colorimetric calibration

7212019 Colorimetric calibration

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2 Mathematical Problems in Engineering

few years with the RGB-ITR (Red Green Blue-ImagingTopological Radar) system [983097] a patented 3D colour laserscanner prototype completely designed and realized at ENEAlaboratories of Frascati(Rome) for cultural heritage purposes

Te advantages of 3D imaging systems in the 1047297eld of culturalheritage are now recognized and widely accepted [10ndash12]Te purely geometric output provided by the majority of currently available laser scanners is not enough for mostcultural heritage applications [13] A common practice is tosuperimpose textures derived from ordinary digital photosonto the 3D models generated by most devices Tis tech-nique hasmany drawbacks andlimitations in terms of achiev-able visual quality and accuracy since the generation andsuperimposition of texture images are realised via sofwareand commercial cameras Diff erently from commercial 3Dscanners the ENEA prototype can simultaneously record foreach investigated point range and RGB colour data Suchdistinguishing feature opens new scenarios and a new classof problems for remote colorimetry [14]

A standardization of colour information detected by RGB-ITR system respecting CIE standards was already per-formed in the past with colour calibrations by distance infor-mation and by MINOLTA spectrophotometer-CM-2600d[983097] Tesecalibrations have assumed a linearrelation betweenthe red green and blue back-re1047298ected signals and scenere1047298ectances at each distance of measurement Te presentstudy shows that due to nonlinearities of the optoelectronicsystem this assumption produces undesired eff ects in theremote colour measurement of certi1047297ed diff usive gray tar-

gets (Labsphere Inc) and proposes a nonlinear calibrationfor compensating the input-output characteristics of thedevice providing meaningful and more accurate 3D RGB-ITR colour images Te improvement gives added value forlaser scanning campaigns providing intensity data muchmoresuitablefor cataloguing disseminationrestoration andremote diagnosis purposes Te eff ects of distance and targetre1047298ectance on the recorded intensity along with the devel-opment of the proposed correction method are analyzedTe model is capable of representing real small and bigsurfaces at short and large distances from the scanner Teimplementation is performed with a fast algorithm that usesa quadratic model with nonpolynomial terms and piecewise

cubic Hermite polynomials with 1047297rst-order accurate deriva-tive approximation method A remote Lambertian model forassessing the accuracy of the algorithm and several experi-mental results that demonstrate the eff ectiveness provided by the novel calibration procedure are reported

2 RGB-ITR Colour Detection

Te RGB-ITR system is a tristimulus laser scanner proto-type based on the amplitude modulation [15 16] of threemonochromatic sources (660 nm 514 nm and 4 40 nm) andis able to simultaneously collect colour and structure infor-mation for any investigated sampled surface point in aworking range of about 3ndash30 meters Tis capability makescolour information as important as range data

Raw colourdata returned by the RGB-ITR simplyconsistsof a triplet of voltage values Each value represents the redgreen or blue light power re1047298ected by a particular surface ascollected by the receiving optics and revealed by the detector

Te RGB triplets are then processed with a calibrationprocedure which consists in illuminating a movable whitetarget with the laser beam at 1047297xed steps (eg 10 cm) along thescanning range Finally calibrated RGB data is merged withrange information to produce high-quality 3D digital models

983090983089 Linear Calibration Ideally a correct colour calibrationprocedure should represent the content of a real target 3Dmodel by re1047298ecting the intrinsic properties of the target sur-face Namely two colorimetrically indistinguishable regionsof the surface should be represented in the model as havingthe same colour independently on the conditions underwhich those regions were sounded during the scan that iswithout being aff ected by scanning geometry measurementambient light eff ects and so forth

Te RGB-ITR colour measuring capabilities are in1047298u-enced by various factors well described in [13] In thisstudy the eff ects produced by the nonlinearities of theoptoelectronic system are illustrated by calibrating certi1047297edLambertian gray targets of 50and 20 re1047298ectances with raw colour measurementsperformedon a white diff usive targetatdiff erent distances from the system Tese raw detected dataform the calibration curves Figures 1 and 2 report the calibra-tion curves and the detected gray target signals as functionsof distance at diff erent ranges from the scanner Te curvesexhibit the same behaviour that is they all have a maximumthat can be 1047297xed by varying some optical parameters Teon-focus distance for these curves corresponds to 1198890 = 44 m(curves in Figure 1) and 1198890 = 208 m (curves in Figure 2)where approximately all the RGB curves have a maximumFor 119889 gt 1198890 the collected power falls down mainly because

of the misalignment of the optical axes and the 11198892 typicaltrend [13] For 119889 lt 1198890 this trend is counterbalanced by thefact thatthe receiver intercepts only part of the re1047298ected lightso the collected power also falls down As a consequenceall the calibration curves their characteristic bell-shape arethe result of the optical parameters tuned up for maximizingthe signal-noise ratio of the detected voltages and their shapeis independent from distance Note that because of theoptoelectronic system features it is not possible to obtainthe same RGB signals on certi1047297ed Lambertian targets at allthe distances of measurement Tis implies that an accuratecalibration with a certi1047297ed reference target is necessary forproducing RGB-ITR colorimetric models

Linear calibration of gray targets is performed by calcu-lating the ratio between the three RGB signals acquired by the instrument for each target and the amplitude value onthe three calibration curves selected by the correspondingdistance 119889 For each channel 120582 the measured re1047298ectancefactor of a gray target is given by

119877gray (120582) = 119881

gray

(120582)119881white (120582) sdot 119877

white (120582) (1)



Mathematical Problems in Engineering 3

Distances (m)

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(a) Red channel amplitudes

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(b) Green channel amplitudes

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(c) Blue channel amplitudes

F983145983143983157983154983141 1 RGB signals measured on 99 50 and 20 re1047298ectance targets at 3983097 sampled distances from the scanner ranging from 28 to 68meters

where 119877gray (120582) is the measured re1047298ectance factor at the

distance 119889 and 119877white(120582) is the certi1047297ed spectral re1047298ectancefactor of the perfect diff user that is the white target usedin this measurement As shown by calibrated signals inFigure 3 linear calibration (1) does not return the samere1047298ectance values for light and dark gray targets at eachdistance Tis nonlinearity generates discontinuities in thecolour representation of gray targets that is the punctualcolour model of the same object at diff erent distances is notexactly represented as having the same colour Figure 3 shows

the result of the linear calibration procedure applied to lightgray and dark gray targets at 3983097 sampled distances

3 Methods

Tis section presents the mathematical models introducedin the proposed calibration procedure Four Spectralon (Lab-sphere Inc) Diff usive Re1047298ectance Standards (SRS) with 220 50 and 99 nominal re1047298ectance have been used asreference targets for the calculations




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F983145983143983157983154983141 2 RGB signals measured on 99 50 and 20 re1047298ectance targets at 24 sampled distances from the scanner ranging from 177 to223 meters

983091983089 Quadratic Calibration Model As shown by linear cali-bration results in Section 21 the back-re1047298ected RGB signalsof light gray and dark gray targets are not constant withdistance A model that considers diff erent certi1047297ed Lam-bertian re1047298ectances is needed for compensating this systemnonlinearity and for producing a more accurate colorimetricrepresentation of real scenes A nonlinear term is introducedin the model in order to both eliminate inaccuracies in theremote colour measurement of gray targets and mimic thenonlinear perceptual response to luminance of human vision[17] More accurate data is much more suitable for restorationand diagnosis aids while adaptation to human perception

makes RGB-ITR images more interesting for disseminationcataloguing and education purposes To address the secondconstraint the nonlinear term is de1047297ned to be comparableto the power function de1047297ned by the Munsell equation of lightness 119871lowast according to the standard CIE 198309731 and 198309764

[18 1983097]

119871lowast

=

9831639831631048699983163983163852091

116 sdot 983080 119884

119884

98308113 minus 16 if 119884

119884

gt 1048616 6

2910486173

119897 sdot 983080 119884119884983081 if 119884119884 lt 1048616 62910486173

(2)




1 3 5 7 9 1 1 1 3 1 5 1 7 1 9 2 1 2 3 2 5 2 7 2 9 3 1 3 3 3 5 3 7 3 9

Distance samples

T r e e - c h a n n e l r e fl e c t a

n c e s

Whitetarget

Lightgray

target

Dark gray

target

(a) Linear calibrated colour images

Distance (m)

2 3 4 5 6 7

R e 1047298 e c t a n c e

015

02

025

03

035

04

045

05

055

06

065

Red channelGreen channel

Blue channel

(b) RGB re1047298ectance plots as function of distance

F983145983143983157983154983141 3 Linear calibration results RGB calibrated values for light gray and dark gray targets at 3983097 sampled distances

with 119897 = (293)3 Te term 119884119884 is called the luminance

factor that is the ratio between the luminances of a specimenand of a perfect diff user when illuminated and viewed underspeci1047297ed geometric conditions By de1047297nition the tristimulus

value 119884 for an object is the luminance factor and can beapproximated as

119884 = 119870 sdot 991761

119878 (120582) sdot 119877 ( 120582) sdot 119910cmf (120582) Δ120582 (3)

where 119878(120582) is a CIE illuminant 119877(120582) is the objectrsquos spectralre1047298ectance factor 119910cmf (120582) is one of the three CIE standardobserver colour-matching functions (119909cmf (120582) 119910cmf (120582) and119911cmf (120582)) sum represents summation across wavelength 120582Δ120582 is the measurement wavelength interval and 119870 is aconventional normalizing constant de1047297ned as [1983097]

119870 = 100

sum 119878 (120582) sdot 119910cmf (120582) sdot Δ120582 (4)

For each distance of measurement and for each channel acurve is constructed with a quadratic model with nonpoly-nomial terms necessary for interpolating at each distancefour points given by lightness values measured with theMINOLTA spectrophotometer-CM-2600d on four SRS Teuse of four targets and the pursuit of the dual objectivementioned above require the model to be nonlinear inthe coefficients and in the data respectively Namely givengeneric 119905 data the model 119910(119905) should be in the form

119910 (119905) = 1198860 + 1198861 sdot 119891 (119905) + 1198862 sdot 119891 (119905) sdot 119905 + 1198863 sdot 119891 (119905) sdot 1199052 (5)

where

119891(

119905) is the nonpolynomial function of the model and

1198860 1198861 1198862 1198863 are the model coefficients Te 2 diff usivetarget is assumed to be the zero (black) for the calibration

measurements Te MINOLTA measurements were per-formed in a spectrum interval of 380 nm divide 740 nm withΔ120582 = 10 nm assuming the 11986365 illuminant and the 10

∘ CIEstandard observer Since re1047298ectance values of gray targetsare almost constant in the visible spectrum RGB-ITR light-nesses related to 440 nm 514 nm and 660 nm can be wellapproximated with the spectrophotometer measurements

Given a set of 119873 calibration distances 1198891 119889 themodel used for each channel and for a generic distance 119889can be de1047297ned with the following compact form

1199101 = 1198860 + 119891 (1198811) sdot 3991761=1

119886 sdot 852008119881minus11 852009

1199102 = 1198860 + 119891 (1198812) sdot 3991761=1

119886 sdot 852008119881minus12 852009

1199103 =

1198860 +

119891 (1198813

) sdot

3

991761=1119886

sdot 852008119881minus1

3

8520091199104 = 1198860 + 119891 (1198814) sdot

3991761=1

119886 sdot 852008119881minus14 852009

(6)

where 1198791 1198792 1198793 and 1198794 correspond to the four Lambertiantargets used in the calibration procedure 119881 is the inputmeasured voltage 119910 is the imposed measured lightness 11988601198861 1198862 and 1198863 are the unknown model coefficients and 119891(119881)

is the nonlinear term de1047297ned as

119891 (119881) = 11988113 (7)

comparable to the lightness measure in (2) Te modelcoefficients are computed by constructing and solving a setof simultaneous equations Tis is accomplished by forming




Input voltages (mV)

C a

l i b r a t e d

i n t e n s i t i e s

0

02

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1

Linear model

Imposed data

Nonlinear model

0 0005 001 0015 002 0025

F983145983143983157983154983141 4 Linear model nonlinear model and the imposed light-ness at a generic distance Input voltages are expressed in mVolts

a design matrix [119883] where each column represents a variableused to predict the response (a term in the model) and eachrow corresponds to one observation of those variables Forgeneric 119879 target and 119889 distance the matrix is given by

[119883

=

9831311

11988113

119881 sdot 11988113

1198812

sdot 11988113

983133 (8)

Te least-square solution of the following system returns themodel coefficients at the distance 119889

(119886) [119883] = (119910) (983097)

Tis procedure is repeated for all 119873 distances Using piece-wise cubic Hermite polynomials 4 times 119873 coefficients areinterpolated at the acquired scenedistances Te evaluation of the design matrix at scene detected voltages V

s for each RGB

channel gives the nonlinear calibrated intensities Ic that is

the approximated lightnesses of the scene

[Ic = (a) 983131X V

s983133 (10)

As an example the nonlinear model for the red channelat a generic distance of measurement and with 25 mV of detected signal measured on the white target is shown inFigure 4

At this distance the black target detected signal is lessthan 10120583V (comparable to the instrument noise signal)thus it does not aff ect the calculation Diff erently near themaximum of the calibration curves the RGB signals can be

very high (tens of mV) and the black signal can be used in thecalibration measure since it can be greater than or equal toabout 1 mV

Te implementation of the proposed model is given inAlgorithm 1

983091983090 Interpolation with Piecewise Cubic Hermite Polynomials

Robust and efficient interpolation of the model coefficients atscene distances is needed for an accurate calibration measurePiecewise cubic Hermite interpolating polynomials 119867(119909)

have properties that meet these requests [20 21] Given aninterval [119886 119887] a function 119891 [119886 119887] rarr R with derivative 119891

[119886 119887] rarr R and a set of partition points 119909 = (11990901199091 119909)

with 119886 = 1199090 lt 1199091 lt sdot sdot sdot lt 119909 = 119887 a 1198621 cubic Hermite splineis de1047297ned by a set of polynomials ℎ0 ℎ1 ℎminus1 with

ℎ (119909) = 119891 (119909)ℎ (119909+1) = 119891 (119909+1)

ℎ (119909) = 119891

(119909)ℎ (119909+1) = 119891 (119909+1)(11)

for 119894 = 0 1 119873minus 1 Te spline formed by this collection of polynomials can be de1047297ned as

119867 (119909) =

991761=0

119891 (119909) ℎ (119909) + 119891 ℎ (119909) (12)

Te Hermite form has two control points and two controltangents for each polynomial Tey are simple to calculate

in terms of time required to determine the interpolant and toevaluate it but at the same time they are too powerful Teslopes at 119909 are chosen in such a way that 119867(119909) preservesthe shape of the data and respects monotonicity [20] Tismeans that on intervals where the data are monotonic119867(119909) is also monotonic at points where the data has a localextremum119867(119909) also has a local extremum Tese propertiesmake monotonic Hermite interpolation extremely local if asingle interpolation point 119909 changes the approximation canonly be aff ected in the two intervals [119909minus1 119909] and [119909119909+1]

which share that partition point An example of Hermiteinterpolation of generic data is shown in Figure 5

Algorithm 983089 (main algorithm steps of the model) Considerthe following

Input Scene intensity matrix V scene distance matrix Dcalibration distance vector d measured targets intensities x iand imposed targets lightnesses y

i at calibration distance 119889

are as follows

(V) 997888rarr (V)timestimes3

(D) 997888rarr (D)times

(d) 997888rarr (d)1times

(x i)1times4 = (1199091 11990921199093 1199094)( y i)1times4

= (1199101119910211991031199104)

(13)




True

Data

Spline

Hermite interpolation

x

f ( x )

(a)

True

Data

Spline


x

f ( x )

(b)

F983145983143983157983154983141 5 An example of data interpolation with cubic Hermite polynomials and spline In (b) the 1047297ner Hermite interpolation is highlighted

Output Calibrated intensity matrix Ic is as follows

(Ic) 997888rarr (Ic)timestimes3 (14)

(1) Form the design matrix at distance 119889

[X i4times4

= 1048667852059852059852059852059852059

1

119909131

1199091

sdot 119909131

11990921

sdot 119909131

1 119909132 1199092 sdot 11990913

2 11990922 sdot 11990913

2

1 119909133 1199093 sdot 11990913

3 11990923 sdot 11990913

3

1 119909134 1199094 sdot 11990913

4 11990924 sdot 11990913

4

1048669852061852061852061852061852061]

(15)

(2) Compute model coefficients (Ai)1times4 =(1198861 1198862 1198863 1198864) with least-square method at

distance 119889(A

i)1times4 [X i4times4 = ( y

i)4times1

(16)

(3) Repeat all for each of the k sampling distances

(Ai)1times4 997888rarr (Ai)times4 (17)

(4) Interpolate model coefficients at scene distances cor-responding to the elements of (D)times with piecewisecubic Hermite polynomials within (d)1times and (A)times4

(Ai)times4 997888rarr (A

i)(sdot)times4

(18)

(5) Evaluate the model for each element of the vectorized(V)(sdot)times intensity channel

(Ic)1times1

= (X V)1times4 (A)4times1 (1983097)

(6) Reshape the calibrated scene intensity matrix

(Ic)(sdot)times3 997888rarr (Ic)timestimes3 (20)

In this work the derivative values of each polynomialare approximated in such a way that the interpolation couldnot be signi1047297cantly aff ected by errors that can occur in thecalibration procedure that is wrong distance measurementsGiving a set of 119899 calibration distances (119889) = (1198891 119889)

and detected voltages (V ) = (V 1 V ) the derivatives 119892 onthe lef and right edges are computed by taking the forwarddiff erences

1198921 = V 2 minus V 11198892 minus 1198891

119892 = V minus V minus1119889 minus 119889minus1

(21)

On interior points V with 1 lt 119894 lt 119899 taking the centraldiff erences

119892 = V +1 minus V minus1119889+1 minus 119889minus1 (22)

Tis simple approximation is 1047297rst-order accurate for evenly

and unevenly spaced points More accurate methods for1047297nite diff erence approximation of derivatives that distinguishbetween evenly and unevenly spaced points do not work wellwith noisy data thus they are not useful in this context Inorder to compare the robustness of this gradient computationmethod to errors occurring in remote measurements withrespect to a more accurate one noise is introduced in thecalibration distance measurements Hermite interpolationsare performed with both approximation methods on dataunevenly perturbated by adding normally distributed ran-dom numbers on acquired distances Numerical simulationsare performed on single channel data detected in a range of [25 divide 75] m for 5 00 diff erent sets of random perturbationsranging from 0 to 5 cm Tree diff erent sets of perturbationsare shown in Figure 6 Te more accurate approximationmethod performs Hermite interpolation by using (21) and




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(a)

Distances (m)

2 3 4 5 6 7 8

Distances (m)

2 3 4 5 6 7 8

Distances (m)

2 3 4 5 6 7 8

p 3

( c m )

012345

p 2

( c m )

012345

p 1

( c m )

012345

(b)

F983145983143983157983154983141 6 (a) Tree sets of normally distributed random perturbations Values range from 0 to 5 cm In the 119894th row 119895th column of the 1047297gureis a scatter plot of the 119894th set against the 119895th set of perturbations A histogram of each set is plotted along the diagonal of the 1047297gure (b)

perturbations p1 p

2 and p

3 plotted as function of calibration distances




Distances (m)

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( m V

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True dataPerturbated data

Method 1Method 2

(a)

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( m V

)

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Method 1Method 2

(b)

Simulation number

M e a n

i n t e r p o

l a t i o n e r r o r

0

1

2

3

4

5

6

7

8

9

Method 1

Method 2

0 100 200 300 400 500

times10minus3

(c)

F983145983143983157983154983141 7 Hermite interpolation on two diff erently perturbated single channel pieces of data with 1047297rst-order accurate (method 1) andsecond-order accurate (method 2) derivative approximations In both cases the maximum perturbation introduced is of 5 cm on distance

measurements In (c) the mean interpolation error produced by 500 diff erent random perturbation sets on true data is reported for bothmethods

(22) for evenly spaced points and replaces (22) for unevenly spaced points with the following formula

119892 = V +1 sdot ℎℎ+1 minus V minus1 sdot ℎ+1ℎℎ + ℎ+1 + V sdot 983080 1

ℎ minus 1

ℎ+1983081 (23)

with ℎ = 119889+1 minus 119889 Contrarily to (22) this approximation issecond-order accurate for both evenly and unevenly spacedcoordinates Some results of Hermite interpolation withboth methods are shown in Figure 7 As can be noted theless accurate approximation method for derivatives has alow mean interpolation error that is almost constant in all

the simulations Contrarily in some cases the more accurateone produces large interpolation errors as can be observed by Figures 7(b) and 7(c) Tus Hermite polynomials with 1047297rst-order accurate method keep good capabilities for interpola-tion and also exhibit low computational complexity

983091983091 Remote Lambertian Model for Experimental ValidationAn accurate quantitative analysis for RGB-ITR colorimetriccalibrations can be carried out by comparing prototyperemote measurements with spectrophotometer measure-ments on certi1047297ed Lambertian targets




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45

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55

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65

7

75

1000 2000 3000 4000

F983145983143983157983154983141 8 Laboratory distance image Te scene distances range

from 3

983097

to 7

7

meters

A remote Lambertian model is computed in labora-tory for assessing the accuracy of the proposed algorithm(Algorithm 1) Te presented laboratory test was processedwith controlled conditions act to simulate all the possiblepower variations of the laser sources equipped inside theRGB-ITR scanner Te model is based on the RGB measuresof the four SRS placed at diff erent distances and illuminatedby RGB-ITR lasers Each RGB signal is collected by movingthe targets at sampling distances of about 10 cm in a rangeof [25 divide 77] m Tree RGB curves as a function of distance

are computed for each target that is three RGB [119899 times 1] vector quantities where 119899 is the number of sampled dis-tances Using Hermite polynomials and 1047297rst-order accuratederivative approximations remote colour models for lightgray (50 re1047298ectance) and dark gray (20 re1047298ectance) SRSare estimated More precisely interpolating the detectedRGB signals of the gray targets at acquired RGB-ITR realscenedistances (Figure 8) andreshaping the computed vectorquantities to scene matrix dimensions colour texture imagesfor both targets are generated Tese interpolated matrixquantities represent the remote Lambertian models whichare then calibrated Linear and nonlinear calibrations of lightgray SRS raw model give the images shown in Figure 983097

Diff

erently from the linear calibrated image the nonlinearone does not show discontinuities that is the same object iscorrectly represented as having the same colour at all scenedistances On the contrary in Figure 983097(a) it is possible toidentify objects located at diff erent distances A quantitativeresult of this correction is given with the RGB histogramsshown in Figure 10 Te histograms are computed using theprobability normalization function with bin width of 005 forboth the Lambertian models

A comparison of these results with MINOLTA measure-ments performed on light and dark gray targets is reportedin Table 1 RGB linear calibrated values are more diff erentbetween them with respect to the re1047298ectances measured by MINOLTA Nonlinear calibration ensures that RGB valuesmeasured on gray targets are much more similar to eachother and to MINOLTA measurements In this way colour

T983137983138983148983141 1 Numerical comparison of the re1047298ectance [119877] and lightness[119871lowast] values measured by MINOLTA spectrophotometer and RGB-ITR with linear and nonlinear calibration methods for the 50re1047298ectance gray target

Instrument 120582 = 440 nm 120582 = 514 nm 120582 = 660 nm

MINOLTA [119877] 046 047 049

RGB-ITR [119877]

mean value 0505 plusmn 002 0521 plusmn 0015 0560 plusmn 002

MINOLTA [119871lowast] 075 075 075

RGB-ITR [119871lowast]


discontinuities caused by nonlinearities of the system canbe reduced Tis quantitative result demonstrates that theproposed model provides signi1047297cantly better colour mapping

in comparison with the conventional linear method whenilluminating certi1047297ed targets

Figure 11 reports raw data normalized by the maximumRGB valuesand linearand nonlinearcalibratedimages forthereal scene acquired in laboratory at distances shown in therange image (Figure 8) used for remote Lambertian modelsgeneration Te raw input image is not colorimetrically correct since each RGB sampled surface point is dependenton the artworkscanner distance and on the level of the RGBsignals that illuminate each point In particular the whitestatue appears greenish while the small object in the lowerlef side of the scene is represented to be too dark and isdifficult to recognize Te wrong gray shades on the white

wall of the linear calibrated image are strongly reduced withthe proposed correction method Te proposed scene withlarge whitish areas well 1047297t the purposeof this validation sincethe small variations of the collected back-re1047298ected signals aremore visible in a neutral (whitish or grayish) region than ina colourful environment and show that two objects havingsimilar colour properties but located at diff erent distancesfrom the scanner such as walland statue are representedwithdiff erent colours in noncalibrated data

4 Applications of the Model

Te nonlinearmodel wastested with success on various RGB-

ITR scans In this paper results concerning the application of the model on two diff erent digitizations are reported

(i) Te fresco of ldquo Amore e Psicherdquo (Villa FarnesinaRome) located at a distance of 7ndash983097 meters from thescanning system

(ii) Te Vault of the Sistine Chapel (Vatican MuseumsRome) located at 17-18 meters of distance from thescanning system

To con1047297rm the eff ectiveness of the proposed colorimetriccalibration a comparison with the corresponding normalizedraw data and linear calibrated models is shown in Figures12 and 13 For illustrative purposes only some portions of these huge digitizations are reported It can be observedthat the raw images are darker than the calibrated ones and




50

100

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200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 50 0 4 00 0 4 500

(a) Linear calibrated remote Lambertian model

50

100

150

200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 500 4 00 0 4 50 0

(b) Nonlinear calibrated remote Lambertian model

F983145983143983157983154983141 983097 Linear and nonlinear calibrations of light gray remote Lambertian model Te model is calculated at the RGB-ITR scene distancesshown in Figure 11(c)

Calibrated intensity

N o r m a l i z e d c o u n

t s

0

01

02

03

04

05

06

07

Linearcalibration

Nonlinearcalibration

0 02 04 0 6 108

(a)

N o r m

a l i z e d

c o u n

t s

0

01

02

03

04

05

06

07

Linearcalibration



0 02 04 06 108

(b)

F983145983143983157983154983141 10 Histograms of the calibrated dark gray (a) and light gray (b) Lambertian colour models Each coloured histogram corresponds toone type of RGB channel data

have higher red and blue intensity components related tohigher red and blue ITR signals that illuminate the scene atthose distances Te processed colour images in union withthe experimental validation in Section 33 demonstrate thatthe nonlinear model can be viewed as a colorimetric trans-formation that visibly reduces inaccuracies occurring whenlinearly calibrating grayishwhitish areas and eliminates thedependence on RGB signalsrsquo level that is the position of theRGB scene signals in the calibration curves as performed by linear calibrations

It is important to underline that linear and nonlinearmodels correspond to two diff erent colour spaces approx-imated re1047298ectance space the linear one and approximatedlightness space the nonlinear one Both images appear tobe visually pleasing to the observer but as quantitatively

demonstrated on certi1047297ed targets the nonlinear one repre-sents a much more accurate laser colour model

Overall these results show that the proposed nonlinearcalibration works well for real scenes located at diff erentdistances from the scanner As con1047297rmed by experts incultural heritage sector the nonlinear calibrated models canbe considered suitable colour images for cultural heritagepurposes

5 Conclusion

Tis paper has presented a nonlinear model for remotecalibration of laser colour images Te problem of linearcalibration forgray diff usive certi1047297ed targets is discussed Tenonlinear model is introduced in the calibration procedure




50

100

150

200

250

1000 2000 3000 4000

(a) Raw colour data

50

100

150

200

250

1000 2000 3000 4000

(b) Linear calibration

50

100

150

200

250

1000 2000 3000 4000

(c) Nonlinear calibration

F983145983143983157983154983141 11 Laboratory real scene raw data and calibrations

(a) Raw colour data (b) Linear calibrated (c) Nonlinear calibrated

(d) Raw colour data (e) Linear calibrated (f) Nonlinear calibrated

F983145983143983157983154983141 12 RGB-ITR raw data and linear and nonlinear calibrated colour textures of real scene at 7ndash983097 meters of distance






F983145983143983157983154983141 13 RGB-ITR raw data and linear and nonlinear calibrated colour textures of real scene at about 17-18 meters of distance

as a postprocessing step for the correction of raw colori-metric data that is for both reducing inaccuracies occur-ring in remote measurement of gray targets and providingmeaningful colour images for cultural heritage purposesTe nonlinearity of the model is the consequence of theprocessing made on data collected by illuminating fourdiff erent calibrated targets An experimental validation of the proposed algorithm is performed with a remote Lamber-tian model computed by using monotonic piecewise cubicHermite polynomials and certi1047297ed diff usive targets Tisquantitatively demonstrates thatthe nonlinear model is muchmore accurate than the linear one when performing remotecolorimetric measurements on diff erent certi1047297ed Lamber-tian targets Hermite polynomials with 1047297rst-order accuratederivative approximation formula have shown robust andefficient interpolation performances in presence of addednoise Results that highlight the eff ectiveness of the proposedmodel in producing meaningful images for disseminationeducation and cataloguing purposes are reported

In conclusion this work has shown that the processing of colorimetric data permits generating more accurate textureimages that if combined with 3D laser models can providehigh-quality information Although remote colour calibra-tion procedure is still object of active research the proposednonlinear correction suggests that an optimal calibrationis possible for the RGB-ITR Tis represents an improve-ment especially in the cultural heritage domain where thecolorimetric characterisation of an artwork has the samesigni1047297cance of the structural determination of the artwork itself

Conflict of Interests

Te authors declare that there is no con1047298ict of interestsregarding the publication of this paper

Acknowledgments

Te authors thank the Vatican in the person of Mons Nicol-ini Professor Paolucci Professor Santamaria and ProfessorDi Pinto for their cooperation as cultural heritage experts andfor having allowed working in the Sistine Chapel

References

[1] A Khandual G Baciu and N Rout ldquoColorimetric processingof digital colour imagerdquo International Journal of Advanced Research in Computer Science and So f ware Engineering vol 3no 7 pp 39830973ndash402 2007

[2] Y-H Liu C-H Chen and P C-P Chao ldquoMathematicalmethods applied to digital image processingrdquo Mathematical Problems in Engineering vol 2014 Article ID 480523 4 pages2014

[3] V Papanikolaou K N Plataniotis and A N VenetsanopoulosldquoAdaptive 1047297lters for color image processingrdquo Mathematical Problems in Engineering vol 4 no 6 pp 52983097ndash538 1983097983097983097

[4] R M Rangayyan B Acha and C Serrano Color Image Pro-cessing with Biomedical Applications Society of Photo-OpticalInstrumentation Engineers (SPIE) 2011

[5] H Altunbasak and H Joel Trussell ldquoColorimetric restoration of digital imagesrdquo IEEE Transactions on Image Processing vol 10no 3 pp 39830973ndash402 2001

[6] N Pfeifer P Dorninger A Haring and H Fan ldquoInvestigatingterrestrial laser scanning intensity data quality and functionalrelationsrdquo in Proceedings of the International Conference onOptical 983091-D Measurement Techniques VIII pp 328ndash337 July 2007

[7] S Kaasalainen A Krooks A Kukko andH Kaartinen ldquoRadio-metric calibration of terrestrial laser scanners with externalreference targetsrdquo Remote Sensing vol 1 no 3 pp 144ndash158200983097

[8] D Garcıa-San-Miguel and J L Lerma ldquoGeometric calibrationof a terrestrial laser scanner with local additional parameters




an automatic strategyrdquo ISPRS Journal of Photogrammetry and Remote Sensing vol 7983097 pp 122ndash136 2013

[983097] M Guarneri M F De Collibus G Fornetti M FrancucciM Nuvoli and R Ricci ldquoRemote colorimetric and structural

diagnosis by RGB-ITR color laser scanner prototyperdquo Advancesin Optical Technologies vol 2012 Article ID 51298309702 6 pages2012

[10] M Barni A Pelagotti and A Piva ldquoImage processing forthe analysis and conservation of paintings opportunities andchallengesrdquo IEEE Signal Processing Magazine vol 22 no 5 pp141ndash144 2005

[11] R Li T Luo and H Zha ldquo3D digitization and its applicationsin cultural heritagerdquo in Digital Heritage Tird International Conference EuroMed 983090983088983089983088 Lemessos Cyprus November 983096-983089983091 983090983088983089983088 Proceedings vol 6436 of Lecture Notes in Computer Science pp 381ndash388 Springer Berlin Germany 2010

[12] M Bacci M Ciatti B Molinas C Oleari U Santamaria andP Soardo ldquoColorimetria e Beni Culturalirdquo in Proceedings of the

Firenze 983089983097983097983097 and Venezia 983090983088983088983088 1983097983097983097-2000

[13] R Ricci L De Dominicis M F De Collibus et al ldquoRGB-ITRan amplitude-modulated 3D colour laser scanner for culturalheritage applications rdquo in Proceedings of the International Con- ference LA- CONA VIII Lasers in the Conservation of ArtworksSeptember 200983097

[14] M Guarneri A Danielis M Francucci M F De CollibusG Fornetti and A Mencattini ldquo3D remote colorimetry andwatershed segmentation techniques for fresco and artwork decay monitoring and preservationrdquo Journal of Archaeological Science vol 46 pp 182ndash19830970 2014

[15] D Nitzan A Brain SIAI Center and R Duda ldquoTe measure-ment and use of registered reectance and range data in sceneanalysisrdquo SRI Project Stanford Research Institute 198309776

[16] L Mullen A Laux B Concannon E P Zege I L Katsev andA S Prikhach ldquoAmplitude-modulated laser imagerrdquo Applied Optics vol 43 no 1983097 pp 3874ndash389830972 2004

[17] R S Berns Ed Billmeyer and Saltzmanrsquos Principles of Color Technology Wiley 2000

[18] E Dubois ldquoTe structure and properties of color spaces andthe representation of color imagesrdquo Synthesis Lectures on ImageVideo and Multimedia Processing vol 4 no 1 pp 1ndash12983097 200983097

[1983097] S Battaglino A Fiorentini W Gerbino et al Misurare il coloreFisiologia della visione a colori fotometria colorimetria e normeinternazionali Hoepli 2008

[20] F N Fritsch and R E Carlson ldquoMonotone piecewise cubicinterpolationrdquo SIAM Journal on Numerical Analysis vol 17 no

2 pp 238ndash246 198309780[21] D J Higham ldquoMonotonic piecewise cubic interpolation with

applications to ODE plottingrdquo Journal of Computational and Applied Mathematics vol 3983097 no 3 pp 287ndash29830974 19830979830972





















119877gray (120582) = 119881

gray

(120582)119881white (120582) sdot 119877

white (120582) (1)




Distances (m)

2 3 4 5 6 7

V o

l t a g e s

( m V

)

0

0005

001

0015

002

0025

003

0035

004

0045

005


20 gray raw data


2 3 4 5 6 7

Distances (m)

V o

l t a g e s

( m V

)

0

0005

001

0015

002

0025

003

0035

004

0045


20 gray raw data


2 3 4 5 6 7

Distances (m)

V o

l t a g e s

( m V

)

0

0005

001

0015

002

0025

003

0035

004

White raw data

50 gray raw data

20 gray raw data






3 Methods





Distances (m)

V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008


20 gray raw data

17 18 19 20 21 22 23


V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007


20 gray raw data

Distances (m)

17 18 19 20 21 22 23


V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008

White raw data

50 gray raw data

20 gray raw data

Distances (m)

17 18 19 20 21 22 23





[18 1983097]

119871lowast

=

9831639831631048699983163983163852091

116 sdot 983080 119884

119884

98308113 minus 16 if 119884

119884

gt 1048616 6

2910486173

119897 sdot 983080 119884119884983081 if 119884119884 lt 1048616 62910486173

(2)




1 3 5 7 9 1 1 1 3 1 5 1 7 1 9 2 1 2 3 2 5 2 7 2 9 3 1 3 3 3 5 3 7 3 9

Distance samples


n c e s

Whitetarget

Lightgray

target

Dark gray

target


Distance (m)

2 3 4 5 6 7

R e 1047298 e c t a n c e

015

02

025

03

035

04

045

05

055

06

065


Blue channel






119884 = 119870 sdot 991761

119878 (120582) sdot 119877 ( 120582) sdot 119910cmf (120582) Δ120582 (3)


119870 = 100




where

119891(






1199101 = 1198860 + 119891 (1198811) sdot 3991761=1

119886 sdot 852008119881minus11 852009

1199102 = 1198860 + 119891 (1198812) sdot 3991761=1

119886 sdot 852008119881minus12 852009

1199103 =

1198860 +

119891 (1198813

) sdot

3

991761=1119886

sdot 852008119881minus1

3

8520091199104 = 1198860 + 119891 (1198814) sdot

3991761=1

119886 sdot 852008119881minus14 852009

(6)



119891 (119881) = 11988113 (7)





Input voltages (mV)

C a

l i b r a t e d


0

02

04

06

08

1

Linear model

Imposed data

Nonlinear model

0 0005 001 0015 002 0025



[119883

=

9831311

11988113

119881 sdot 11988113

1198812

sdot 11988113

983133 (8)


(119886) [119883] = (119910) (983097)


s for each RGB



[Ic = (a) 983131X V

s983133 (10)










ℎ (119909) = 119891 (119909)ℎ (119909+1) = 119891 (119909+1)

ℎ (119909) = 119891

(119909)ℎ (119909+1) = 119891 (119909+1)(11)


119867 (119909) =

991761=0

119891 (119909) ℎ (119909) + 119891 ℎ (119909) (12)







are as follows




(x i)1times4 = (1199091 11990921199093 1199094)( y i)1times4

= (1199101119910211991031199104)

(13)




True

Data

Spline


x

f ( x )

(a)

True

Data

Spline


x

f ( x )

(b)





[X i4times4

= 1048667852059852059852059852059852059

1

119909131

1199091

sdot 119909131

11990921

sdot 119909131

1 119909132 1199092 sdot 11990913

2 11990922 sdot 11990913

2

1 119909133 1199093 sdot 11990913

3 11990923 sdot 11990913

3

1 119909134 1199094 sdot 11990913

4 11990924 sdot 11990913

4

1048669852061852061852061852061852061]

(15)


distance 119889(A


i)4times1

(16)





i)(sdot)times4

(18)


(Ic)1times1

= (X V)1times4 (A)4times1 (1983097)





1198921 = V 2 minus V 11198892 minus 1198891


(21)








0

1

2

3

4

5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

(a)

Distances (m)

2 3 4 5 6 7 8

Distances (m)

2 3 4 5 6 7 8

Distances (m)

2 3 4 5 6 7 8

p 3

( c m )

012345

p 2

( c m )

012345

p 1

( c m )

012345

(b)


perturbations p1 p

2 and p





Distances (m)

2 3 4 5 6 7 8

V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008


Method 1Method 2

(a)

Distances (m)

2 3 4 5 6 7 8

V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008


Method 1Method 2

(b)

Simulation number

M e a n

i n t e r p o


0

1

2

3

4

5

6

7

8

9

Method 1

Method 2

0 100 200 300 400 500

times10minus3

(c)





ℎ minus 1

ℎ+1983081 (23)







50

100

150

200

250

4

45

5

55

6

65

7

75

1000 2000 3000 4000


from 3

983097

to 7

7

meters



Diff





MINOLTA [119877] 046 047 049

RGB-ITR [119877]


MINOLTA [119871lowast] 075 075 075
















50

100

150

200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 50 0 4 00 0 4 500


50

100

150

200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 500 4 00 0 4 50 0





t s

0

01

02

03

04

05

06

07

Linearcalibration


0 02 04 0 6 108

(a)

N o r m

a l i z e d

c o u n

t s

0

01

02

03

04

05

06

07

Linearcalibration



0 02 04 06 108

(b)






5 Conclusion





50

100

150

200

250

1000 2000 3000 4000

(a) Raw colour data

50

100

150

200

250

1000 2000 3000 4000


50

100

150

200

250

1000 2000 3000 4000
















Acknowledgments


References


































Distances (m)

2 3 4 5 6 7

V o

l t a g e s

( m V

)

0

0005

001

0015

002

0025

003

0035

004

0045

005


20 gray raw data


2 3 4 5 6 7

Distances (m)

V o

l t a g e s

( m V

)

0

0005

001

0015

002

0025

003

0035

004

0045


20 gray raw data


2 3 4 5 6 7

Distances (m)

V o

l t a g e s

( m V

)

0

0005

001

0015

002

0025

003

0035

004

White raw data

50 gray raw data

20 gray raw data






3 Methods





Distances (m)

V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008


20 gray raw data

17 18 19 20 21 22 23


V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007


20 gray raw data

Distances (m)

17 18 19 20 21 22 23


V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008

White raw data

50 gray raw data

20 gray raw data

Distances (m)

17 18 19 20 21 22 23





[18 1983097]

119871lowast

=

9831639831631048699983163983163852091

116 sdot 983080 119884

119884

98308113 minus 16 if 119884

119884

gt 1048616 6

2910486173

119897 sdot 983080 119884119884983081 if 119884119884 lt 1048616 62910486173

(2)




1 3 5 7 9 1 1 1 3 1 5 1 7 1 9 2 1 2 3 2 5 2 7 2 9 3 1 3 3 3 5 3 7 3 9

Distance samples


n c e s

Whitetarget

Lightgray

target

Dark gray

target


Distance (m)

2 3 4 5 6 7

R e 1047298 e c t a n c e

015

02

025

03

035

04

045

05

055

06

065


Blue channel






119884 = 119870 sdot 991761

119878 (120582) sdot 119877 ( 120582) sdot 119910cmf (120582) Δ120582 (3)


119870 = 100




where

119891(






1199101 = 1198860 + 119891 (1198811) sdot 3991761=1

119886 sdot 852008119881minus11 852009

1199102 = 1198860 + 119891 (1198812) sdot 3991761=1

119886 sdot 852008119881minus12 852009

1199103 =

1198860 +

119891 (1198813

) sdot

3

991761=1119886

sdot 852008119881minus1

3

8520091199104 = 1198860 + 119891 (1198814) sdot

3991761=1

119886 sdot 852008119881minus14 852009

(6)



119891 (119881) = 11988113 (7)





Input voltages (mV)

C a

l i b r a t e d


0

02

04

06

08

1

Linear model

Imposed data

Nonlinear model

0 0005 001 0015 002 0025



[119883

=

9831311

11988113

119881 sdot 11988113

1198812

sdot 11988113

983133 (8)


(119886) [119883] = (119910) (983097)


s for each RGB



[Ic = (a) 983131X V

s983133 (10)










ℎ (119909) = 119891 (119909)ℎ (119909+1) = 119891 (119909+1)

ℎ (119909) = 119891

(119909)ℎ (119909+1) = 119891 (119909+1)(11)


119867 (119909) =

991761=0

119891 (119909) ℎ (119909) + 119891 ℎ (119909) (12)







are as follows




(x i)1times4 = (1199091 11990921199093 1199094)( y i)1times4

= (1199101119910211991031199104)

(13)




True

Data

Spline


x

f ( x )

(a)

True

Data

Spline


x

f ( x )

(b)





[X i4times4

= 1048667852059852059852059852059852059

1

119909131

1199091

sdot 119909131

11990921

sdot 119909131

1 119909132 1199092 sdot 11990913

2 11990922 sdot 11990913

2

1 119909133 1199093 sdot 11990913

3 11990923 sdot 11990913

3

1 119909134 1199094 sdot 11990913

4 11990924 sdot 11990913

4

1048669852061852061852061852061852061]

(15)


distance 119889(A


i)4times1

(16)





i)(sdot)times4

(18)


(Ic)1times1

= (X V)1times4 (A)4times1 (1983097)





1198921 = V 2 minus V 11198892 minus 1198891


(21)








0

1

2

3

4

5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

(a)

Distances (m)

2 3 4 5 6 7 8

Distances (m)

2 3 4 5 6 7 8

Distances (m)

2 3 4 5 6 7 8

p 3

( c m )

012345

p 2

( c m )

012345

p 1

( c m )

012345

(b)


perturbations p1 p

2 and p





Distances (m)

2 3 4 5 6 7 8

V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008


Method 1Method 2

(a)

Distances (m)

2 3 4 5 6 7 8

V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008


Method 1Method 2

(b)

Simulation number

M e a n

i n t e r p o


0

1

2

3

4

5

6

7

8

9

Method 1

Method 2

0 100 200 300 400 500

times10minus3

(c)





ℎ minus 1

ℎ+1983081 (23)







50

100

150

200

250

4

45

5

55

6

65

7

75

1000 2000 3000 4000


from 3

983097

to 7

7

meters



Diff





MINOLTA [119877] 046 047 049

RGB-ITR [119877]


MINOLTA [119871lowast] 075 075 075
















50

100

150

200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 50 0 4 00 0 4 500


50

100

150

200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 500 4 00 0 4 50 0





t s

0

01

02

03

04

05

06

07

Linearcalibration


0 02 04 0 6 108

(a)

N o r m

a l i z e d

c o u n

t s

0

01

02

03

04

05

06

07

Linearcalibration



0 02 04 06 108

(b)






5 Conclusion





50

100

150

200

250

1000 2000 3000 4000

(a) Raw colour data

50

100

150

200

250

1000 2000 3000 4000


50

100

150

200

250

1000 2000 3000 4000
















Acknowledgments


References


































Distances (m)

V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008


20 gray raw data

17 18 19 20 21 22 23


V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007


20 gray raw data

Distances (m)

17 18 19 20 21 22 23


V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008

White raw data

50 gray raw data

20 gray raw data

Distances (m)

17 18 19 20 21 22 23





[18 1983097]

119871lowast

=

9831639831631048699983163983163852091

116 sdot 983080 119884

119884

98308113 minus 16 if 119884

119884

gt 1048616 6

2910486173

119897 sdot 983080 119884119884983081 if 119884119884 lt 1048616 62910486173

(2)




1 3 5 7 9 1 1 1 3 1 5 1 7 1 9 2 1 2 3 2 5 2 7 2 9 3 1 3 3 3 5 3 7 3 9

Distance samples


n c e s

Whitetarget

Lightgray

target

Dark gray

target


Distance (m)

2 3 4 5 6 7

R e 1047298 e c t a n c e

015

02

025

03

035

04

045

05

055

06

065


Blue channel






119884 = 119870 sdot 991761

119878 (120582) sdot 119877 ( 120582) sdot 119910cmf (120582) Δ120582 (3)


119870 = 100




where

119891(






1199101 = 1198860 + 119891 (1198811) sdot 3991761=1

119886 sdot 852008119881minus11 852009

1199102 = 1198860 + 119891 (1198812) sdot 3991761=1

119886 sdot 852008119881minus12 852009

1199103 =

1198860 +

119891 (1198813

) sdot

3

991761=1119886

sdot 852008119881minus1

3

8520091199104 = 1198860 + 119891 (1198814) sdot

3991761=1

119886 sdot 852008119881minus14 852009

(6)



119891 (119881) = 11988113 (7)





Input voltages (mV)

C a

l i b r a t e d


0

02

04

06

08

1

Linear model

Imposed data

Nonlinear model

0 0005 001 0015 002 0025



[119883

=

9831311

11988113

119881 sdot 11988113

1198812

sdot 11988113

983133 (8)


(119886) [119883] = (119910) (983097)


s for each RGB



[Ic = (a) 983131X V

s983133 (10)










ℎ (119909) = 119891 (119909)ℎ (119909+1) = 119891 (119909+1)

ℎ (119909) = 119891

(119909)ℎ (119909+1) = 119891 (119909+1)(11)


119867 (119909) =

991761=0

119891 (119909) ℎ (119909) + 119891 ℎ (119909) (12)







are as follows




(x i)1times4 = (1199091 11990921199093 1199094)( y i)1times4

= (1199101119910211991031199104)

(13)




True

Data

Spline


x

f ( x )

(a)

True

Data

Spline


x

f ( x )

(b)





[X i4times4

= 1048667852059852059852059852059852059

1

119909131

1199091

sdot 119909131

11990921

sdot 119909131

1 119909132 1199092 sdot 11990913

2 11990922 sdot 11990913

2

1 119909133 1199093 sdot 11990913

3 11990923 sdot 11990913

3

1 119909134 1199094 sdot 11990913

4 11990924 sdot 11990913

4

1048669852061852061852061852061852061]

(15)


distance 119889(A


i)4times1

(16)





i)(sdot)times4

(18)


(Ic)1times1

= (X V)1times4 (A)4times1 (1983097)





1198921 = V 2 minus V 11198892 minus 1198891


(21)








0

1

2

3

4

5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

(a)

Distances (m)

2 3 4 5 6 7 8

Distances (m)

2 3 4 5 6 7 8

Distances (m)

2 3 4 5 6 7 8

p 3

( c m )

012345

p 2

( c m )

012345

p 1

( c m )

012345

(b)


perturbations p1 p

2 and p





Distances (m)

2 3 4 5 6 7 8

V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008


Method 1Method 2

(a)

Distances (m)

2 3 4 5 6 7 8

V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008


Method 1Method 2

(b)

Simulation number

M e a n

i n t e r p o


0

1

2

3

4

5

6

7

8

9

Method 1

Method 2

0 100 200 300 400 500

times10minus3

(c)





ℎ minus 1

ℎ+1983081 (23)







50

100

150

200

250

4

45

5

55

6

65

7

75

1000 2000 3000 4000


from 3

983097

to 7

7

meters



Diff





MINOLTA [119877] 046 047 049

RGB-ITR [119877]


MINOLTA [119871lowast] 075 075 075
















50

100

150

200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 50 0 4 00 0 4 500


50

100

150

200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 500 4 00 0 4 50 0





t s

0

01

02

03

04

05

06

07

Linearcalibration


0 02 04 0 6 108

(a)

N o r m

a l i z e d

c o u n

t s

0

01

02

03

04

05

06

07

Linearcalibration



0 02 04 06 108

(b)






5 Conclusion





50

100

150

200

250

1000 2000 3000 4000

(a) Raw colour data

50

100

150

200

250

1000 2000 3000 4000


50

100

150

200

250

1000 2000 3000 4000
















Acknowledgments


References


































1 3 5 7 9 1 1 1 3 1 5 1 7 1 9 2 1 2 3 2 5 2 7 2 9 3 1 3 3 3 5 3 7 3 9

Distance samples


n c e s

Whitetarget

Lightgray

target

Dark gray

target


Distance (m)

2 3 4 5 6 7

R e 1047298 e c t a n c e

015

02

025

03

035

04

045

05

055

06

065


Blue channel






119884 = 119870 sdot 991761

119878 (120582) sdot 119877 ( 120582) sdot 119910cmf (120582) Δ120582 (3)


119870 = 100




where

119891(






1199101 = 1198860 + 119891 (1198811) sdot 3991761=1

119886 sdot 852008119881minus11 852009

1199102 = 1198860 + 119891 (1198812) sdot 3991761=1

119886 sdot 852008119881minus12 852009

1199103 =

1198860 +

119891 (1198813

) sdot

3

991761=1119886

sdot 852008119881minus1

3

8520091199104 = 1198860 + 119891 (1198814) sdot

3991761=1

119886 sdot 852008119881minus14 852009

(6)



119891 (119881) = 11988113 (7)





Input voltages (mV)

C a

l i b r a t e d


0

02

04

06

08

1

Linear model

Imposed data

Nonlinear model

0 0005 001 0015 002 0025



[119883

=

9831311

11988113

119881 sdot 11988113

1198812

sdot 11988113

983133 (8)


(119886) [119883] = (119910) (983097)


s for each RGB



[Ic = (a) 983131X V

s983133 (10)










ℎ (119909) = 119891 (119909)ℎ (119909+1) = 119891 (119909+1)

ℎ (119909) = 119891

(119909)ℎ (119909+1) = 119891 (119909+1)(11)


119867 (119909) =

991761=0

119891 (119909) ℎ (119909) + 119891 ℎ (119909) (12)







are as follows




(x i)1times4 = (1199091 11990921199093 1199094)( y i)1times4

= (1199101119910211991031199104)

(13)




True

Data

Spline


x

f ( x )

(a)

True

Data

Spline


x

f ( x )

(b)





[X i4times4

= 1048667852059852059852059852059852059

1

119909131

1199091

sdot 119909131

11990921

sdot 119909131

1 119909132 1199092 sdot 11990913

2 11990922 sdot 11990913

2

1 119909133 1199093 sdot 11990913

3 11990923 sdot 11990913

3

1 119909134 1199094 sdot 11990913

4 11990924 sdot 11990913

4

1048669852061852061852061852061852061]

(15)


distance 119889(A


i)4times1

(16)





i)(sdot)times4

(18)


(Ic)1times1

= (X V)1times4 (A)4times1 (1983097)





1198921 = V 2 minus V 11198892 minus 1198891


(21)








0

1

2

3

4

5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

(a)

Distances (m)

2 3 4 5 6 7 8

Distances (m)

2 3 4 5 6 7 8

Distances (m)

2 3 4 5 6 7 8

p 3

( c m )

012345

p 2

( c m )

012345

p 1

( c m )

012345

(b)


perturbations p1 p

2 and p





Distances (m)

2 3 4 5 6 7 8

V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008


Method 1Method 2

(a)

Distances (m)

2 3 4 5 6 7 8

V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008


Method 1Method 2

(b)

Simulation number

M e a n

i n t e r p o


0

1

2

3

4

5

6

7

8

9

Method 1

Method 2

0 100 200 300 400 500

times10minus3

(c)





ℎ minus 1

ℎ+1983081 (23)







50

100

150

200

250

4

45

5

55

6

65

7

75

1000 2000 3000 4000


from 3

983097

to 7

7

meters



Diff





MINOLTA [119877] 046 047 049

RGB-ITR [119877]


MINOLTA [119871lowast] 075 075 075
















50

100

150

200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 50 0 4 00 0 4 500


50

100

150

200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 500 4 00 0 4 50 0





t s

0

01

02

03

04

05

06

07

Linearcalibration


0 02 04 0 6 108

(a)

N o r m

a l i z e d

c o u n

t s

0

01

02

03

04

05

06

07

Linearcalibration



0 02 04 06 108

(b)






5 Conclusion





50

100

150

200

250

1000 2000 3000 4000

(a) Raw colour data

50

100

150

200

250

1000 2000 3000 4000


50

100

150

200

250

1000 2000 3000 4000
















Acknowledgments


References


































Input voltages (mV)

C a

l i b r a t e d


0

02

04

06

08

1

Linear model

Imposed data

Nonlinear model

0 0005 001 0015 002 0025



[119883

=

9831311

11988113

119881 sdot 11988113

1198812

sdot 11988113

983133 (8)


(119886) [119883] = (119910) (983097)


s for each RGB



[Ic = (a) 983131X V

s983133 (10)










ℎ (119909) = 119891 (119909)ℎ (119909+1) = 119891 (119909+1)

ℎ (119909) = 119891

(119909)ℎ (119909+1) = 119891 (119909+1)(11)


119867 (119909) =

991761=0

119891 (119909) ℎ (119909) + 119891 ℎ (119909) (12)







are as follows




(x i)1times4 = (1199091 11990921199093 1199094)( y i)1times4

= (1199101119910211991031199104)

(13)




True

Data

Spline


x

f ( x )

(a)

True

Data

Spline


x

f ( x )

(b)





[X i4times4

= 1048667852059852059852059852059852059

1

119909131

1199091

sdot 119909131

11990921

sdot 119909131

1 119909132 1199092 sdot 11990913

2 11990922 sdot 11990913

2

1 119909133 1199093 sdot 11990913

3 11990923 sdot 11990913

3

1 119909134 1199094 sdot 11990913

4 11990924 sdot 11990913

4

1048669852061852061852061852061852061]

(15)


distance 119889(A


i)4times1

(16)





i)(sdot)times4

(18)


(Ic)1times1

= (X V)1times4 (A)4times1 (1983097)





1198921 = V 2 minus V 11198892 minus 1198891


(21)








0

1

2

3

4

5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

(a)

Distances (m)

2 3 4 5 6 7 8

Distances (m)

2 3 4 5 6 7 8

Distances (m)

2 3 4 5 6 7 8

p 3

( c m )

012345

p 2

( c m )

012345

p 1

( c m )

012345

(b)


perturbations p1 p

2 and p





Distances (m)

2 3 4 5 6 7 8

V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008


Method 1Method 2

(a)

Distances (m)

2 3 4 5 6 7 8

V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008


Method 1Method 2

(b)

Simulation number

M e a n

i n t e r p o


0

1

2

3

4

5

6

7

8

9

Method 1

Method 2

0 100 200 300 400 500

times10minus3

(c)





ℎ minus 1

ℎ+1983081 (23)







50

100

150

200

250

4

45

5

55

6

65

7

75

1000 2000 3000 4000


from 3

983097

to 7

7

meters



Diff





MINOLTA [119877] 046 047 049

RGB-ITR [119877]


MINOLTA [119871lowast] 075 075 075
















50

100

150

200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 50 0 4 00 0 4 500


50

100

150

200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 500 4 00 0 4 50 0





t s

0

01

02

03

04

05

06

07

Linearcalibration


0 02 04 0 6 108

(a)

N o r m

a l i z e d

c o u n

t s

0

01

02

03

04

05

06

07

Linearcalibration



0 02 04 06 108

(b)






5 Conclusion





50

100

150

200

250

1000 2000 3000 4000

(a) Raw colour data

50

100

150

200

250

1000 2000 3000 4000


50

100

150

200

250

1000 2000 3000 4000
















Acknowledgments


References


































True

Data

Spline


x

f ( x )

(a)

True

Data

Spline


x

f ( x )

(b)





[X i4times4

= 1048667852059852059852059852059852059

1

119909131

1199091

sdot 119909131

11990921

sdot 119909131

1 119909132 1199092 sdot 11990913

2 11990922 sdot 11990913

2

1 119909133 1199093 sdot 11990913

3 11990923 sdot 11990913

3

1 119909134 1199094 sdot 11990913

4 11990924 sdot 11990913

4

1048669852061852061852061852061852061]

(15)


distance 119889(A


i)4times1

(16)





i)(sdot)times4

(18)


(Ic)1times1

= (X V)1times4 (A)4times1 (1983097)





1198921 = V 2 minus V 11198892 minus 1198891


(21)








0

1

2

3

4

5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

(a)

Distances (m)

2 3 4 5 6 7 8

Distances (m)

2 3 4 5 6 7 8

Distances (m)

2 3 4 5 6 7 8

p 3

( c m )

012345

p 2

( c m )

012345

p 1

( c m )

012345

(b)


perturbations p1 p

2 and p





Distances (m)

2 3 4 5 6 7 8

V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008


Method 1Method 2

(a)

Distances (m)

2 3 4 5 6 7 8

V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008


Method 1Method 2

(b)

Simulation number

M e a n

i n t e r p o


0

1

2

3

4

5

6

7

8

9

Method 1

Method 2

0 100 200 300 400 500

times10minus3

(c)





ℎ minus 1

ℎ+1983081 (23)







50

100

150

200

250

4

45

5

55

6

65

7

75

1000 2000 3000 4000


from 3

983097

to 7

7

meters



Diff





MINOLTA [119877] 046 047 049

RGB-ITR [119877]


MINOLTA [119871lowast] 075 075 075
















50

100

150

200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 50 0 4 00 0 4 500


50

100

150

200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 500 4 00 0 4 50 0





t s

0

01

02

03

04

05

06

07

Linearcalibration


0 02 04 0 6 108

(a)

N o r m

a l i z e d

c o u n

t s

0

01

02

03

04

05

06

07

Linearcalibration



0 02 04 06 108

(b)






5 Conclusion





50

100

150

200

250

1000 2000 3000 4000

(a) Raw colour data

50

100

150

200

250

1000 2000 3000 4000


50

100

150

200

250

1000 2000 3000 4000
















Acknowledgments


References


































0

1

2

3

4

5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

0

1

2

3

4

5

0 1 2 3 4 5

(a)

Distances (m)

2 3 4 5 6 7 8

Distances (m)

2 3 4 5 6 7 8

Distances (m)

2 3 4 5 6 7 8

p 3

( c m )

012345

p 2

( c m )

012345

p 1

( c m )

012345

(b)


perturbations p1 p

2 and p





Distances (m)

2 3 4 5 6 7 8

V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008


Method 1Method 2

(a)

Distances (m)

2 3 4 5 6 7 8

V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008


Method 1Method 2

(b)

Simulation number

M e a n

i n t e r p o


0

1

2

3

4

5

6

7

8

9

Method 1

Method 2

0 100 200 300 400 500

times10minus3

(c)





ℎ minus 1

ℎ+1983081 (23)







50

100

150

200

250

4

45

5

55

6

65

7

75

1000 2000 3000 4000


from 3

983097

to 7

7

meters



Diff





MINOLTA [119877] 046 047 049

RGB-ITR [119877]


MINOLTA [119871lowast] 075 075 075
















50

100

150

200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 50 0 4 00 0 4 500


50

100

150

200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 500 4 00 0 4 50 0





t s

0

01

02

03

04

05

06

07

Linearcalibration


0 02 04 0 6 108

(a)

N o r m

a l i z e d

c o u n

t s

0

01

02

03

04

05

06

07

Linearcalibration



0 02 04 06 108

(b)






5 Conclusion





50

100

150

200

250

1000 2000 3000 4000

(a) Raw colour data

50

100

150

200

250

1000 2000 3000 4000


50

100

150

200

250

1000 2000 3000 4000
















Acknowledgments


References


































Distances (m)

2 3 4 5 6 7 8

V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008


Method 1Method 2

(a)

Distances (m)

2 3 4 5 6 7 8

V o

l t a g e s

( m V

)

0

001

002

003

004

005

006

007

008


Method 1Method 2

(b)

Simulation number

M e a n

i n t e r p o


0

1

2

3

4

5

6

7

8

9

Method 1

Method 2

0 100 200 300 400 500

times10minus3

(c)





ℎ minus 1

ℎ+1983081 (23)







50

100

150

200

250

4

45

5

55

6

65

7

75

1000 2000 3000 4000


from 3

983097

to 7

7

meters



Diff





MINOLTA [119877] 046 047 049

RGB-ITR [119877]


MINOLTA [119871lowast] 075 075 075
















50

100

150

200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 50 0 4 00 0 4 500


50

100

150

200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 500 4 00 0 4 50 0





t s

0

01

02

03

04

05

06

07

Linearcalibration


0 02 04 0 6 108

(a)

N o r m

a l i z e d

c o u n

t s

0

01

02

03

04

05

06

07

Linearcalibration



0 02 04 06 108

(b)






5 Conclusion





50

100

150

200

250

1000 2000 3000 4000

(a) Raw colour data

50

100

150

200

250

1000 2000 3000 4000


50

100

150

200

250

1000 2000 3000 4000
















Acknowledgments


References


































50

100

150

200

250

4

45

5

55

6

65

7

75

1000 2000 3000 4000


from 3

983097

to 7

7

meters



Diff





MINOLTA [119877] 046 047 049

RGB-ITR [119877]


MINOLTA [119871lowast] 075 075 075
















50

100

150

200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 50 0 4 00 0 4 500


50

100

150

200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 500 4 00 0 4 50 0





t s

0

01

02

03

04

05

06

07

Linearcalibration


0 02 04 0 6 108

(a)

N o r m

a l i z e d

c o u n

t s

0

01

02

03

04

05

06

07

Linearcalibration



0 02 04 06 108

(b)






5 Conclusion





50

100

150

200

250

1000 2000 3000 4000

(a) Raw colour data

50

100

150

200

250

1000 2000 3000 4000


50

100

150

200

250

1000 2000 3000 4000
















Acknowledgments


References


































50

100

150

200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 50 0 4 00 0 4 500


50

100

150

200

250

5 00 1 00 0 1 50 0 2 00 0 2 50 0 3 00 0 3 500 4 00 0 4 50 0





t s

0

01

02

03

04

05

06

07

Linearcalibration


0 02 04 0 6 108

(a)

N o r m

a l i z e d

c o u n

t s

0

01

02

03

04

05

06

07

Linearcalibration



0 02 04 06 108

(b)






5 Conclusion





50

100

150

200

250

1000 2000 3000 4000

(a) Raw colour data

50

100

150

200

250

1000 2000 3000 4000


50

100

150

200

250

1000 2000 3000 4000
















Acknowledgments


References


































50

100

150

200

250

1000 2000 3000 4000

(a) Raw colour data

50

100

150

200

250

1000 2000 3000 4000


50

100

150

200

250

1000 2000 3000 4000
















Acknowledgments


References









































Acknowledgments


References































colorimetric calibration

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