Developing an optimum computer-designedmultispectral system comprising amonochrome CCD camera anda liquid-crystal tunable filter
Miguel A. López-Álvarez,1,* Javier Hernández-Andrés,2 and Javier Romero2
1Hewlett-Packard Spain, Large Format Printing Division, 08174 Sant Cugat del Vallès, Barcelona, Spain2Colour Imaging Laboratory, Departamento de Óptica, Universidad de Granada, 18071 Granada, Spain
Over the past ten years several authors have contrib-uted to the development of the theory behind multi-spectral imaging systems [1–6], and in doing so haveproved the reliability of these devices for makingaccurate estimations of spectral-power distributions(SPDs) in every pixel of the imaged scene [1]. Mostof these works focus on recovering the spectral reflec-tance of objects or the combined color signal [7–9], butlittle information has been published about spectralestimationofnatural illuminants [10–12].Theadvan-tagesofusingsuchsystemsinsteadof traditionalspec-troradiometers are numerous. For example, we canobtain a radiance spectrum for each pixel of the ima-ging matrix, typically a charge-coupled device (CCD)or a complementary-metal-oxide-semiconductor de-vice (CMOS) [13]. Moreover, multispectral systems
are cheaper, lighter, and more portable than classicalspectroradiometers.
Here we focus our interest on studying skylight, animportant natural illuminant [14,15], from the spec-tral curves of which we can extract information aboutclimate parameters such as the optical depth or theAngstrom exponent [15], which inform us about thesize and concentration of aerosol particles. Recentlywe also suggested the possibility of developing com-puter algorithms for automatic cloud detection andclassification based on the spectral information esti-mated from multispectral images of the sky. A hugedatabase of multispectral skylight images could beinteresting for scientists of many disciplines, sinceit would provide images of the entire skydome withhigh spatial and spectral resolution that could beused in many areas of research.
In previous works [10,11] we published a completetheoretical study about a planned optimum multi-spectral system for the spectral imaging of sky-light. By developing computational simulations we
obtained very interesting results concerning the in-fluence on the behavior of the multispectral systemof several parameters, such as the spectral responsiv-ity of its sensors, the number and type of sensors, thespectral estimation method and linear bases chosen,the number and quality of training spectra, and thenoise that always affects any electronic device. As aresult we found that when certain values for theseparameters were set we could make very accuratespectral reconstructions. Thus, we now intend touse these previous results to build a prototype of thisoptimum multispectral system by using a cooled,12 bit, monochrome CCD camera (Model Retiga QI-maging SRV1340) and an LCTF (Model Varispec,from CRi). Given the possibility of making a completecalibration [16] of the elements of the multispectralsystem, we also obtained spectral measurements ofskylight radiance by direct radiometric wavelengthsampling. We show how the optimum configurationfound for the system in the computational study pro-vides better spectral recoveries and results muchfaster.InSection2weresumetheresultsofpreviouspapers
[10,11] by showing the optimum configuration of thesystem that can be implemented when we use the de-vices available in our laboratory. We also describe thetraining and test sets of experimental measurementsof spectral skylight used in this study. In Section 3 weshow how to obtain spectral measurements with theCCDcameraandtheLCTFwhenthesedevicesarecor-rectly calibrated (themeasurement process is done bydevelopingadirect radiometricwavelengthsampling,which is here described in detail), and we plot the re-sults when using the system in this configuration. InSection 4we obtain spectral estimations byusing a re-gressionmodelwith the 33 channels or transmittancemodes [1] of the LCTFand compare it with the resultsobtained when we use the optimum configuration forthe system (Section 5). Finally, we give some ideasaboutfutureworkanddiscusstheprosandconsofeachof the configurations proposed for the multispectralsystem.
2. Computational Results
We used a model proposed by Maloney and Wandell[17], which is a widely accepted [1–9] theoreticalmodel of the responses of the camera, ρ (a columnvector of k rows corresponding to the k channels orsensors available), when a given radiance spectrum,E (a column vector of N rows corresponding to the Ndifferent wavelengths sampled in the visible spec-trum), impinges on it. Thus,
ρ ¼ RtEþ σ; ð1Þ
where R is an N × k matrix containing the spectralresponsivities of the k sensors at N sampled wave-lengths (superscript t denotes its transpose), and σis a k row vector of uncorrelated components of noisethat affect each sensor separately [1,10].
The goal here is to recover spectrum E from re-sponses ρ of the sensors. Different spectral estimationmethods [3–5,10,17,18] have been used to try to solvethis problem. Our previous study [10] was focused oncomparing the accuracy of the spectral reconstruc-tions obtained with each of these methods when theoptimum sensors found for them in each case wereused. We found that the Linear Pseudoinverse meth-od (sometimes erroneously called the Wiener methodbecause of their mathematical similarity) and theImai–Berns method [3] provided good spectral recon-structions,with the additional advantage—comparedto other spectral estimation methods—that it is notnecessary to know the spectral responsivity of cameraR in practical situations where computer simulationsare not involved. These twomethodswere fast in theircalculations and also very robust against noise. Theonly drawback of the Imai–Berns method is that itneeds a representative linear basis of spectra fortraining, but this may be an advantage in situationswithhighnoise, because a reduction of the dimension-ality can be achieved by using basis vectors and thismay help to reduce the influence of noise [5,9,10]. Thetask of calculating a basis is fairly easy by means ofprincipal component analysis [1] (PCA), nonnegativematrix factorization (NMF) [19] or the independentcomponent analysis (ICA) [20] of a training set of spec-tral measurements. These mathematical tools coin-cide in providing a set of vectors that can be used toexpress a given spectrum as a linear combination:
E ¼ Vϵ; ð2Þ
where V is an N × n matrix containing the first nvectors used for reconstructing N wavelengths (n isalways less than or equal to N and is usually chosento equal k, the number of sensors, which oftengives the best results [3,10]). Vector ϵ is an n rowedvector that contains the coefficients of the linearcombination.
We use the Imai–Berns method here because withourLCTFwe canbuild the optimumset of five sensorsfound with this method when the noise simulated atthe camera gives a value of 26dB for the signal-to-noise ratio (SNR). This noise level was really closeto the real noise in our camera, which was estimatedbymeasuring the variancewhen imaging an integrat-ing sphere [16] that serves as a perfectly homoge-neous and constant object. As some authors haveindicated [5], the closest the estimated noise usedto find the optimum sensors is to the real noise level,thebetter those filterswill bewhen implementedwiththe real system.Wemust point out that using only theImai–Berns method does not mean that we cannotuse other spectral estimation methods with our sys-tem, just that the optimumsensors found for the othermethods studied are not achievable with our LCTF.
In Fig. 1 we show the 33 transmittancemodesmea-sured for our LCTF in the laboratory. Figure 2 showsthe five optimumsensors thatwe intend to implementand how these sensors can be obtained with seven
modes of our filter by adjusting the exposure timesand summing up the contributions of modes 3, 4,and 5 (corresponding to the third optimum sensor),where the spectral responsitivity of the CCD (shownin Fig. 3) has been taken into account.We also use theImai–Berns method and the Linear Pseudoinversemethod with the 33 channels (Section 4) alreadyavailable with our multispectral system to comparethe quality of the spectral reconstructions when weuse only five optimum sensors (Section 5).We need to train the system before using the two
spectral estimation methods mentioned above, sincethese algorithms use the information obtained from atraining set of spectra to provide good spectral recon-structions from the sensors’ responses. For the Imai–Berns method we directly establish a relationshipbetween the sensors’ responses ρ and coefficients ϵ,which now includes a column in ρts and ϵts for eachof the m training spectra (subscript ts stands fortraining spectra), and we obtain
εts ¼ Gρts; ð3Þ
where matrix G is an n × kmatrix that is determinedempirically by a least-squares analysis of the train-ing-spectra measurements. Hence it is not necessaryto measure spectral sensitivities R of the camera touse this method with real sensor-response measure-ments [10]. We can estimate G via a least-squaresanalysis by pseudoinverting the k ×m matrix, ρts :
G ¼ εtsρþts: ð4Þ
In our case, the recovered skylight spectrum is sim-ply calculated in this method from sensors’ responsesρ by
ER ¼ VGρ: ð5Þ
Here the information provided by the training spec-tra is included in V and in G.
If we use the Linear Pseudoinverse method, wemust establish a relationship directly between sen-sors’ responses of the training set ρts and trainingspectra Ets:
W ¼ Etsρþts: ð6Þ
Thus we can obtain spectral estimations with theLinear Pseudoinverse method exactly in the wayshown in Eq. (5) by replacing VG with the matrix W.
It is desirable to use different sets of spectral mea-surements as training and test sets. As the trainingset inthisstudyweuseadatabaseof1567spectralsky-lightmeasurements takenby our group between 1997and 1999 [21] in Granada (Spain, 37:16°N, 3:60°W,680 m.a.s.l.) with a LICOR spectrorradiometer, atmanydifferentsolarelevations,withdifferentrelativeazimuths towards the sun and during different sea-sons of the year; each spectrum ranged from 380 to780nm in 5nmsteps. By that timewehadnot yet con-structed our multispectral system and so we have noexperimental information on responses of the camera
Fig. 1. (Color online) Spectral transmittance of the 33 modes ofthe Varispec liquid-crystal tunable filter measured three times inour laboratory (error bars show the standard deviation obtained ateach sampled wavelength).
Fig. 2. (Color online) Five optimum sensors found in the compu-tational simulations (solid line) for the Imai–Berns method with aSNR equal to 26dB. Seven transmittance modes of the LCTF(dashed line) used to implement the theoretical optimum sensors.
Fig. 3. (Color online) Spectral responsivity of the CCD camera(Model Retiga QImaging SRV1394) measured at our laboratory.Error bars show the uncertainty in these measurements.
ρts butwecancalculatethetheoretical responsesofourmultispectral system (which is correctly calibrated) tothese spectra anduse them inEq. (4) to obtain thema-trixG. This step is explained indetail inSection4.Asatestset,weuseasetof125spectralmeasurementsalsotaken in Granada in 2007 over a period of 7 months,which now does include the information of the experi-mental camera responses, ρ. This set of spectra wasmeasured simultaneouslywith ourmultispectral sys-tem and a SpectraScan PR650 spectroradiometer be-tween 380 and 780nm in 4nmsteps.We try to recoverspectraEof the test set byusing the information regis-tered with camera ρ and the information provided bythe training set (Ets, ϵts, and the calculated ρts; theseparametersareneededtocalculatematrixG). InFig.4we show the chromaticity coordinates, in the CIE-31space, of the 1567 skylight spectra belonging to thetraining set (its correlated color temperatures (CCT)ranging from 3500K to infinity; we must say thattwo measurements of this set do not have an assoc-iatedCCTbecause their chromaticities lie too far from
the Planckian locus [21]), while in Fig. 5 we show thesame diagram for the test set of 125 measurements(with CCTs ranging from 8300K to 32; 000K).
3. Spectral Measurements of Skylight by DirectRadiometric Sampling
In Figs. 1 and 3 we show the results of calibrating theLCTF and the CCD camera. Thus we have two pre-cise devices that can be used together to obtain spec-troradiometric measurements by taking advantageof two things: the radiometric information providedby the CCD camera and the narrow spectral band inthe visible range selected with the LCTF. The mainassumptions here are two. First, each of the trans-mittance modes of the LCTF is narrow enough to as-sume that the radiance information received by thecamera when a filter mode is tuned corresponds tothe central wavelength alone, i.e., we assume thatthe modes of the LCTF are equivalent to monochro-matic filters. Since the typical spectral accuracy ofspectroradiometric devices is 4nm, monochromaticin this context means to use a spectral width of aboutthat range. The full width at half-maximum (FWHM)of the modes of the LCTF is between 7 and 15nm,
Fig. 4. (a) CIE-31 chromaticity diagram for the 1567 spectralmeasurements of skylight belonging to the training set. (b) Detailof (a).
Fig. 5. (a) CIE-31 chromaticity diagram for the 125 spectral mea-surements of skylight belonging to the test set. (b) Detail of (a).
depending on the central wavelength chosen, andthus we can accept the assumption of monochroma-ticity. Second, the radiometric information given bythe CCD camera is accurate enough to guaranteethat it does not depend on the wavelength, the expo-sure time, or other external factors. Ferrero et al. [16]described a precise procedure to assure this bymeans of a complete radiometric calibration elimi-nating the influence of noise and thus we havefollowed their recommendations.The direct radiometric sampling of the visible spec-
trum consists of tuning the LCTF into a centralwavelength between 400 and 720nm (in 10nm steps,which are the available modes of our filter) and thentaking a picture (corrected for noise influence [16]), ofthe sky in this case. By doing this, we obtain a radio-metric sample of the selected wavelength for everymode of the LCTF, hence covering the whole visiblespectrum. This method has the advantage of notneeding a training set of spectra, while spectral es-timationmethods do. Nevertheless, a complete radio-metric calibration of the CCD and the filter must bemade before using this procedure, and the spectralrange covered is reduced to the maximum and mini-mum central wavelengths achievable with the LCTF.While using the spectral estimation methods, we canobtain information in the whole spectral rangecovered by the training set, which is usually a littlelarger, as can be seen in Section 4.Spectral responsivity RðλÞ of our monochrome
CCD camera was calculated from a proposed modelby Ferrero et al. [16]:
RðλÞ ¼ Cc
EðλÞtexp; ð7Þ
whereCc refers to the corrected pixel value (eliminat-ing all the possible noise and correcting from spatialnonuniformity [22]), EðλÞ is the spectral radiance im-pinging on the CCD, and texp is the exposure timeused for imaging. In this procedure of radiometricsampling, we take one picture for every wavelengthselected by the LCTF, which we denote by subindexk. Thus, if we include the effect of filtering at everywavelength by using the LCTFand assume that EðλÞremains constant during all the imaging process, wecan rewrite Eq. (7) as
Fk ¼ Cck
Ektexp;k; k ¼ 1;……33; ð8Þ
where Fk is equal to the product of the spectral re-sponsivity of the CCD and the LCTF transmittancewhen mode k is selected and Ek is the value of EðλÞ atthe wavelength chosen by the filter when mode k istuned. Finally, texp;k and Ck
c are the exposure timeused and the pixel level registered for the corre-sponding picture, respectively. We can easily calcu-
late the spectral radiance from Eq. (8) as
Ek ¼ Cck
Fktexp;k; k ¼ 1;……33: ð9Þ
In Table 1 we show the mean values (�standarddeviations) for the various quality metrics used tocompare the similarity between each pair of simulta-neous spectrameasuredwith the PR650 and ourmul-tispectral system over the set of 125 spectralmeasurements taken in Granada in 2007 by usingthe radiometric sampling procedure. Since the testset was acquired with the PR650 spectroradiometerbetween380and780nmin steps of4nm,and themul-tispectral system in this radiometric sampling config-uration gets spectral information between 400 and720nm every 10nm, a conversion of the data fromthe PR650 was made prior to comparing the spectrafrom both instruments. Hence, we discarded the databelow400nmandabove720nm,andwemadea linearinterpolation to get spectral data every 10nm (someintermediate data were also discarded). The metricsshown [11,23] are the goodness-fit-coefficient (GFC)(which is the cosine of the angle between two spectraif these are intended to be vectors in a Hilbert space),the colorimetric CIELAB ΔE�
ab distance, the percen-tage of the integrated-radiance-error metric [IRE(%)] (which is a relative measure of the difference inthe total energy of the two spectral curves compared),and the colorimetric and spectral combined metric(CSCM)proposed [10,11,23] to comparespectraofnat-ural illuminants fromcolorimetric andspectral pointsof view, which has also been used by other researchers[24]. The equations defining these four metrics areshown here (EðλÞ represents the original spectrumwhile ERðλÞ stands for the recovered spectrum).
ab metricis intended to compare two reflectance spectra undera given illuminant and not for comparing illuminantsdirectly. Nevertheless, we can assume that the twoSPD of the illuminants we want to compare impingeon a perfectly reflective white patch, and that we areseeing these two assumed patches under the equie-nergetic (spectrally flat) illuminant. By doing this,we can obtain the CIELAB ΔE�
ab error betweentwo illuminants even though this metric is not in-tended for this aim.Three examples of spectral reconstructions made
by using this method, corresponding to (a) the 10th,(b) the 50th, and (c) the 90th percentiles of the CSCMmetric over the test set of 125 skylightmeasurementsare shown in Fig. 6, where it can be seen how the spec-tralmeasurements given by themultispectral systemare quite similar to those given by the spectro-radiometer PR650, although there is a tendency tooverestimate the total energy of the spectra, whichimplies that high values are obtained for the IRE(%) metric. This could be due to a systematic differ-ence between the theoretically expected pixel valuesand the real ones,Cc, registered at the camera, whichis brought about by the inexact assumption of mono-chromaticity of the LCTF transmittance modes.Nevertheless, the quality of the spectral measure-ments taken with the multispectral system in thisconfiguration of radiometric sampling may be accu-rate enough for certain purposes when studying sky-light, where the total energy estimation is not ofparamount importance and we may only need therelative SPD.
4. Spectral Estimation Using a Regression Model with33 Channels
Here we use the information provided by the CCDcamera in every one of the 33 available channels ofthe LCTF to obtain spectral estimations of skylightusing both the Imai–Berns and the Linear Pseudoin-verse methods referred to in Section 2. The maindrawback of using such methods is the need to trainthe system, i.e., to establish a relationship betweenthe training set of spectra and their known sensors’responses. If we intend to use different test and train-ing setswe cannot also use the set of 125 spectralmea-surements taken in 2007 as a training set. Thus weshould use the set of 1567 skylight spectra measuredbetween1997and1999asa training set.Wemust facetwo drawbacks when using this approach. First, theset of 1567 measurements have a sampling intervalof 5nm, while the test set of 125 measurements havea sampling interval of4nm.Wesolved this problembymaking a linear interpolation of the data at 5nmdown to 4nm, which is not a major source of errorsince both sampling intervals are accurate enoughto register spectral skylight information. The secondproblem is that we do not have the information re-
lated to the sensors’ responses corresponding to those1567 spectra because our multispectral system wasnot available at that time. This problem can be solvedby simulating the sensors’ responses of our system to
Fig. 6. (Color online) (a) 10th percentile (CSCM ¼ 6:00), (b) 50thpercentile (CSCM ¼ 15:35), and (c) 90th percentile (CSCM ¼25:05) over the test set of 125 spectral measurements taken inGranada in 2007 when using the radiometric sampling method.
the training set of 1567 spectra [see Eq. (7)], keepinginmind that the spectral responsivity of the cameraRis already corrected from noise influence during thecalibration [16] (i.e., noise is taken into account inR as explained in Eq. (7) above):
Cck;ts ¼ REtstexp;k ð14Þ
and identifying the kth component of the vector ρ ofthe sensors’ responses with the corresponding cor-rected pixel value divided by the exposure time usedfor it. Thus,
ρk ¼ Cck
texp;kk ¼ 1;……33: ð15Þ
If we look at Eq. (5), the spectral estimations arecalculated as
ER ¼ Xρ; ð16Þ
whereX ¼ VG for the Imai–Bernsmethod andX ¼ Wfor theLinear Pseudoinversemethod [seeEqs. (4) and(6)]. Hence, for the Imai–Berns method we must alsoselect linear basis V of representative vectors, whichis unnecessary with the Linear Pseudoinverse meth-od.We choose PCA for constructing this basis becauseit is the most widely used strategy [1–5,7–11,17,18].We use different numbers of PCA basis vectors to findthe optimumnumber of themtobeusedwith thismul-tispectral system of 33 channels.In Table 2 we show the results obtained when we
recover the 125 skylight spectra of the test set byusing the Linear Pseudoinverse method with the 33channels available. These results are significativelybetter than those for the radiometric sampling meth-od in Table 1 (only the GFC metric is slightly worse,probably due to the higher spectral resolutionachieved now, as we show below), thus proving thatthe training of the system by simulating the sensors’responses to the set of 1567 spectra is correct. Figure 7shows the 10, 50, and 90 percentiles of the CSCMme-tric when recovering the 125 spectra of the test setwith the Linear Pseudoinverse method with 33 chan-nels (just as Fig. 6 did for the radiometric samplingmethod). Now the spectral range of these estimationsis seen to extend to the interval between 380 and780nm, which corresponds to the specifications ofthe spectroradiometer used to measure the trainingset in 1997 (as mentioned in Section 2), but with aspectral resolution of 4nmbecause of the linear inter-polationweperformed—aswe explained above in this
same section—from the original training set sampledat 5nm. It should also be remembered that the spec-tral range covered in Section 3 was from 400nm up to720nm, with a spectral resolution of 10nm, corre-
Table 2. Mean � Standard Deviation Values of Various Metricsa
GFC CIELAB ΔE�ab IRE (%) CSCM
0:998� 0:001 0:85� 0:21 6:30� 5:43 8:51� 5:57
aOver the test set of 125 spectral measurements taken in Gran-ada in 2007 when using the Linear Pseudoinverse method with 33channels
Fig. 7. (Color online) (a) 10th percentile (CSCM ¼ 3:28), (b) 50thpercentile (CSCM ¼ 6:73), and (c) 90th percentile (CSCM ¼ 15:65)over the test set of 125 spectral measurements taken in Granadain 2007 when using the Linear Pseudoinverse method with k ¼ 33.
sponding to the minimum and maximum centralwavelengths tunable with the LCTF (each of thesecentral tunable wavelengths was 10nm apart fromits neighbors). This extension of the spectral rangecovered will also occur with the Imai–Berns methodlater; it does not mean that the CCD provides moreinformation now, but that the spectral estimationmethods are capable of predicting the spectral shapeof the curves throughout the whole range covered bythe training set, even if there are no sensors in somespectral regions. This can be done by taking advan-tage of the statistical information obtained by train-ing the system. Nevertheless, it can be seen in Fig. 7that the spectral estimations achieved are extremelyaccurate when compared with the measurementsmade simultaneously with the PR650, thus provingthe Linear Pseudoinverse method’s reliability in ob-taining good spectral reconstructions of skylight.In Table 3 we show the results of the Imai–Berns
methodwhenusing the33channels of our system.Dif-ferent numbers of basis vectors n were used to findthe optimum value of this parameter, which turnedout toben ¼ 6.Other studiesofmultispectral systems[2,3,10,11] showed that the best results are foundwhen we use the same number of vectors n as sensorsk, but this seems tobe trueonlywithasmallnumberofsensors. In Table 3 we have included the case wheren ¼ 101 because this means that we use all the avail-able PCA vectors, and then the Imai–Berns and theLinearPseudoinversemethodsare formally thesame,as can be seen in Eq. (17), which is easily derived fromEqs. (2) and (4)–(6). Thus,
ER ¼ VGρ ¼ Vεtsρþtsρ ¼ Etsρþtsρ ¼ Wρ; ð17Þ
in which case the results for the Imai–Berns methodwith n ¼ 101 and the Linear Pseudoinverse methodare exactly the same (see Tables 2 and 3). The corre-sponding 10th, 50th, and 90th percentile curves forthe Imai–Berns method with k ¼ 33 and n ¼ 6 areshown in Fig. 8. If we compare Tables 2 and 3 wecan see that the Imai–Berns method is slightly betterthan theLinearPseudoinversemethodwhen33 chan-nels are involved because the reduction in dimension-
ality achieved byusingn ¼ 6helps to reduce the effectof noise.
5. Spectral Estimation Using Five Optimum Sensors
In Fig. 2 we showed how to implement the five opti-mumsensors found for the Imai–Bernsmethod (usingfive PCA vectors) in computational simulations [10]with our LCTF (see Fig. 1) by adjusting the exposuretime of each transmittance mode. In this section weshow the results obtainedwhenweuse ourmultispec-tral system in this optimum configuration with only
Table 3. Mean � Standard Deviation Values of Various Metricsa
aOver the test set of 125 spectral measurements taken in Gran-ada in 2007 when using the Imai–Berns method with 33 channels
Fig. 8. (Color online) (a) 10th percentile (CSCM ¼ 3:00), (b) 50thpercentile (CSCM ¼ 6:44), and (c) 90th percentile (CSCM ¼ 15:48)over the test set of 125 spectral measurements taken in Granadain 2007when using the Imai–Bernsmethodwith k ¼ 33 and n ¼ 6.
seven channels implementing the five intended opti-mum sensors. We also trained the system here withthe 1567 spectra measured in Granada between1997 and 1999 by simulating their sensors’ responses,as shown in Eq. (14). Table 4 shows that the values forthemetrics used are very similar to the ones obtainedin Section 4 with a larger number of channels. It canalso be seen that the spectral curves obtained are veryaccurate (Fig. 9). The advantage of using this config-uration is important, since the spectral estimationsare of about the same quality as those in previous sec-tions, but at a fivefold lower cost in time (we use only 7modes of the LCTF instead of 33, whichmeans a totalprocessing time of 13 s against 1 min). Hence, westrongly recommend an optimization study prior tousing the multispectral imaging system.
6. Conclusions
We have proved that accurate multispectral estima-tions of skylight can be obtained by using a mono-chrome CCD camera attached to a liquid-crystaltunable filter. The spectral curves of skylight ob-tained with such a system are very similar to thosemeasured simultaneously with a spectroradiometerbut have the several advantages of price, weight, spa-tial resolution, and portability. Given the spectral si-milarity of skylight SPDs with any kind of naturalilluminant, the multispectral information providedby our multispectral system could be used for manyscientific purposes related to the climatology or at-mospheric physics.We tested different configurations of our multi-
spectral system. First, we made use of a complete ca-libration of the CCD camera and the LCTF to developa direct radiometric sampling in the visible range ofthe spectrum. Second, we compared the Linear Pseu-doinverse and the Imai–Berns methods when thesewere used as spectral estimation methods in ourmultispectral system with 33 channels, and they im-proved the results obtained with the direct radio-metric sampling method. Finally, we implementedthe five optimum sensors found in a previous compu-tational study [10] for the Imai–Berns method andshowed that they can be implemented by using justseven transmittance modes of the LCTF and adjust-ing their exposure times. We demonstrated thatusing a small number of optimum sensors providesalmost the same spectral results as using all theavailable channels of the system, with a significantsaving in time. Thus we can recommend developingan optimization procedure prior to building a multi-spectral system.
To conclude, we have demonstrated that thespectroradiometric model proposed theoretically todescribe our system does describe its behavior
Table 4. Mean � Standard Deviation Values of Various Metricsa
GFC CIELAB ΔE�ab IRE (%) CSCM
0:998� 0:001 0:87� 0:21 7:05� 5:85 8:97� 6:00
aOver the test set of 125 spectral measurements taken in Gran-ada in 2007whenusing the five optimumsensors of the Imai–Bernsmethod with five PCA vectors.
Fig. 9. (Color online) (a) 10th percentile (CSCM ¼ 3:12), (b) 50thpercentile (CSCM ¼ 7:48), and (c) 90th percentile (CSCM ¼ 16:76)over the test set of 125 spectral measurements taken in Granadain 2007 when using the Imai–Berns method with five optimumsensors and five PCA vectors.
accurately since the training process using simulatedsensors’ responses provides spectral reconstructionsof high qualitywhenadifferent test set of 125 spectralcurves is recovered.
This research was supported by the SpanishMinistry of Education and Science and the EuropeanFund for Regional Development (FEDER) throughgrant FIS2007-60736. The authors thank A. L. Tatefor revising the text.
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