emitted ﬂuorescence from thesame material under different monochromatic light remain unchanged up to a certain scale.

represented in a similar manner. For the sake of simplicity,
we hereafter derive equations for red channel and omit thosefor green and blue channels if they are trivial.
Substituting Eq. (2) into Eq. (3), we obtainf(4)
Rp = kp ep (λ)cR (λ)dλ.
Note that the scale factor kp is the same for all three channels. Thus, the chromaticity (rpf , gpf , bfp )T of the emitted
ﬂuorescence is described as
rpf

=

Rpf
f
Rp + Gfp + Bpf=

ep (λ)cR (λ)dλ.
ep (λ)(cR (λ) + cG (λ) + cB (λ))dλ(5)

The other two components gpf and bfp can be described in asimilar manner. Clearly, the scale factor kp is eliminated.
As a result, the chromaticity of the emitted ﬂuorescence isinvariant with respect to the spectrum of the incoming light.
Using the chromaticity invariance of emitted ﬂuorescence is the key idea of our method. It enables us to estimate the spectral sensitivity of a camera without having toknow the illumination spectrum.

4. Separating Fluorescent and Reﬂective Components
As discussed in Sec. 3, the chromaticity of emitted ﬂuorescence is invariant with respect to changes in illumination. Unfortunately, however, ﬂuorescent materials often not only emit ﬂuorescence but also reﬂect incident light. Thus, the observed pixel intensity of the p-th ﬂuorescent material, (Rp , Gp , Bp )T , contains a ﬂuorescent component (Rpf , Gfp , Bpf )T as well as a reﬂective component(Rpr , Grp , Bpr )T :(6)
Rp = Rpf + Rpr .
Therefore, we need to separate these two components in order to make use of the chromaticity invariance of the ﬂuorescent component.

Recently, Zhang and Sato [23] proposed a method forseparating ﬂuorescent and reﬂective components by using
independent component analysis, but their method requires
at least two images captured under different illuminations. In this section, we show how to separate the ﬂuorescentand reﬂective components from a single image. The key
idea is to use non-ﬂuorescent materials with known spectral
reﬂectance as a reference (Fig. 2).
We ﬁrst estimate the reﬂective components of ﬂuorescent materials under an unknown illumination. It can be
described as
(7)
Rpr = sp (λ)cR (λ)l(λ)dλ,where sp (λ) is the spectral reﬂectance of the p-th ﬂuorescent material. Grp and Bpr can be described in a similarmanner. According to a previous study [18], the spectral
reﬂectance of various materials can be approximately represented by a linear combination of a small number of basis
functions. Thus, we havesp (λ) =N

αp,n brn (λ),

(8)

n=1

where brn (λ) (n = 1, 2, 3, ..., N ) are the basis functions forspectral reﬂectance that are available in [18], αp,n are thecorresponding coefﬁcients. Substituting Eq. (8) into Eq. (7),
we obtain
Rpr =

N



αp,n

brn (λ)cR (λ)l(λ)dλ.

(9)

n=1

compute Rpr , we just need to know αp,n and To
rbn (λ)cR (λ)l(λ)dλ. Because our objective is to design
a calibration target, corresponding sp (λ) can be measured
once in advance.
Then, coefﬁcients can be computedas αp,n = sp (λ)brn (λ)dλ since the basis functions are
known and orthogonal to each other. Hence, the remaining problem is how to get brn (λ)cR (λ)l(λ)dλ. To solve
this problem, a reference containing a number of nonﬂuorescent materials with known spectral reﬂectance isused.
As shown in Fig. 2, we put the reference as well as theﬂuorescent chart together. Under the same illumination,
the observed pixel intensity of the q-th reference material,(Rq , Gq , Bq )T (q = 1, 2, 3, ..., Q), is represented by
Rq =

N

n=1


αq,n

brn (λ)cR (λ)l(λ)dλ.

(10)Here, αq,n are known and brn (λ)cR (λ)l(λ)dλ are unknown. For all materials on the reference, we obtain a set of
linear equations that are similar to Eq. (10). The number ofinto a sum of sines and cosines ascR (λ) =M

βR,m bcm (λ),

(12)

m=1

Reference Fluorescentchart
Figure 2. Under spatial uniformly distributed illumination, a reference containing a number of non-ﬂuorescent patches with known
spectral reﬂectance is used to help separating the ﬂuorescent component from the reﬂective component of the captured images of
the ﬂuorescent chart.

equations is Q (reference materials) × 3 (channels), and thenumber of unknowns is N (basis functions)× 3 (channels).
When Q ≥ N , the unknown integral brn (λ)cR (λ)l(λ)dλcan be estimated. Because we set N = 8 according to theprevious study [18], the reference chart should contain atleast eight materials whose spectral reﬂectance is linearly
independent.
Once we obtain the value of the integral in Eq. (10), thereﬂective components of the ﬂuorescent materials can becalculated by Eq. (9). Finally, the ﬂuorescent components(Rpf , Gfp , Bpf )T are computed by subtracting the calculated
reﬂective components from the observed pixel intensities as
Rpf = Rp − Rpr .

(11)

5. Estimating Camera Spectral Sensitivity
In the previous sections, we described that the chromaticity of ﬂuorescence is invariant under different illuminations and that the ﬂuorescent components can be extracted by using non-ﬂuorescent materials as a reference evenunder an unknown illumination. In this section, we show
how to use the illumination-invariant chromaticity of theﬂuorescent components for estimating the camera spectral
sensitivity.
Although different cameras have different spectral sensitivities, their spectral sensitivities should not deviate a lot
from each other. Thus, it becomes possible to use a limited
number of parameters or basis functions to express camera
spectral sensitivity [7, 20]. To estimate spectral sensitivity,
we just need to estimate the parameters or corresponding
coefﬁcients. To guarantee the general applicability of our
method, we adopt Fourier basis functions2 . Accordingly,
the spectral sensitivity (cR , cG , cB )T can be decomposed
2 Assuming the range of spectral sensitivity is [λl , λu ], the ﬁrst few
Fourier basis functions are bc1 (λ) = 1, bc2 (λ) = sin[2π(λ − λl )/(λu −λl )], bc3 (λ) = cos[2π(λ − λl )/(λu − λl )], and etc. We could also use
eigenvectors computed from a set of camera spectral sensitivities as basis
functions if such data are available.

where bcm (λ) are the m-th Fourier bases for representing
the camera spectral sensitivity. βR,m are the corresponding
coefﬁcients for the red channel. The spectral sensitivities of
the green and blue channels, cG and cB , can be decomposedin a similar manner.
Substituting Eq. (12) into Eq. (5), we obtain
βR,m ep (λ)bcm (λ)dλfmrp = 
.
cm (βR,m + βG,m + βB,m ) ep (λ)bm (λ)dλ(13)
Denoting the integral ep (λ)bcm (λ)dλ by tp,m , we canrewrite Eq. (13) as
⎡ ⎤
βR
 f
(rp − 1)tp rpf tp rpf tp ⎣β G ⎦ = 0,(14)
βB
where tp = (tp,1 , . . . , tp,M ) is a 1 × M row vector, β R =(βR,1 , . . . , βR,M )T is a M × 1 column vector, β G and β Bare deﬁned in a similar manner. Similar equations can beobtained for the green and blue channels. Since there are P
ﬂuorescent materials, we have P (ﬂuorescent materials) ×
3 (channels) equations in total:⎡(r1f − 1)t1
⎢ gf t⎢1 1⎢
f⎢ b1 t 1
⎢
..
⎢
.
⎢
⎢ f
⎢(rP − 1)tP⎢
⎣ gPf tP
bfP tP

r1f t1f(g1 − 1)t1
bf1 t1..
.

rPf tP(gPf − 1)tP
bfP tP⎤
r1f t1
g1f t1 ⎥⎥⎥(bf1 − 1)t1 ⎥ ⎡β ⎤R⎥..
⎥ ⎣β G ⎦ = 0.
.
⎥⎥
rPf tP ⎥ β B⎥
gPf tP ⎦(bfP − 1)tP

(15)
Eq. (15) can be expressed in the form of AX = 0, where A
is a 3P × 3M matrix and X is a 3M × 1 vector. A nonzero
solution of this linear equation is found as the eigenvectorof the square matrix AT A corresponding to the smallest
eigenvalue.
In experiments, we found that the estimated coefﬁcients(β R , β G , β B , )T are sometimes sensitive to noise. Therefore, to make the computation more stable, we incorporate asmoothness constraint on spectral sensitivity. Here, the second derivative of spectral sensitivity with respect to wavelength is used. As a result, we obtain⎤⎡A⎢w R φ00 ⎥⎥ X = 0,
⎢
(16)
⎣ 0
0 ⎦
wG φ
0
0
wB φ

6.1. Result of spectral sensitivity estimation

Intensity

1

0
380

Wavelength

780

Figure 3. Left: ﬂuorescent chart used in this work. Right: spectral
distributions of ﬂuorescence emitted from its 16 patches.

where (wR , wG , wB )T is the weight of the smoothness term and φ = (d2 bc1 (λ)/dλ2 , . . . , d2 bcM (λ)/dλ2 ). Substituting the solution of Eq. (16) into Eq. (12), camera spectral
sensitivity can be obtained up to a scale.
The steps for camera spectral sensitivity is summarizedas (1) take a picture of the ﬂuorescent chart and the reference; (2) separate the ﬂuorescent and the reﬂective components of the ﬂuorescent chart in the captured image; (3)compute the chromaticity of the ﬂuorescent component; (4)estimate the camera spectral sensitivity as described in thissection; (5) normalize the estimated camera spectral sensitivity, by which the largest absolute value of the estimatedresults in RGB channels is set to one.

6. Experiments
In this section, we show experimental results to evaluate
our method for estimating camera spectral sensitivity andtwo applications, namely, daylight spectrum estimation andcolor correction.
A chart containing 16 patches, shown in Fig. 3, was usedin our experiments. Those patches were made by smearing 16 different types of ﬂuorescent paint (P = 16) on a
black board. To measure the emission spectra of these ﬂuorescent patches, they were lit by 320nm UV light generated
by a monochromator (SHIMADZUTM SPG-120) one byone, and the spectral distributions of light from these patches were measured by a spectrometer (PhotoResearchTM
PR-670). Because no light in the visible wavelength range
was emitted from the monochromator, the measured results(Fig. 3) are the spectral distributions of ﬂuorescence fp (λ).
As stated in Sec. 4, a non-ﬂuorescent reference is used
for separating ﬂuorescent and reﬂective components. In
our experiments, we used a Macbeth ColorChecker with 24
patches whose spectral reﬂectance is publicly available [1].
For the spectral reﬂectance of ﬂuorescent patches, we use
the method proposed in [5]. To each ﬂuorescent patch, webuild its bispectral radiance factor matrix (Donaldson matrix). Spectral reﬂectance of the ﬂuorescent patch can be
represented by the diagonal elements of the matrix.

We estimated the spectral sensitivities of three different
cameras by taking pictures under four common illuminations, sun, blue sky, cloudy sky, and ﬂuorescent-lamp. The
spectral distributions of these illuminations were unknown.
Based on the previous study [7], 9 Fourier basis functions
are used for representing camera spectral sensitivity. The
weight of the smoothness term is set to (8, 8, 3). The Estimated results using our method are shown as continuous
curves in Fig. 4. The Results by using monochromatic
lights are shown as dotted curves. To evaluate the difference between them, for each estimated result, we computed
the average value of the root mean square errors (RMSE)
of RGB 3 channels. The numbers are also shown in Fig. 4.
These results show the accuracy of our method even without
knowing the illumination spectra.
Observing the estimated results of our method, we cansee that the results under sunlight have a little bigger errors
than those under the other illuminations. The reason behind
this is the fact that ﬂuorescent materials absorb light withhigher energy than their emitted ﬂuorescence. As shownin Fig. 5, to light from the blue sky or the cloudy sky, its intensity in the shorter wavelength range (higher energy)
is stronger than that in the longer wavelength range (lower
energy). On the contrary, sunlight is stronger in the longer
wavelength range. Hence, the ﬂuorescent component of theﬂuorescent chart under the sunlight is darker than those under the other illuminations. This results in that the estimatedresults under sunlight by our method are more easily affected by noise and errors. Therefore, when using our method,
illuminations which are strong in high energy wavelength
range, e.g., skylight, are recommended.
Another observation about the estimated camera spectral
sensitivities is that results of the blue channel show larger deviations when compared with those of the green or red
channel. Our explanation is that ﬂuorescent materials whichemit blue ﬂuorescence require stronger UV light and absorb
less visible light, thus their ﬂuorescent components always
appears darker than the other patches under common illuminations. Our estimate for the blue channel are thus moreeasily affected by noise or errors than the other two channels.

6.2. Daylight spectrum estimation
Daylight includes all direct and indirect sunlight during
the daytime. The spectrum of daylight is important for dealing with various imaging problems, such as color correction
and color constancy in outdoor environments. In this subsection we show how to estimate the spectrum of daylightby the camera spectral sensitivity.
According to the previous studies [14, 22, 19], daylight
spectrum can be well approximated with a small number ofbasis functions. To reconstruct daylight, we need to esti-

RMSE: 0.0942

700

Wavelength

RMSE: 0.0938

0
400

700

Wavelength

700

Wavelength

RMSE: 0.0981

0
400

1

700

Wavelength

RMSE: 0.1354

700

Wavelength

RMSE: 0.0813

0
400

0
400

1

700

Wavelength

RMSE: 0.0829

Sensitivity

Sensitivity

Sensitivity

0
400

700

Wavelength

RMSE: 0.1219

0
400

1

Sensitivity

1

0
400

1

Sensitivity

RMSE: 0.0877

1

Sensitivity

700

Wavelength

0
400

700

Wavelength

RMSE: 0.1241

Sensitivity

1

Fluorescent Lamp

Sensitivity

1

Sensitivity

RMSE: 0.0752

0
400

1

Sun

RMSE: 0.0996

0
400

700

Wavelength

Sensitivity

1

OlympusE-PL1s1

Sensitivity

Canon
5D

0
400

Casio
EX-F1

RMSE: 0.1063

Ours
Monochro1

Cloudy Sky

Sensitivity

Blue Sky

700

Wavelength

0
400

700

Wavelength

Figure 4. Estimated spectral sensitivities of cameras under different illuminations whose spectra are unknown. Results of our method aresimilar to those using monochromatic light, but ours have less high frequency because of the use of Fourier bases and the smooth constraint.

Blue Sky

Cloudy Sky

1.0

Sun

1.0

Estimated

RMSE: 0.0403

1.0

RMSE: 0.0150

RMSE: 0.0192

0400

Intensity

Intensity

Intensity

Ground truth

Wavelength

700

0
400

Wavelength

700

0
400

Wavelength

700

Figure 5. The spectral distributions of daylight can be accurately estimated from the estimated camera spectral sensitivity of a Canon 5D
and the appearance of the Macbeth ColorChecker in the captured images. Note that each spectrum is normalized to ensure the area of theregion below the corresponding curve is a constant. Thus, the intensities are relative values for the two distributions in the same ﬁgure.

mate its corresponding coefﬁcients. Here, we adopt a widely used three-basis model [22]:l(λ) =

3Rq =γj blj (λ),

(17)

j=1

where blj (λ) is the j-th basis for daylight that is available
in [22], and γj is the corresponding coefﬁcient. Recall that we used the Macbeth ColorChecker as the reference
for separating the ﬂuorescent component in Sec. 4, the appearance of its q-th patch under daylight, (Rq , Gq , Bq )T(q = 1, ..., 24), can be represented asRq =Substituting Eq. (17) into Eq. (18), we obtainsq (λ)cR (λ)l(λ)dλ.

(18)

3



γj

sq (λ)cR (λ)blj (λ)dλ.

(19)

j=1

Three factors in the integral of Eq. (19) are known, and only γj are unknown. For all 24 patches on the ColorChecker,
we have 24(patch number)× 3 (channels), i.e., 72, linear equations that are similar to Eq. (19). The three unknown coefﬁcients γj can be calculated in terms of the least-squares
error. With the calculated results, the spectrum of daylight
can be estimated by Eq. (17).
The estimates for different daylight conditions are shownin Fig. 5. The estimated spectra are indicated by red curves,and ground truths measured by the spectrometer are indicated by black curves. We can see that they are very similar toFigure 6. A synthetic image of a postcard composed from three
real captured images. Cameras and illuminations are shown inparentheses.

Figure 7. The calibration targets are placed in the scene.

each other. The root mean square errors (RMSE) of the estimated results are very small. Through these comparisons,
we can see that the spectra of daylight can be accurately
estimated by the camera spectral sensitivity.
Based on the above discussion, it is apparent that we arecapable of estimating not only camera spectral sensitivity
but also the daylight spectrum from a single captured image.

6.3. Color correction
It is well known that the appearance of the same scene
varies a lot under different illuminations or using different cameras. One example is shown in Fig. 6. This difference can be seen as multiplying albedo of scene points withdifferent scales in the RGB channels. The scales can becalculated by multiplying the obtained camera spectral sensitivities
and daylight spectra in the spectral domain, i.e.,cR (λ)l(λ)dλ. By making use of these scales, the color of
captured images under different illuminations or by different cameras can be corrected.
Let us suppose that the spectral sensitivity of a camera is (cR (λ), cG (λ), cB (λ))T , daylight is l(λ), and the observed intensity of a scene point is (R, G, B)T . For a different camera (cR (λ), cG (λ), cB (λ))T or under different
daylight l(λ), the observed intensity of the same scene pointis (R, G, B)T . The relationship between these two observations can be described ascR (λ)l(λ)dλ
R= 
R,
(20)
cR (λ)l(λ)dλfor the red channel.The relationship for the blue and green
channels can be described in a similar manner.
As discussed in previous sections, appearance of the ﬂuorescent chart and the Macbeth ColorChecker is required

Figure 9. Difference between two Macbeth ColorChecker images
captured under sunlight and blue sky conditions by the Canon 5D.
Left: Macbeth ColorChecker. Middle: difference (×5) before correction. Right: difference (×5) after correction.

for estimating camera spectral sensitivity and daylight spectrum. If they can be placed in the scene, a single image cancapture their appearance as well as the scene (as shown in
Fig. 7). Otherwise, two images need to be taken, i.e., one of
the calibration targets, and another one of the scene.
Once the corresponding camera spectral sensitives andthe daylight spectra are obtained, the color of a scene’s appearance can be corrected using Eq. (20). Here, we took
images about 2 scenes under different daylights with different cameras, a building and a postcard. These images
are framed in blue in Fig. 8. With the estimated camera spectral sensitivities and daylight spectra, we calculated
the scenes’ appearances under different daylight conditions
with different cameras by Eq. (20). The calculated results
are framed in red. Each column shows a real captured image and a color corrected image. We can see that, although
the corrected results are not identical to the real captured
images, the difference is signiﬁcantly decreased, by which
the effectiveness of our method can be demonstrated.
To evaluate how well our method works on different colors, we captured two Macbeth ColorChecker images usinga Canon 5D under the sun and blue sky conditions. The differences between these images before and after correction
are shown in Fig. 9. The darker the patches are, the smaller
the differences are. From the middle image, we can see that
because sunlight is more reddish than light from the bluesky, for patches which have high reﬂectivity to red light,
their differences in the red channel are obvious before color correction. After color correction, as shown in the rightimage, those obvious differences are greatly reduced.

7. Conclusions
In this work, we proposed an estimation method for camera spectral sensitivity under unknown illumination. Making use of chromaticity invariance of ﬂuorescence, ourmethod is capable of estimating camera spectral sensitivity from a single image of a calibration chart made withﬂuorescent paint and a Macbeth ColorChecker taken under
unknown illumination. The effectiveness of the proposed
method was successfully demonstrated with experiments
using real images taken under various illumination conditions. In addition, two applications of our method were in-(EPL1s,Cloudy Sky)

(5D, Sun)(EX-F1,Sun)(EPL1s,Blue Sky)

Figure 8. The color in the captured images can be corrected to match other images captured under different illuminations or by different
cameras with estimated camera spectral sensitivities and daylight spectra. Here, we show two scenes, buildings (left) and a postcard (right).
Cameras and illuminations are shown in parentheses.

troduced: daylight spectrum estimation and color correction
for outdoor images. In the future, we are planning to workon how to select a set of ﬂuorescent paint to achieve better estimation accuracy under a wide range of illumination
conditions.

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