. INTRODUCTIONolorimetric device calibration is the problem of estimat-
ng the responses of a device A in the space of a secondevice B. As an example, if device A is a red–green–blueRGB) camera and its responses to a given scene are con-ained in an RGB image, then colorimetric calibration ishe problem of estimating a mathematical transformationf the image data to the space of device B. The latteright be the human’s XYZ space or a monitor’s RGB
pace such as sRGB.The process of integrating the continuous light signal
eflected off an object to a device response is linear. Thisinearity has, for a long time, been the motivation for ap-lying linear transforms to the problem of colorimetric de-ice calibration. It is, however, known that for a linearransform to perfectly map the responses of device A tohe space of B the sensor curves of A must span the samepace as those of B. A camera whose spectral sensitivitiesre a linear combination of the color-matching functions1] is said to satisfy the Luther condition [2] and is re-erred to as colorimetric.
Although color scanner and camera manufacturerstrive to achieve colorimetric color reproduction, theyave to take into account other design factors such asoise amplification and manufacturing limitations [3].he result of the design compromise is that cameras andcanners are not perfectly colorimetric, and linear trans-orms fall short from being the ideal solution to the prob-em of color correction.
Several methods have been suggested to improve uponhe performance of linear transforms. These methods in-lude constrained linear transforms [4], polynomial fitting5], look-up tables [5], and constrained local linear trans-orms [6]. Unfortunately, all these techniques have theirwn limitations, some of which are practical, i.e., relate tohe technique’s performance, while others are theoretical.
s an example of the practical limitations, polynomialransformations are known to perform well for the cali-ration target but may not generalize adequately to otherata. Our main criticism of these methods is, however,hat theoretically they represent, albeit useful, heuristicsnd fail to shed light on the underlying problem.Finlayson and Morovic [7–10] developed algorithms for
he calculation of metamer sets that provided a strong ba-is for color correction. They showed that, given a singleGB value, there is a corresponding metamer set of re-ectances that might have induced the RGB value. Inheir algorithms, the authors assume the spectral sensi-ivities of the device are known, as are the prevailingighting conditions. This metamer set, when projectednto the target sensitivities, e.g., XYZ color-matchingunctions, is usually nonsingular: each set, which maps tosingle RGB value, maps to a convex set in XYZ space ofYZs. Algorithms were presented for selecting the most
ikely metamer to represent the set [9]. It is this single re-ectance that, when projected onto the target color space,
s the end point of color correction. Experiments [9] vali-ated the usefulness of calculating the metamer sets, forolor correction, with significantly improved correction re-orted, especially for saturated colors.Unfortunately, the calculation of metamers [11] re-
uires knowledge about the camera’s spectral sensitivityurves. These are typically not known and difficult to es-imate reliably. In certain restricted cases a monochro-ator might be used to measure the system response to
ifferent monochromatic spectra. However, monochro-aters are expensive, and the experiment is tedious to
arry out. Alternatively, given a reference target, such asMacbeth ColorChecker, one can infer the spectral sensi-
ivities [12–15]. While this approach can work, it is sub-ect to significant numerical errors and dependent on theoise level; the estimated sensitivities can be quite differ-
007 Optical Society of America
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2506 J. Opt. Soc. Am. A/Vol. 24, No. 9 /September 2007 A. Alsam and G. Finlayson
nt from the true sensor curves [16]. In a recent study bylsam and Lenz [17], a measure to estimate the goodnessf sensor recovery from calibration data was developed.or experiments based on real data it was shown that thet between the camera’s unknown sensor and the esti-ated set is around 85%. In other words, there is typi-
ally 15% uncertainty in the recovery.A metamer set is the set of all spectra that result in an
xact match to a given RGB value when integrated by thepectral sensitivities of the camera [11]. In the presence ofoise, it is not possible to define a set that results in thexact RGB value without knowledge of the noise level.ue to the various uncorrelated noise sources in a digital
amera, such as dark-current noise, shot noise, and digi-ization noise, estimating the noise level from RGB datas not trivial. To overcome this limitation, Finlayson and
orovic [18] suggested relaxing the constraint thatetamers should result in the exact RGB value and, in-
tead, required that the metamer set results in the RGBalue within a user-defined noise limit. In the presence ofoise the set of metamers is instead referred to as thearamer set.In this paper, we show how metamer sets can be calcu-
ated in the presence of noise when the device’s spectralensitivities are not known. The result is built on two ob-ervations. First, the set of all reflectance spectra is con-ex combinations of certain basic colors that tend to beery bright, or dark, and have high chroma. Second, theonvex combinations that describe reflectance spectra re-ult in convex combinations of RGB values. Thus, givenn RGB value, we find the set of all convex combinationsor the basic colors that generate the response. The corre-ponding set of convex combinations of the basic spectras the metamer set.
To elucidate: starting with an RGB value that is insidehe gamut of RGB values for a calibration set, we writehe three components of the RGB value as a convex com-ination of the neighboring colors’ vector triplets. Foronsaturated colors there are more than one set of neigh-ors that can be used, and for each set a new combinationf corresponding convex weights is obtained. We showhat, when the weights are applied to the higher-imensional spectral data, the resulting vectors areetamers. Thus by calculating all such weight combina-
ions we are able to find a set of metamers that is math-matically guaranteed to result in the same RGB value.he difference from previous methods is that we do nottart by finding a fundamental metamer and a blackpace [19] but assume that any set of convex weights isqually likely to be that of the actual spectrum. Finally,ur approach assumes that noise in the RGB data resultsn subtle perturbations of the RGB value from its real,.e., noise-free, location. Because these perturbations areccounted for by the geometry of the data, the influence ofoise will manifest itself in slight changes of the convexeights. In other words, if the data used are noise free,
hen the recovered set is the metamer set. If, on the otherand, the data are real, i.e., include noise, then theetamer set can be obtained without knowledge of theoise statistics. Finally, similar to previous methods, it isossible to include a user-defined noise limit to recover aarger paramer set.
. BACKGROUNDhe noise-free response of a linear sensor to a spectraltimulus can be modeled as
pi = �Es�Tri, i = 1,2,3, �1�
here E is a diagonal matrix whose diagonal elementsre the intensity of the scene’s illumination at each dis-rete wavelength, i.e., E=diag�e�; s is the surface reflec-ance; ri is the camera sensitivity vector at channel i; and
is the matrix transpose operator. Spectral functions aredequately represented by sampling at 10 nm intervalscross the visible spectrum: 400 to 700 nm [20]. Hence e,, and r are 31�1 vectors. We write the product Es as
c = Es, �2�
hich allows us to write the sensor response to a spectraltimulus as
p = cTR, �3�
here R is a �31�3� matrix whose columns are the red,reen, and blue sensitivities of the camera and p is a 13 camera response vector.In Eq. (3) the color signal c is a 31�1 dimensional vec-
or, while the response of the device is 1�3. This propertyeans that the sensors of a camera collapse the informa-
ion in the color signal from a 31-dimensional (31-D)pace onto a much lower-dimensional space, normally 3-Dpace. As a result of this projection, it is impossible to ex-ctly recover the spectral information of a surface basedn the 3-D camera response, as many spectrally differentignals can integrate to a single response triplet whenrojected down to the 3-D space.Spectral signals that integrate to the same camera re-
ponse are said to be metameric to one another [11]. Fur-her, as Horn pointed out [19], the metamers of one devicere different from those of another unless both devices areithin an exact linear combination of each other.Let us consider a camera’s response to a single color
ignal, cT, such as that in Eq. (2). It is possible to decom-ose cT into two components: (1) one in the range of R;.e., it integrates to a nonzero response; and (2) another ints null space; i.e., it integrates to 0:
cT = craT + cnu
T . �4�
hus for a vector craT in the range of R we have
craT R = p, �5�
hile for cnuT in the null space of R we have
cnuT R = 0. �6�
he vector craT in Eq. (4) is sometimes called the funda-
ental metamer, while cnuT is known as a metameric black
21].For a given RGB response, the fundamental metamer
s unique. On the other hand, because the black metam-rs project to a null response, they can be added arbi-rarily to create new metamers.
The set Q�p� of all metamers that induce a given re-ponse is defined as
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A. Alsam and G. Finlayson Vol. 24, No. 9 /September 2007 /J. Opt. Soc. Am. A 2507
Q�p� = �craT and cnu
T � Rn, subject to craT R = p,
cnuT R = 0,
craT � ran�R�,
cnuT � null�R��, �7�
here ran�R� and null�R� are the range and null spacesf R, respectively. As is evident from the set defined in Eq.7), the calculation of the metamers requires knowledge ofhe range and null spaces of the sensors. Thus to calculatehe metamers, the sensor curves must be known. We notehat additional constraints such as those related to theonnegativity of the color signals, smoothness, natural-ess, and noise statistics are normally incorporated, as
inear inequalities, in the definition of the set [9] in Eq.7).
In the remaining sections of this paper we show that its possible to solve for the set of all metamers associatedith a given RGB camera response without having tonow the device’s spectral sensitivities.
. CONVEXITY IN THE SPECTRAL ANDGB SPACESur method is based on the idea that different sensor sets
esult in different geometries in the RGB space and thathe projection from spectra to RGB space preserves con-exity. The result is based on two insights. First, the set ofll reflectance spectra can be written as convex combina-ions of a set of basic reflectances, e.g., such as thoseound on a reference color chart. Second, the convex com-inations that model spectral interactions map to theame combinations in the RGB domain: 0.5* red+0.5* yel-ow, in the spectral domain, results in an RGB value thats 0.5* RGB (for red) +0.5* RGB (for yellow).
We next give the required background concerning con-exity. This is then used to develop a method to solve foretamer sets when the spectral sensitivities of the device
re not known.Definition. A set Q in Rn is said to be convex if for ev-
ry x and y in Q the line segment joining x and y also liesn Q.
A line segment going from point x to y can be defined as
�x,y� = ��y + �1 − ��x:0 � � � 1�. �8�
herefore, a set Q in Rn is convex if and only if for every xnd y in Q and every � with 0���1 the vector �y+ �1��x is also in Q [22].Let us consider a color signal, c, which is defined as a
onvex combination of two signals, namely, c1 and c2, tohich we have measured camera responses p1 and p2. We
an write c as
c = �c1 + �1 − ��c2. �9�
rom Eq. (3) we know the camera response to c1 can beritten as
p1 = c1TR. �10�
f we scale c1 by �, then the response is a scalar multiplef p , i.e.,
1
�p1 = �c1TR. �11�
If the second color signal, c2, was multiplied by 1−�,hen its corresponding response can be written as
�1 − ��p2 = �1 − ��c2TR. �12�
y making use of additivity, a fundamental property of ainear system [23], we can group Eqs. (11) and (12) as
�p1 + �1 − ��p2 = ��c1T + �1 − ��c2
T�R. �13�
y substituting c from Eq. (9) in Eq. (13), we get
�p1 + �1 − ��p2 = cTR. �14�
he response,
p = �p1 + �1 − ��p2, �15�
s a point in the 3-D RGB space that lies on the line con-ecting p1 and p2. Further, the weights � and �1−�� thatelate c to c1 and c2 are identical to those that relate p to1 and p2.Finally, we remark that � is constrained to be in the in-
erval �� �0,1�. This constraint is based on the spectraleugebauer equations [24] where a color signal is ex-ressed as a convex combination of the basic colors: theeugebauer primaries. Thus if �=0.5, we can simulate
he formation of a new color signal by painting half of aanvas c1 and the other half c2. Viewed from a far enoughistance this canvas will map to a single measurementoint, and it would be as if the scene contained a singleolor signal, c=0.5c1+0.5c2.
. SOLVING FOR THE METAMER SETITHOUT SENSOR KNOWLEDGE
imilar to existing metamer characterization algorithms,ur algorithm requires a set of calibration data. Com-only, such a set is obtained by taking a linear, raw im-
ge of a calibration chart such as the Macbeth Color-hecker under a given illuminant. This set offers us aumber of RGB triplets and their corresponding color sig-als. Given such a set, we define the calculation of theetamer set as the problem of expressing an RGB re-
ponse triplet that is not a member of the calibration datas the set of all possible convex combinations of the RGBalues from the calibration set.
To clarify, we consider a simple example depicted inig. 1. We note that point f in the example can be writtens a convex combination of a and d, a and c, b, and d, or bnd c. All these combinations would result in the samexed coordinates for point f on the line defined by a and b.It is clear from the example in Fig. 1 that the convex
eights that relate f to a and d are different from thosehat relate f to a and c. Having said that, the point f willlways be fixed. The example in Fig. 1 allows us to inferhat, unless there are only two points on a line, the
ig. 1. Point f on the line ad can be defined as a convex combi-ation of other points on the line.
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2508 J. Opt. Soc. Am. A/Vol. 24, No. 9 /September 2007 A. Alsam and G. Finlayson
eights that can fix the value of a point need not benique. We can extend our example to two dimensions toemonstrate the effect of estimating a color signal usingifferent convex weights. Let us assume that we havehree color signals, A, B, and C, which are defined in alane (shown in Fig. 2).Further, let us assume that their corresponding re-
ponses, the RGB data in the case of a camera, are theoints a, b, and c, all of which lie on a line. For a givenoint d, we know that if we estimate d as a convex com-ination of a and c, then our estimate of the color signalhen we apply the weights to the spectral space will be aoint D on the line AC. On the other hand, if the weightsor d are those associated with points a and b, then thestimated color signal will be a point D� of the line AB.he relation between D and D� is that both D and D� areetamers when seen in relation to the projection operator
hat takes A, B, and C to their response spaces a, b, and c.f course, in the case of a camera system, such as the oneefined in Eq. (1), the projection operator, which takes thepectral data from Rn to RGB space, is the spectral sensi-ivities of the device.
So far, we have concluded that it is likely for a givenGB value to have more than one combination of convexeights, all of which are equally valid. We have also dem-nstrated that the weights that relate a given response toset of measured calibration responses can be used to de-ne a color signal. The latter is either the original colorignal that gave rise to the response or a metamer of theriginal. Thus it is possible to account for the set of fea-ible metamers by recovering all the convex combinationsn the response space.
To elucidate this concept, let us consider the set of RGBesponses shown in Fig. 3. Further, let us assume that fornew point, such as the one shown in Fig. 3, we would
ike to calculate all the possible convex weights. Math-matically, we wish to solve for all the weights �i suchhat
p = �1p1 + �2p2 + ¯ + �mpm, subject to
�i=1
m
�i = 1,
∀i�i � 0,
∀ � � 1. �16�
ig. 2. Two-dimensional pictorial representation of the projec-ion from spectral to response space.
i i
n our example, m=24, which corresponds to the 24 coloratches from the Macbeth ColorChecker. In matrix for-at, it is possible to write the system in Eq. (16) as
p = P�, subject to
1T� = 1,
I� � 0,
I� � 1, �17�
here p is a 3�1 vector, P is a 3�m matrix, � is a m1 vector, and 1 and 0 are m�1 vectors of ones and ze-
os, respectively. Further, p is the 3�1 RGB vector for aovel surface, and P is a 3�m matrix whose columns con-ain the RGB responses to the calibration chart.
To solve for the set of all vectors � that would satisfyhe system in Eq. (17), we note that each inequality in Eq.17) defines a hyperplane. A hyperplane, defined by an in-quality of the form ax�b, divides the space into threearts: the first contains the vectors x that satisfy the in-quality, i.e., ax�b; the second is the space of all theeights that violate the inequality, i.e., ax�b; and the
hird satisfies the equality, i.e., ax=b. For a linear systemf equalities and inequalities, such as the one defined inq. (17), intersecting all the hyperplanes results in alosed and convex region, which is the space of all feasibleolutions to the system. Using computational algorithmsuch as quick hull [25], it is possible to solve for the regionf all feasible solutions to a system such as the one de-cribed in Eq. (17). Having done that, we need only applyhe weights to the set of surfaces available from the cali-ration data.We note that, although it is possible to exactly define
n RGB point using only the extreme points on the RGBamut, for the calculation of the weights it is important toeep all the calibration points, both internal and ex-remes. This is true due to the idea demonstrated in thexample of Fig. 2, where we saw that different combina-ions would lead to different surface estimates. For the
ig. 3. Set of 24 RGB responses calculated based on the re-ponses of the Sony DX camera to the surfaces of the MacbetholorChecker is shown as the solid black disks. Further, a point
nside is shown as a black ring.
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A. Alsam and G. Finlayson Vol. 24, No. 9 /September 2007 /J. Opt. Soc. Am. A 2509
ake of completeness, we can state that if the extremeoints in the n-dimensional spectral space were available,hen it would be sufficient to base the calibration of theetamer set on those extremes and their corresponding
amera RGB values.To summarize, we can list the steps involved in our ap-
roach for calculating metamer sets. (1) Find the RGBalues for a set of calibration surfaces with known spec-ral characteristics (e.g., by taking a picture of a calibra-ion chart). (2) Given a new response, express that re-ponse in the form of Eq. (17). (3) Use the quick-hulllgorithm to solve for all the weights that satisfy theiven constraints. (4) Apply the resulting weights to thepectra of the calibration surfaces to obtain the metameret.
We note that in the traditional metamer set formula-ion [8,9,26] additional constraints need to be imposed onhe feasible metamer space. Specifically, the metamersre constrained to be nonnegative, smooth, and naturalnd are constrained to force the maximum value of a colorignal to be less than one. In our proposed formulation, nodditional constraints are needed. The proposed formula-ion constrains a color signal to be defined inside thepace of the available calibration signals. Thus the recov-red signal is guaranteed to satisfy all the previouslyentioned constraints.
. EXPERIMENTS AND RESULTSn this section we present results of two experiments: therst is based on the calculation of the metamer set foreal data captured with a Nikon D70 camera with un-nown sensitivities; the second is based on synthetic databtained based on the set of sensors shown in Fig. 8. Inoth experiments, the 24-patch Macbeth ColorCheckeras used to provide the spectral and RGB data. Further,
he spectral data were measured using a Minolta CS-1000pectroradiometer under the daylight simulator of theacbeth Verda viewing booth. In the first experiment, the
amera responses were captured in the Nikon raw-imageormat. The response, RGB, data were checked for linear-ty, and the dark noise was subtracted. Spatial uniformityas accounted for by averaging the spectral and RGBata over the center of each color patch.
. Real Data: Nikon D70his experiment was designed to show the usefulness andower of the method for real data with unknown levels ofoise and unknown sensitivities. Furthermore, in this ex-eriment we include an example where three differentensor recovery methods were used to estimate the sensoret and, as a second step, the metamers associated with aingle surface were recovered. The sensor recovery algo-ithms used in this experiment were: principal eigenvec-ors [13], quadratic programming [12], and Tikhonovegularization [14].
Using the algorithm proposed in this paper, we calcu-ated the metamers of six RGB triplets. In the RGB space,4 points were extremes while the rest (the first fouratches, the tenth, and the gray colors) were internal. Byefinition, the proposed algorithm will result in a singleetamer to the extreme points: the actual spectrum of
he surface. For each of the other surfaces, a set of convexeight combinations was obtained. This set was then ap-lied, with no modification, to the calibration spectra toesult in the sought-after metamer sets.
In Fig. 4 we plot the metamer sets for four of the sur-aces. Further, in Fig. 5, we plot the convex hull of the
etamers in the CIExy chromaticity space. For clarity, weid not include the gray surfaces. In this figure we havelso superimposed the chromaticity values of the originalata. We note, from the figure, that the metamer set in-ludes the original point as one of its members. We alsoote that the location of the original point is fairly arbi-rary in the set, which is exactly why color correction isifficult the more metameristic the sensors are.As an indication of the level of metamerism that would
e experienced when the camera metamers are projectednto the CIEXYZ space, we measured the CIELab differ-nces between the Lab of the actual surfaces and those ofhe camera metamers. In Table 1 we report the delta Labifferences for the first, second, third, fourth, and twenti-th patches of the Macbeth Color Checker under D50.
ig. 4. (Color online) Four examples of metamer sets for the Ni-on D70 camera based on the Macbeth ColorChecker. Althoughot shown in the figures, the original spectrum is always a mem-er of the set. Note that the x axis of the figures represents theavelength in 400–700 nm and the y axis represents the per-
entage reflectivity 0–1.
ig. 5. (Color online) Metamer sets in the CIExy space for theikon D70 camera based on the Macbeth ColorChecker. We note
hat the original value is always a member of the set. Note thathe x axis of the figure represents x=X / �X+Y+Z� and the y axisepresents y=X / �X+Y+Z�.
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2510 J. Opt. Soc. Am. A/Vol. 24, No. 9 /September 2007 A. Alsam and G. Finlayson
Using three different sensor recovery algorithms, prin-ipal eigenvectors [13], quadratic programming [12], andikhonov regularization [14], we estimated the sensorurves shown in Fig. 6.
Although, as is clear in Fig. 6, the three sensor esti-ates show very similar trends, they are noticeably dif-
erent in details. In the background section, we men-ioned that the calculation of metamer sets based on theethods proposed in the literature [9] requires the calcu-
ation of the range and null spaces of the sensor sets.hus differences in the sensor estimates will manifesthemselves in the calculation of metamers, where theetamers predicted with one sensor estimate need not
ecessarily be metamers for the other estimates. An ex-mple of this is shown in Fig. 7, where we have calculatedhe metamers associated with a single surface based onhe sensor estimate achieved by quadratic programmingsolid curves in Fig. 6) and, as a second step, integratedhe metamers with the sensors predicted by principaligenvectors and Tikhonov regularization. The metamer
Table 1. Delta Lab Error between the Actual ColorSignal and the Set of Metamers Predicted with
aThe camera in this experiment is a Nikon D70, and the illuminant is D50.
ig. 6. (Color online) Three sets of estimates to the Nikon D70amera. The methods are principal eigenvectors (dotted curve),uadratic programming (solid curve with circles), and Tikhonovegularization (dashed–dotted curve).
alculation algorithm [9] results in zero error; i.e., theetamers predicted with quadratic programming result
n exactly the same RGB value. Thus the differences andistribution seen in Fig. 7 are due solely to the discrepan-ies between the sensor estimates achieved by the differ-nt methods.
. Synthetic Data: Sony DX960he objective of this second experiment is to offer com-arison with the well-established algorithm of Finlaysonnd Morovic [9]. We started by calculating the metameret of a surface based on the proposed method (the SonyX sensors shown in Fig. 8 were used) and compared the
esult with the metamer set obtained by Finlayson’s algo-ithm with the following constraints: nonnegativity,
ig. 7. Metamer clouds in the RGB space. The gray disks arehe result of integrating the metamers calculated with the qua-ratic programming sensor estimate with the sensors predictedith Tikhonov regularization. The black stars are the RGB val-es achieved by integrating the same metamers with the sensorurves estimated by principal eigenvectors.
ig. 8. Spectral sensitivities of a Sony DX camera. Note that theaxis of the figure represents the wavelength in 400–700 nm
nd the y axis represents the sensitivity.
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A. Alsam and G. Finlayson Vol. 24, No. 9 /September 2007 /J. Opt. Soc. Am. A 2511
moothness, and that all the metamers should have aaximum value no greater than unity. An example of the
esults is shown in Fig. 9.In the previous example we notice that although the
roposed set, shown in solid curves, is inside the metameret obtained with Finlayson’s algorithm, shown in dottedurves, the latter set is greater than that arrived at withhe proposed algorithm. In other words, the proposed sets a subset of Finlayson’s set. This was true for all the sur-aces. We noticed, however, that while the proposedethod results in a single solution for each of the satu-
ated colors (those on the boundary of the RGB gamut)inlayson’s algorithm returned a set greater than unity.hus we calculated the set with the additional constraintf naturalness, which restricts the solution to be insidehe convex hull of the calibration spectra. In doing so, weound that the two sets were an exact match. An example
ig. 9. Example of the proposed metamer set method comparedith Finlayson’s algorithm. The calculations are based on theony DX camera and the Macbeth ColorChecker. Note that the xxis of the figure represents the wavelength in 400–700 nm andhe y axis represents the percentage reflectivity 0–1.
ig. 10. (Color online) Example of the proposed metamer setethod (right), compared with Finlayson’s algorithm (left), with
he added constraint of naturalness. The calculations are basedn the Sony DX camera and the Macbeth ColorChecker. Notehat the x axis of the figure represents the wavelength in00–700 nm and the y axis represents the percentage reflectivity–1.
s shown in Fig. 10. Notice that the reason why Finlay-on’s algorithm appears to result in fewer metamers ishat the algorithm returns the extreme points only.
. DISCUSSIONhe algorithm introduced in this paper has a number ofimilarities and differences when compared with previousechniques [9]. In this section, we address a number ofhese characteristics.
First, we examine the dependency of the results on theamut: in the metamer set algorithm by Finlayson andorovic [9], the spectral space of metamers is character-
zed by the data’s principal components. In this work, weharacterize the spectral space by the data points them-elves. Thus metamers generated with the proposed algo-ithm are constrained to be inside the gamut of the cali-ration data. In [9], metamers can be outside this gamutnless the constraint of naturalness is imposed. Based onhis, care must be taken in the choice of calibration data.his aspect would, however, be taken care of if an emis-ive calibration chart such as the one marketed by HP issed.Another important issue is the computational complex-
ty of the algorithm: in this paper we argued that all thealibration points, both internal and external, need to bencluded in the calculation of the metamer set. Given thatlarge-gamut calibration chart, such as the new emissive
alibration charts, is available, the metamer calculationould result in adequate results, i.e., sufficient for theurpose of device characterization, based on around 30oints. This same aspect, however, means that the com-lexity of the calculation is greater than that in Finlay-on’s previous work where four to five principal vectorsre used. On the other hand, when the metamer set algo-ithm in [9] is used, the dimension of the principal vectorss taken as 31; i.e., in the case of four vectors the bases are1�4. In the proposed algorithm, the weight calculations carried out in the 3-D RGB space. Thus the dimensionf the bases is related to the number of calibration pointsather than the dimension of the sampled spectral space.
The main difference between the two algorithms is,owever, that the proposed algorithm does not requirenowledge about the sensor. This aspect of the algorithms, in fact, not only useful in calculating metamers butlso in using knowledge of the metamers to address otherhallenges. Alsam and Lenz [17] have shown that the cal-ulation of metamers without sensor knowledge can besed to robustly estimate the unknown device sensitivi-ies. Further, it is possible, using the proposed algorithm,o solve for a single reflectance inside the metamer set. Inther words, solve for a reflectance without solving for thentire set. In [18] Finlayson and Morovic showed thataving a reflectance that is guaranteed to be a member ofhe metamer set improves upon color calibration. How-ver, the implementation in [18] requires both knowledgef the sensor sets and the calculation of all the metamerets corresponding to the patches in a calibration target.sing our algorithm, it is possible to solve for a member
f the metamer set for little computational cost. This can,gain, be achieved without knowledge of the sensors.
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2512 J. Opt. Soc. Am. A/Vol. 24, No. 9 /September 2007 A. Alsam and G. Finlayson
. CONCLUSIONSn this paper we presented a method to calculate theetamer set with the advantage that knowledge of device
pectral sensitivities is not required.The result is built on two observations. First, the set of
ll reflectance spectra is convex combinations of certainasic colors that tend to be very bright (or dark) and haveigh chroma. Second, the convex combinations that de-cribe reflectance spectra result in convex combinations ofGB values. Thus, given an RGB value, it is possible tond the set of convex combinations of the RGB values ofhe basic colors that generate the same RGB value. Theorresponding set of convex combinations of the basicpectra is precisely the metamer set.
Two experiments were presented. In the first, we dem-nstrated the usefulness of the method in calculating theetamer sets in the presence of noise and when the sen-
itivities are not known. In the second experiment, wehowed that when the naturalness constraint is incorpo-ated, the proposed algorithm results in a match exact tohat achieved by Finlayson and Morovic. This leads us totate that when our method is used no additional con-traints are needed; constraints relating to nonnegativity,moothness, and naturalness are naturally incorporatedn the solution. Knowing that (a) previous studies showedhat using metamer sets greatly improves upon the accu-acy of color correction, (b) the calculation requires knowl-dge of the spectral sensitivities, and (c) estimating theensor curves is an ill-posed problem, we believe that ouresults offer a solid foundation to solving the problem ofetamer set calculation without the need for spectral
alibration.
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