Reflectance spectra recovery from tristimulusvalues by adaptive estimation with
metameric shape correction
Simone Bianco
DISCo-Dipartimento di Informatica, Sistemistica e Comunicazione, Università degli Studi di Milano-Bicocca,Viale Sarca 336, Edificio U14, 20126 Milano, Italy ([email protected])
Received February 16, 2010; revised June 23, 2010; accepted June 23, 2010;posted June 28, 2010 (Doc. ID 124119); published July 30, 2010
. INTRODUCTIONhe most used representation of colors is with a standardrichromatic color coordinate system. There are cases inhich this representation is not enough. This representa-
ion, besides being illuminant and observer-dependent, islso affected by metameric issues. The most informativend complete way to describe a color is to give its reflec-ance spectrum, which is defined as the ratio of the re-ected light to the incident light. The reflectance spectraan be directly measured using spectrophotometers; un-ortunately the most used color acquisition devices cap-ure the color signal by acquiring only three channels.ince the reflectance spectra are essential for many appli-ations, the problem of recovering the reflectance spectrarom triplets in a trichromatic color coordinate systemas been extensively studied. Dupont in his comparativetudy [1] considered several optimization methods to re-over the spectra reflectances, including the simplexethod, the simulated annealing method, the Hawkyardethod [8], genetic algorithms, and neural networks. All
he methods considered are global methods, which give alobal solution. Recent works have improved the spectraecovery accuracy using local methods [2,3].
The method proposed in this work is a localptimization-based method which is able to recover theeflectance spectra with the desired tristimulus values,hoosing the metamer with the shape most similar to theeflectances available in the training set. The novelty ofhe paper comes from the introduction of what is to theuthor’s knowledge a new optimization function com-osed of heterogeneous terms: a colorimetric error term, apectral error term, and two shape feasibility terms.
The performance assessment of the proposed methodnd of the benchmarking algorithms considered has beenone on four different spectral datasets in terms of differ-nt colorimetric errors, different spectral similarity met-ics, and spectral residuals plots. The sensitivity analysis
f the proposed optimization function with respect to thearious terms of which it is composed is also reported. Ac-ording to all the error metrics considered, the proposedlgorithm was able to recover the spectral reflectancesith a higher accuracy than all the benchmarking meth-ds considered, which are described in the next section.
. REFLECTANCE SPECTRA RECOVERY:ROBLEM FORMULATION AND RELATEDORKS
he CIE XYZ tristimulus values of a surface with spec-ral reflectance r��� that is viewed under an illuminantith spectral power distribution I��� can be determineds
X = k� r���I���x���d�,
Y = k� r���I���y���d�,
Z = k� r���I���z���d�, �1�
ith
k =100
� I���y���d�
, �2�
here x���, y���, and z��� are the CIE color matchingunctions, and the integral is computed over the visiblepectrum. In practice, spectral functions can be repre-ented by their samples, and spectral integrals may bepproximated by sums. If spectral functions are evenly
ampled at N wavelengths, then Eqs. (1) can be writtens
�X
Y
Z� = Mr, �3�
here M is a 3�N matrix with the samples of I���x���,���y���, and I���z��� stacked row-wise, and r is a N�1pectral reflectance vector made of the samples of r���.
The problem of recovering the reflectance spectra fromristimulus values is that of estimating r from the outputf Eq. (3). The most immediate and straightforward solu-ion is to use the pseudo-inverse algorithm (PINV) [4],
r = �MTM�−1MT�X
Y
Z� = M+�
X
Y
Z� . �4�
he performance of the pseudo-inverse algorithm in thease of an inadequate number of data points is not tooood and could yield atypical spiky estimates of the spec-rum, which are due to the inversion of a N�N bad con-itioned matrix.The most successful approaches for the estimation of
he spectral reflectance from the CIE XYZ values areased on applying dimensionality reduction techniques.he algorithm was originally formulated by Fairman andrill (PINV-PCA) [5]. This method exploits linear models
o represent each reflectance through the weighted sum ofsmall number k of basis functions vi,
r � v + �i=1
k
aivi, �5�
here v denotes the mean of the spectral reflectances ofhe training set, vi is the ith basis vector of the principalomponent analysis (PCA) and is calculated from thepectral reflectances of the training set after subtracting, and ai denotes the coefficient of the corresponding PCAasis vector. Since we have only one set of CIE XYZ tris-imulus values for each reflectance we want to recover, kn Eq. (5) takes the value of 3. Substituting Eq. (5) intoq. (3) yields
�X
Y
Z� = Mv + M�v1 v2 v3�
a1
a2
a3� = Mv + MV�
a1
a2
a3� , �6�
hich can be easily inverted to find the coefficients aiwith i=1, . . . ,3),
�a1
a2
a3� = �MV�−1�
X
Y
Z� − Mv� . �7�
he main advantage of this method is that now we haveo invert a 3�3 matrix that is better conditioned and pro-uces less spiky estimates of the spectrum. The final es-imated spectrum is then obtained by substituting theutput of Eq. (7) into Eq. (5).
In order to use more PCA basis vectors, Harifi et al. [6]roposed a modified version (PCAemRe) of the previous
ethod in order to estimate the CIE XYZ values under aecond illuminant. This permits one to have twice the in-ormation and thus to use k=6 PCA basis vectors. TheIE XYZ values under a second illuminant are estimated
rom the ones under the given illuminant by a 3�11 poly-omial transformation matrix. Equation (7) then becomes
�a1
a2
a3
a4
a5
a6
� = �MV�−1�Xill1
Yill1
Zill1
Xill2
Yill2
Zill2
� − Mv� , �8�
here now V= �v1 v2 v3 v4 v5 v6 .In order to improve the reconstruction accuracy of the
CA method, Zhang and Xu [2] proposed a local PCAethod (PCAmuBa). They used different sets of principal
omponents, one for each of the tonal subgroups they con-idered. They defined 11 tonal subgroups including tenifferent hues and a near gray region. If the chroma ofach reflectance was less than 15, it was heuristically de-ned as belonging to the near gray subgroup. If thehroma was greater than 15 it was assigned to one of theen hue subgroups into which the spectral space was di-ided according to ten separate hue angles in the Munsellpace.
Mansouri et al. [3] proposed an adaptive PCA algorithmadaPCA) for reflectance estimation. The algorithm firstomputes a global PCA basis for all the training sets andstimates the reflectance using Eqs. (7) and (5). Then iteasures the likelihood between the estimated reflec-
ance and each element of the training set. A new traininget is build by keeping only those elements whose similar-ty falls in the range [95%, 100%]. Then a new PCA analy-is is performed from this new set to derive a new basisith which the final estimation is done. The likelihood is
alculated using the noncentered correlation coefficient,ommonly known as the goodness of fit coefficient (GFC)7]:
GFC =�j=1
N
rjrj
��j=1
N
�rj�2 1/2��j=1
N
�rj�2 1/2 , �9�
here rj and rj are the jth samples of the original and re-onstructed spectra, respectively.
A different approach that does not use any dimensionaleduction technique is the Hawkyard method (HAW) [8].t is an iterative method, and its principle states that theeflectance spectrum is the weighted sum of the primaryunctions. The primary functions are defined as the rowsm1 m2 m3 T of the matrix M of Eq. (3). Given the tri-timulus values X ,Y ,Z of the reflectance we want to re-over, the initial estimate is computed as
here �Xo Yo Zo = �X Y Z . From this reflectanceurve the corresponding tristimulus values X� ,Y� ,Z� arealculated using Eq. (3). Then the differences,
�X = X� − X,
�Y = Y� − Y,
�Z = Z� − Z, �11�
re calculated and Xo ,Yo ,Zo in Eq. (10) are replaced as
Xo = Xo − �X,
Yo = Yo − �Y,
Zo = Zo − �Z. �12�
he process is repeated until the required accuracy or theaximum number of iterations has been reached.All the algorithms described have the problem that the
ecovered reflectance spectra can be unfeasible: in fact theecovered reflectance can present lobes outside the range0%, 100%] or can present atypical spikes due to the re-onstruction method or to the clipping of the recovered re-ectance in the admissible range. To address these prob-
ems Zuffi et al. [9] proposed an algorithm (ZSS) to obtaineasible reflectance spectra. The algorithm starts with es-imating the recovered reflectance r0 through Eqs. (7) and5) and then decomposing it as
r0 = r0,f + r0,+ + r0,−, �13�
here r0,f ,r0,+ ,r0,− are the parts of the spectrum inside,bove, and under the admissible range, respectively. Theristimulus values of r0,+ and −r0,− are computed andiven as input to Eqs. (7) and (5) to obtain r0,+ and r0,−. Aew reflectance is defined as
r1 = r0,f + r0,+ + r0,−. �14�
his new reflectance is decomposed into r1,f, r1,+, and r1,−.smoothing procedure can be run on r1,+ and r1,− which
re then given as input to Eqs. (7) and (5). All the opera-ions described are looped until the required accuracy orhe maximum number of iteration has been reached.
. PROPOSED ALGORITHMhe proposed algorithm starts with using a dimensional-
ty reduction technique on the measured training reflec-ances r to have a linear base V0 for them by first sub-racting the average value v0 of the spectral reflectancesf the training set. An initial estimate r0 of the recon-tructed spectra is then obtained using Eq. (7).
The spectral residuals b between the reconstructedpectra r0 and the measured spectra r are then computed,.e., b=r−r . The average spectral residual b is computed
0
nd subtracted, and a dimensionality reduction techniques employed to extract a linear basis B for b− b.
The likelihood between the initial estimate r0 of thepectra and each element of the training set is then mea-ured using the GFC in Eq. (9). The samples with a nor-alized likelihood in the range �p ,1 are retained and a
imensionality reduction technique is employed to extractlinear basis V1 for them by first subtracting their aver-
ge reflectance value v1.An optimization routine is then run to estimate the co-
fficients �a1 , . . . , a6 to give to the bases V1 and B in or-er to give the final reflectance estimation r. The optimi-ation function used is composed of four different terms: aolorimetric error term, two shape feasibility terms, and apectral error term.
Before introducing the optimization function, we haveo first define the terms of which it is composed. Let usefine �E94�r , r� as the �E94 colorimetric error under theIE D65 illuminant between the computed L*a*b* valuesf the estimated r and the measured r; as a colorimetricrror, �E94 is chosen as it is a refinement of the �E76 er-or. It is computationally simpler than successive erroretrics �E2000 and �ECMC and permits one to leaveE2000 and �ECMC as test metrics. For the shape feasibil-
ty, let us define u−�r� as the sum of the values assumedy the recovered reflectance below the normalized admis-ible range [0,1], i.e.,
u−�r� = �j=1
N
rj:rj � 0. �15�
et us define u+�r� as the sum of the values assumed byhe recovered reflectance above the admissible range0,1], i.e.,
u+�r� = �j=1
N
rj:rj � 1. �16�
or the spectral error, let us define GFCmax as the maxi-um GFC between the recovered r and all the reflec-
ances in the training set.Then the recovered reflectance r can be found by
re the values found by optimization. Equation (18) ishen the optimization function adopted. It is possible tootice that it is composed of four different terms: the firstne is a colorimetric error, which is used to find the mostolorimetrically similar solution. The second and thirdnes are shape feasibility terms: they are used to find aolution which has values in the admissible range [0,1].he fourth term is a spectral error, which is used to find
he solution which is the most spectrally similar to the re-ectances in the training set. The use of the colorimetricrror in the optimization function is a common choice forptimization-based algorithms for the recovery of thepectra reflectances [1]. The use of the shape feasibilityerms has been inspired by the work of Zuffi et al. [9], al-hough here they are used in a different way. Finally, these of the spectral error term, proposed by Mansouri et al.3] as a criterion to choose a subset of the training set onhich to compute the refined PCA basis, is used here toroduce the solution which is the most spectrally similaro the reflectances in the training set, and thus the mostpectrally likely. The algorithm is reported in a pseudo-ode in Table 1.
The dimensionality reduction technique employed forhis algorithm is the independent component analysisICA) [10]. The ICA was developed to find statistically in-ependent components in the general case where the datare non-gaussian, which makes it different from otheractor analysis techniques where the data are modeled asinear mixtures of some underlying factors [11].
The weights in Eq. (18) are heuristically chosen as� ,� ,� ,= �1,100,100,1 in order to be sure that the op-imization algorithm first finds feasible spectra and thenptimizes simultaneously the colorimetric and the spec-ral errors. The weights � ,� are chosen 2 orders of mag-itude higher than � , to prevent unfeasible solutions.bviously a different choice for the weights could beade, and different terms could be used in the optimiza-
ion function. As a further analysis, the sensitivity of thelgorithm with respect to the weights �� ,� ,� , is re-orted in Appendix A.The optimization algorithm used is the pattern search
12] which is a direct search method for nonlinear optimi-ation that does not require any explicit estimate of de-ivatives. The general form of a pattern search methodan be described in the following way. At each step k, weave the current iterate xk, a set Dk of search directions,nd a step-length parameter �k. Usually the set Dk is theame for all iterations. For each direction dk�Dk, we set+=xk+�kdk (the “pattern”) and we examine f�x+�, where
Table 1. Pseudo-Code of the Proposed Algorithm
eginDerive a basis V0 for the whole training setEstimate the recovered spectraCalculate the spectral residuals between the recoveredand the training set spectraDerive a basis B for the spectral residualsUse the basis V0 to give an initial estimate of the recoveredspectra using Eq. (5) for each initial estimate of therecovered spectraDo
Calculate the GFC with the spectra in the training setDerive a basis V1 for the spectra with GFC� �p ,1Minimize Eq. (18)Compute the final recovered spectra usingEq. (17)
Endnd
is the function to be minimized. If ∃dk�Dk : f�x+�� f�xk�,e set xk+1=x+ and �k+1=�k�k with �k�1; otherwise, we
et xk+1=xk and �k+1=�k�k with �k�1. The algorithmtops when step �k is smaller than a fixed threshold orhen the maximum number of iterations has been
eached. In this work we have chosen �k=2, �k=0.5, Dk�±e1 , ±e2 , ±e3 , ±e4 , ±e5 , ±e6� (i.e., the six versors of theartesian coordinate system of the Euclidean space R6
aken with both positive and negative directions), and0=0.1. Finally, the likelihood range to select the reflec-
ances used to extract the local basis V1 is set to [0.95,1],.e., p=0.95.
. EXPERIMENTAL SETUPhe experimental data include five different data sets:ne was used as a training set for all the methods of Sec-ions 2 and 3 and the other four were used as test sets.he training set is composed of the odd samples of theunsell Atlas. The even samples of the Munsell Atlas, the
70 samples of the Vhrel dataset [13], the 24 samples ofhe GretagMacBeth Color Checker CC (MCC), and theentral 172 samples of the GretagMacBeth Color CheckerC (MDC) (except the eight glossy samples) were used as
est sets. All the data were sampled from 400 to 700 nm in0 nm steps.Five different error metrics are used to assess the per-
ormances of the considered algorithms: the �E94 colori-etric error under the training CIE D65 illuminant andnder the CIE A and CIE F2 illuminants, the peak signal-o-noise ratio (PSNR), which for two n-dimensional reflec-ances r and r is defined as
PSNR�r, r� = 20 log10
1
�1
n�i=1
n
�ri − ri�2
, �19�
nd the GFC [see Eq. (9)].
. EXPERIMENTAL RESULTShe experimental results are reported in Table 2. Threeifferent statistics are reported for �E94: the averagealue, the 95% percentile, and the standard deviation.ollowing [14] the PSNR values have been subdivided
nto three intervals: PSNR�34 dB, 34 dBPSNR40 dB, and PSNR�40 dB which correspond to poor, ac-
urate, and good spectral estimations, respectively. Fol-owing [14] the GFC values have been subdivided intoour intervals: GFC�0.995, 0.995GFC�0.999, 0.999
GFC�0.9999, and GFC�0.9999 which correspond tooor, accurate, good, and excellent spectral estimations,espectively. For each of the considered intervals the per-entage of spectra within it is reported. Although the GFCas already been used as a spectral similarity term in theptimization function reported in Eq. (18), it is used hereifferently: in the optimization function it was used tond the most likely solution, i.e., the most spectrally simi-
ar solution to the spectra reflectances in the training set;ere it is used as a similarity measure between the recon-tructed and measured spectra.
It can be seen from Table 2 that only two out of theeven benchmarking methods considered were able to al-ays recover a spectra with almost the same tristimulusalues under the CIE D65 training illuminant: they arehe HAW [8] and the ZSS [9]. The colorimetric errors un-er the CIE A and F2 illuminants show that none of themas able to perfectly reconstruct the spectra. Among theenchmarking algorithms considered, the ones with theighest overall performances are the adaPCA [3] and the
Table 2. Algorithm Performances on All the DatasDeviation (std) of the �E94 Colorimetric Error undReconstructed Spectra with Poor, Accurate, and G
CAmuBa [2]. Both of them were unable to give a perfectetameric solution under the CIE D65 training illumi-ant, but were able to obtain lower colorimetric errors un-er the CIE A and F2 test illuminants with respect to theAW and ZSS algorithms. Better results are obtainedlso for the spectral similarity measures of PSNR andFC.It is possible to notice that the proposed method is able
o always recover spectra with almost the same tristimu-
onsidered: Average, 95% Percentile, and Standarde CIE D65, A, and F2 Illuminants; Percentages ofeconstructions Judged by PSNR; Percentages of
nd Excellent Reconstructions Judged by GFC
er Ill. F2 PSNR (%) GFC (%)
% std Poor Accurate Good Poor Accurate Good Excellent
us values under the CIE D65 training illuminant, almostalving the colorimetric errors under the CIE A and F2esting illuminants with respect to the HAW and ZSSethods. Furthermore the proposed method is also able
o lower the colorimetric errors obtained by the best per-orming benchmarking algorithms, i.e., adaPCA and PCA-uBa. For what concerns the PSNR and GFC error met-
Table 3. Algorithm Performances on All the Dataunder D65, A,
ics, it can be noticed that the proposed method reacheshe highest good/excellent percentage of the reconstruc-ion with respect to all the benchmarking algorithms con-idered.
In Table 3 the average �E2000 and �ECMC2:1 colorimet-ic errors between the measured and reconstructed spec-ra under the CIE D65, A, and F2 illuminants are re-
Considered: Average �E2000 and �ECMC2:1 Errors2 Illuminants
orted. It is possible to notice that even using differentolorimetric errors, the ranking of the considered methodsemains unchanged with respect to Table 2.
In order to objectively evaluate the performance of theethods considered, and to assess if they are statistically
ignificant, the Wilcoxon sign test (WST) [15] is used. TheST is a statistical test able to compare the whole error
istributions of two methods without making any as-umptions about the underlying error distribution. Mak-ng all the pairwise comparisons among the �E94 erroristributions obtained by the considered algorithms onach different dataset, it is possible to generate a scoreepresentative of the number of times that each algorithmas been considered better or equivalent to the others. Allhe pairwise comparisons are done with a significanceevel of �=0.05. The scores obtained for the �E94 erroristributions under the CIE D64, A, and F2 illuminantsre reported in Table 4. The best score for each column iseported in bold font. It can be noticed that for all theataset-illuminant combinations considered, the proposedlgorithm always gives the best spectral reconstruction.he scores obtained for the �E2000 and �ECMC2:1 error dis-
ributions are not reported as being identical to the oneseported in Table 4.
The initial ICA basis V0 obtained for the proposed al-orithm on the training set is reported in Fig. 1, while theCA basis B of the spectral residuals is reported in Fig. 2.he computation of the tristimulus values shows that theasis V0 and then V1 are the ones that are used by theroposed algorithm in trying to reconstruct spectra withhe desired tristimulus values. The basis B, having tris-imulus values with magnitudes on the order of 10−15, issed by the proposed algorithm to correct the shape of theeconstructed spectra by adding a linear combination ofetameric blacks, thus identifying the metameric solu-
ion which minimizes Eq. (18).In Figs. 3–7 the average spectral residuals betweeneasured and reconstructed spectra are reported as func-
ions of wavelength. In each figure all the consideredethods are compared on a different dataset: the half ofunsell dataset used as the training set (Fig. 3), the
ther half of Munsell dataset (Fig. 4), the Vhrel datasetFig. 5), the MCC dataset (Fig. 6), and the MDC dataset
Table 4. WST Scores, Evaluated on the �E94Subdivided for Each Datas
aThe best score for each dataset–illuminant combination is reported in bold font.lgorithm gives statistically better or equivalent results.
Fig. 7), which are all used as test sets. From the analysisf Figs. 3–7 it is possible to notice how the proposedethod tends to have lower and flatter spectral residual
lots.
. CONCLUSIONSn this work a local optimization-based method which isble to recover the reflectance spectra with the desiredristimulus values, choosing the metamer with the mostimilar shape to the reflectances available in the traininget, is proposed. The reflectance spectra of the Munselltlas, the Vhrel dataset, the GretagMacBeth Colorhecker CC, and the GretagMacBeth Color Checker DCere used as samples in this study. Different error met-
ics have been considered to assess the performances ofhe proposed methods: the �E94 colorimetric error underhree different illuminants (CIE D65, A, and F2), theE2000 colorimetric error, the �ECMC2:1 colorimetric error,
he peak signal-to-noise ratio (PSNR), and the goodness oft coefficient (GFC). According to all the error metricsonsidered, the proposed algorithm was able to recover
Distributions, Obtained by the Algorithmsd Illuminant Considereda
he spectral reflectances with a higher accuracy than allhe state of the art methods considered.
PPENDIX An Tables 5–7 the results of the sensitivity analysis of theptimization function of Eq. (18) with respect to the vari-us terms of which it is composed are reported. The analy-is is simplified changing one weight at a time and as-uming �=�. Nine different values for each weight areonsidered: 0 (i.e. the corresponding term is not consid-red in the optimization), 0.001, 0.01, 0.1, 1, 10, 100,000, and setting all the other weights to 0 (i.e. the corre-ponding term is the only one considered in the optimiza-ion). The analysis results are reported in terms of the av-rage �E94 colorimetric error under the CIE D65lluminant, and the average GFC, for all the datasets con-idered. It is possible to notice that the colorimetric andpectral errors are both needed in the optimization func-ion of Eq. (18) (see the first and last columns of Tables 5nd 7). The shape feasibility terms, given the two-step na-ure of the proposed algorithm, do not seem to have a
Table 5. Sensitivity Analysis of the Optimization F�E94, i.e.,
eep impact on the final solution. When only the shapeeasibility terms are used (last column of Table 6) the fi-al solution is similar to the ones obtained by the adaPCAnd PCAmuBa algorithms (see Table 2).
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