Research ArticleSpectral Reflectance Recovery from TristimulusValues under Multi-Illuminants

Guangyuan Wu 12 Linyong Qian3 Guichun Hu14 and Xiaozhou Li1

1School of Light Industry and Engineering Qilu University of Technology (Shandong Academy of Sciences) Jinan 250353 China2National-Local Joint Engineering Laboratory for Digitalized Electrical Design Technology Wenzhou 325035 China3School of Physics and Electronic Engineering Jiangsu Normal University Xuzhou 221116 China4Key Laboratory of Auxiliary Chemistry and Technology for Chemical Industry Ministry of EducationShaanxi University of Science and Technology Xirsquoan 710021 China

Correspondence should be addressed to Guangyuan Wu wgy19882000163com

Received 23 February 2019 Accepted 9 October 2019 Published 3 November 2019

Academic Editor Khalique Ahmed

Copyright copy 2019 Guangyuan Wu et al +is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

An efficient procedure for recovering spectral reflectance using an objectrsquos tristimulus values under multi-illuminants is proposedby adapting with the characteristics of the testing sample to obtain the transformation matrix of pseudoinverse Specifically wepropose the reference illuminants selection strategy and local sample weighted strategy to obtain the optimal transformationmatrix under multi-illuminants condition Selecting the reference illuminants are based on the result of the spectral angle mapper(SAM) statistics +e number of the selected local training samples and the weighted local samples can be determined by using themulticolor space Euclidean distance To compare the experimental results the proposed method significantly increases thespectral and colorimetric accuracy for the spectral reflectance recovery process

1 Introduction

+e objectrsquos spectral reflectance is almost the definition ofldquofingerprintingrdquo that predicts accurately the object appear-ance under arbitrary illuminants and observers +e mostaccurate and effective information to represent an objectcolor by its spectral reflectance is highly desirable since it isessential in common application scenarios for exampleprinting inspection disease diagnosis textile color match-ing and computer vision [1ndash4] Normally the spectral re-flectance could be directed to acquire from thespectrophotometers and spectral cameras Unfortunatelyportability complexity and expensiveness of these deviceslimit their own availability +e most used colorimetricvalues instead can be easily procurable by digital stillcameras smartphone and colorimeters

+e colorimetric values such as RGB or CIE XYZ tri-stimulus values obtaining from three channels only recordcolor information under fixed viewing conditions +ere are

some cases in which this representation is not enoughbecause it is heavily dependent on the illuminant So thecolorimetric values still vary significantly with the illumi-nants Although the computation of colorimetric values isuniquely made from the objectrsquos spectral reflectance theproblem in the spectral reflectance recovery calculating fromthe colorimetric values is usually ill-posed Many differentmathematical methods are still widely studied to recover thespectrum for example the pseudoinverse method (PI) [5]principal component analysis (PCA) [6] matrix R method[7] non-negative matrix transformation (NNMF) [8]simulated annealing [9] compressive sensing [10] simplexmethod [9] and so on Of all these methods the PI method isan uncomplicated and straightforward solution for the ill-posed inverse problem which shows more clearly theconnection between colorimetric values and the corre-sponding spectrum

In fact spectral recovery from a set of the correspondingcolorimetric values under a specified illuminant-observer

HindawiJournal of SpectroscopyVolume 2019 Article ID 3538265 9 pageshttpsdoiorg10115520193538265

condition limits only three available dimensions whichleads to a noticeable insufficiency for spectral reflectancerecovery +e colorimetric values under multi-illuminantshave inspired researchers to start using several different setsof the corresponding values for the spectral recovery processSchettini and Zuffi [11] exploited genetic algorithms tospectrum recovery from the CIE XYZ tristimulus valuesunder one or multi-illuminants on the premise that basisfunctions and cardinalities were investigated Abed et al [12]proposed the different color spaces of lookup tables (LUTs)using a novel interpolation strategy for spectrum recoveryunder illuminate D65 and illuminate A they stated that themore color coordinates reconstructed LUTs the better thespectrum recovery results could be obtained Harifi et al [13]initially applied the nonlinear regression method to virtuallyincrease the number of tristmulus values and then used sixeigenvectors in the spectral recovery process Amiri andAmirshahi [8] increased virtually the CIE XYZ tristimulusvalues under another illuminant and then recovered thespectral reflectance using of six eigenvectors by adopting thePCA or NNMF method respectively Zhang et al [14]initially predicted CIE XYZ tristimulus values from cameraresponses values according to different illuminants and thenrecovered the spectral reflectance through the PI methodHowever the aforementioned methods are available forspectral reflectance recovery from tristimulus values underpredefined reference illuminants (such as illuminant A andilluminant D65) and treat equally each of the trainingsamples to effect on the spectral recovery process undermultireference illuminants

+is study proposes a more accurate recovering spectralreflectance method from tristimulus values under multi-illuminants that solves the problems mentioned above

+e novelty of our proposed method is concerned withthe reference illuminant selection strategy and local sampleweighted strategy under the multi-illuminants condition+e experimental results of the proposed and traditionalmethods are compared to evaluate the spectral and colori-metric accuracy between the actual and estimated spectra

2 Mathematic Background and Method

+e CIE XYZ tristimulus values under multi-illuminants aresimply computed by the following equations

Xi K 1113946 r(λ)Ii(λ)x(λ)dλ

Yi K 1113946 r(λ)Ii(λ)y(λ)dλ

Zi K 1113946 r(λ)Ii(λ)z(λ)dλ

(1)

with

K 100

1113946 Ii(λ)ydλ

(2)

where r(λ) denotes the objectrsquos reflectance spectrum K

represents the factor of standardization x(λ) y(λ) and

z(λ) denote the color matching functions of the CIEstandard colorimetric observer Ii(λ) denotes the referenceilluminants of index i(i 1 m) ti [Xi Yi Zi]

T de-notes the object color tristimulus values under a givenreference illuminant Ii(λ) and the superscript T denotes thematrix transpose Equation (1) can then be represented inmatrix notation as follows

t ATr (3)

where t [t1 tn]T is the combining matrix of tri-stimulus values under multi-illuminants and AT is the co-efficient matrix involving the reference illuminants and theCIE color matching functions For arbitrary color of thecombining matrix 1113954t from tristimulus values under multi-illuminants the recovered spectra 1113954r can be calculated toobtain from the inverse matrix AT directly So this spectralrecovery process immediately implements transformationfrom tristimulus value space to spectral reflectance spacecalled the direct spectral recovery method [15] Since AT isan underdetermined matrix the recovered spectra are ob-tained from the pseudoinverse matrix (AT)+ which in-evitably causes large calculation error on ill-posed inverseproblems +e recovery of spectral reflectance 1113954r using the PImethod is simply calculated by the following equation

1113954r AT

1113872 1113873+1113954t (4)

where the superscript ldquo+rdquo is the matrix pseudoinverseEquation (4) creates a linear conversion between tri-

stimulus values under multi-illuminants and the spectralreflectance so it is assumed to be a straightforward andresultful solution When the transformation matrix (AT)+

is calculated the recovery process directly yields a singlelinear relation It must be accepted that for the spectralrecovery process by the implementation of the PI methodeach of the training samples has made an equal difference inthe formation of (AT)+ so it leads to optimization for thesamples fully but not for each individual sample Hencethe standard pseudoinverse method leads to large calcu-lation error between the actual and estimated spectra andgenerates an imprecise result Obviously the more similarthe spectral reflectance between the testing samples and thetraining samples is the more accurate the results can begenerated due to the more linearly relation [16 17] +e-oretically speaking if the matrix (AT)+ can be formed bythe testing samplersquos own characteristics the performance ofthe recovered spectra would be calculated optimally In thiswork to obtain the matrix (AT)+ adapting with thecharacteristics of the testing sample the local sampleweighted strategy is proposed for calculating the spectralreflectance recovery +e local sample weighted strategyinvolves selection of the optimal training samples andweighting of the local selected samples +e local optimaltraining samples based on the testing samplersquos owncharacteristics are selected which use the multicolor spaceEuclidean distance between the testing sample and thetraining samples Specifically the multicolor space Eu-clidean distance between each testing sample and thetraining samples can be calculated as follows

2 Journal of Spectroscopy

dj

1113944

m

i1Xitest minus Xij1113872 1113873

2+ Yitest minus Yij1113872 1113873

2+ Zitest minus Zij1113872 1113873

21113876 1113877

11139741113972

j 1 2 n

(5)

where Xitest Yitest and Zitest denote the tristimulus values ofthe testing sample under the ith reference illuminant n

demonstrates the number of the training samples Xij Yijand Zij denote the jth training samplersquos tristimulus valuesunder the ith reference illuminant and dj refers to themulticolor space Euclidean distance between the testingsample and the jth training samples under the multi-illu-minants condition After that the training samples arrangein the increasing order based on the dj values +e localoptimal training samples have been obtain to select thep (1leple n) neighboring samples from the whole trainingsamples Meanwhile different local training samples shouldchoose different weighting efficients since the larger theweighting efficient a certain training sample chooses thestronger the influence the matrix (AT)+ determinates Sothe weight coefficient depending on the inverse multicolorspace Euclidean distance can be calculated as

wk 1

dk + ε k 1 2 p (6)

where the subscript k denotes the kth local optimal trainingsamples dk denotes the multicolor space Euclidean distancebetween the testing sample and the kth sample of the localoptimal training samples and ε 0001 is used in this studyClearly more neighboring training samples to the testingsample would generate the larger wk value +is mathe-matical symbol wk is shown in the galley proof +eweighting matrix W is a diagonal matrix defined as

W

w1 0 middot middot middot 0

0 w2 0 ⋮⋮

0

0

middot middot middot

⋱ 00 wp

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimesp

(7)

AT

1113872 1113873+

rW(tW)+ (8)

Indeed the designing method has selectively controlledthe influence on the formation of the matrix (AT)+ based onthe characteristics of the testing sample which varies fromone testing sample to the other one Each of the testingsamples should have its own the transformation matrixrather than a unique transformation matrix for the wholetesting samples which generates a more precise estimationresult

3 Experiment and Procedure

In this study three different datasets were selected thatinclude theMunsell Chips [18] ColorChecker SG and Vrheldataset [19] +e Munsell Chips consist 1269 spectral colorchips in Munsell Matte Color Book +e 140 chips of the

ColorChecher SG use a X-rite i1 pro spectrophotometer tomeasure personally +e Vrhel dataset contains 354 samplesof reflectance spectra +e spectral reflectance function isproved to sample at 10 nm intervals without impactinggreatly the mathematical precision [20] So the spectralreflectance of three different dataset samples range from400 nm to 700 nm at 10 nm intervals +e Munsell Chipswere selected as the training samples in this study and theirtransformation matrix was used to recover three differenttesting samples

To illuminate the advantages of the proposed methodthe standard pseudoinverse method (method PI) andprincipal component analysis embedded weight regressiontechnique suggested by Amiri and Amirshahi [8] werecompared with the results of the spectral recovery fromtristimulus values under multireference illuminants +eroot mean square error (RMSE) goodness of fit coefficient(GFC) and CIELAB color difference (∆Eab) were selected asthe evaluation of the spectral recovery accuracy [21] whileall the methods of the spectral reflectance recovery performexperimental simulation by the Matlab software

+e dominant spectral power distributions (SPDs) of theselected reference illuminants and light sources were ofvarious distributions and were relatively smooth which didnot have considerable spiky radiance So the CIE illuminants(A B C D50 D55 and D65) and the two actual light-emitting diode (LED) light sources (LED1 Sylvania Concord2048794 and LED2 Photon Star CS5) [22] were selected as thereference illuminants so as to evaluate spectral recovery ac-curacy in this study All the CIE illuminants and LEDs weresampled at a range from 400nm to 700 nm at 10 nm intervalsTo evaluate the performance of the proposed method it iscritical to determine the type of the reference illuminants thatis what is the optimal type of reference illuminants to im-prove the spectral reflectance recovery accuracy As Zhanget al [14] discussed previously a trend can be observed themore similar the spectral power distributions (SPDs) of twoselected reference illuminants are the lower the spectral re-covery accuracy is However this point lacks of a clear nu-merical explanation to verify +is study adopted the spectralangle mapper (SAM) algorithm [23 24] to calculate thesimilarity of the reference illuminants as the reference illu-minants selection strategy shown as

Sαβ accosIα middot Iε

Iα1113868111386811138681113868

1113868111386811138681113868 middot Iε1113868111386811138681113868

1113868111386811138681113868 (9)

where Iα and Iε denote the SPDs value vectors of the ref-erence illuminant of index α(α 1 m) and the referenceilluminant of index ε(ε 1 m) respectively +ismethod determines the SPDs similarity to treat the tworeference illuminants as vectors with treating equally to eachdimension and calculate the angle between the two referenceilluminants which do not adopt the vector length but ratherthe vector direction [15]

4 Results and Discussion

+e proposed spectral reflectance recovery methodmainly involves two optimal parameters namely optimal

Journal of Spectroscopy 3

parameter of the reference illuminants and optimal pa-rameter of the local sample weighted method +is studyfirst investigated two optimal parameters of the spectralrecovery process and finally evaluated the spectral re-flectance recovery accuracy of the proposed comparedwith the traditional methods

41 Optimal Parameter of the Reference Illuminants +espectral reflectance recovery accuracy can be affected by thenumber and type of selected reference illuminants +eo-retically increasing the number of reference illuminants canincrease the spectral recovery accuracy but obtaining extratristimulus values by the colorimetric device would be noteasily procurable Meanwhile according to the conclusionmade by Schettini and Zuffi [11] the result of the spectralrecovery under the three reference illuminants is less toimprove the computing accuracy than under the two illu-minants In this study the number of reference illuminantsdetermined to two illuminants +e type of reference illu-minants selected optimally are the relatively smooth illu-minants rather than the considerable spiky illuminants thisis because spectral recovery accuracy was low when one ortwo spiky illuminants were adopted as reference illuminants[14] First all the samples of the Munsell Chips Color-Checker SG and Vrhel dataset were calculated numericallyto obtain the CIE 1964 XYZ tristimulus values under the CIEilluminants (A B C D50 D55 and D65) and the two actuallight-emitting diode (LED) light sources (LED1 SylvaniaConcord 2048794 LED2 Photon Star CS5) which illustratedto select optimally the type of reference illuminants In thisstudy we considered the Munsell Chips as the trainingsamples to estimate the spectral reflectance of the MunsellChips ColorChecker SG and Vrhel dataset under twodifferent illuminants with the corresponding CIE 1964 XYZtristimulus values Table 1 shows the mean root-mean-square error (RMSE) between the actual and reconstructedspectra for the Munsell Chips ColorChecker SG and Vrheldataset under two illuminant combinations First it is notedthat the CIE 1964XYZ tristimulus values under two differentilluminants have a very significant influence on the spectrallyrecovered result Second Table 1 illustrates the spectralreflectance accuracy for the Munsell Chips presented betterthan for the ColorChecker SG and Vrhel dataset As Babaeiet al [5] proposed in a previous study the best optimumcondition would be obtained when the testing sample is anumber of the training samples +ird the similarity of thetwo reference illuminants also affects spectral estimationaccuracy Table 2 shows the spectral angle mapper (SAM)statistics for two reference illuminants As the spectral curveshape of the two reference illuminants is more dissimilarbetween each other the spectral recovery accuracy will bebetter as shown in Tables 1 and 2 Combining all the ex-perimental results of Tables 1 and 2 this study selects finallythe two illuminants D65 and A as the reference illuminants

42Optimal Parameter of the Local SampleWeightedMethod+e number of the selected local training samples and theweighted local samples can also influence the accuracy of the

spectral reflectance recovery To analyse these issues thespectral reflectance of the Munsell Chips ColorChecker SGand Vrhel dataset was recovered by using the multicolorspace Euclidean distance dj and the weighting matrix W

under the reference illuminants A and D65 with differentnumbers of the local Munsell Chips as the training samples+e mean RMSE between the actual and recovered spectraunder different numbers of the local Munsell Chips wascomputed It is noted that in selecting dynamically thesuitable local samples we always adopt the strategy that ifthe testing sample itself is included in the Munsell Chipsthen this reflectance is removed from the training samples+e relationship between the mean RMSE of the MunsellChips ColorChecker SG and Vrhel dataset and the numberof local training samples is shown in Figure 1 It is found thatthe mean RMSE initially decreases with the increase in thenumber of the local training samples and trends basicallystable in the end To achieve the better spectral reflectanceaccuracy sufficient training samples can be selected So weselected adaptively 100 local training samples from theMunsell Chips for the spectral recovery process

43AccuracyEvaluationof theProposedMethod To evaluatethe colorimetric and spectral performance of the proposedmethod the proposed method was implemented to comparewith the pseudoinverse method (method PI) and theMorteza Maali Amirirsquos method (method WRPCA) [8] +estatistic comparison results among PI WRPCA and theproposed method are summarized in Table 3 First it is easyfind that the proposed method obviously outperforms the PIandWRPCA for the three testing datasets Second results inTable 3 suggest that all the methods for Munsell Chipsimplemented better than for the ColorChecker SG and Vrheldataset +is finding shows that the training sample has thebetter performance to recover itself than other testingsamples Second Table 3 indicates that the proposed methodpresents the spectral reflectance with the highest accuracycompared with the other methods

To further accurately evaluate the performance of theproposed method the average spectral residuals betweenrecovered and measured spectra for the three datasets areshown in Figures 2ndash4 In each figure PI WRPCA and theproposed methods are contrasted for the three differentdatasets the Munsell Chips (Figure 2) the ColorChecker SG(Figure 3) and the Vrhel dataset (Figure 4) By the analysisof Figures 2ndash4 it is easy to note that the average spectralresiduals are more accurate to the middle of the wavelengthsthan both ends +e phenomenon is consistent with theconclusion made byWu et al and is attributed mainly to thehuman visual system that obtains the XYZ tristimulus valuesfor three methods being calculated by the CIE 1964 standardobserver [15]

In order to illustrate the performance of the proposedmethod the recovered results of the spectral reflectance fortwo randomly selected samples from the Munsell Chips areshown in Figures 5(a) and 5(b) ColorChecker SG are shownin Figures 5(c) and 5(d) and Vrhel dataset are shown inFigures 5(e) and 5(f) Compared with the PI and WRPCA

4 Journal of Spectroscopy

Table 1 Mean RMSE of the Munsell chips ColorChecker SG and Vrhel dataset under two illuminant combinations

A B C D50 D55 D65 LED1 LED2

Munsell chips

A 00218 00112 00113 00097 00096 00095 00140 00123B 00112 00227 00115 00126 00134 00113 00123 00129C 00113 00115 00232 00128 00136 00149 00121 00125

D50 00097 00126 00128 00228 00102 00102 00119 00128D55 00096 00134 00136 00102 00230 00101 00116 00112D65 00095 00113 00149 00102 00101 00233 00113 00117LED1 00140 00123 00121 00119 00116 00113 00221 00107LED2 00123 00129 00125 00128 00112 00117 00107 00221

ColorChecker SG

A 00348 00188 00191 00175 00171 00165 00250 00178B 00188 00369 00194 00213 00241 00228 00197 00211C 00191 00194 00380 00228 00250 00267 00195 00206

D50 00175 00213 00228 00370 00160 00159 00181 00208D55 00171 00241 00250 00160 00374 00158 00177 00195D65 00165 00228 00267 00159 00158 00380 00172 00182LED1 00250 00197 00195 00181 00177 00172 00355 00176LED2 00178 00211 00206 00208 00195 00182 00176 00356

Vrhel dataset

A 00398 00201 00207 00186 00179 00170 00293 00195B 00201 00425 00215 00251 00264 00237 00251 00277C 00207 00215 00439 00266 00287 00273 00244 00265

D50 00186 00251 00266 00427 00175 00173 00235 00277D55 00179 00264 00287 00175 00432 00173 00226 00256D65 00170 00237 00273 00173 00173 00440 00217 00234LED1 00293 00251 00244 00235 00226 00217 00404 00196LED2 00195 00277 00265 00277 00256 00234 00196 00405

Table 2 Spectral angle mapper (SAM) statistics for two reference illuminants

A B C D50 D55 D65 LED1 LED2

SAM

A 00000 04276 06339 04552 05280 06406 03053 03185B 04276 00000 02141 00620 01137 02249 03891 03558C 06339 02141 00000 02087 01417 00831 05748 05392

D50 04552 00620 02087 00000 00774 01986 03990 03659D55 05280 01137 01417 00774 00000 01215 04653 04313D65 06406 02249 00831 01986 01215 00000 05751 05410LED1 03053 03891 05748 03990 04653 05751 00000 00619LED2 03185 03558 05392 03659 04313 05410 00619 00000

0

001

002

003

0 200 400 600 800 1000 1200 1400

MunsellColorChecker SGVrhel dataset

Number

Mea

n RM

SE

Figure 1 Relationship between the number of local training samples and mean RMSE of the Munsell chips ColorChecker SG and Vrheldataset

Journal of Spectroscopy 5

Table 3 Statistics of the comparison results among PI WRPCA and the proposed method

MethodTesting samples

Munsell chips ColorChecker SG Vrhel datasetPI WRPCA Proposed PI WRPCA Proposed PI WRPCA Proposed

Mean RMSE 00095 00112 00034 00165 00296 00120 00170 00365 00152Max RMSE 00714 00637 00370 00536 01361 00373 01278 01690 01454Min GFC 09239 09088 09850 09708 07997 09803 09246 06299 09061Mean GFC 09989 09983 09998 09980 09731 09988 09955 09831 09959Max GFC 10000 10000 10000 09999 09998 10000 09999 09999 10000CIELAB color difference under testing illuminant F2Mean ∆Eab 02520 03719 00892 04135 12824 02648 05012 14597 03547Vara 00723 02046 00054 01962 12150 00403 02096 14668 00817Max ∆Eab 28043 49694 06701 27253 58017 12775 30989 53056 18061∆Eabgt 3b 00000 06304 00000 00000 92857 00000 05650 132768 00000CIELAB color difference under testing illuminant F11Mean ∆Eab 08966 12880 03095 13238 27523 07869 14677 27012 12836Var 09990 32857 00836 31423 49938 05667 19217 53176 14947Max ∆Eab 115112 285594 24955 127500 97594 37572 158069 131744 67993∆Eabgt 3 39401 87470 00000 142857 421429 14286 104520 336158 90395aldquoVarrdquo means the variance of the the CIELAB color difference bldquo∆Eabgt 3rdquo means the percentage of testing samples with a color difference greater than 3CIELAB units

000

001

002

003

400 450 500 550 600 650 700

PIWRPCAProposed

Wavelength (nm)

Ave

rage

spec

tral

resid

ual

Figure 2 Average spectral residuals between recovered and measured spectra on the Munsell chips

000

002

004

006

400 450 500 550 600 650 700

PIWRPCAProposed

Wavelength (nm)

Ave

rage

spec

tral

resid

ual

Figure 3 Average spectral residuals between recovered and measured spectra on the ColorChecker SG

6 Journal of Spectroscopy

000

002

004

006

008

400 450 500 550 600 650 700

PIWRPCAProposed

Wavelength (nm)

Ave

rage

spec

tral

resid

ual

Figure 4 Average spectral residuals between recovered and measured spectra on the Vrhel dataset

0

01

02

03

04

05

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(a)

0

02

04

06

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(b)

0

02

04

06

08

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(c)

0

02

04

06

08

1

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(d)

Figure 5 Continued

Journal of Spectroscopy 7

methods the performance of the proposed method is moreaccurately illustrated in Figure 5

5 Conclusions

A method for the spectral reflectance recovery from tri-stimulus values under multi-illuminants was presented byadapting with the characteristics of the testing sample toobtain the transformation matrix of pseudoinverse Toobtain the optimal transformation matrix the proposedmethod for spectral reflectance recovery mainly involves twooptimal parameters optimal parameter of the reference il-luminants and optimal parameter of the local sampleweighted method Selecting the two illuminants A and D65as the reference illuminants are based on the result of thespectral angle mapper (SAM) statistics for two referenceilluminants +e number of the selected local trainingsamples and the weighted local samples can be also de-termined for the spectral reflectance recovery

+e reflectance of the testing samples from the MunsellChips ColorChecker SG and Vrhel dataset were used toevaluate the spectral reflectance accuracy for different methodsin this study +e Munsell Chips were selected as the trainingsamples Meanwhile the performance of three differentmethods namely PI WRPCA and the proposed method wasassessed by the root-mean-square (RMSE) the goodness of fitcoefficient (GFC) and CIE LAB color differences under illu-minants F2 and F11 +e spectral and colorimetrical recoveryaccuracy of the proposedmethod were compared with those ofPI and WRPCA and the results showed the proposed methodis an effective spectral reflectance recovery method

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is study was supported by the Shandong ProvincialNatural Science Foundation China (ZR2017LF027ZR2017PC020) Key Research and Development Program ofShandong Province (2018GGX106009 2019GGX105016)Project of Shandong Province Higher Educational Scienceand Technology Program (J17KA178 J18KA332) OpenFund of National-Local Joint Engineering Laboratory forDigitalize Electrical Design Technology (NEL-DED2017K003) National Natural Science Foundation ofChina (NSFC) (11704162) Foundation of Xuzhou City(KC18002) and the Key Laboratory of Auxiliary Chemistryand Technology for Chemical Industry Ministry of Edu-cation (KFKT2019-03)

References

[1] E M Valero Y Hu J Hernandez-Andres et al ldquoCompar-ative performance analysis of spectral estimation algorithmsand computational optimization of a multispectral imagingsystem for print inspectionrdquo Color Research amp Applicationvol 39 no 1 pp 16ndash27 2014

[2] G Wu ldquoStatistical characterization of skin color spectrumand its application to dermatologic diagnosisrdquo Basic ampClinical Pharmacology amp Toxicology vol 124 pp 69-70 2019

[3] J Mohtasham A S Nateri and H Khalili ldquoTextile colourmatching using linear and exponential weighted principalcomponent analysisrdquo Coloration Technology vol 128 no 3pp 199ndash203 2012

[4] G Wu ldquoCharamer mismatch-based spectral gamut map-pingrdquo Laser Physics Letters vol 16 no 9 Article ID 0952062019

[5] V Babaei S H Amirshahi and F Agahian ldquoUsing weightedpseudo-inverse method for reconstruction of reflectancespectra and analyzing the dataset in terms of normalityrdquoColorResearch amp Application vol 36 no 4 pp 295ndash305 2011

[6] G Wu X Shen and Z Liu ldquoWavelength-sensitive-function-based spectral reconstruction using segmented principalcomponent analysisrdquo Optica Applicata vol 46 no 3pp 365ndash374 2016

0

02

04

06

08

1

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(e)

0

02

04

06

08

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(f )

Figure 5 +e measured spectral reflectance curve of two randomly selected samples from (a) and (b) the Munsell Chips (c) and (d) theColorChecker SG and (e) and (f) the Vrhel dataset by using PI WRPCA and the proposed method

8 Journal of Spectroscopy

[7] Y Zhao and R S Berns ldquoImage-based spectral reflectancereconstruction using the matrix R methodrdquo Color Research ampApplication vol 32 no 5 pp 343ndash351 2007

[8] M M Amiri and S H Amirshahi ldquoA hybrid of weightedregression and linear models for extraction of reflectancespectra from CIEXYZ tristimulus valuesrdquo Optical Reviewvol 21 no 6 pp 816ndash825 2014

[9] D Dupont ldquoStudy of the reconstruction of reflectance curvesbased on tristimulus values comparison of methods of op-timizationrdquo Color Research amp Application vol 27 no 2pp 88ndash99 2002

[10] G Wu ldquoReflectance spectra recovery from a single RGBimage by adaptive compressive sensingrdquo Laser Physics Lettersvol 16 no 8 Article ID 085208 2019

[11] R Schettini and S Zuffi ldquoA computational strategy exploitinggenetic algorithms to recover color surface reflectancefunctionsrdquo Neural Computing and Applications vol 16 no 1pp 69ndash79 2007

[12] F M Abed S H Amirshahi and M R M Abed ldquoRe-construction of reflectance data using an interpolationtechniquerdquo Journal of the Optical Society of America A vol 26no 3 pp 613ndash624 2009

[13] T Harifi S H Amirshahi and F Agahian ldquoRecovery ofreflectance spectra from colorimetric data using principalcomponent analysis embedded regression techniquerdquo OpticalReview vol 15 no 6 pp 302ndash308 2008

[14] X Zhang Q Wang J Li X Zhou Y Yang and H XuldquoEstimating spectral reflectance from camera responses basedon CIEXYZtristimulus values under multi-illuminantsrdquo ColorResearch amp Application vol 42 no 1 pp 68ndash77 2016

[15] G Wu X Shen Z Liu S Yang and M Zhu ldquoReflectancespectra recovery from tristimulus values by extraction of colorfeature matchrdquo Optical and Quantum Electronics vol 48no 1 64 pages 2016

[16] J Liang and X Wan ldquoOptimized method for spectral re-flectance reconstruction from camera responsesrdquo OpticsExpress vol 25 no 23 pp 28273ndash28287 2017

[17] J Liang K Xiao M R Pointer X Wan and C Li ldquoSpectraestimation from raw camera responses based on adaptivelocal-weighted linear regressionrdquo Optics Express vol 27no 4 pp 5165ndash5180 2019

[18] Spectral Database University of Eastern Finland KuopioFinland httpwww2ueffifispectral8

[19] M J Vrhel R Gershon and L S Iwan ldquoMeasurement andanalysis of object reflectance spectrardquo Color Research ampApplication vol 19 no 1 pp 4ndash9 1994

[20] GWu Z Liu E Fang and H Yu ldquoReconstruction of spectralcolor information using weighted principal componentanalysisrdquo Optik-International Journal for Light and ElectronOptics vol 126 no 11-12 pp 1249ndash1253 2015

[21] G Wu Z Liu and J Zhang ldquoSpectral color reproductionfrom CIE tristimulus values using a node address array se-lection techniquerdquo International Journal of Signal ProcessingImage Processing and Pattern Recognition vol 8 no 9pp 141ndash150 2015

[22] Spectral Power Distribution (SPD) Curves National GalleryLondon UK httpresearchng-londonorgukscientificspd

[23] F A Kruse A B Lefkoff J W Boardman et al ldquo+e spectralimage processing system (SIPS)-interactive visualization andanalysis of imaging spectrometer datardquo Remote Sensing ofEnvironment vol 44 no 2-3 pp 145ndash163 1993

[24] A Pelagotti A Mastio A Rosa and A Piva ldquoMultispectralimaging of paintingsrdquo IEEE Signal Processing Magazinevol 25 no 4 pp 27ndash36 2008

Journal of Spectroscopy 9

dj

1113944

m

i1Xitest minus Xij1113872 1113873

2+ Yitest minus Yij1113872 1113873

2+ Zitest minus Zij1113872 1113873

21113876 1113877

11139741113972

j 1 2 n

(5)

where Xitest Yitest and Zitest denote the tristimulus values ofthe testing sample under the ith reference illuminant n

demonstrates the number of the training samples Xij Yijand Zij denote the jth training samplersquos tristimulus valuesunder the ith reference illuminant and dj refers to themulticolor space Euclidean distance between the testingsample and the jth training samples under the multi-illu-minants condition After that the training samples arrangein the increasing order based on the dj values +e localoptimal training samples have been obtain to select thep (1leple n) neighboring samples from the whole trainingsamples Meanwhile different local training samples shouldchoose different weighting efficients since the larger theweighting efficient a certain training sample chooses thestronger the influence the matrix (AT)+ determinates Sothe weight coefficient depending on the inverse multicolorspace Euclidean distance can be calculated as

wk 1

dk + ε k 1 2 p (6)

where the subscript k denotes the kth local optimal trainingsamples dk denotes the multicolor space Euclidean distancebetween the testing sample and the kth sample of the localoptimal training samples and ε 0001 is used in this studyClearly more neighboring training samples to the testingsample would generate the larger wk value +is mathe-matical symbol wk is shown in the galley proof +eweighting matrix W is a diagonal matrix defined as

W

w1 0 middot middot middot 0

0 w2 0 ⋮⋮

0

0

middot middot middot

⋱ 00 wp

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ptimesp

(7)

AT

1113872 1113873+

rW(tW)+ (8)

Indeed the designing method has selectively controlledthe influence on the formation of the matrix (AT)+ based onthe characteristics of the testing sample which varies fromone testing sample to the other one Each of the testingsamples should have its own the transformation matrixrather than a unique transformation matrix for the wholetesting samples which generates a more precise estimationresult

3 Experiment and Procedure

In this study three different datasets were selected thatinclude theMunsell Chips [18] ColorChecker SG and Vrheldataset [19] +e Munsell Chips consist 1269 spectral colorchips in Munsell Matte Color Book +e 140 chips of the

ColorChecher SG use a X-rite i1 pro spectrophotometer tomeasure personally +e Vrhel dataset contains 354 samplesof reflectance spectra +e spectral reflectance function isproved to sample at 10 nm intervals without impactinggreatly the mathematical precision [20] So the spectralreflectance of three different dataset samples range from400 nm to 700 nm at 10 nm intervals +e Munsell Chipswere selected as the training samples in this study and theirtransformation matrix was used to recover three differenttesting samples

To illuminate the advantages of the proposed methodthe standard pseudoinverse method (method PI) andprincipal component analysis embedded weight regressiontechnique suggested by Amiri and Amirshahi [8] werecompared with the results of the spectral recovery fromtristimulus values under multireference illuminants +eroot mean square error (RMSE) goodness of fit coefficient(GFC) and CIELAB color difference (∆Eab) were selected asthe evaluation of the spectral recovery accuracy [21] whileall the methods of the spectral reflectance recovery performexperimental simulation by the Matlab software

+e dominant spectral power distributions (SPDs) of theselected reference illuminants and light sources were ofvarious distributions and were relatively smooth which didnot have considerable spiky radiance So the CIE illuminants(A B C D50 D55 and D65) and the two actual light-emitting diode (LED) light sources (LED1 Sylvania Concord2048794 and LED2 Photon Star CS5) [22] were selected as thereference illuminants so as to evaluate spectral recovery ac-curacy in this study All the CIE illuminants and LEDs weresampled at a range from 400nm to 700 nm at 10 nm intervalsTo evaluate the performance of the proposed method it iscritical to determine the type of the reference illuminants thatis what is the optimal type of reference illuminants to im-prove the spectral reflectance recovery accuracy As Zhanget al [14] discussed previously a trend can be observed themore similar the spectral power distributions (SPDs) of twoselected reference illuminants are the lower the spectral re-covery accuracy is However this point lacks of a clear nu-merical explanation to verify +is study adopted the spectralangle mapper (SAM) algorithm [23 24] to calculate thesimilarity of the reference illuminants as the reference illu-minants selection strategy shown as

Sαβ accosIα middot Iε

Iα1113868111386811138681113868

1113868111386811138681113868 middot Iε1113868111386811138681113868

1113868111386811138681113868 (9)

where Iα and Iε denote the SPDs value vectors of the ref-erence illuminant of index α(α 1 m) and the referenceilluminant of index ε(ε 1 m) respectively +ismethod determines the SPDs similarity to treat the tworeference illuminants as vectors with treating equally to eachdimension and calculate the angle between the two referenceilluminants which do not adopt the vector length but ratherthe vector direction [15]

4 Results and Discussion

+e proposed spectral reflectance recovery methodmainly involves two optimal parameters namely optimal

Journal of Spectroscopy 3

parameter of the reference illuminants and optimal pa-rameter of the local sample weighted method +is studyfirst investigated two optimal parameters of the spectralrecovery process and finally evaluated the spectral re-flectance recovery accuracy of the proposed comparedwith the traditional methods

41 Optimal Parameter of the Reference Illuminants +espectral reflectance recovery accuracy can be affected by thenumber and type of selected reference illuminants +eo-retically increasing the number of reference illuminants canincrease the spectral recovery accuracy but obtaining extratristimulus values by the colorimetric device would be noteasily procurable Meanwhile according to the conclusionmade by Schettini and Zuffi [11] the result of the spectralrecovery under the three reference illuminants is less toimprove the computing accuracy than under the two illu-minants In this study the number of reference illuminantsdetermined to two illuminants +e type of reference illu-minants selected optimally are the relatively smooth illu-minants rather than the considerable spiky illuminants thisis because spectral recovery accuracy was low when one ortwo spiky illuminants were adopted as reference illuminants[14] First all the samples of the Munsell Chips Color-Checker SG and Vrhel dataset were calculated numericallyto obtain the CIE 1964 XYZ tristimulus values under the CIEilluminants (A B C D50 D55 and D65) and the two actuallight-emitting diode (LED) light sources (LED1 SylvaniaConcord 2048794 LED2 Photon Star CS5) which illustratedto select optimally the type of reference illuminants In thisstudy we considered the Munsell Chips as the trainingsamples to estimate the spectral reflectance of the MunsellChips ColorChecker SG and Vrhel dataset under twodifferent illuminants with the corresponding CIE 1964 XYZtristimulus values Table 1 shows the mean root-mean-square error (RMSE) between the actual and reconstructedspectra for the Munsell Chips ColorChecker SG and Vrheldataset under two illuminant combinations First it is notedthat the CIE 1964XYZ tristimulus values under two differentilluminants have a very significant influence on the spectrallyrecovered result Second Table 1 illustrates the spectralreflectance accuracy for the Munsell Chips presented betterthan for the ColorChecker SG and Vrhel dataset As Babaeiet al [5] proposed in a previous study the best optimumcondition would be obtained when the testing sample is anumber of the training samples +ird the similarity of thetwo reference illuminants also affects spectral estimationaccuracy Table 2 shows the spectral angle mapper (SAM)statistics for two reference illuminants As the spectral curveshape of the two reference illuminants is more dissimilarbetween each other the spectral recovery accuracy will bebetter as shown in Tables 1 and 2 Combining all the ex-perimental results of Tables 1 and 2 this study selects finallythe two illuminants D65 and A as the reference illuminants

42Optimal Parameter of the Local SampleWeightedMethod+e number of the selected local training samples and theweighted local samples can also influence the accuracy of the

spectral reflectance recovery To analyse these issues thespectral reflectance of the Munsell Chips ColorChecker SGand Vrhel dataset was recovered by using the multicolorspace Euclidean distance dj and the weighting matrix W

under the reference illuminants A and D65 with differentnumbers of the local Munsell Chips as the training samples+e mean RMSE between the actual and recovered spectraunder different numbers of the local Munsell Chips wascomputed It is noted that in selecting dynamically thesuitable local samples we always adopt the strategy that ifthe testing sample itself is included in the Munsell Chipsthen this reflectance is removed from the training samples+e relationship between the mean RMSE of the MunsellChips ColorChecker SG and Vrhel dataset and the numberof local training samples is shown in Figure 1 It is found thatthe mean RMSE initially decreases with the increase in thenumber of the local training samples and trends basicallystable in the end To achieve the better spectral reflectanceaccuracy sufficient training samples can be selected So weselected adaptively 100 local training samples from theMunsell Chips for the spectral recovery process

43AccuracyEvaluationof theProposedMethod To evaluatethe colorimetric and spectral performance of the proposedmethod the proposed method was implemented to comparewith the pseudoinverse method (method PI) and theMorteza Maali Amirirsquos method (method WRPCA) [8] +estatistic comparison results among PI WRPCA and theproposed method are summarized in Table 3 First it is easyfind that the proposed method obviously outperforms the PIandWRPCA for the three testing datasets Second results inTable 3 suggest that all the methods for Munsell Chipsimplemented better than for the ColorChecker SG and Vrheldataset +is finding shows that the training sample has thebetter performance to recover itself than other testingsamples Second Table 3 indicates that the proposed methodpresents the spectral reflectance with the highest accuracycompared with the other methods

To further accurately evaluate the performance of theproposed method the average spectral residuals betweenrecovered and measured spectra for the three datasets areshown in Figures 2ndash4 In each figure PI WRPCA and theproposed methods are contrasted for the three differentdatasets the Munsell Chips (Figure 2) the ColorChecker SG(Figure 3) and the Vrhel dataset (Figure 4) By the analysisof Figures 2ndash4 it is easy to note that the average spectralresiduals are more accurate to the middle of the wavelengthsthan both ends +e phenomenon is consistent with theconclusion made byWu et al and is attributed mainly to thehuman visual system that obtains the XYZ tristimulus valuesfor three methods being calculated by the CIE 1964 standardobserver [15]

In order to illustrate the performance of the proposedmethod the recovered results of the spectral reflectance fortwo randomly selected samples from the Munsell Chips areshown in Figures 5(a) and 5(b) ColorChecker SG are shownin Figures 5(c) and 5(d) and Vrhel dataset are shown inFigures 5(e) and 5(f) Compared with the PI and WRPCA

4 Journal of Spectroscopy

Table 1 Mean RMSE of the Munsell chips ColorChecker SG and Vrhel dataset under two illuminant combinations

A B C D50 D55 D65 LED1 LED2

Munsell chips

A 00218 00112 00113 00097 00096 00095 00140 00123B 00112 00227 00115 00126 00134 00113 00123 00129C 00113 00115 00232 00128 00136 00149 00121 00125

D50 00097 00126 00128 00228 00102 00102 00119 00128D55 00096 00134 00136 00102 00230 00101 00116 00112D65 00095 00113 00149 00102 00101 00233 00113 00117LED1 00140 00123 00121 00119 00116 00113 00221 00107LED2 00123 00129 00125 00128 00112 00117 00107 00221

ColorChecker SG

A 00348 00188 00191 00175 00171 00165 00250 00178B 00188 00369 00194 00213 00241 00228 00197 00211C 00191 00194 00380 00228 00250 00267 00195 00206

D50 00175 00213 00228 00370 00160 00159 00181 00208D55 00171 00241 00250 00160 00374 00158 00177 00195D65 00165 00228 00267 00159 00158 00380 00172 00182LED1 00250 00197 00195 00181 00177 00172 00355 00176LED2 00178 00211 00206 00208 00195 00182 00176 00356

Vrhel dataset

A 00398 00201 00207 00186 00179 00170 00293 00195B 00201 00425 00215 00251 00264 00237 00251 00277C 00207 00215 00439 00266 00287 00273 00244 00265

D50 00186 00251 00266 00427 00175 00173 00235 00277D55 00179 00264 00287 00175 00432 00173 00226 00256D65 00170 00237 00273 00173 00173 00440 00217 00234LED1 00293 00251 00244 00235 00226 00217 00404 00196LED2 00195 00277 00265 00277 00256 00234 00196 00405

Table 2 Spectral angle mapper (SAM) statistics for two reference illuminants

A B C D50 D55 D65 LED1 LED2

SAM

A 00000 04276 06339 04552 05280 06406 03053 03185B 04276 00000 02141 00620 01137 02249 03891 03558C 06339 02141 00000 02087 01417 00831 05748 05392

D50 04552 00620 02087 00000 00774 01986 03990 03659D55 05280 01137 01417 00774 00000 01215 04653 04313D65 06406 02249 00831 01986 01215 00000 05751 05410LED1 03053 03891 05748 03990 04653 05751 00000 00619LED2 03185 03558 05392 03659 04313 05410 00619 00000

0

001

002

003

0 200 400 600 800 1000 1200 1400

MunsellColorChecker SGVrhel dataset

Number

Mea

n RM

SE

Figure 1 Relationship between the number of local training samples and mean RMSE of the Munsell chips ColorChecker SG and Vrheldataset

Journal of Spectroscopy 5

Table 3 Statistics of the comparison results among PI WRPCA and the proposed method

MethodTesting samples

Munsell chips ColorChecker SG Vrhel datasetPI WRPCA Proposed PI WRPCA Proposed PI WRPCA Proposed

Mean RMSE 00095 00112 00034 00165 00296 00120 00170 00365 00152Max RMSE 00714 00637 00370 00536 01361 00373 01278 01690 01454Min GFC 09239 09088 09850 09708 07997 09803 09246 06299 09061Mean GFC 09989 09983 09998 09980 09731 09988 09955 09831 09959Max GFC 10000 10000 10000 09999 09998 10000 09999 09999 10000CIELAB color difference under testing illuminant F2Mean ∆Eab 02520 03719 00892 04135 12824 02648 05012 14597 03547Vara 00723 02046 00054 01962 12150 00403 02096 14668 00817Max ∆Eab 28043 49694 06701 27253 58017 12775 30989 53056 18061∆Eabgt 3b 00000 06304 00000 00000 92857 00000 05650 132768 00000CIELAB color difference under testing illuminant F11Mean ∆Eab 08966 12880 03095 13238 27523 07869 14677 27012 12836Var 09990 32857 00836 31423 49938 05667 19217 53176 14947Max ∆Eab 115112 285594 24955 127500 97594 37572 158069 131744 67993∆Eabgt 3 39401 87470 00000 142857 421429 14286 104520 336158 90395aldquoVarrdquo means the variance of the the CIELAB color difference bldquo∆Eabgt 3rdquo means the percentage of testing samples with a color difference greater than 3CIELAB units

000

001

002

003

400 450 500 550 600 650 700

PIWRPCAProposed

Wavelength (nm)

Ave

rage

spec

tral

resid

ual

Figure 2 Average spectral residuals between recovered and measured spectra on the Munsell chips

000

002

004

006

400 450 500 550 600 650 700

PIWRPCAProposed

Wavelength (nm)

Ave

rage

spec

tral

resid

ual

Figure 3 Average spectral residuals between recovered and measured spectra on the ColorChecker SG

6 Journal of Spectroscopy

000

002

004

006

008

400 450 500 550 600 650 700

PIWRPCAProposed

Wavelength (nm)

Ave

rage

spec

tral

resid

ual

Figure 4 Average spectral residuals between recovered and measured spectra on the Vrhel dataset

0

01

02

03

04

05

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(a)

0

02

04

06

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(b)

0

02

04

06

08

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(c)

0

02

04

06

08

1

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(d)

Figure 5 Continued

Journal of Spectroscopy 7

methods the performance of the proposed method is moreaccurately illustrated in Figure 5

5 Conclusions

A method for the spectral reflectance recovery from tri-stimulus values under multi-illuminants was presented byadapting with the characteristics of the testing sample toobtain the transformation matrix of pseudoinverse Toobtain the optimal transformation matrix the proposedmethod for spectral reflectance recovery mainly involves twooptimal parameters optimal parameter of the reference il-luminants and optimal parameter of the local sampleweighted method Selecting the two illuminants A and D65as the reference illuminants are based on the result of thespectral angle mapper (SAM) statistics for two referenceilluminants +e number of the selected local trainingsamples and the weighted local samples can be also de-termined for the spectral reflectance recovery

+e reflectance of the testing samples from the MunsellChips ColorChecker SG and Vrhel dataset were used toevaluate the spectral reflectance accuracy for different methodsin this study +e Munsell Chips were selected as the trainingsamples Meanwhile the performance of three differentmethods namely PI WRPCA and the proposed method wasassessed by the root-mean-square (RMSE) the goodness of fitcoefficient (GFC) and CIE LAB color differences under illu-minants F2 and F11 +e spectral and colorimetrical recoveryaccuracy of the proposedmethod were compared with those ofPI and WRPCA and the results showed the proposed methodis an effective spectral reflectance recovery method

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is study was supported by the Shandong ProvincialNatural Science Foundation China (ZR2017LF027ZR2017PC020) Key Research and Development Program ofShandong Province (2018GGX106009 2019GGX105016)Project of Shandong Province Higher Educational Scienceand Technology Program (J17KA178 J18KA332) OpenFund of National-Local Joint Engineering Laboratory forDigitalize Electrical Design Technology (NEL-DED2017K003) National Natural Science Foundation ofChina (NSFC) (11704162) Foundation of Xuzhou City(KC18002) and the Key Laboratory of Auxiliary Chemistryand Technology for Chemical Industry Ministry of Edu-cation (KFKT2019-03)

References

[1] E M Valero Y Hu J Hernandez-Andres et al ldquoCompar-ative performance analysis of spectral estimation algorithmsand computational optimization of a multispectral imagingsystem for print inspectionrdquo Color Research amp Applicationvol 39 no 1 pp 16ndash27 2014

[2] G Wu ldquoStatistical characterization of skin color spectrumand its application to dermatologic diagnosisrdquo Basic ampClinical Pharmacology amp Toxicology vol 124 pp 69-70 2019

[3] J Mohtasham A S Nateri and H Khalili ldquoTextile colourmatching using linear and exponential weighted principalcomponent analysisrdquo Coloration Technology vol 128 no 3pp 199ndash203 2012

[4] G Wu ldquoCharamer mismatch-based spectral gamut map-pingrdquo Laser Physics Letters vol 16 no 9 Article ID 0952062019

[5] V Babaei S H Amirshahi and F Agahian ldquoUsing weightedpseudo-inverse method for reconstruction of reflectancespectra and analyzing the dataset in terms of normalityrdquoColorResearch amp Application vol 36 no 4 pp 295ndash305 2011

[6] G Wu X Shen and Z Liu ldquoWavelength-sensitive-function-based spectral reconstruction using segmented principalcomponent analysisrdquo Optica Applicata vol 46 no 3pp 365ndash374 2016

0

02

04

06

08

1

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(e)

0

02

04

06

08

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(f )

Figure 5 +e measured spectral reflectance curve of two randomly selected samples from (a) and (b) the Munsell Chips (c) and (d) theColorChecker SG and (e) and (f) the Vrhel dataset by using PI WRPCA and the proposed method

8 Journal of Spectroscopy

[7] Y Zhao and R S Berns ldquoImage-based spectral reflectancereconstruction using the matrix R methodrdquo Color Research ampApplication vol 32 no 5 pp 343ndash351 2007

[8] M M Amiri and S H Amirshahi ldquoA hybrid of weightedregression and linear models for extraction of reflectancespectra from CIEXYZ tristimulus valuesrdquo Optical Reviewvol 21 no 6 pp 816ndash825 2014

[9] D Dupont ldquoStudy of the reconstruction of reflectance curvesbased on tristimulus values comparison of methods of op-timizationrdquo Color Research amp Application vol 27 no 2pp 88ndash99 2002

[10] G Wu ldquoReflectance spectra recovery from a single RGBimage by adaptive compressive sensingrdquo Laser Physics Lettersvol 16 no 8 Article ID 085208 2019

[11] R Schettini and S Zuffi ldquoA computational strategy exploitinggenetic algorithms to recover color surface reflectancefunctionsrdquo Neural Computing and Applications vol 16 no 1pp 69ndash79 2007

[12] F M Abed S H Amirshahi and M R M Abed ldquoRe-construction of reflectance data using an interpolationtechniquerdquo Journal of the Optical Society of America A vol 26no 3 pp 613ndash624 2009

[13] T Harifi S H Amirshahi and F Agahian ldquoRecovery ofreflectance spectra from colorimetric data using principalcomponent analysis embedded regression techniquerdquo OpticalReview vol 15 no 6 pp 302ndash308 2008

[14] X Zhang Q Wang J Li X Zhou Y Yang and H XuldquoEstimating spectral reflectance from camera responses basedon CIEXYZtristimulus values under multi-illuminantsrdquo ColorResearch amp Application vol 42 no 1 pp 68ndash77 2016

[15] G Wu X Shen Z Liu S Yang and M Zhu ldquoReflectancespectra recovery from tristimulus values by extraction of colorfeature matchrdquo Optical and Quantum Electronics vol 48no 1 64 pages 2016

[16] J Liang and X Wan ldquoOptimized method for spectral re-flectance reconstruction from camera responsesrdquo OpticsExpress vol 25 no 23 pp 28273ndash28287 2017

[17] J Liang K Xiao M R Pointer X Wan and C Li ldquoSpectraestimation from raw camera responses based on adaptivelocal-weighted linear regressionrdquo Optics Express vol 27no 4 pp 5165ndash5180 2019

[18] Spectral Database University of Eastern Finland KuopioFinland httpwww2ueffifispectral8

[19] M J Vrhel R Gershon and L S Iwan ldquoMeasurement andanalysis of object reflectance spectrardquo Color Research ampApplication vol 19 no 1 pp 4ndash9 1994

[20] GWu Z Liu E Fang and H Yu ldquoReconstruction of spectralcolor information using weighted principal componentanalysisrdquo Optik-International Journal for Light and ElectronOptics vol 126 no 11-12 pp 1249ndash1253 2015

[21] G Wu Z Liu and J Zhang ldquoSpectral color reproductionfrom CIE tristimulus values using a node address array se-lection techniquerdquo International Journal of Signal ProcessingImage Processing and Pattern Recognition vol 8 no 9pp 141ndash150 2015

[22] Spectral Power Distribution (SPD) Curves National GalleryLondon UK httpresearchng-londonorgukscientificspd

[23] F A Kruse A B Lefkoff J W Boardman et al ldquo+e spectralimage processing system (SIPS)-interactive visualization andanalysis of imaging spectrometer datardquo Remote Sensing ofEnvironment vol 44 no 2-3 pp 145ndash163 1993

[24] A Pelagotti A Mastio A Rosa and A Piva ldquoMultispectralimaging of paintingsrdquo IEEE Signal Processing Magazinevol 25 no 4 pp 27ndash36 2008

Journal of Spectroscopy 9

parameter of the reference illuminants and optimal pa-rameter of the local sample weighted method +is studyfirst investigated two optimal parameters of the spectralrecovery process and finally evaluated the spectral re-flectance recovery accuracy of the proposed comparedwith the traditional methods

41 Optimal Parameter of the Reference Illuminants +espectral reflectance recovery accuracy can be affected by thenumber and type of selected reference illuminants +eo-retically increasing the number of reference illuminants canincrease the spectral recovery accuracy but obtaining extratristimulus values by the colorimetric device would be noteasily procurable Meanwhile according to the conclusionmade by Schettini and Zuffi [11] the result of the spectralrecovery under the three reference illuminants is less toimprove the computing accuracy than under the two illu-minants In this study the number of reference illuminantsdetermined to two illuminants +e type of reference illu-minants selected optimally are the relatively smooth illu-minants rather than the considerable spiky illuminants thisis because spectral recovery accuracy was low when one ortwo spiky illuminants were adopted as reference illuminants[14] First all the samples of the Munsell Chips Color-Checker SG and Vrhel dataset were calculated numericallyto obtain the CIE 1964 XYZ tristimulus values under the CIEilluminants (A B C D50 D55 and D65) and the two actuallight-emitting diode (LED) light sources (LED1 SylvaniaConcord 2048794 LED2 Photon Star CS5) which illustratedto select optimally the type of reference illuminants In thisstudy we considered the Munsell Chips as the trainingsamples to estimate the spectral reflectance of the MunsellChips ColorChecker SG and Vrhel dataset under twodifferent illuminants with the corresponding CIE 1964 XYZtristimulus values Table 1 shows the mean root-mean-square error (RMSE) between the actual and reconstructedspectra for the Munsell Chips ColorChecker SG and Vrheldataset under two illuminant combinations First it is notedthat the CIE 1964XYZ tristimulus values under two differentilluminants have a very significant influence on the spectrallyrecovered result Second Table 1 illustrates the spectralreflectance accuracy for the Munsell Chips presented betterthan for the ColorChecker SG and Vrhel dataset As Babaeiet al [5] proposed in a previous study the best optimumcondition would be obtained when the testing sample is anumber of the training samples +ird the similarity of thetwo reference illuminants also affects spectral estimationaccuracy Table 2 shows the spectral angle mapper (SAM)statistics for two reference illuminants As the spectral curveshape of the two reference illuminants is more dissimilarbetween each other the spectral recovery accuracy will bebetter as shown in Tables 1 and 2 Combining all the ex-perimental results of Tables 1 and 2 this study selects finallythe two illuminants D65 and A as the reference illuminants

42Optimal Parameter of the Local SampleWeightedMethod+e number of the selected local training samples and theweighted local samples can also influence the accuracy of the

spectral reflectance recovery To analyse these issues thespectral reflectance of the Munsell Chips ColorChecker SGand Vrhel dataset was recovered by using the multicolorspace Euclidean distance dj and the weighting matrix W

under the reference illuminants A and D65 with differentnumbers of the local Munsell Chips as the training samples+e mean RMSE between the actual and recovered spectraunder different numbers of the local Munsell Chips wascomputed It is noted that in selecting dynamically thesuitable local samples we always adopt the strategy that ifthe testing sample itself is included in the Munsell Chipsthen this reflectance is removed from the training samples+e relationship between the mean RMSE of the MunsellChips ColorChecker SG and Vrhel dataset and the numberof local training samples is shown in Figure 1 It is found thatthe mean RMSE initially decreases with the increase in thenumber of the local training samples and trends basicallystable in the end To achieve the better spectral reflectanceaccuracy sufficient training samples can be selected So weselected adaptively 100 local training samples from theMunsell Chips for the spectral recovery process

43AccuracyEvaluationof theProposedMethod To evaluatethe colorimetric and spectral performance of the proposedmethod the proposed method was implemented to comparewith the pseudoinverse method (method PI) and theMorteza Maali Amirirsquos method (method WRPCA) [8] +estatistic comparison results among PI WRPCA and theproposed method are summarized in Table 3 First it is easyfind that the proposed method obviously outperforms the PIandWRPCA for the three testing datasets Second results inTable 3 suggest that all the methods for Munsell Chipsimplemented better than for the ColorChecker SG and Vrheldataset +is finding shows that the training sample has thebetter performance to recover itself than other testingsamples Second Table 3 indicates that the proposed methodpresents the spectral reflectance with the highest accuracycompared with the other methods

To further accurately evaluate the performance of theproposed method the average spectral residuals betweenrecovered and measured spectra for the three datasets areshown in Figures 2ndash4 In each figure PI WRPCA and theproposed methods are contrasted for the three differentdatasets the Munsell Chips (Figure 2) the ColorChecker SG(Figure 3) and the Vrhel dataset (Figure 4) By the analysisof Figures 2ndash4 it is easy to note that the average spectralresiduals are more accurate to the middle of the wavelengthsthan both ends +e phenomenon is consistent with theconclusion made byWu et al and is attributed mainly to thehuman visual system that obtains the XYZ tristimulus valuesfor three methods being calculated by the CIE 1964 standardobserver [15]

In order to illustrate the performance of the proposedmethod the recovered results of the spectral reflectance fortwo randomly selected samples from the Munsell Chips areshown in Figures 5(a) and 5(b) ColorChecker SG are shownin Figures 5(c) and 5(d) and Vrhel dataset are shown inFigures 5(e) and 5(f) Compared with the PI and WRPCA

4 Journal of Spectroscopy

Table 1 Mean RMSE of the Munsell chips ColorChecker SG and Vrhel dataset under two illuminant combinations

A B C D50 D55 D65 LED1 LED2

Munsell chips

A 00218 00112 00113 00097 00096 00095 00140 00123B 00112 00227 00115 00126 00134 00113 00123 00129C 00113 00115 00232 00128 00136 00149 00121 00125

D50 00097 00126 00128 00228 00102 00102 00119 00128D55 00096 00134 00136 00102 00230 00101 00116 00112D65 00095 00113 00149 00102 00101 00233 00113 00117LED1 00140 00123 00121 00119 00116 00113 00221 00107LED2 00123 00129 00125 00128 00112 00117 00107 00221

ColorChecker SG

A 00348 00188 00191 00175 00171 00165 00250 00178B 00188 00369 00194 00213 00241 00228 00197 00211C 00191 00194 00380 00228 00250 00267 00195 00206

D50 00175 00213 00228 00370 00160 00159 00181 00208D55 00171 00241 00250 00160 00374 00158 00177 00195D65 00165 00228 00267 00159 00158 00380 00172 00182LED1 00250 00197 00195 00181 00177 00172 00355 00176LED2 00178 00211 00206 00208 00195 00182 00176 00356

Vrhel dataset

A 00398 00201 00207 00186 00179 00170 00293 00195B 00201 00425 00215 00251 00264 00237 00251 00277C 00207 00215 00439 00266 00287 00273 00244 00265

D50 00186 00251 00266 00427 00175 00173 00235 00277D55 00179 00264 00287 00175 00432 00173 00226 00256D65 00170 00237 00273 00173 00173 00440 00217 00234LED1 00293 00251 00244 00235 00226 00217 00404 00196LED2 00195 00277 00265 00277 00256 00234 00196 00405

Table 2 Spectral angle mapper (SAM) statistics for two reference illuminants

A B C D50 D55 D65 LED1 LED2

SAM

A 00000 04276 06339 04552 05280 06406 03053 03185B 04276 00000 02141 00620 01137 02249 03891 03558C 06339 02141 00000 02087 01417 00831 05748 05392

D50 04552 00620 02087 00000 00774 01986 03990 03659D55 05280 01137 01417 00774 00000 01215 04653 04313D65 06406 02249 00831 01986 01215 00000 05751 05410LED1 03053 03891 05748 03990 04653 05751 00000 00619LED2 03185 03558 05392 03659 04313 05410 00619 00000

0

001

002

003

0 200 400 600 800 1000 1200 1400

MunsellColorChecker SGVrhel dataset

Number

Mea

n RM

SE

Figure 1 Relationship between the number of local training samples and mean RMSE of the Munsell chips ColorChecker SG and Vrheldataset

Journal of Spectroscopy 5

Table 3 Statistics of the comparison results among PI WRPCA and the proposed method

MethodTesting samples

Munsell chips ColorChecker SG Vrhel datasetPI WRPCA Proposed PI WRPCA Proposed PI WRPCA Proposed

Mean RMSE 00095 00112 00034 00165 00296 00120 00170 00365 00152Max RMSE 00714 00637 00370 00536 01361 00373 01278 01690 01454Min GFC 09239 09088 09850 09708 07997 09803 09246 06299 09061Mean GFC 09989 09983 09998 09980 09731 09988 09955 09831 09959Max GFC 10000 10000 10000 09999 09998 10000 09999 09999 10000CIELAB color difference under testing illuminant F2Mean ∆Eab 02520 03719 00892 04135 12824 02648 05012 14597 03547Vara 00723 02046 00054 01962 12150 00403 02096 14668 00817Max ∆Eab 28043 49694 06701 27253 58017 12775 30989 53056 18061∆Eabgt 3b 00000 06304 00000 00000 92857 00000 05650 132768 00000CIELAB color difference under testing illuminant F11Mean ∆Eab 08966 12880 03095 13238 27523 07869 14677 27012 12836Var 09990 32857 00836 31423 49938 05667 19217 53176 14947Max ∆Eab 115112 285594 24955 127500 97594 37572 158069 131744 67993∆Eabgt 3 39401 87470 00000 142857 421429 14286 104520 336158 90395aldquoVarrdquo means the variance of the the CIELAB color difference bldquo∆Eabgt 3rdquo means the percentage of testing samples with a color difference greater than 3CIELAB units

000

001

002

003

400 450 500 550 600 650 700

PIWRPCAProposed

Wavelength (nm)

Ave

rage

spec

tral

resid

ual

Figure 2 Average spectral residuals between recovered and measured spectra on the Munsell chips

000

002

004

006

400 450 500 550 600 650 700

PIWRPCAProposed

Wavelength (nm)

Ave

rage

spec

tral

resid

ual

Figure 3 Average spectral residuals between recovered and measured spectra on the ColorChecker SG

6 Journal of Spectroscopy

000

002

004

006

008

400 450 500 550 600 650 700

PIWRPCAProposed

Wavelength (nm)

Ave

rage

spec

tral

resid

ual

Figure 4 Average spectral residuals between recovered and measured spectra on the Vrhel dataset

0

01

02

03

04

05

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(a)

0

02

04

06

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(b)

0

02

04

06

08

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(c)

0

02

04

06

08

1

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(d)

Figure 5 Continued

Journal of Spectroscopy 7

methods the performance of the proposed method is moreaccurately illustrated in Figure 5

5 Conclusions

A method for the spectral reflectance recovery from tri-stimulus values under multi-illuminants was presented byadapting with the characteristics of the testing sample toobtain the transformation matrix of pseudoinverse Toobtain the optimal transformation matrix the proposedmethod for spectral reflectance recovery mainly involves twooptimal parameters optimal parameter of the reference il-luminants and optimal parameter of the local sampleweighted method Selecting the two illuminants A and D65as the reference illuminants are based on the result of thespectral angle mapper (SAM) statistics for two referenceilluminants +e number of the selected local trainingsamples and the weighted local samples can be also de-termined for the spectral reflectance recovery

+e reflectance of the testing samples from the MunsellChips ColorChecker SG and Vrhel dataset were used toevaluate the spectral reflectance accuracy for different methodsin this study +e Munsell Chips were selected as the trainingsamples Meanwhile the performance of three differentmethods namely PI WRPCA and the proposed method wasassessed by the root-mean-square (RMSE) the goodness of fitcoefficient (GFC) and CIE LAB color differences under illu-minants F2 and F11 +e spectral and colorimetrical recoveryaccuracy of the proposedmethod were compared with those ofPI and WRPCA and the results showed the proposed methodis an effective spectral reflectance recovery method

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is study was supported by the Shandong ProvincialNatural Science Foundation China (ZR2017LF027ZR2017PC020) Key Research and Development Program ofShandong Province (2018GGX106009 2019GGX105016)Project of Shandong Province Higher Educational Scienceand Technology Program (J17KA178 J18KA332) OpenFund of National-Local Joint Engineering Laboratory forDigitalize Electrical Design Technology (NEL-DED2017K003) National Natural Science Foundation ofChina (NSFC) (11704162) Foundation of Xuzhou City(KC18002) and the Key Laboratory of Auxiliary Chemistryand Technology for Chemical Industry Ministry of Edu-cation (KFKT2019-03)

References

[1] E M Valero Y Hu J Hernandez-Andres et al ldquoCompar-ative performance analysis of spectral estimation algorithmsand computational optimization of a multispectral imagingsystem for print inspectionrdquo Color Research amp Applicationvol 39 no 1 pp 16ndash27 2014

[2] G Wu ldquoStatistical characterization of skin color spectrumand its application to dermatologic diagnosisrdquo Basic ampClinical Pharmacology amp Toxicology vol 124 pp 69-70 2019

[3] J Mohtasham A S Nateri and H Khalili ldquoTextile colourmatching using linear and exponential weighted principalcomponent analysisrdquo Coloration Technology vol 128 no 3pp 199ndash203 2012

[4] G Wu ldquoCharamer mismatch-based spectral gamut map-pingrdquo Laser Physics Letters vol 16 no 9 Article ID 0952062019

[5] V Babaei S H Amirshahi and F Agahian ldquoUsing weightedpseudo-inverse method for reconstruction of reflectancespectra and analyzing the dataset in terms of normalityrdquoColorResearch amp Application vol 36 no 4 pp 295ndash305 2011

[6] G Wu X Shen and Z Liu ldquoWavelength-sensitive-function-based spectral reconstruction using segmented principalcomponent analysisrdquo Optica Applicata vol 46 no 3pp 365ndash374 2016

0

02

04

06

08

1

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(e)

0

02

04

06

08

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(f )

Figure 5 +e measured spectral reflectance curve of two randomly selected samples from (a) and (b) the Munsell Chips (c) and (d) theColorChecker SG and (e) and (f) the Vrhel dataset by using PI WRPCA and the proposed method

8 Journal of Spectroscopy

[7] Y Zhao and R S Berns ldquoImage-based spectral reflectancereconstruction using the matrix R methodrdquo Color Research ampApplication vol 32 no 5 pp 343ndash351 2007

[8] M M Amiri and S H Amirshahi ldquoA hybrid of weightedregression and linear models for extraction of reflectancespectra from CIEXYZ tristimulus valuesrdquo Optical Reviewvol 21 no 6 pp 816ndash825 2014

[9] D Dupont ldquoStudy of the reconstruction of reflectance curvesbased on tristimulus values comparison of methods of op-timizationrdquo Color Research amp Application vol 27 no 2pp 88ndash99 2002

[10] G Wu ldquoReflectance spectra recovery from a single RGBimage by adaptive compressive sensingrdquo Laser Physics Lettersvol 16 no 8 Article ID 085208 2019

[11] R Schettini and S Zuffi ldquoA computational strategy exploitinggenetic algorithms to recover color surface reflectancefunctionsrdquo Neural Computing and Applications vol 16 no 1pp 69ndash79 2007

[12] F M Abed S H Amirshahi and M R M Abed ldquoRe-construction of reflectance data using an interpolationtechniquerdquo Journal of the Optical Society of America A vol 26no 3 pp 613ndash624 2009

[13] T Harifi S H Amirshahi and F Agahian ldquoRecovery ofreflectance spectra from colorimetric data using principalcomponent analysis embedded regression techniquerdquo OpticalReview vol 15 no 6 pp 302ndash308 2008

[14] X Zhang Q Wang J Li X Zhou Y Yang and H XuldquoEstimating spectral reflectance from camera responses basedon CIEXYZtristimulus values under multi-illuminantsrdquo ColorResearch amp Application vol 42 no 1 pp 68ndash77 2016

[15] G Wu X Shen Z Liu S Yang and M Zhu ldquoReflectancespectra recovery from tristimulus values by extraction of colorfeature matchrdquo Optical and Quantum Electronics vol 48no 1 64 pages 2016

[16] J Liang and X Wan ldquoOptimized method for spectral re-flectance reconstruction from camera responsesrdquo OpticsExpress vol 25 no 23 pp 28273ndash28287 2017

[17] J Liang K Xiao M R Pointer X Wan and C Li ldquoSpectraestimation from raw camera responses based on adaptivelocal-weighted linear regressionrdquo Optics Express vol 27no 4 pp 5165ndash5180 2019

[18] Spectral Database University of Eastern Finland KuopioFinland httpwww2ueffifispectral8

[19] M J Vrhel R Gershon and L S Iwan ldquoMeasurement andanalysis of object reflectance spectrardquo Color Research ampApplication vol 19 no 1 pp 4ndash9 1994

[20] GWu Z Liu E Fang and H Yu ldquoReconstruction of spectralcolor information using weighted principal componentanalysisrdquo Optik-International Journal for Light and ElectronOptics vol 126 no 11-12 pp 1249ndash1253 2015

[21] G Wu Z Liu and J Zhang ldquoSpectral color reproductionfrom CIE tristimulus values using a node address array se-lection techniquerdquo International Journal of Signal ProcessingImage Processing and Pattern Recognition vol 8 no 9pp 141ndash150 2015

[22] Spectral Power Distribution (SPD) Curves National GalleryLondon UK httpresearchng-londonorgukscientificspd

[23] F A Kruse A B Lefkoff J W Boardman et al ldquo+e spectralimage processing system (SIPS)-interactive visualization andanalysis of imaging spectrometer datardquo Remote Sensing ofEnvironment vol 44 no 2-3 pp 145ndash163 1993

[24] A Pelagotti A Mastio A Rosa and A Piva ldquoMultispectralimaging of paintingsrdquo IEEE Signal Processing Magazinevol 25 no 4 pp 27ndash36 2008

Journal of Spectroscopy 9

Table 1 Mean RMSE of the Munsell chips ColorChecker SG and Vrhel dataset under two illuminant combinations

A B C D50 D55 D65 LED1 LED2

Munsell chips

A 00218 00112 00113 00097 00096 00095 00140 00123B 00112 00227 00115 00126 00134 00113 00123 00129C 00113 00115 00232 00128 00136 00149 00121 00125

D50 00097 00126 00128 00228 00102 00102 00119 00128D55 00096 00134 00136 00102 00230 00101 00116 00112D65 00095 00113 00149 00102 00101 00233 00113 00117LED1 00140 00123 00121 00119 00116 00113 00221 00107LED2 00123 00129 00125 00128 00112 00117 00107 00221

ColorChecker SG

A 00348 00188 00191 00175 00171 00165 00250 00178B 00188 00369 00194 00213 00241 00228 00197 00211C 00191 00194 00380 00228 00250 00267 00195 00206

D50 00175 00213 00228 00370 00160 00159 00181 00208D55 00171 00241 00250 00160 00374 00158 00177 00195D65 00165 00228 00267 00159 00158 00380 00172 00182LED1 00250 00197 00195 00181 00177 00172 00355 00176LED2 00178 00211 00206 00208 00195 00182 00176 00356

Vrhel dataset

A 00398 00201 00207 00186 00179 00170 00293 00195B 00201 00425 00215 00251 00264 00237 00251 00277C 00207 00215 00439 00266 00287 00273 00244 00265

D50 00186 00251 00266 00427 00175 00173 00235 00277D55 00179 00264 00287 00175 00432 00173 00226 00256D65 00170 00237 00273 00173 00173 00440 00217 00234LED1 00293 00251 00244 00235 00226 00217 00404 00196LED2 00195 00277 00265 00277 00256 00234 00196 00405

Table 2 Spectral angle mapper (SAM) statistics for two reference illuminants

A B C D50 D55 D65 LED1 LED2

SAM

A 00000 04276 06339 04552 05280 06406 03053 03185B 04276 00000 02141 00620 01137 02249 03891 03558C 06339 02141 00000 02087 01417 00831 05748 05392

D50 04552 00620 02087 00000 00774 01986 03990 03659D55 05280 01137 01417 00774 00000 01215 04653 04313D65 06406 02249 00831 01986 01215 00000 05751 05410LED1 03053 03891 05748 03990 04653 05751 00000 00619LED2 03185 03558 05392 03659 04313 05410 00619 00000

0

001

002

003

0 200 400 600 800 1000 1200 1400

MunsellColorChecker SGVrhel dataset

Number

Mea

n RM

SE

Figure 1 Relationship between the number of local training samples and mean RMSE of the Munsell chips ColorChecker SG and Vrheldataset

Journal of Spectroscopy 5

Table 3 Statistics of the comparison results among PI WRPCA and the proposed method

MethodTesting samples

Munsell chips ColorChecker SG Vrhel datasetPI WRPCA Proposed PI WRPCA Proposed PI WRPCA Proposed

Mean RMSE 00095 00112 00034 00165 00296 00120 00170 00365 00152Max RMSE 00714 00637 00370 00536 01361 00373 01278 01690 01454Min GFC 09239 09088 09850 09708 07997 09803 09246 06299 09061Mean GFC 09989 09983 09998 09980 09731 09988 09955 09831 09959Max GFC 10000 10000 10000 09999 09998 10000 09999 09999 10000CIELAB color difference under testing illuminant F2Mean ∆Eab 02520 03719 00892 04135 12824 02648 05012 14597 03547Vara 00723 02046 00054 01962 12150 00403 02096 14668 00817Max ∆Eab 28043 49694 06701 27253 58017 12775 30989 53056 18061∆Eabgt 3b 00000 06304 00000 00000 92857 00000 05650 132768 00000CIELAB color difference under testing illuminant F11Mean ∆Eab 08966 12880 03095 13238 27523 07869 14677 27012 12836Var 09990 32857 00836 31423 49938 05667 19217 53176 14947Max ∆Eab 115112 285594 24955 127500 97594 37572 158069 131744 67993∆Eabgt 3 39401 87470 00000 142857 421429 14286 104520 336158 90395aldquoVarrdquo means the variance of the the CIELAB color difference bldquo∆Eabgt 3rdquo means the percentage of testing samples with a color difference greater than 3CIELAB units

000

001

002

003

400 450 500 550 600 650 700

PIWRPCAProposed

Wavelength (nm)

Ave

rage

spec

tral

resid

ual

Figure 2 Average spectral residuals between recovered and measured spectra on the Munsell chips

000

002

004

006

400 450 500 550 600 650 700

PIWRPCAProposed

Wavelength (nm)

Ave

rage

spec

tral

resid

ual

Figure 3 Average spectral residuals between recovered and measured spectra on the ColorChecker SG

6 Journal of Spectroscopy

000

002

004

006

008

400 450 500 550 600 650 700

PIWRPCAProposed

Wavelength (nm)

Ave

rage

spec

tral

resid

ual

Figure 4 Average spectral residuals between recovered and measured spectra on the Vrhel dataset

0

01

02

03

04

05

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(a)

0

02

04

06

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(b)

0

02

04

06

08

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(c)

0

02

04

06

08

1

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(d)

Figure 5 Continued

Journal of Spectroscopy 7

methods the performance of the proposed method is moreaccurately illustrated in Figure 5

5 Conclusions

A method for the spectral reflectance recovery from tri-stimulus values under multi-illuminants was presented byadapting with the characteristics of the testing sample toobtain the transformation matrix of pseudoinverse Toobtain the optimal transformation matrix the proposedmethod for spectral reflectance recovery mainly involves twooptimal parameters optimal parameter of the reference il-luminants and optimal parameter of the local sampleweighted method Selecting the two illuminants A and D65as the reference illuminants are based on the result of thespectral angle mapper (SAM) statistics for two referenceilluminants +e number of the selected local trainingsamples and the weighted local samples can be also de-termined for the spectral reflectance recovery

+e reflectance of the testing samples from the MunsellChips ColorChecker SG and Vrhel dataset were used toevaluate the spectral reflectance accuracy for different methodsin this study +e Munsell Chips were selected as the trainingsamples Meanwhile the performance of three differentmethods namely PI WRPCA and the proposed method wasassessed by the root-mean-square (RMSE) the goodness of fitcoefficient (GFC) and CIE LAB color differences under illu-minants F2 and F11 +e spectral and colorimetrical recoveryaccuracy of the proposedmethod were compared with those ofPI and WRPCA and the results showed the proposed methodis an effective spectral reflectance recovery method

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is study was supported by the Shandong ProvincialNatural Science Foundation China (ZR2017LF027ZR2017PC020) Key Research and Development Program ofShandong Province (2018GGX106009 2019GGX105016)Project of Shandong Province Higher Educational Scienceand Technology Program (J17KA178 J18KA332) OpenFund of National-Local Joint Engineering Laboratory forDigitalize Electrical Design Technology (NEL-DED2017K003) National Natural Science Foundation ofChina (NSFC) (11704162) Foundation of Xuzhou City(KC18002) and the Key Laboratory of Auxiliary Chemistryand Technology for Chemical Industry Ministry of Edu-cation (KFKT2019-03)

References

[1] E M Valero Y Hu J Hernandez-Andres et al ldquoCompar-ative performance analysis of spectral estimation algorithmsand computational optimization of a multispectral imagingsystem for print inspectionrdquo Color Research amp Applicationvol 39 no 1 pp 16ndash27 2014

[2] G Wu ldquoStatistical characterization of skin color spectrumand its application to dermatologic diagnosisrdquo Basic ampClinical Pharmacology amp Toxicology vol 124 pp 69-70 2019

[3] J Mohtasham A S Nateri and H Khalili ldquoTextile colourmatching using linear and exponential weighted principalcomponent analysisrdquo Coloration Technology vol 128 no 3pp 199ndash203 2012

[4] G Wu ldquoCharamer mismatch-based spectral gamut map-pingrdquo Laser Physics Letters vol 16 no 9 Article ID 0952062019

[5] V Babaei S H Amirshahi and F Agahian ldquoUsing weightedpseudo-inverse method for reconstruction of reflectancespectra and analyzing the dataset in terms of normalityrdquoColorResearch amp Application vol 36 no 4 pp 295ndash305 2011

[6] G Wu X Shen and Z Liu ldquoWavelength-sensitive-function-based spectral reconstruction using segmented principalcomponent analysisrdquo Optica Applicata vol 46 no 3pp 365ndash374 2016

0

02

04

06

08

1

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(e)

0

02

04

06

08

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(f )

Figure 5 +e measured spectral reflectance curve of two randomly selected samples from (a) and (b) the Munsell Chips (c) and (d) theColorChecker SG and (e) and (f) the Vrhel dataset by using PI WRPCA and the proposed method

8 Journal of Spectroscopy

[7] Y Zhao and R S Berns ldquoImage-based spectral reflectancereconstruction using the matrix R methodrdquo Color Research ampApplication vol 32 no 5 pp 343ndash351 2007

[8] M M Amiri and S H Amirshahi ldquoA hybrid of weightedregression and linear models for extraction of reflectancespectra from CIEXYZ tristimulus valuesrdquo Optical Reviewvol 21 no 6 pp 816ndash825 2014

[9] D Dupont ldquoStudy of the reconstruction of reflectance curvesbased on tristimulus values comparison of methods of op-timizationrdquo Color Research amp Application vol 27 no 2pp 88ndash99 2002

[10] G Wu ldquoReflectance spectra recovery from a single RGBimage by adaptive compressive sensingrdquo Laser Physics Lettersvol 16 no 8 Article ID 085208 2019

[11] R Schettini and S Zuffi ldquoA computational strategy exploitinggenetic algorithms to recover color surface reflectancefunctionsrdquo Neural Computing and Applications vol 16 no 1pp 69ndash79 2007

[12] F M Abed S H Amirshahi and M R M Abed ldquoRe-construction of reflectance data using an interpolationtechniquerdquo Journal of the Optical Society of America A vol 26no 3 pp 613ndash624 2009

[13] T Harifi S H Amirshahi and F Agahian ldquoRecovery ofreflectance spectra from colorimetric data using principalcomponent analysis embedded regression techniquerdquo OpticalReview vol 15 no 6 pp 302ndash308 2008

[14] X Zhang Q Wang J Li X Zhou Y Yang and H XuldquoEstimating spectral reflectance from camera responses basedon CIEXYZtristimulus values under multi-illuminantsrdquo ColorResearch amp Application vol 42 no 1 pp 68ndash77 2016

[15] G Wu X Shen Z Liu S Yang and M Zhu ldquoReflectancespectra recovery from tristimulus values by extraction of colorfeature matchrdquo Optical and Quantum Electronics vol 48no 1 64 pages 2016

[16] J Liang and X Wan ldquoOptimized method for spectral re-flectance reconstruction from camera responsesrdquo OpticsExpress vol 25 no 23 pp 28273ndash28287 2017

[17] J Liang K Xiao M R Pointer X Wan and C Li ldquoSpectraestimation from raw camera responses based on adaptivelocal-weighted linear regressionrdquo Optics Express vol 27no 4 pp 5165ndash5180 2019

[18] Spectral Database University of Eastern Finland KuopioFinland httpwww2ueffifispectral8

[19] M J Vrhel R Gershon and L S Iwan ldquoMeasurement andanalysis of object reflectance spectrardquo Color Research ampApplication vol 19 no 1 pp 4ndash9 1994

[20] GWu Z Liu E Fang and H Yu ldquoReconstruction of spectralcolor information using weighted principal componentanalysisrdquo Optik-International Journal for Light and ElectronOptics vol 126 no 11-12 pp 1249ndash1253 2015

[21] G Wu Z Liu and J Zhang ldquoSpectral color reproductionfrom CIE tristimulus values using a node address array se-lection techniquerdquo International Journal of Signal ProcessingImage Processing and Pattern Recognition vol 8 no 9pp 141ndash150 2015

[22] Spectral Power Distribution (SPD) Curves National GalleryLondon UK httpresearchng-londonorgukscientificspd

[23] F A Kruse A B Lefkoff J W Boardman et al ldquo+e spectralimage processing system (SIPS)-interactive visualization andanalysis of imaging spectrometer datardquo Remote Sensing ofEnvironment vol 44 no 2-3 pp 145ndash163 1993

[24] A Pelagotti A Mastio A Rosa and A Piva ldquoMultispectralimaging of paintingsrdquo IEEE Signal Processing Magazinevol 25 no 4 pp 27ndash36 2008

Journal of Spectroscopy 9

Table 3 Statistics of the comparison results among PI WRPCA and the proposed method

MethodTesting samples

Munsell chips ColorChecker SG Vrhel datasetPI WRPCA Proposed PI WRPCA Proposed PI WRPCA Proposed

Mean RMSE 00095 00112 00034 00165 00296 00120 00170 00365 00152Max RMSE 00714 00637 00370 00536 01361 00373 01278 01690 01454Min GFC 09239 09088 09850 09708 07997 09803 09246 06299 09061Mean GFC 09989 09983 09998 09980 09731 09988 09955 09831 09959Max GFC 10000 10000 10000 09999 09998 10000 09999 09999 10000CIELAB color difference under testing illuminant F2Mean ∆Eab 02520 03719 00892 04135 12824 02648 05012 14597 03547Vara 00723 02046 00054 01962 12150 00403 02096 14668 00817Max ∆Eab 28043 49694 06701 27253 58017 12775 30989 53056 18061∆Eabgt 3b 00000 06304 00000 00000 92857 00000 05650 132768 00000CIELAB color difference under testing illuminant F11Mean ∆Eab 08966 12880 03095 13238 27523 07869 14677 27012 12836Var 09990 32857 00836 31423 49938 05667 19217 53176 14947Max ∆Eab 115112 285594 24955 127500 97594 37572 158069 131744 67993∆Eabgt 3 39401 87470 00000 142857 421429 14286 104520 336158 90395aldquoVarrdquo means the variance of the the CIELAB color difference bldquo∆Eabgt 3rdquo means the percentage of testing samples with a color difference greater than 3CIELAB units

000

001

002

003

400 450 500 550 600 650 700

PIWRPCAProposed

Wavelength (nm)

Ave

rage

spec

tral

resid

ual

Figure 2 Average spectral residuals between recovered and measured spectra on the Munsell chips

000

002

004

006

400 450 500 550 600 650 700

PIWRPCAProposed

Wavelength (nm)

Ave

rage

spec

tral

resid

ual

Figure 3 Average spectral residuals between recovered and measured spectra on the ColorChecker SG

6 Journal of Spectroscopy

000

002

004

006

008

400 450 500 550 600 650 700

PIWRPCAProposed

Wavelength (nm)

Ave

rage

spec

tral

resid

ual

Figure 4 Average spectral residuals between recovered and measured spectra on the Vrhel dataset

0

01

02

03

04

05

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(a)

0

02

04

06

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(b)

0

02

04

06

08

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(c)

0

02

04

06

08

1

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(d)

Figure 5 Continued

Journal of Spectroscopy 7

methods the performance of the proposed method is moreaccurately illustrated in Figure 5

5 Conclusions

A method for the spectral reflectance recovery from tri-stimulus values under multi-illuminants was presented byadapting with the characteristics of the testing sample toobtain the transformation matrix of pseudoinverse Toobtain the optimal transformation matrix the proposedmethod for spectral reflectance recovery mainly involves twooptimal parameters optimal parameter of the reference il-luminants and optimal parameter of the local sampleweighted method Selecting the two illuminants A and D65as the reference illuminants are based on the result of thespectral angle mapper (SAM) statistics for two referenceilluminants +e number of the selected local trainingsamples and the weighted local samples can be also de-termined for the spectral reflectance recovery

+e reflectance of the testing samples from the MunsellChips ColorChecker SG and Vrhel dataset were used toevaluate the spectral reflectance accuracy for different methodsin this study +e Munsell Chips were selected as the trainingsamples Meanwhile the performance of three differentmethods namely PI WRPCA and the proposed method wasassessed by the root-mean-square (RMSE) the goodness of fitcoefficient (GFC) and CIE LAB color differences under illu-minants F2 and F11 +e spectral and colorimetrical recoveryaccuracy of the proposedmethod were compared with those ofPI and WRPCA and the results showed the proposed methodis an effective spectral reflectance recovery method

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is study was supported by the Shandong ProvincialNatural Science Foundation China (ZR2017LF027ZR2017PC020) Key Research and Development Program ofShandong Province (2018GGX106009 2019GGX105016)Project of Shandong Province Higher Educational Scienceand Technology Program (J17KA178 J18KA332) OpenFund of National-Local Joint Engineering Laboratory forDigitalize Electrical Design Technology (NEL-DED2017K003) National Natural Science Foundation ofChina (NSFC) (11704162) Foundation of Xuzhou City(KC18002) and the Key Laboratory of Auxiliary Chemistryand Technology for Chemical Industry Ministry of Edu-cation (KFKT2019-03)

References

[1] E M Valero Y Hu J Hernandez-Andres et al ldquoCompar-ative performance analysis of spectral estimation algorithmsand computational optimization of a multispectral imagingsystem for print inspectionrdquo Color Research amp Applicationvol 39 no 1 pp 16ndash27 2014

[2] G Wu ldquoStatistical characterization of skin color spectrumand its application to dermatologic diagnosisrdquo Basic ampClinical Pharmacology amp Toxicology vol 124 pp 69-70 2019

[3] J Mohtasham A S Nateri and H Khalili ldquoTextile colourmatching using linear and exponential weighted principalcomponent analysisrdquo Coloration Technology vol 128 no 3pp 199ndash203 2012

[4] G Wu ldquoCharamer mismatch-based spectral gamut map-pingrdquo Laser Physics Letters vol 16 no 9 Article ID 0952062019

[5] V Babaei S H Amirshahi and F Agahian ldquoUsing weightedpseudo-inverse method for reconstruction of reflectancespectra and analyzing the dataset in terms of normalityrdquoColorResearch amp Application vol 36 no 4 pp 295ndash305 2011

[6] G Wu X Shen and Z Liu ldquoWavelength-sensitive-function-based spectral reconstruction using segmented principalcomponent analysisrdquo Optica Applicata vol 46 no 3pp 365ndash374 2016

0

02

04

06

08

1

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(e)

0

02

04

06

08

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(f )

Figure 5 +e measured spectral reflectance curve of two randomly selected samples from (a) and (b) the Munsell Chips (c) and (d) theColorChecker SG and (e) and (f) the Vrhel dataset by using PI WRPCA and the proposed method

8 Journal of Spectroscopy

[7] Y Zhao and R S Berns ldquoImage-based spectral reflectancereconstruction using the matrix R methodrdquo Color Research ampApplication vol 32 no 5 pp 343ndash351 2007

[8] M M Amiri and S H Amirshahi ldquoA hybrid of weightedregression and linear models for extraction of reflectancespectra from CIEXYZ tristimulus valuesrdquo Optical Reviewvol 21 no 6 pp 816ndash825 2014

[9] D Dupont ldquoStudy of the reconstruction of reflectance curvesbased on tristimulus values comparison of methods of op-timizationrdquo Color Research amp Application vol 27 no 2pp 88ndash99 2002

[10] G Wu ldquoReflectance spectra recovery from a single RGBimage by adaptive compressive sensingrdquo Laser Physics Lettersvol 16 no 8 Article ID 085208 2019

[11] R Schettini and S Zuffi ldquoA computational strategy exploitinggenetic algorithms to recover color surface reflectancefunctionsrdquo Neural Computing and Applications vol 16 no 1pp 69ndash79 2007

[12] F M Abed S H Amirshahi and M R M Abed ldquoRe-construction of reflectance data using an interpolationtechniquerdquo Journal of the Optical Society of America A vol 26no 3 pp 613ndash624 2009

[13] T Harifi S H Amirshahi and F Agahian ldquoRecovery ofreflectance spectra from colorimetric data using principalcomponent analysis embedded regression techniquerdquo OpticalReview vol 15 no 6 pp 302ndash308 2008

[14] X Zhang Q Wang J Li X Zhou Y Yang and H XuldquoEstimating spectral reflectance from camera responses basedon CIEXYZtristimulus values under multi-illuminantsrdquo ColorResearch amp Application vol 42 no 1 pp 68ndash77 2016

[15] G Wu X Shen Z Liu S Yang and M Zhu ldquoReflectancespectra recovery from tristimulus values by extraction of colorfeature matchrdquo Optical and Quantum Electronics vol 48no 1 64 pages 2016

[16] J Liang and X Wan ldquoOptimized method for spectral re-flectance reconstruction from camera responsesrdquo OpticsExpress vol 25 no 23 pp 28273ndash28287 2017

[17] J Liang K Xiao M R Pointer X Wan and C Li ldquoSpectraestimation from raw camera responses based on adaptivelocal-weighted linear regressionrdquo Optics Express vol 27no 4 pp 5165ndash5180 2019

[18] Spectral Database University of Eastern Finland KuopioFinland httpwww2ueffifispectral8

[19] M J Vrhel R Gershon and L S Iwan ldquoMeasurement andanalysis of object reflectance spectrardquo Color Research ampApplication vol 19 no 1 pp 4ndash9 1994

[20] GWu Z Liu E Fang and H Yu ldquoReconstruction of spectralcolor information using weighted principal componentanalysisrdquo Optik-International Journal for Light and ElectronOptics vol 126 no 11-12 pp 1249ndash1253 2015

[21] G Wu Z Liu and J Zhang ldquoSpectral color reproductionfrom CIE tristimulus values using a node address array se-lection techniquerdquo International Journal of Signal ProcessingImage Processing and Pattern Recognition vol 8 no 9pp 141ndash150 2015

[22] Spectral Power Distribution (SPD) Curves National GalleryLondon UK httpresearchng-londonorgukscientificspd

[23] F A Kruse A B Lefkoff J W Boardman et al ldquo+e spectralimage processing system (SIPS)-interactive visualization andanalysis of imaging spectrometer datardquo Remote Sensing ofEnvironment vol 44 no 2-3 pp 145ndash163 1993

[24] A Pelagotti A Mastio A Rosa and A Piva ldquoMultispectralimaging of paintingsrdquo IEEE Signal Processing Magazinevol 25 no 4 pp 27ndash36 2008

Journal of Spectroscopy 9

000

002

004

006

008

400 450 500 550 600 650 700

PIWRPCAProposed

Wavelength (nm)

Ave

rage

spec

tral

resid

ual

Figure 4 Average spectral residuals between recovered and measured spectra on the Vrhel dataset

0

01

02

03

04

05

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(a)

0

02

04

06

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(b)

0

02

04

06

08

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(c)

0

02

04

06

08

1

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(d)

Figure 5 Continued

Journal of Spectroscopy 7

methods the performance of the proposed method is moreaccurately illustrated in Figure 5

5 Conclusions

A method for the spectral reflectance recovery from tri-stimulus values under multi-illuminants was presented byadapting with the characteristics of the testing sample toobtain the transformation matrix of pseudoinverse Toobtain the optimal transformation matrix the proposedmethod for spectral reflectance recovery mainly involves twooptimal parameters optimal parameter of the reference il-luminants and optimal parameter of the local sampleweighted method Selecting the two illuminants A and D65as the reference illuminants are based on the result of thespectral angle mapper (SAM) statistics for two referenceilluminants +e number of the selected local trainingsamples and the weighted local samples can be also de-termined for the spectral reflectance recovery

+e reflectance of the testing samples from the MunsellChips ColorChecker SG and Vrhel dataset were used toevaluate the spectral reflectance accuracy for different methodsin this study +e Munsell Chips were selected as the trainingsamples Meanwhile the performance of three differentmethods namely PI WRPCA and the proposed method wasassessed by the root-mean-square (RMSE) the goodness of fitcoefficient (GFC) and CIE LAB color differences under illu-minants F2 and F11 +e spectral and colorimetrical recoveryaccuracy of the proposedmethod were compared with those ofPI and WRPCA and the results showed the proposed methodis an effective spectral reflectance recovery method

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is study was supported by the Shandong ProvincialNatural Science Foundation China (ZR2017LF027ZR2017PC020) Key Research and Development Program ofShandong Province (2018GGX106009 2019GGX105016)Project of Shandong Province Higher Educational Scienceand Technology Program (J17KA178 J18KA332) OpenFund of National-Local Joint Engineering Laboratory forDigitalize Electrical Design Technology (NEL-DED2017K003) National Natural Science Foundation ofChina (NSFC) (11704162) Foundation of Xuzhou City(KC18002) and the Key Laboratory of Auxiliary Chemistryand Technology for Chemical Industry Ministry of Edu-cation (KFKT2019-03)

References

[1] E M Valero Y Hu J Hernandez-Andres et al ldquoCompar-ative performance analysis of spectral estimation algorithmsand computational optimization of a multispectral imagingsystem for print inspectionrdquo Color Research amp Applicationvol 39 no 1 pp 16ndash27 2014

[2] G Wu ldquoStatistical characterization of skin color spectrumand its application to dermatologic diagnosisrdquo Basic ampClinical Pharmacology amp Toxicology vol 124 pp 69-70 2019

[3] J Mohtasham A S Nateri and H Khalili ldquoTextile colourmatching using linear and exponential weighted principalcomponent analysisrdquo Coloration Technology vol 128 no 3pp 199ndash203 2012

[4] G Wu ldquoCharamer mismatch-based spectral gamut map-pingrdquo Laser Physics Letters vol 16 no 9 Article ID 0952062019

[5] V Babaei S H Amirshahi and F Agahian ldquoUsing weightedpseudo-inverse method for reconstruction of reflectancespectra and analyzing the dataset in terms of normalityrdquoColorResearch amp Application vol 36 no 4 pp 295ndash305 2011

[6] G Wu X Shen and Z Liu ldquoWavelength-sensitive-function-based spectral reconstruction using segmented principalcomponent analysisrdquo Optica Applicata vol 46 no 3pp 365ndash374 2016

0

02

04

06

08

1

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(e)

0

02

04

06

08

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(f )

Figure 5 +e measured spectral reflectance curve of two randomly selected samples from (a) and (b) the Munsell Chips (c) and (d) theColorChecker SG and (e) and (f) the Vrhel dataset by using PI WRPCA and the proposed method

8 Journal of Spectroscopy

[7] Y Zhao and R S Berns ldquoImage-based spectral reflectancereconstruction using the matrix R methodrdquo Color Research ampApplication vol 32 no 5 pp 343ndash351 2007

[8] M M Amiri and S H Amirshahi ldquoA hybrid of weightedregression and linear models for extraction of reflectancespectra from CIEXYZ tristimulus valuesrdquo Optical Reviewvol 21 no 6 pp 816ndash825 2014

[9] D Dupont ldquoStudy of the reconstruction of reflectance curvesbased on tristimulus values comparison of methods of op-timizationrdquo Color Research amp Application vol 27 no 2pp 88ndash99 2002

[10] G Wu ldquoReflectance spectra recovery from a single RGBimage by adaptive compressive sensingrdquo Laser Physics Lettersvol 16 no 8 Article ID 085208 2019

[11] R Schettini and S Zuffi ldquoA computational strategy exploitinggenetic algorithms to recover color surface reflectancefunctionsrdquo Neural Computing and Applications vol 16 no 1pp 69ndash79 2007

[12] F M Abed S H Amirshahi and M R M Abed ldquoRe-construction of reflectance data using an interpolationtechniquerdquo Journal of the Optical Society of America A vol 26no 3 pp 613ndash624 2009

[13] T Harifi S H Amirshahi and F Agahian ldquoRecovery ofreflectance spectra from colorimetric data using principalcomponent analysis embedded regression techniquerdquo OpticalReview vol 15 no 6 pp 302ndash308 2008

[14] X Zhang Q Wang J Li X Zhou Y Yang and H XuldquoEstimating spectral reflectance from camera responses basedon CIEXYZtristimulus values under multi-illuminantsrdquo ColorResearch amp Application vol 42 no 1 pp 68ndash77 2016

[15] G Wu X Shen Z Liu S Yang and M Zhu ldquoReflectancespectra recovery from tristimulus values by extraction of colorfeature matchrdquo Optical and Quantum Electronics vol 48no 1 64 pages 2016

[16] J Liang and X Wan ldquoOptimized method for spectral re-flectance reconstruction from camera responsesrdquo OpticsExpress vol 25 no 23 pp 28273ndash28287 2017

[17] J Liang K Xiao M R Pointer X Wan and C Li ldquoSpectraestimation from raw camera responses based on adaptivelocal-weighted linear regressionrdquo Optics Express vol 27no 4 pp 5165ndash5180 2019

[18] Spectral Database University of Eastern Finland KuopioFinland httpwww2ueffifispectral8

[19] M J Vrhel R Gershon and L S Iwan ldquoMeasurement andanalysis of object reflectance spectrardquo Color Research ampApplication vol 19 no 1 pp 4ndash9 1994

[20] GWu Z Liu E Fang and H Yu ldquoReconstruction of spectralcolor information using weighted principal componentanalysisrdquo Optik-International Journal for Light and ElectronOptics vol 126 no 11-12 pp 1249ndash1253 2015

[21] G Wu Z Liu and J Zhang ldquoSpectral color reproductionfrom CIE tristimulus values using a node address array se-lection techniquerdquo International Journal of Signal ProcessingImage Processing and Pattern Recognition vol 8 no 9pp 141ndash150 2015

[22] Spectral Power Distribution (SPD) Curves National GalleryLondon UK httpresearchng-londonorgukscientificspd

[23] F A Kruse A B Lefkoff J W Boardman et al ldquo+e spectralimage processing system (SIPS)-interactive visualization andanalysis of imaging spectrometer datardquo Remote Sensing ofEnvironment vol 44 no 2-3 pp 145ndash163 1993

[24] A Pelagotti A Mastio A Rosa and A Piva ldquoMultispectralimaging of paintingsrdquo IEEE Signal Processing Magazinevol 25 no 4 pp 27ndash36 2008

Journal of Spectroscopy 9

methods the performance of the proposed method is moreaccurately illustrated in Figure 5

5 Conclusions

A method for the spectral reflectance recovery from tri-stimulus values under multi-illuminants was presented byadapting with the characteristics of the testing sample toobtain the transformation matrix of pseudoinverse Toobtain the optimal transformation matrix the proposedmethod for spectral reflectance recovery mainly involves twooptimal parameters optimal parameter of the reference il-luminants and optimal parameter of the local sampleweighted method Selecting the two illuminants A and D65as the reference illuminants are based on the result of thespectral angle mapper (SAM) statistics for two referenceilluminants +e number of the selected local trainingsamples and the weighted local samples can be also de-termined for the spectral reflectance recovery

+e reflectance of the testing samples from the MunsellChips ColorChecker SG and Vrhel dataset were used toevaluate the spectral reflectance accuracy for different methodsin this study +e Munsell Chips were selected as the trainingsamples Meanwhile the performance of three differentmethods namely PI WRPCA and the proposed method wasassessed by the root-mean-square (RMSE) the goodness of fitcoefficient (GFC) and CIE LAB color differences under illu-minants F2 and F11 +e spectral and colorimetrical recoveryaccuracy of the proposedmethod were compared with those ofPI and WRPCA and the results showed the proposed methodis an effective spectral reflectance recovery method

Data Availability

No data were used to support this study

Conflicts of Interest

+e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is study was supported by the Shandong ProvincialNatural Science Foundation China (ZR2017LF027ZR2017PC020) Key Research and Development Program ofShandong Province (2018GGX106009 2019GGX105016)Project of Shandong Province Higher Educational Scienceand Technology Program (J17KA178 J18KA332) OpenFund of National-Local Joint Engineering Laboratory forDigitalize Electrical Design Technology (NEL-DED2017K003) National Natural Science Foundation ofChina (NSFC) (11704162) Foundation of Xuzhou City(KC18002) and the Key Laboratory of Auxiliary Chemistryand Technology for Chemical Industry Ministry of Edu-cation (KFKT2019-03)

References

[1] E M Valero Y Hu J Hernandez-Andres et al ldquoCompar-ative performance analysis of spectral estimation algorithmsand computational optimization of a multispectral imagingsystem for print inspectionrdquo Color Research amp Applicationvol 39 no 1 pp 16ndash27 2014

[2] G Wu ldquoStatistical characterization of skin color spectrumand its application to dermatologic diagnosisrdquo Basic ampClinical Pharmacology amp Toxicology vol 124 pp 69-70 2019

[3] J Mohtasham A S Nateri and H Khalili ldquoTextile colourmatching using linear and exponential weighted principalcomponent analysisrdquo Coloration Technology vol 128 no 3pp 199ndash203 2012

[4] G Wu ldquoCharamer mismatch-based spectral gamut map-pingrdquo Laser Physics Letters vol 16 no 9 Article ID 0952062019

[5] V Babaei S H Amirshahi and F Agahian ldquoUsing weightedpseudo-inverse method for reconstruction of reflectancespectra and analyzing the dataset in terms of normalityrdquoColorResearch amp Application vol 36 no 4 pp 295ndash305 2011

[6] G Wu X Shen and Z Liu ldquoWavelength-sensitive-function-based spectral reconstruction using segmented principalcomponent analysisrdquo Optica Applicata vol 46 no 3pp 365ndash374 2016

0

02

04

06

08

1

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(e)

0

02

04

06

08

400 450 500 550 600 650 700

OriginalPI

WRPCAProposed

Wavelength (nm)

Refle

ctan

ce

(f )

Figure 5 +e measured spectral reflectance curve of two randomly selected samples from (a) and (b) the Munsell Chips (c) and (d) theColorChecker SG and (e) and (f) the Vrhel dataset by using PI WRPCA and the proposed method

8 Journal of Spectroscopy

[7] Y Zhao and R S Berns ldquoImage-based spectral reflectancereconstruction using the matrix R methodrdquo Color Research ampApplication vol 32 no 5 pp 343ndash351 2007

[8] M M Amiri and S H Amirshahi ldquoA hybrid of weightedregression and linear models for extraction of reflectancespectra from CIEXYZ tristimulus valuesrdquo Optical Reviewvol 21 no 6 pp 816ndash825 2014

[9] D Dupont ldquoStudy of the reconstruction of reflectance curvesbased on tristimulus values comparison of methods of op-timizationrdquo Color Research amp Application vol 27 no 2pp 88ndash99 2002

[10] G Wu ldquoReflectance spectra recovery from a single RGBimage by adaptive compressive sensingrdquo Laser Physics Lettersvol 16 no 8 Article ID 085208 2019

[11] R Schettini and S Zuffi ldquoA computational strategy exploitinggenetic algorithms to recover color surface reflectancefunctionsrdquo Neural Computing and Applications vol 16 no 1pp 69ndash79 2007

[12] F M Abed S H Amirshahi and M R M Abed ldquoRe-construction of reflectance data using an interpolationtechniquerdquo Journal of the Optical Society of America A vol 26no 3 pp 613ndash624 2009

[13] T Harifi S H Amirshahi and F Agahian ldquoRecovery ofreflectance spectra from colorimetric data using principalcomponent analysis embedded regression techniquerdquo OpticalReview vol 15 no 6 pp 302ndash308 2008

[14] X Zhang Q Wang J Li X Zhou Y Yang and H XuldquoEstimating spectral reflectance from camera responses basedon CIEXYZtristimulus values under multi-illuminantsrdquo ColorResearch amp Application vol 42 no 1 pp 68ndash77 2016

[15] G Wu X Shen Z Liu S Yang and M Zhu ldquoReflectancespectra recovery from tristimulus values by extraction of colorfeature matchrdquo Optical and Quantum Electronics vol 48no 1 64 pages 2016

[16] J Liang and X Wan ldquoOptimized method for spectral re-flectance reconstruction from camera responsesrdquo OpticsExpress vol 25 no 23 pp 28273ndash28287 2017

[17] J Liang K Xiao M R Pointer X Wan and C Li ldquoSpectraestimation from raw camera responses based on adaptivelocal-weighted linear regressionrdquo Optics Express vol 27no 4 pp 5165ndash5180 2019

[18] Spectral Database University of Eastern Finland KuopioFinland httpwww2ueffifispectral8

[19] M J Vrhel R Gershon and L S Iwan ldquoMeasurement andanalysis of object reflectance spectrardquo Color Research ampApplication vol 19 no 1 pp 4ndash9 1994

[20] GWu Z Liu E Fang and H Yu ldquoReconstruction of spectralcolor information using weighted principal componentanalysisrdquo Optik-International Journal for Light and ElectronOptics vol 126 no 11-12 pp 1249ndash1253 2015

[21] G Wu Z Liu and J Zhang ldquoSpectral color reproductionfrom CIE tristimulus values using a node address array se-lection techniquerdquo International Journal of Signal ProcessingImage Processing and Pattern Recognition vol 8 no 9pp 141ndash150 2015

[22] Spectral Power Distribution (SPD) Curves National GalleryLondon UK httpresearchng-londonorgukscientificspd

[23] F A Kruse A B Lefkoff J W Boardman et al ldquo+e spectralimage processing system (SIPS)-interactive visualization andanalysis of imaging spectrometer datardquo Remote Sensing ofEnvironment vol 44 no 2-3 pp 145ndash163 1993

[24] A Pelagotti A Mastio A Rosa and A Piva ldquoMultispectralimaging of paintingsrdquo IEEE Signal Processing Magazinevol 25 no 4 pp 27ndash36 2008

Journal of Spectroscopy 9

[7] Y Zhao and R S Berns ldquoImage-based spectral reflectancereconstruction using the matrix R methodrdquo Color Research ampApplication vol 32 no 5 pp 343ndash351 2007

[8] M M Amiri and S H Amirshahi ldquoA hybrid of weightedregression and linear models for extraction of reflectancespectra from CIEXYZ tristimulus valuesrdquo Optical Reviewvol 21 no 6 pp 816ndash825 2014

[9] D Dupont ldquoStudy of the reconstruction of reflectance curvesbased on tristimulus values comparison of methods of op-timizationrdquo Color Research amp Application vol 27 no 2pp 88ndash99 2002

[10] G Wu ldquoReflectance spectra recovery from a single RGBimage by adaptive compressive sensingrdquo Laser Physics Lettersvol 16 no 8 Article ID 085208 2019

[11] R Schettini and S Zuffi ldquoA computational strategy exploitinggenetic algorithms to recover color surface reflectancefunctionsrdquo Neural Computing and Applications vol 16 no 1pp 69ndash79 2007

[12] F M Abed S H Amirshahi and M R M Abed ldquoRe-construction of reflectance data using an interpolationtechniquerdquo Journal of the Optical Society of America A vol 26no 3 pp 613ndash624 2009

[13] T Harifi S H Amirshahi and F Agahian ldquoRecovery ofreflectance spectra from colorimetric data using principalcomponent analysis embedded regression techniquerdquo OpticalReview vol 15 no 6 pp 302ndash308 2008

[14] X Zhang Q Wang J Li X Zhou Y Yang and H XuldquoEstimating spectral reflectance from camera responses basedon CIEXYZtristimulus values under multi-illuminantsrdquo ColorResearch amp Application vol 42 no 1 pp 68ndash77 2016

[15] G Wu X Shen Z Liu S Yang and M Zhu ldquoReflectancespectra recovery from tristimulus values by extraction of colorfeature matchrdquo Optical and Quantum Electronics vol 48no 1 64 pages 2016

[16] J Liang and X Wan ldquoOptimized method for spectral re-flectance reconstruction from camera responsesrdquo OpticsExpress vol 25 no 23 pp 28273ndash28287 2017

[17] J Liang K Xiao M R Pointer X Wan and C Li ldquoSpectraestimation from raw camera responses based on adaptivelocal-weighted linear regressionrdquo Optics Express vol 27no 4 pp 5165ndash5180 2019

[18] Spectral Database University of Eastern Finland KuopioFinland httpwww2ueffifispectral8

[19] M J Vrhel R Gershon and L S Iwan ldquoMeasurement andanalysis of object reflectance spectrardquo Color Research ampApplication vol 19 no 1 pp 4ndash9 1994

[20] GWu Z Liu E Fang and H Yu ldquoReconstruction of spectralcolor information using weighted principal componentanalysisrdquo Optik-International Journal for Light and ElectronOptics vol 126 no 11-12 pp 1249ndash1253 2015

[21] G Wu Z Liu and J Zhang ldquoSpectral color reproductionfrom CIE tristimulus values using a node address array se-lection techniquerdquo International Journal of Signal ProcessingImage Processing and Pattern Recognition vol 8 no 9pp 141ndash150 2015

[22] Spectral Power Distribution (SPD) Curves National GalleryLondon UK httpresearchng-londonorgukscientificspd

[23] F A Kruse A B Lefkoff J W Boardman et al ldquo+e spectralimage processing system (SIPS)-interactive visualization andanalysis of imaging spectrometer datardquo Remote Sensing ofEnvironment vol 44 no 2-3 pp 145ndash163 1993

[24] A Pelagotti A Mastio A Rosa and A Piva ldquoMultispectralimaging of paintingsrdquo IEEE Signal Processing Magazinevol 25 no 4 pp 27ndash36 2008

Journal of Spectroscopy 9

TribologyAdvances in

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

International Journal ofInternational Journal ofPhotoenergy

Hindawiwwwhindawicom Volume 2018

Journal of

Chemistry

Hindawiwwwhindawicom Volume 2018

Advances inPhysical Chemistry

Hindawiwwwhindawicom

Analytical Methods in Chemistry

Journal of

Volume 2018

Bioinorganic Chemistry and ApplicationsHindawiwwwhindawicom Volume 2018

SpectroscopyInternational Journal of

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Medicinal ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

NanotechnologyHindawiwwwhindawicom Volume 2018

Journal of

Applied ChemistryJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Biochemistry Research International

Hindawiwwwhindawicom Volume 2018

Enzyme Research

Hindawiwwwhindawicom Volume 2018

Journal of

SpectroscopyAnalytical ChemistryInternational Journal of

Hindawiwwwhindawicom Volume 2018

MaterialsJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

BioMed Research International Electrochemistry

International Journal of

Hindawiwwwhindawicom Volume 2018

Na

nom

ate

ria

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Hindawiwwwhindawicom Volume 2018

Journal ofNanomaterials

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