color mixing and color separation of pigments with concentration prediction

Color Mixing and Color Separation ofPigments with Concentration Prediction

Pesal Koirala,* Markku Hauta-Kasari,Birgitta Martinkauppi, Jouni HiltunenDepartment of Computer Science and Statistics, University of Joensuu, Joensuu, Finland

Received 19 July 2007; revised 28 November 2007; accepted 15 February 2008

Abstract: In this study, we propose a color mixing andcolor separation method for opaque surface made of thepigments dispersed in filling materials. The method isbased on Kubelka–Munk model. Eleven different pigmentswith seven different concentrations have been used astraining sets. The amount of concentration of each pig-ment in the mixture is estimated from the training sets byusing the least-square pseudo-inverse calculation. Theresult depends on the number and type of pigmentsselected for calculation. At most we can select all pig-ments. The combinations resulted with negative concen-trations or unusual high concentrations are discardedfrom the list of candidate combination. The optimal pig-ment’s set and its concentrations are estimated by mini-mizing the reflectance difference of given reflectance andpredicted reflectance. � 2008 Wiley Periodicals, Inc. Col Res

Appl, 33, 461 – 469, 2008; Published online in Wiley InterScience

(www.interscience.wiley.com). DOI 10.1002/col.20441

Key words: color prediction; Kubelka–Munk method; re-flectance correction; reflectance; least-square pseudo-inverse calculation

INTRODUCTION

Originally Kubelka–Munk (KM) theory is a model of the

light traveling in two directions in the materials.1 The ba-

sic KM theory is admissible to the diffuse illumination of

particular coating. The KM theory is of great importance

in many areas of applied research and has been used for

the optical properties of decorative and protective coat-

ings, paints, papers, pigmented polymers, fibers and

wools, thermal insulations, biological systems, and in

medical physics.2 KM method assumes a linear relation-

ship between the concentration and the scattering coeffi-

cient S and the absorption coefficient K, and this makes

the computation process faster. Further improvement in

this method was achieved by Saunderson correction.3

Saunderson correction converts the measured reflection by

integrating sphere spectrophotometer including the specu-

lar component to the internal reflection on which the KM

theory works. Marcus and Pierce4 extend the reflection

correction to different measuring geometries. The revised

KM theory5 has been used for ink, paper, and dyed paper.

Monte Carlo simulations, Expert systems or Neural net-

works, and Mie theory have been emerged as the alterna-

tive as well as collaborative method of KM model. Inde-

pendent component analysis6 may be used as the reflec-

tance separation but further research is required.

In this study, we have implemented single constant KM

theory for the pigments dispersed in the filling materials

and finally located on the plastic. Our method predicts the

reflectance of mixture from the given set of pigments with

different concentration. In additions our method is capable

of predicting the accurate concentrations and reflectance

of mixed pigments from the given reflectance of mixture.

This color separation method also uses the color mixing

method as a sub-problem because the given mixture is

compared with predicted reflectance of mixture to mini-

mize the reflectance difference. There are different meth-

ods for evaluating these differences. CIE color difference

equations (CIELAB, CIE LUV CIE94, etc.), Spectral

curve difference metrics [root mean square error (RMS),

goodness of fit coefficient (GFC)], metamerism indices,

and weighted RMS metrics7 can be used to calculate

color differences during minimizing reflectance process.

In proposed method CIELAB error, GFC, and mean

square error (MSE) were computed to calculate quantita-

tive value of reflectance matching. The reflectance of

training sets and test sets were measured by AvaMouse

handheld reflection spectrometer with annular measuring

*Correspondence to: Pesal Koirala (e-mail: [email protected]).

VVC 2008 Wiley Periodicals, Inc.

Volume 33, Number 6, December 2008 461

geometry of 458/08 under circular illumination in the visi-

ble range of 380–750 nm with 5 nm resolution. The col-

ors are accurately measured regardless of the gloss and

orientation of the texture.8

The spectrometer like AvaMouse with small aperture

was chosen because the specimens are circular pipe with

small radius of about 1.5 cm. In that particular case, a

bigger aperture for device would lead a wrong measure-

ment results. All the measured samples were converted to

internal reflection Eq. (6). In total eleven different sam-

ples were used as training sets. Seven different concentra-

tions of each sample were prepared. Every sample con-

sists of single pigment dispersed in the filling material.

The training samples at 4 g concentration are displayed in

Fig. 1.

CORRECTION FOR MEASURED REFLECTIONS

The key assumption in applying the KM theory is that the

light within the colorant layer is completely diffused. In

addition, there cannot be change in refractive index in the

samples boundaries.9 The KM theory is applicable to the

internal reflectance. Many modern spectrometers are capa-

ble of measuring reflectance factor without changing the

refractive index in the samples boundaries.10 However, if

the available spectrometer can measure only total reflec-

tance, the measured reflectance should be corrected

depending on the measuring geometry because the refrac-

tive index changes in the film/air boundaries influence

significantly the color and appearance of the coating.

These complications were first corrected by the Saunderson3

for the color matching of plastics. The measured reflectance

is converted to internal reflectance by Saunderson correction

as shown in Eq. (1).

Rl ¼ rl � K1

1� K1 � K2ð1� rlÞ (1)

where, rl is the measured reflectance normalized between

[0, 1] in each wavelength l, K1 is the Fresnel reflection coef-

ficient for the collimated light, and K2 is the Fresnel reflec-

tion coefficient for the diffused light striking the surface

from inside. The value of K1 is 0.04 for plastic material

because plastic has the refractive index of 1.5.9 The value of

K2 usually lies between 0.4 and 0.6.3 The optimized value of

K2 should be calculated practically. Once the internal reflec-

tance is calculated by the KM mixing law, the measured

reflectance is computed as shown in Eq. (2) by reversing

Eq. (1).

rl ¼ K1 þ ð1� K1Þð1� K2ÞRl

1� K2Rl: (2)

The mentioned Saunderson equation is applicable to

the integrating sphere spectrophotometer including the

specular component in the measurement. Marcus and

Pierce4 proposed the surface correction for different meas-

uring geometries. Here correction for 458/diffuse8 and

458/08 has been presented. The measured reflectance by

458/diffuse8 geometry is converted to internal reflectance


Rl ¼ rl � ðK1 � GÞð1� K1Þð1� K2Þ þ K2ðrl � ðK1 � GÞÞ (3)

Here G is the gloss factor that varies from 0 for perfect

glossy surface to K1 for perfect matt surface. The meas-

ured reflectance by 458/diffuse8 is calculated from internal

reflectance by reversing Eq. (3) as shown in Eq. (4).

rl ¼ ðK1 � GÞ þ ð1� K1Þð1� K2ÞRl

1� K2Rl(4)

Similarly the measured reflectance by 458/08 measuring

geometry is converted to the internal reflectance as shown

in Eq. (5).

Rl ¼ rlT1T2 þ K2rl

(5)

Here T1 ¼ 0.095 and T2 ¼ 0.43 are the transmittance

across the air film boundary for the film of refractive

index of 1.5. The other symbols carry the same meaning

as mentioned before. The measured reflectance by 458/08

FIG. 1. Reflectance (right) and its RGB color (left) of eleven different sets at 4 g concentration. T1, T2 , . . . ,T10 are thenames of training sets.

462 COLOR research and application

FIG. 2. Reflectance and K/S ratio of a sample pigment at different concentrations [0.2, 0.5, 1, 2, 4, 6, 10] g in 1 L of fillingmaterial.

is calculated from internal reflectance by reversing Eq. (5)


rl ¼ T1T2Rl

1� K2Rl(6)

KM THEORY

Originally KM method is a two flux radiative transfer

approach. Some rules were made for the acceptable result.

The rules are as follows: light striking on a surface should

diffuse, should be scattered in two vertical up and down

direction, and should not be polarized. The KMmethod does

not consider the reflectance difference in the boundary

because of difference of refractive index between air and

medium. Therefore the reflectance measured should be cor-

rected depending on the measuring device. Reflected light

from the surface of the film depends on its thickness, absorp-

tion and scattering coefficient of the material of the film, and

the reflection from the substrate. The basic representation of

the reflectance from the film is shown in Eq. (7).

R ¼ 1� Rgða� b cothðbSXÞÞa� Rg þ b cothðbSXÞ (7)

where a ¼ 1 þ (K/S); b ¼ (a2 2 1)1/2; X is the thickness; Rg,

is the reflectance of substrate; K is the absorption coefficient;

and S is the scattering coefficient.For complete hiding,11 opaque materials9; the reflec-

tance of substrate does not have the effect on the reflec-

tance of the film. In complete hiding, thickness (X) is

assumed to be infinity and reflectance of substrate Rg is

zero. The reflectance of opaque film is obtained from Eq.

(7) as the function of ratio of absorption coefficient K to

scattering coefficient S as shown in Eq. (8).

R ¼ 1þ K

S

� ��

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiK

S

� �2

þ2K

S

� �s(8)

The K over S ratio is obtained by reversing Eq. (8).

K

S

� �¼ ð1� RÞ2

2R(9)

Figure 2 illustrates the measured internal reflectance

factor and its conversion to K over S values obtained

from Eq. (9) for the known concentration of [0.2, 0.5, 1,

2, 4, 6, 10] g in 1 L filling material.

The scattering and absorption coefficients of mixture

are described as the linear combination of scattering and

absorption coefficients of mixed pigments scaled by its

concentrations as shown in Eq. (10). This method is well

known as two constant KM model.

K

S

� �mix

¼Pni¼1

CiKi

Pni¼1

CiSi

(10)

The individual absorption and scattering coefficients

required for Eq. (10) can be calculated by using the white

set (setting scattering 1 in every wavelengths), masstone

(100% relative percentage pigment), and tint (pigment

mixed with white).12 This method is called the relative

two constant theory.13 In the case of single colorant in

the mixture Eq. (10) is reduced to Eq. (11).

Kmix

Smix

¼ K1 þ Kw

S1 þ Sw(11)

Dividing numerator and denominator in Eq. (11) by the

scattering coefficient Sw of the white pigment, Eq. (12) is

obtained

Kmix

Smix

¼K1

Swþ Kw

SwS1Swþ 1

(12)

The K/S ratio of the mixture, white, and colorant are

represented by Wmix, Ww, and W1, respectively. Then the

Eq. (12) is represented in Eq. (13).

Wmix ¼W1

S1Sw

� �þWw

S1Swþ 1

(13)

The Eq. (13) is rearranged to get the S1/Sw, the scatter-

ing coefficient of the colorant relative to the scattering

coefficient of the white.


S1Sw

¼ Wmix �Ww

W1 �Wmix

(14)

Finally the K1/Kw, the absorption coefficient of the col-

orant relative to the scattering coefficient of the white is

shown in Eq. (15) because K1 ¼ S1W1.

K1

Kw

¼ W1

Wmix �Ww

W1 �Wmix

� �(15)

From Eqs. (14) and (15), it can be seen that the absorp-

tion and scattering coefficient of colorant relative to scat-

tering coefficient of white can be calculated if we know

the K/S value of mixture, white, and colorant. In the Eqs.

(14) and (15), only known information is K/S value of

mixture. The K/S value of the white and colorant can be

obtained from the internal reflectance from the 100%

white opaque sample and 100% colorant opaque sample

using Eq. (9).

The inversion technique from KM analysis14 can be

exploited to calculate the optical characteristics of the

semitransparent materials. In that method the coating

layer or sample is measured over black and white

substrate separately. When it is measured over black

substrate the reflectance from substrate Rg tends to zero.

Similarly, when it is measured over white substrate, the

reflectance from substrate Rg tends to one. Under these

conditions two expressions of reflectance, one for over

white Rw and another for over black Rb are calculated

from Eq. (7) as shown in Eqs. (16) and (17).

Rw ¼ 1� aþ y

aþ y� 1(16)

Rb ¼ 1

aþ y(17)

where y ¼ X coth (X Sh).From Eqs. (16) and (17), the K/S ratio is calculated as

shown in Eq. (18).

K

S¼ ð1� RbÞð1� RwÞ

2Rb

(18)

Once the K/S is calculated, then the value of a and bcan be calculated and scattering coefficient is calculated

by rearranging the Eq. (17) as shown in Eq. (19).

S ¼ 1

bXcoth�1 1� aRb

bRb

� �(19)

Finally the absorption coefficient is as shown in Eq.

(20).

K ¼ K

S

� �S (20)

In our experiment the pigments are completely dis-

persed in filling materials. The relationship between K/Sratio and concentration is approximately linear. The rela-

tionship between K/S ratio of red pigment and its concen-

trations at different wavelengths have been displayed in

Fig. 3. The linear relationship of K/S ratio to concentra-

tion suggests that pigments itself have little scattering

power of their own in comparison with the filling materials.

Therefore any scattering of the light can be regarded as from

the filling materials. In that case the Eq. (10) is reduced to

more simple form called the single constant KM model see

Eq. (21). The ratio of K/S is used instead of calculating indi-vidual K and S in single constant KMmethod.

K

S

� �mix

¼Xni¼1

Cik

s

� �i

(21)

where (K/S)mix is the ratio of absorption and scattering of

pigment mixture; n is the number of pigments in mixture; Ci

is the concentration of ith pigment in mixture by weight of

dry pigment; and (k/s)i is the ratio of absorption and scatter-

ing of ith pigment per unit concentration.

COLOR MIXING

Given a set of pigments with reflectance curve, we can

get the reflectance curve of specified mixture of these pig-

ments using Eqs. (8), (9), and (21). In single constant KM

method unit k/s and concentration of each pigment are

required to predict the specified mixture of pigments.

Equation (22) gives the method to calculate unit k/s valueof single colorant because number of colorants in the

mixture is one (n ¼ 1).

k

s

� �l;1¼

KS

� �l;mix

�CWks

� �l;w

C1

(22)

where C1 is the concentration of pigments and Cw is the

concentration of filling material.

Each pigment in the training sets may have different

concentrations, in this study there are seven different con-

centrations for each pigment (see Fig. 2). Ideally the unit

k/s of sample pigment at different concentrations should

be the same but it does not happen in practical case. From

different concentrations of each pigment one representa-

tive unit k/s should be estimated by using least-square

FIG. 3. Relation between K over S and concentration ofpigment of red color. [Color figure can be viewed in the onlineissue, which is available at www.interscience.wiley.com.]


pseudo-inverse calculation as shown in Eq. (25) (see Fig.

4). The Eq. (21) is arranged in matrix form in Eq. (23)

because mixture of pigments is made from one pigment

and filling material. Each pigment is represented by dif-

ferent concentrations, here n is the number of different

concentrations of each pigment. Concentration of mixture

is set as 1.

KS

� �380;1

� ks

� �380;W

:: :: KS

� �750;1

� ks

� �750;W

: : : :

: : : :

: : : :KS

� �380;n

� ks

� �380;W

:: :: KS

� �750;n

� ks

� �750;W

266664

377775

¼c1:

:

cn

2664

3775 k

s

� �380

:: :: ks

� �750

� (23)

where

C ¼c1:

:

cn

2664

3775X ¼ k

s

� �380

:: :: ks

� �750

� Y

¼

KS

� �380;1

� ks

� �380;W

:: :: KS

� �750;1

� ks

� �750;W

: : : :

: : : :

: : : :KS

� �380;n

� ks

� �380;W

:: :: KS

� �750;n

� ks

� �750;W

266664

377775

Equation (23) is represented as:

Y ¼ CX (24)

The value of X (unit k/s in all wavelength) is solved as:

X ¼ �CC� ��1 �CY (25)

Figure 4 illustrates the unit k/s ratio and its normalized

spectrum in wavelength 650 nm. The normalized spec-

trums of unit k/s ratio of same pigment from different

concentrations should be almost the same for proper

selection of sample sets. The difference in unit k/s of

same sample shows that K/S is not exactly the linear

function of concentration. These nonlinearity can be

approximated by using K/S as a power series function of

concentration C as shown in Eq. (26).

K

S¼ Caþ C2bþ C3c (26)

The best coefficients, a, b, and c can be calculated by

using least square calculation from the set of same pig-

ments of different concentrations. These coefficients can

be used instead of single unit k/s.Figure 5 illustrates the reflectance curves obtained by

mixing three samples with the concentration of [0.5, 0.7,

3.0] g in 1 L of filling material. The color of reflectance

of samples and mixture is visualized in monitors by cal-

culating the tristimulus values X, Y, and Z from reflec-

tance and then converting them to device RGB coordinate

system by using linear transformation.9,15

COLOR SEPARATION ANDCONCENTRATION PREDICTION

The color separation and concentration prediction mainly

relies on matching function. The computational method

for the colorimetric matching function was proposed by

Allen16,17 based on Park and Stearn’s method.18 The

method for three colorants was formulated considering

the single constant KM method.16 The method was

improved to four colorants using two constant KM

method.17 These methods are limited to the nonmetameric

samples because the differences of the tristimulus values

of the test and predicted samples were minimized under a

particular light source. Our goal is to apply the color

matching formulation to all the set of colorants available

in the training sets. The spectral color matching was per-

formed by using the least square method.

In single constant KM theory, the concentrations of

the pigments can be estimated from Eq. (29) if the K/Svalue of mixture and the unit ratio of absorption to scat-

tering coefficient k/s value of mixed pigments are known.

In two constant methods the concentration of the

FIG. 4. The unit k/s (solid red line is the representative unit k/s) and normalized unit k/s at 650 nm wavelength calculatedfrom the samples at different concentrations.


TABLE I. Computational steps to predict concen-trations of mixing pigments

1. Compute unit k/s ratio of each training set.2. Convert reflectance of test set Rmix to K/S ratio using

Eq. (9).3. Choose n number of pigments in mixture.

Repeat step 4 to 8 for all combinationsNn

� ��

4. Predict concentrations using Eq. (29) and store row wise inmatrix concentration.

5. The negative concentrations and unexpected highconcentrations are neglected.

6. Predict (K/S)p ratio of mixture using predicted concentrationsand unit k/s ratio from training sets, see Eqs. (21) or (27).

7. Determine reflectance Rp using (K/S)p, see Eq. (8).8. Calculate difference DE between Rmix, and Rp and store DE in

array error.9. Order the matrix concentration according to array error sorted

in ascending order for Lab difference and MSE, and descendingorder for GFC.

FIG. 5. Three different colorants reconstructed with 0.5, 0.7, and 3.0 g pigment concentration. The right image is theresultant reflectance computed mixing those three pigments. [Color figure can be viewed in the online issue, which isavailable at www.interscience.wiley.com.]

pigments can be estimated if the K/S value of mixture

and the absolute or relative unit absorption coefficient kand scattering coefficient s are known separately. The

mixture of pigments is made by mixing n different pig-

ments and the number of pigments used is far less than

the number of wavelengths sampled to present the reflec-

tance curve. Therefore only the n number of wavelengths

can be selected to solve the n number of concentrations.9

However choosing n number of different wavelengths

results the different concentrations, therefore for more sta-

ble result least-square pseudo-inverse calculation is used

to calculate concentration considering all visible range

wavelengths [see Eqs. (27)–(29)]. After knowing the con-

centration and unit k/s value, the reflectance of the pig-

ments is predicted by using Eqs. (21) and (8), consecu-

tively. The Eq. (21) is represented in matrix form in Eq.

(27) extending for all wavelengths.

KS

� �380;mix

� ks

� �380;W

:

:

:KS

� �750;mix

� ks

� �750;W

266664

377775 ¼

ks

� �380;1

:: :: ks

� �380;n

: : : :

: : : :

: : : :ks

� �750;1

:: :: ks

� �750;n

266664

377775

c1:

:

cn

2664

3775

(27)

where,

C ¼c1:

:

cn

2664

3775X ¼

ks

� �380;1

:: :: ks

� �380;n

: : : :

: : : :

: : : :ks

� �750;1

:: :: ks

� �750;n

266664

377775Y

¼

KS

� �380;mix

� ks

� �380;W

:

:

:KS

� �750;mix

� ks

� �750;W

266664

377775

Equation (27) is represented as:

Y ¼ XC (28)

The least-square pseudo-inverse calculation [see Eq.

(29)] is used to find the concentrations of mixed pig-

ments. The number of pigments mixed (n) should be less

than number of wavelengths. Therefore the problem can

be solved by choosing the n number of different wave-

lengths, alternatively. Nevertheless the entire wavelength

calculation gives more robust result.

C ¼ �XXð Þ�1 �XY (29)

Considering a more complex case where we have only

been given the reflectance of mixture and our task is to

estimate the concentration and the reflectance of the pig-

ments used in the mixture. The problem is solved using

the unit k/s values of each pigment of the training sets.

The predicted concentrations and used unit k/s of each

iteration are employed to estimate the reflectance [see

Eqs. (21) and (8)]. This process is repeated for all possi-

ble combinations. Equation (30) shows the total number

of combinations to be computed.


Nn

� �¼ N!

ðN � nÞ!n! (30)

where N is the number of pigments in training sets and nis the number of pigments used in mixture.

In this experiment N ¼ 11 and n may vary from 1 to

N. The unit k/s and predicted concentrations are chosen

so that estimated reflectance using this concentration and

unit k/s has minimum differences with given reflectance

of mixture. The differences of the reflectance are calcu-

lated by using color difference of CIELAB color space9,15

GFC and MSE.7 The computation step predicting optimal

concentrations of pigments used in the mixture is shown

in Table I. This method provides the different choices of

selection of mixing pigments depending on the error

between real and predicted reflectance of mixture. The re-

flectance of the mixture is calculated from predicted K/Svalue. In each iteration the K/S value of mixture is calcu-

lated from the predicted concentration and known unit k/svalue of training sets. The best colorants in the mixture

and its corresponding concentrations are found according

to minimum error values between predicted and original

reflectance of mixture. The colorants and concentrations

with second minimum error between original and pre-

dicted reflectance are the second best choice. The choice

of the result depends on the requirement of the accuracy.

The provided choices may reduce the cost on pigments in

industrial application. Figure 6 illustrates the first and

tenth best results optimized by MSE, the concentrations

and reflectance of mixing pigments, and the price of mix-

ture per kilogram.

The real concentrations of pigments used in mixture

and corresponding predicted concentrations by our method

are illustrated along X and Y axes in Fig. 7. The predicted

concentrations of colorants can be corrected by fitting the

predicted concentration with real concentration by using

interpolation methods. However, in advance we should

have the relation between real concentration and predicted

concentrations of each pigment of the training sets. The

Table II shows the differences between the measured re-

flectance of training sets and the reconstructed reflectance

FIG. 6. Color separation of mixture and concentration prediction of mixing colors. Left image shows the first best resultand right image shows the tenth best result. The price of the mixture has reduced drastically. T5, T9, T10, T3, and T8 arethe name of training sets.

FIG. 7. Real versus predicted concentrations. Concentra-tions are in grams. The solid line is the average of all dot-ted lines. [Color figure can be viewed in the online issue,which is available at www.interscience.wiley.com.]

TABLE II. Error calculated between original samplesand reconstructed samples

Sample no MSE GFC CIELAB DE

T1 0.0001 0.9999 0.7813T2 0.0003 0.9999 0.9052T3 0.0002 1.0 0.81T4 0.0006 0.9996 2.35T5 0.0002 1.0 1.26T6 0.0001 0.9998 1.248T7 0.0004 1.0 0.68T8 0.00020 0.9998 1.97T9 0.0001 0.9999 1.52T10 0.0001 1.0 0.72T11 0.0023 0.9991 2.7

The results are the average values of all samples reconstructedin all seven different concentration by single constant method.


FIG. 8. The top figure is the reflectance of mixture. The bottom figures are the reflectances of the predicted colorants.The MSE, CIELAB DE, GFC between original and predicted reflectance are 0.000128, 1.6811, and 0.991. The original con-centrations of samples T1 and T6 are 2.0 an 6 g and predicted concentrations are 2.62 and 5.77 g. [Color figure can beviewed in the online issue, which is available at www.interscience.wiley.com.]

FIG. 9. The top figure is the reflectance of mixture. The bottom figures are the reflectances of the predicted colorants.The MSE, CIELAB DE GFC between original and predicted reflectance are 0.000488, 1.5497, and 0.9897. The original con-centrations of samples T1 and T5 and T6 are 4.0, 4.0, and 6.0 g and predicted concentrations are 4.23, 412, and 5.67 g.[Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]


of training sets by setting the number of pigments one in

mixture. The errors shown in each row are the average

reconstruction error of all seven different concentrations

of each pigment in the training sets. The CIELAB color

difference formula was calculated using daylight source

(D65) and CIE 1931 standard observer.

Figures 8–10 show the reflectance of mixture that have

been separated to the reflectance of individual colorants.

CONCLUSIONS

The basic theory of KM method was discussed. The

method to predict the reflectance of mixture made from

the pigments with arbitrary concentration was described.

Computation process for the concentration prediction and

separated color prediction was described. Our future work

will consider more accurate color separation and concen-

tration prediction from the given transparent and translu-

cent object by KM methods and revised KM methods5

and independent component analysis.6

1. Kubelka P. New contributions to the optics of intensity light scatter-

ing materials, Part 1. J Opt Soc Am 1948;38:448–457.

2. Vargas W, Niklasson G. Applicability conditions of the Kubelka-

Munk theory. Appl Opt 1997;36:5580–5586.

3. Saunderson J. Calculation of the color pigmented plastics. J Opt Soc

Am 1941;32:727–736.

4. Marcus R, Pierce P. An analysis of the first surface correction for

the color matching of organic coatings from the viewpoint of radia-

tive transfer theory. Prog Org Coat 1994;23:239–264.

5. Yang L, Kruse B. Revised Kubelka-Munk theory. I. Theory and

application. J Opt Soc Am A 2004;21:1933–1940.

6. Hyvarinen A, Karhunen J, Oja E. Independent Component Analysis.

New York: Wiley; 2001.

7. Imai F, Rosen M, Berns R. Comparative study of metrics for spec-

tral match quality. In: The First European Conference on Colour

Graphics, Imaging, and Vision. France: University of Poitiers; 2002.

p 492–496.

8. http://www.avantes.com/news/avamouse.pdf 3-3-2007.

9. Berns R. Principles of Color Technology, 3rd edition. New York:

Wiley; 2000.

10. Bondioli F, Manfredini T, Romagnoli M. Color matching algorithms

in ceramic tile production, J Eur Ceram Soc 2006;26:311–316.

11. Haase C, Meyer G. Modelling pigmented materials for realistic

image synthesis. ACM Trans Graphics 1992;11:305–335.

12. Davinson H, Hemmendinger H. Color prediction using the two-con-

stant turbid-media theory. J Opt Soc Am 1966;56:1102–1110.

13. Nobbs J. Colour-match prediction for pigmented materials. J Soc

Dyers Colourists 1997;34:112–113.

14. Vargas W. Inversion methods from Kubelka-Munk analysis. J Opt

Soc Am A 2002;4:452–456.

15. Kuehni R. Color an Introduction to Practice and Principles. New

York: Wiley; 1997.

16. Allen E. Basic equations used in computer color matching. J Opt

Soc Am 1966;56:1256–1259.

17. Allen E. Basic equations used in computer color matching. II.

Tristimulus match, two-constant theory. J Opt Soc Am 1974;64:991–

993.

18. Park R, Stearns E. Spectrophotometric formulation. J Opt Soc Am

1944;34:112–113.

FIG. 10. The top figure is the reflectance of mixture. The bottom figures are the reflectances of the predicted colorants.The MSE, CIELAB DE GFC between original and predicted reflectance are 0.000032, 1.0265, and 0.9993. The original con-centrations of samples T2 and T5, T8 and T11 are 2.0, 2.0, 0.5, and 6.0 g and predicted concentrations are 2.37, 2.438,0.4022, and 6.671 g. [Color figure can be viewed in the online issue, which is available at www.interscience.wiley.com.]


color mixing and color separation of pigments with concentration prediction

Documents