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A. Alsam and R. Lenz Vol. 24, No. 1 /January 2007 /J. Opt. Soc. Am. A 11

Calibrating color cameras using metameric blacks

Ali Alsam

Gjøvik University College, P.O. Box 191, N-2802 Gjøvik, Norway

Reiner Lenz

Linköping University, SE-60174 Norrköping, Sweden

Received February 21, 2006; revised May 30, 2006; accepted June 30, 2006;posted August 21, 2006 (Doc. ID 68216); published December 13, 2006

Spectral calibration of digital cameras based on the spectral data of commercially available calibration chartsis an ill-conditioned problem that has an infinite number of solutions. We introduce a method to estimate thesensor’s spectral sensitivity function based on metamers. For a given patch on the calibration chart we con-struct numerical metamers by computing convex linear combinations of spectra from calibration chips withlower and higher sensor response values. The difference between the measured reflectance spectrum and thenumerical metamer lies in the null space of the sensor. For each measured spectrum we use this procedure tocompute a collection of color signals that lie in the null space of the sensor. For a collection of such spaces wecompute the robust principal components, and we obtain an estimate of the sensor by computing the commonnull space spanned by these vectors. Our approach has a number of advantages over standard techniques: It isrobust to outliers and is not dominated by larger response values, and it offers the ability to evaluate the good-ness of the solution where it is possible to show that the solution is optimal, given the data, if the calculatedrange is one dimensional. © 2006 Optical Society of America

OCIS codes: 330.1690, 330.1710, 110.2990, 100.3190.

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. INTRODUCTIONamera sensor calibration consists in the problem of esti-ating the device’s spectral sensitivities from its re-

ponses to a number of spectrally different surfaces. Gen-rally, there are two approaches to solving the spectralalibration problem: one based on physical measurementsnd one based on a theoretical model. The physical ap-roach, using a monochrometer, gives an accurate esti-ate of the spectral sensitivities, but it is expensive and

ime-consuming to use. The model-based approach isheaper and provides insight into the characteristics ofhe camera system. It is based on solving a linear equa-ion system of the form

Y� = Y − � = AX. �1�

et l be the number of sensor sensitivity functions of theamera, m the number of surfaces used, and n the dimen-ion of the spectral data. Typical values are n=31, corre-ponding to a 10 nm sampling of the wavelength range00 nm–700 nm, and l=3 for an RGB camera. For the ma-rices involved we have the following: A is the m�n ma-rix of measured color signals, X is an n� l dimensionalatrix whose elements are the spectral sensitivities, Y is

f size m� l and contains the measured camera re-ponses, and � is the acquisition noise matrix. The colorignals are the componentwise products of the illumina-ion spectrum and the reflectance spectra. The goodnessf the solution for the spectral sensitivities based on Eqs.1) depends on two main factors: the noise level in the re-ponse data �� and the statistical properties of the spec-ral data available from the calibration chart (the matrix). Estimation of X from the RGB measurements Y and
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is a typical inverse problem, and standard methodsrom linear algebra are often used to solve it.

From the general theory, it is also known, however, thathe quality of the estimation can be substantially im-roved by use of additional constraints. The findings inef. 1 indicate that the uncertainty surrounding spectralecovery is proportional to the size of the recovered setnd is governed by factors such as the noise level, the di-ensionality of the spectral data, and the constraints im-

osed on the solution space. In Ref. 2 the authors con-trained the sensors to be positive, smooth and to predicthe responses within an acceptable noise bound. In Ref. 3he authors added a constraint on the number of peaks al-owed in the recovered sensor, and the authors in Refs.–6 constrained the sensor’s magnitude to be small. Allhese methods2–6 require that the recovered sensor mini-ize the difference between the measured and the esti-ated responses.The particular contribution of this paper is to useetamerism to construct new color signals that areetameric blacks. These metameric black color signals

re then used to construct projection operators that char-cterize, fully or partially, the null space of the camera.his knowledge about the properties of the null space of

he camera is then used to reduce the noise sensitivity ofhe estimator. Different from standard methods, the pro-osed algorithm is based on estimating the space of theensor without resorting to minimization. Said differ-ntly, our method does not require that the estimated sen-or result in the best data fit between the measured andhe estimated response values. Furthermore, combiningifferent spectra to generate metameric black signals ischieved by the calculation of weights in the response

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12 J. Opt. Soc. Am. A/Vol. 24, No. 1 /January 2007 A. Alsam and R. Lenz

pace. Doing so, we introduce variations in the metamericignals that are due to the noise statistics of the camera’sesponses. Further, when the noise statistics are trans-ormed from the response to the spectral space, it is pos-ible to reduce the sensitivity of the estimated sensor toutliers in the response data by using standard robusttatistics algorithms. This property results in a methodhat is robust to outliers and is not driven by large re-ponse values.

In the experimental part of the paper we test the pro-osed method by calibrating two digital cameras: Nikon70 and MegaVision. We compare the result obtained by

he proposed methods with standard estimation methodsuch as truncated singular-value decomposition (TSVD)nd Tikhonov regularization (TR).

. BACKGROUND. Notation and Model Descriptionhe device’s response � to a given color signal A�� over

he visible spectrum � based on an illuminant E��, a sur-ace S��, and a sensor X��, can be written as

� =��

E��S��X��d�. �2�

e combine the illumination and reflectance spectra toefine the color signal A��=E��S��:

� =��

A��X��d�. �3�

n the calibration process we measure a set of color sig-als Ai��i=1,2. . . ,m� with a spectroradiometer and theorresponding camera responses �i.

In practice, functions are replaced by vectors since it isufficient to represent the data at uniformly spaced dis-rete points between 400 and 700 nm with 4–10 nmntervals.7 Thus we introduce the vectors

e = �E��1�,E��2�, . . . ,E��n��T, �4�

s = �S��1�,S��2�, . . . ,S��n��T, �5�

x = �X��1�,X��2�, . . . ,X��n��T, �6�

here T is the transpose operator. Equation (3) can nowe rewritten as

� = �j=1

n

ejsjxj��, �7�

here �� is the sampling distance.Equation (7) can be written in matrix form as

� = sT diag�e�x, �8�

here diag�e� is the diagonal matrix with E��j� in the di-gonal.Let us now consider the sensor response to a set of m

pectral stimuli. We collect the surface reflectance spectran the n�m matrix S and compute the matrix A of colorignals as

A = �diag�e�S�T. �9�

he m device responses form a vector y, and we obtain ouriscrete model of the measurement process,

y = Ax + �, �10�

here y is the red, green, or blue response vector and � ishe vector of measurements noise at that channel. Com-ared with Eq. (1), � is a column vector in the matrix �.The central task in spectrally calibrating an imaging

evice is to solve for x based on the measured device’s re-ponses y and the corresponding color signals A.

Grouping the response and noise vectors, Eq. (10) cane rewritten as

y� = Ax, �11�

here y� is equal to y−�.

. Sensor Recoveryumerically, there are two approaches to solving Eq. (11)

or the sensor x; direct and iterative methods.8 Directethods are those that solve for the spectral sensitivities

n one step by inverting Eq. (11), or a variation of it, for x.nder this approach we include the TSVD9 method alsonown as the principal eigenvectors PE2 and TR.4–6 Onhe other hand, iterative methods search for a solution toq. (11) that satisfies a number of constraints. In the lit-rature various iterative methods have been used to solvehe spectral calibration problem, including projection ontoonvex sets �POCS�,2 quadratic programming �QP�,3 andinear programming (LP).10 These methods are alsonown as constrained optimization techniques. In the re-aining part of this section, we briefly outline some of

hose methods.It is well documented that solving for a sensor using

he least-squares formulation results in poorstimates.4,11 This is true because the color signal matrix

obtained from commercially available calibrationharts is ill-conditioned. More formally, using the singularalue decomposition,12 we can write A as

A = U�VT, �12�

here U and V the orthonormal basis vectors associatedith the left and right eigenvectors of A and � is a diag-nal matrix whose diagonal elements � are the singularalues of A. It is known13 that for an ill-conditioned ma-rix such as A the singular values drop very rapidly from

large value to zero. Further, using the singular-valueecomposition, sensor estimation can be written as

x = �i=1

n viuiT

�iy�. �13�

large volume of statistical studies based on various col-ections of surfaces14,15 confirm that the singular values ofatural reflectances diminish to zero as the number of di-ensions is increased beyond a small number of basis

ectors (3–11). Thus it is clear from Eq. (13) that includ-ng those very small singular values especially whenoise is present would lead to an unstable solution.

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To stabilize the solutions two methods are widely used:he first is TSVD (Ref. 9), in which Eq. (13) is rewrittens

x = �i=1

r viuiT

�iy�, �14�

here r is less than the rank of A.The second regularization method is TR, in which Eq.

13) is rewritten as

x = �i=1

n viuiT

�i + �2y�, �15�

here � is a small positive value that can be estimatedsing the L-curve criterion.16,17 The L-curve is a log–loglot of the norm of a regularized solution versus the normf the corresponding residual norm. It is a convenientraphical tool for displaying the trade-off between theize of a regularized solution and its fit to the given datas the regularization parameter varies.Equation (15) is traditionally written as

x = �ATA + I�2�−1Ay�, �16�

here I is the identity matrix.

. Constrained Optimizationo further control the estimate of the sensors, Finlaysont al.3 used QP to solve the spectral recovery problem sub-ect to a set of linear constraints. Generally, such a formu-ation can be stated as

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�Ax − y��

subject to Bx t, �17�

here the linear constraints defined by Bx t can includeonstraints such as nonnegativity, smoothness, andnimodality.2,3,5 A similar approach has been taken byharma et al.3 who used POCS and Barnard et al.,5 whosed a constrained TR.

. PROJECTION OPERATORS FORALIBRATIONe consider first the role of the calibration chips, i.e., the

roperties of the matrix A. As mentioned in Sec. 2, theeast-squares estimate can be written X= �ATA�−1ATY�.ften, �ATA�−1 does not exist. But we can find an orthogo-al matrix U such that

AUT = �A1 0�, �18�

ith A1 of full rank. For the product AX we get

AX = �AUT��UX� = �A1 0��X1

X2 = A1X1.

We summarize this result as follows: We can find an op-rator Q defined as the projection operator that maps X to1=QX, where X1 is defined as above. Replacing A with1, we can work in the subspace of the original space,here we can assume that A has a pseudo-inverse A†

�ATA�−1AT.

. METAMERS AND PROJECTIONPERATORSe now apply the following procedure to construct projec-

ion operators to the null space of the camera:

• Collect the color-signal measurement pairs �am ,ym�,here am is the color signal and ym is the camera outputelonging to the mth surface: ym=amx. The calculation isarried out on the individual channels.

• For the response value ym, 0ym1, in the set ofamera output vectors (we assume that the outputs areormalized such that the maximum output is one), findwo points y�� ,y�� such that y��ymy��. Now we havem=�y��+ �1−��y��, where 0 � 1 and given as

� =ym − y��

y�� − y��. �19�

• By applying the convex weight � from the responsepace in the spectral space, we obtain a “numerical” colorignal z=�a��+ �1−��a��, where a�� and a�� are the colorignals corresponding to y�� and y��, respectively. Now weave two metameric signals, namely, am and z, resulting

n the same camera output, ym. For a three-dimensionalase, this procedure is discussed in detail in Ref. 18.

• Depending on the number of surfaces available forhe calibration, a number of metameric surfaces z can bealculated for the same response value ym by simply usinguitable pairs of points y�� ,y��.

• By definition we know that two metameric surfaces zi

nd zj result in the same camera response ym, i.e., zixzjx=ym. Thus it follows that zix−zjx=0. In other words,

he difference between two metamers is in the null spacef the sensor x.

• Thus if we collect all the numerical metamers for aingle camera response ym in a matrix Z and subtract theean of the set from each surface, then the result is a ma-

rix Zn where Znx=0. The mean is subtracted from theata to define the vectors in a plane. In theory, anyetamer can be used instead of the mean point, but we

hoose the mean as it is, statistically, the most represen-ative vector.

From the construction of Zn we know that the dot prod-ct of any vector in Zn with the sensor x is zero. Further,e know that the points in Zn define a hyperplane in the

pace of the color signal data in A. The planes associatedith different response values y have to be parallel to

ach other and intersect the sensor x at a point y. A three-imensional example is shown in Fig. 1. Where we noticehat for a number of responses we have calculated the as-ociated black planes defined as Zn and that they are allrthogonal to a single vector.

. ESTIMATING THE SENSOR FROM THELACK METAMERShus far, we have shown that for each response value ym

rom a color filter, x it is possible to solve for a set ofetameric surfaces Z and calculate their orthogonal ele-

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ents on the sensor, i.e., the black metamers Zn. The sen-or is defined as the component orthogonal to Zn.

In practice there are, however, two difficulties thatave to be taken into account: (1) The measured reflec-ance spectra from the calibration chart do not span thehole space, and (2) the measured RGB vectors are cor-

upted by noise. In the implementation, we will thereforese the following procedure to obtain a robust estimationf the sensor function: We calculate a basis of the spacepanned by metameric black color signals. For calculatingset of basis vectors in the spectral space, principal com-

onents analysis (PCA) is commonly employed, but sincehe color signals are obtained from the noisy RGB vectorst is advantageous to use a robust version of PCA. Usingobust statistics, we are able to filter the data outliers,hus making our estimate less sensitive to noise. One ofhe most compelling advantages of using the proposedethod, and indeed one of the unique features of the al-

orithm, is the use of metamers to transfer the variabilitynoise) from the response data into the spectral space. Us-ng metamerism, it is possible to state that the metamersf a sensor, at a given response value, should be points onhyperplane orthogonal to the sensor. Thus any deviation

rom this plane is due to noise.Among the available robust PCA methods, we choose

he method by Hubert et al. described in Ref. 19. Thisethod is based on the observation that the first eigen-

ector defines the direction in the space that explainsost of the variance of the data. The most important

teps in this method are the following: First, the medianf the data is computed, and the data are centered aroundhat point. Next, one defines a robust estimate of the vari-nce of a scalar-valued stochastic variable, and then allentered data vectors are projected onto the coordinatexis by one of the data vectors. The robust estimate of theariance of these projected values is computed and is usedo measure the contribution of this projection direction.fter all the data vectors are examined, the direction with

he highest robust variation is selected as the first robustigenvector. Then the original data vectors are projectednto the orthogonal complement of this eigenvector, and

ig. 1. Three-dimensional example of the metameric blacklanes, shown to be orthogonal to a single vector. The axes a1, a2,nd a3 are the three-dimensional axes of the input data.

he estimation procedure is repeated for the new data vec-ors. In this way a series of orthogonal vectors is con-tructed in such a way that in each iteration the directionf the maximum robust variance is estimated. (A detailedescription of the method can be found in the originalaper.19)For our sensor estimation procedure we define the set

f robust PCA vectors that span the space of theetameric plane as UZn

. Since UZnis orthonormal, we de-

ne the projection onto the black hyperplane as

P0 = UZnUZn

T . �20�

ere, we note that consideration must be given to the di-ensionality of the original data, and thus only a limitedumber of basis vectors can be included in the calculationf P0 (in the experiments we used 7–9 basis vectors).

The definition of the projection matrix P0 in Eq. (20) al-ows us to determine the portion on the color signals thatre in the direction of the sensor x. This portion is defineds

A� = A − AP0. �21�

n this paper, we define a sensor as the first principal vec-or of A�. If the decomposition defined in Eq. (21) is exact,hich is the case for noise-free data, then the vectors in� will be perfectly parallel to each other. Thus choosing

he first principal vector is only necessary when A� hasimensions higher than one.

. MEASURING THE GOODNESSF THE ESTIMATE

hus far, we have divided the spectral space into two com-onents, A� and its orthogonal complement A0. Theoreti-ally, given that the original data has dimension n, weish to divide the space such that A� has dimension 1nd A0 has dimension n−1. If such a decomposition ischieved, then we have calibrated the sensor. However, inhe general case, when noise is present in the calibrationata, it is not possible to estimate the black planes per-ectly. Thus there will be no unique orthogonal sensor butather a number of possible vectors. One of the novel fea-ures of the proposed algorithm is that it allows us tolearly estimate the goodness of the recovery in the spec-ral space. This is different from the goodness estimationsed for other methods, where goodness is normally de-ned in the least-squares sense between the measurednd the estimated responses.We have stated that the sensor is perfectly calibrated if

he dimensionality of the color signals in matrix A� isne. Thus we propose a goodness measure that is basedn the rank of A�. Such a measure can be defined by con-idering the singular-value decomposition of A�:

A� = UA�DA�

VA�

T . �22�

t is known that in the case where the vectors in A� arearallel, the first element is the only nonzero element onhe diagonal of DA�

. It is further understood that the di-ensionality of A can be estimated by studying the ratio
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. EXPERIMENTS AND RESULTSo test the performance of the proposed algorithm andompare it with standard methods, we spectrally cali-rated a Nikon D70 digital camera and a MegaVisionamera. For the Nikon D70 the actual sensitivities wereot available, whereas for the Mega Vision the sensorurves were measured using a monochrometer.

In the first experiment, with the Nikon D70 camera,wo calibration charts were used: the Esser chart with82 colored patches and the 24 patches of the Macbetholor Checker. For both charts, the spectral data wereeasured using a Minolta CS-1000 spectroradiometer un-

er the daylight simulator of the Macbeth Verda viewingooth. The camera responses were captured in the Nikonaw image format; the response data were checked for lin-arity and the dark noise was subtracted.

For numerical data comparison we used the absoluterror between the estimated and the measured responses,efined as

AE = aix̃ − yi, �24�

here x̃ is the estimated sensor. To allow meaningful com-arison in terms of the absolute error metric, the data inand Ax̃, for all channels, were normalized such that theaximum value was set 1 to 100. Thus a difference of 1 is

quivalent to 1% error.In the second experiment, with the MegaVision cam-

ra, the spectral data were those of the Macbeth Colorhecker measured under a daylight simulator. Further,ecause the actual sensitivities were available, it was pos-ible to compare the estimated with the measured sensi-ivities in the spectral space.

. Nikon D70n the first part of this experiment we performed a spec-ral calibration of the Nikon D70 based on the camera’sesponses to the spectral data of the Esser calibrationhart. The validity of the sensors estimate was checked byalculating the responses to the spectral data of the Mac-eth Color Checker. When calculating the sensitivities us-ng the proposed method, we used only a subsection of thesser data. This subset was chosen such that the dataoints were the extremes of three-dimensional RGBpace. For the Nikon data we got 38 out of the 282 dataoints. Further, for comparison we calculated the sensi-ivities using TSVD and TR. For both, the training dataere the whole of the Esser chart.The results of the comparison based on the Esser data

re tabulated in Table 1, and those based on the Macbetholor Checker test set are tabulated in Table 2. The spec-

ral sensitivities obtained with the new method, TSVD,nd TR are plotted in Fig. 2.The value of the proposed goodness measure is also

abulated in Table 1, where we find that the red channelchieves a value of 81.4%, the green channel 81.6%, andhe blue channel 79.8%. From our experiments with dif-erent sensor sets we conclude that adding an increasingumber of basis vectors to the black space gradually im-roves the results until a maximum value is obtained.his property means that the proposed goodness measurean be used to automate the choice of number of basis vec-ors included in the calibration. Indeed, for the purpose ofhis paper we used the optimal values reported in Table 1.

hen calculating the goodness measure, for data used in

Table 1. Absolute Error between Measured andEstimated Responses for the Red, Green, and

Blue Channels of the Nikon D70 Camera: EsserCalibration Charta

Absolute Error

ethod Red Green Blue

SVDMean 1.34 1.00 0.96Median 0.94 0.62 0.75Max 8.40 8.08 6.61

RMean 1.82 1.26 1.53Median 1.35 0.83 1.07Max 8.92 5.75 6.13BGMb 0.816 0.834 0.798Mean 1.354 1.131 1.044Median 0.886 0.775 0.752Max 9.085 8.464 5.785

aThe results are based on the TVSD, TR, and MB methods. Data were from thesser calibration chart.

bGM, goodness measure.

Table 2. Absolute Error between Measured andEstimated Responses for the Red, Green, and

Blue Chanels of the Nikon D70 Camera: MacbethColor Checkera

Absolute Error


SVDMean 2.64 2.58 2.74Median 1.66 2.34 2.04Max 9.21 8.34 8.84

RMean 2.07 2.49 2.80Median 1.54 1.45 1.77Max 8.09 8.61 10.49BMean 1.637 1.600 1.153Median 0.903 1.231 0.684Max 9.247 7.442 8.331

aThe results are based on the TVSD, TR, and MB methods. Data were from theacbeth Color Checker.

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he experiment, using an increasing number of black ba-is vectors we obtained the values plotted in Fig. 3. As weee those values increase almost linearly until a maxi-um value is achieved.From Table 1 we note that the new metamer-based

MB) algorithm performs slightly better than TSVD. Heree remind the reader that the MB method is not aimed atinimizing the difference between the measured and the

stimated responses. Thus the fact that the results out-erform those achieved with minimization-based ap-roaches such as TSVD and TR is clearly significant. Fi-ally, when the estimated sensors from the three methodsere used to predict the camera’s responses to the spec-

ral data of the Macbeth Color Checker, which was useds a test set, the MB method showed a clear advantagever both TSVD and TR. These results are tabulated inable 2, where we find that similar to the results achievedor the training set, Table 1, the error between the esti-ated and the measured responses is within 1%.

ig. 2. (Color online) Estimated spectral sensitivities of the Ni-on D70 camera as a function of wavelength 380–750 nm. Esti-ation methods used the proposed MB (circles), TR (diamonds),

nd TSVD (asterisks).

ig. 3. (Color online) Estimates of the red (solid curve), greendashed curve), and blue (dotted curve).

. MegaVisionn this experiment, we estimated the spectral sensitivi-ies of the MegaVision camera using the Macbeth Colorhecker. Measurements of the camera’s spectral sensitivi-

ies using a monochrometer were available to us. The es-imated sensitivities are plotted in Fig. 4. To estimate theimilarity between the two sets, we used the Vora value,20

hich is a measure comparing the norm of the sensor setn its original space and the norm of its projection ontohe space of the second sensor. A Vora value of 1 indicateshat the sensors are within a linear transform of eachther, and a value of 0 means that the sensors are or-hogonal. For the estimates shown in Fig. 4 we found thathe Vora value is 0.96, which indicates a very close fit.

When the estimated sensitivities were used to estimatehe responses, the results were comparable to thosechieved with the measured filters. These results areabulated in Table 3. In Table 3 we note that the goodnesseasure is higher for the red and blue channels than that

chieved with the Nikon D70 camera; however, we pointut that the goodness measure is dependent on the

Table 3. Absolute Error between Measured andEstimated Responses for the Red, Green, and Blue

Channels of the MegaVision Camera: MacbethColor Checkera

Absolute Error


ean 0.73 0.62 0.67edian 0.63 0.44 0.65ax 1.58 1.72 1.64BM 0.82 0.87 0.95ean 0.50 0.50 0.42edian 0.47 0.37 0.37ax 1.01 1.89 1.22

aThe results are based on the MB method and the actual sensitivities. Data wererom the Macbeth Color Checker.

ig. 4. (Color online) Estimated spectral sensitivities (dashedurves) of the MegaVision camera as a function of wavelength00–700 nm. Actual sensitivities are shown by solid curves.

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umber of calibration surfaces used, where using one sur-ace is guaranteed to result in a goodness value of one,hereas including more surfaces would result in a valuef unity only if all the vectors in the range were perfectlyarallel to each other.

. CONCLUSIONn this paper, we introduced a method to estimate theolor sensitivity curves of a camera. The method is basedn the observation that the difference between twoetameric spectra lies in the black space of the sensor. In

he first step of the estimation, we use selected triples ofalibration colors to construct numerical spectral distri-utions that are in the black space of the sensor. Afteronstructing a set of metameric blacks, we found the sen-or sensitivity function as the orthogonal complement ofhe black space. This construction is based on the con-truction of subspaces of the color signal space only. Itoes not use approximation errors as do conventionalethods and therefore avoids the sensitivity to outliers.We tested the basic properties of the new algorithm by

pectrally calibrating a Nikon D70 and a MegaVisionamera. In the case of the Nikon D70, we evaluated theerformance with the help of the estimation errors be-ween the measured and the estimated RGB vectors. Forhe MegaVision camera, measurements of the sensitivityurves were available, and we could therefore comparehese measured curves with the curves obtained by ouralibration method.

Our first implementation of the method showed that forhe training set the results were comparable to those ob-ained by standard techniques (TSVD and TR). For coloratches that were not used in the calibration, the perfor-ance of our method was clearly better than the perfor-ance of TSVD and TR.The experiments described in this paper are only a first

est of the properties of our improved estimation method.urther improvements can be obtained by a more detailedvaluation of the noise characteristics.

Corresponding author Ali Alsam’s e-mail address [email protected].

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sensitivities with uncertainty,” in Proceedings of the FirstEuropean Conference on Color in Graphics, Imaging andVision (CG1V 2002) (Society for Imaging Science andTechnology, 2002), pp. 22–26.

2. G. Sharma and H. Trussell, “Characterization of scanner
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Calibrating color cameras using metameric blacks