1

Technical Report

Te s tin g Ins tr ume ntal-Bas e d Co lo r match ing fo r

Ar tis t Acr ylic Paints

Summer 2006

Mahnaz Mohammadi

Roy S. Berns

Spectral Color Imaging Laboratory Group

Munsell Color Science Laboratory

Chester F. Carlson Center for Imaging Science

Rochester Institute of Technology

mm0252@cis.rit.edu, berns@cis.rit.edu

http://www.art-si.org/

2

Summary

Colorant selection and recipe prediction for restorative inpainting (retouching) using

spectrophotometry and computer processing requires developing a database of the optical

properties of candidate colorants. Typically, tints are prepared of each colorant mixed

with titanium dioxide white at a number of levels in addition to the colorant by itself, the

“masstone.” The tint recipes must be determined accurately, applied uniformly to a

support until opacity is achieved, and measured with a reflection spectrophotometer.

Given the large number of colorants used in inpainting, database development requires a

considerable time commitment. Accordingly, it was of interest to determine the minimum

number of samples required to characterize a colorant’s optical properties. Two optical

models were considered, two-constant Kubelka-Munk (K-M) turbid media theory and its

single-constant simplification. This was tested using four acrylic emulsion paints with a

range of absorption and scattering properties: cobalt blue, cadmium yellow medium,

green gold (a three-pigment mixture), and phthalocyanine green. For colorant selection, a

single tint, preferably near the maximum chroma achievable by mixing with white, was

sufficient. For both colorant selection and recipe prediction, two-constant K-M theory

was required; in this case, a single tint preferably near the maximum chroma and the

masstone were sufficient. Additionally, the effect of accounting for the refractive index

discontinuity between air and the paint film was considered, known as the “Saunderson

correction,” and found to be important in order to achieve best performance.

Introduction Restorative inpainting, or retouching, is an important and common treatment for works of

art. One inpainting goal is creating an appearance match, that is, an “invisible” restoration

3

under typical illumination and viewing conditions. This goal requires a methodology for

inpainting that minimizes metamerism. This is a challenge for conservators. General

background about metamerism is described by Berns [1]. A review of metamerism and

inpainting specifically for blue pigments was given by Staniforth [2].

Conservators are less interested in identifying the exact components of losses

when inpainting. Hence, the problem of inpainting is simplified to colorant selection

rather than colorant identification. The choice of colorants from a large pool of colorants

is the critical issue in inpainting. Also a non-destructive technique is generally desirable.

Visible reflectance spectrophotometry along with Kubelka-Munk (hereafter abbreviated

K-M) turbid media theory [3] is an appealing technique for colorant selection. Berns, et

al. [4] developed a simplified instrumental-based system to aid in colorant selection.

They tested the system for colorant selection for Barnett Newman’s Dionysius and

Sanford Robinson Gifford’s Siout, Egypt. They adopted the single-constant simplification

of K-M theory and multiple-linear regression to select a set of colorants most closely

matching the spectral properties of the original paintings.

As a tool for colorant selection and colorant recipe prediction, the simplified K-M

model using the ratio of absorption and scattering coefficients of colorants, the so-called

single-constant K-M theory, or a more precise and complex model using the absorption

and scattering coefficients, called two-constant K-M theory, are used. In order to use the

computer calculation, one needs to have concentration-independent quantities as a

function of wavelength: (k/s)! in single-constant theory or (k)! and (s)! in two-constant

theory, where k and s define absorption and scattering, respectively. Ideally, these

quantities are characteristic of a colorant regardless of the employed media. Therefore,

4

any colorant with the certain Color Index Number could have a defined (k/s)! or (k)! and

(s)! and the user would have a database of these quantities for all available colorants. This

kind of database is not available for users since these quantities for a colorant vary in

different media. In the literature [5-8] making a tint ladder with binary colorants was

suggested to derive these quantities, referred to as a tint ladder. Furthermore, the spectral

signature of any colorant is poorly revealed without the addition of a white colorant [9].

A black colorant is also used to characterize a very light colorant, which might not be

differentiated from white at long wavelengths where two-constant K-M theory is

employed, for example some yellows. Walowit, et al. [10, 11] made several mixtures of

primaries with each other at known concentration levels in order to calculate their (k)!

and (s)!. Today, most available color-matching software is based on tint ladders. The

question is whether having a tint ladder, which is time consuming and costly when

having a large number of colorants to be characterized, is required for the purpose of

inpainting? If the answer is negative, how many tints and at what ratio is required to

develop a colorant database for instrumental-based inpainting? In order to answer these

questions, single-constant and two-constant K-M theory along with a non-negative least

square technique were adopted to match the spectral properties of all mixtures of tint

ladders of four acrylic emulsion paints.

Kubelka-Munk Turbid Media Theory Overview Materials modify light differently. This fact categorizes materials into transparent,

translucent, and opaque. Based on ASTM [12] definitions, opaque means transmitting no

optical radiation, translucent is transmitting light diffusely, but not permitting a clear

view of objects beyond the specimen and not in contact with it, and transparent is

5

transmitting radiant energy without diffusion. K-M turbid media theory [3] is a simplified

model of light traveling within a material. This model is mostly applied to characterize

and formulate translucent and opaque colored materials. More detail on the derivation of

K-M theory is available in the literature [13-16].

The ratio of absorption and scattering coefficients of a film of a colorant is related

to the reflectivity or internal diffuse reflectance of the film, equation (1). In the later

section, a method to calculate the internal diffuse reflectance will be discussed. This

relationship between absorption, scattering, and internal diffuse reflectance assumes that

increasing the thickness of the film does not change the reflectivity, R!i. In other words,

the film should be opaque.

!

R" ,,i = 1+K

S

#

$ %

&

' ( "

)K

S

#

$ %

&

' ( "

2

+ 2K

S

#

$ %

&

' ( "

*

+ , ,

-

. / /

1 2

, (1)

where, R!,i is the internal diffuse reflectance of a film of a colorant and (K/S)! is the

spectral absorption (K) and scattering (S) ratio of the film.

K-M theory and Linearity With Concentration Additivity and linearity of the scattering and absorption coefficients of the individual

colorants, k!i and s!i, to that of the mixture, K!,m and S!,m was expressed by Duncan [17]

as

!

K" ,m = cik" ,i

i

# , (2)

!

S" ,m = cis" ,i

i

# , (3)

where ci represents the concentration of each ith colorant. The subscript m stands for the

mixture sample. Following Duncan’s assumption, the ratio of the absorption and

coefficients are written as

6

!

K

S

"

# $

%

& ' ( ,m

=c1k( ,1 + c

2k( ,2 + ...+ c

wk( ,w

c1s( ,1 + c

2s( ,2 + ...+ c

ws( ,w

. (4)

The subscript w represents a white, highly scattering colorant, such as titanium dioxide.

When a colorant, such as white, contributes the majority of scattering in the mixture, the

scattering of the other colorants would be negligible and equation (4) simplifies as:

!

K

S

"

# $

%

& ' ( ,m

=c1k( ,1 + c

2k( ,2 + ...+ c

wk( ,w

cws( ,w

=c1

cw

k

sw

"

# $

%

& ' ( ,1

+c2

cw

k

sw

"

# $

%

& ' ( ,2

+ ...+k

sw

"

# $

%

& ' ( ,w

. (5)

The ratio (k/sw)!i (hereafter noted (k/s)!i) is the so-called “unit k over s” of the colorant.

This quantity describes the absorption power of the colorant at unit amount in the

employed medium but ideally it can be employed as the general characteristic of the

colorant. Having at least one mixture of the colored and white colorants is sufficient to

derive (k/s)!. This simplified equation, called single-constant K-M theory, is a solution

for characterizing and formulating the mixture of colorants of an opaque film.

Saunderson [18] proposed that the unit scattering and absorption coefficients of

the colorants can be obtained relatively by setting the scattering coefficient of white equal

to unity. This model is called the relative two-constant theory, which is applicable for

colorant layers at opacity. By this assumption, equation (4) for a mixture of two

colorants, colored plus white, will be simplified as

!

K

S

"

# $

%

& ' ( ,m

=c1k( ,1 + c

wk( ,w

c1s( ,1 + c( ,w

, (6)

where

s! ,w = 1

k! ,w =K

S

"#$

%&'! ,w

. (7)

7

In this case, at least two mixtures of the colored and white colorants at known

concentrations are needed to solve for (k)! and (s)! of the colorant. The matrix notation of

equation (6) to solve these quantities having a tint ladder is described in Appendix I.

Saunderson Correction In K-M theory, internal diffuse reflectance is considered in all the calculations, which

means the reflections at the sample surface are not accounted. Ignoring the surface

reflection correction can be valid for textile and paper samples since the fibers that are

scattering the light are immersed in air and the correction for refractive index

discontinuity is not necessary. In a paint sample, the scattering materials are immersed in

resins and vehicles with different refractive indices than air. A correction for this surface

reflection has to be considered for such a sample. The measured reflectance with a

spectrophotometer and the calculated reflectance by K-M theory are related as the

Saunderson correction [18]:

R! ,i =R! ,m " K1

1" K1" K

2+ K

2R! ,m

, (8)

where R!,m and R!,i are the measured and the calculated reflectance, respectively. (In fact,

this correction was derived by Ryde [19], as noted by Saunderson [18] and Berns [20]).

The Fresnel reflection coefficients for collimated and diffused light are defined as K1 and

K2, respectively. In this expression, K1 is the fraction of the incident light that is reflected

from the front surface of the sample. Since the refractive index of most resins in the paint

industry is in the range of 1.4-1.5, K1 would be 0.02-0.04 based on the Fresnel equation,

equation (9), for depolarized incident radiation normal to the surface plane. Saunderson

[18] suggested a value between 0.4 - 0.6 for the reflection coefficients of the diffused

light, K2.

8

!

K1

=n "1

n +1

#

$ %

&

' (

2

, (9)

where n is the refractive index of the paint film. Equation (8) is valid for integrating

sphere spectrophotometric measurements with specular component included. For

specular component excluded or bidirectional geometries, the K1 in the numerator is

removed from this equation [1]. Extensive detail on the refractive index and Fresnel

equations applied in oil paintings including various varnishes with different refractive

indices and different molecular weights was addressed by Berns, et al.[20-21]

Rewriting equation (5) for a chromatic colorant and white, we obtain

!

K

S

"

# $

%

& ' ( ,m

=c

cw

k

s

"

# $ %

& ' (

+k

s

"

# $ %

& ' ( ,w

, (10)

in which a plot of [(K/S)!,m-(k/s)!,w] against concentration, c/cw , would yield a straight

line with slope equal to (k/s)!. This is analogous to determining the extinction coefficient

of a dye in solution using Beer’s law [1].

In order to show this relationship, an ideal tint ladder was simulated using the

K/S’s of a phthalocyanine green masstone and titanium white. At each level of

concentration the (K/S)!,m was calculated using equation (10). The calculated (K/S)!,m

value was converted to internal reflectance factor for each theoretical mixture. Figures 1a

and 1b shows the spectral reflectance factor and the (K/S)!,m-(k/s)!,w of the ideal ramp,

respectively. In any color mixing application, the scalability of the tint ladder should be

investigated [1,22]. Scalability means changes in amount of colorant should not affect

the spectral shape, only the level of it. If this requirement is met, the spectra of the entire

tint ladder can be predicted based on knowing the spectrum at one level. The scalability

requirement is met when the normalized (k/s)! curves ((k/s)!/(k/s)! at wavelength

9

maximum) are nearly coincident, seen in Figure 1c where all the curves are coincident,

by definition for this ideal tint ladder. Hence, [(K/S)!,m-(k/s)!,w] against concentration,

c/cw , yields a straight line with slope equal to (k/s)!, Figure 1d.

(a)

(b)

( c )

(d)

Figure 1 Ideal tint ladder of Phthalocyanine Green. (a)- Spectral Reflectance factor, (b)-

Spectral [(K/S)!-(k/s)!,w], (c)- Spectral normalized (k/s)!, and (d)- [(K/S)!-(k/s)!,w] versus

concentration, c/cw, at 590 nm.

The other method to investigate the scalability is plotting 1/(K/S)!,m versus

wavelength using a logarithmic scale. This requirement would be met if all the curves

had the same shape. Derby [23]

demonstrated that when performing colorant

identification using spectral data, the logarithm of absorption and scattering ratio was the

most invariant to changes in concentration, not reflectance.

10

Theoretically, such a straight line gives the (k/s)! completely independent of

concentration, but in practice [(K/S)!,m-(k/s)!,w] against concentration, c/cw, usually results

in a concave curve. One reason might be attributed to ignoring the refractive index

discontinuity. Secondly, as concentration increases, the light-absorptive capacity of the

paint would approach its utmost degree. Therefore, the absorption would not be

increasing linearly. The former reason is largely corrected using the Saunderson

correction, equation (8). A plot of [(K/S)!,m-(k/s)!,w] against concentration, c/cw, of the

real ramp of Golden Matte Fluid Phthalocyanine Green (Blue shade) is shown in Figure

2. The data of this plot are based on 630 nm. The effect of the Saunderson correction to

linearize [(K/S)!,m-(k/s)!,w] and concentration, c/cw, is evident. Theoretically, the K1 and

K2 should be optimized at each wavelength to achieve best performance since these

optical effects are wavelength dependent. In practice, it is assumed that the wavelength

dependency is negligible. In this research optimized [24] K1 and K2 coefficients were

employed for all wavelengths.

11

Figure 2 The upper curve is a plot of [(K/S)!,m-(k/s)!,w] against concentration, c/cw, for

Phthalocyanine Green. The bottom curve shows the same data applying the Saunderson

correction.

An important distinction has been made between spectral scalability, verified by

plotting normalized spectra, for example Figure 1c, and the common nonlinearity

between actual concentration, c/cw, and effective concentration, for example Figure 2 [1].

The term “effective” refers to the optical effect of a colorant within the mixing system.

The concavity means that with increasing actual concentration, there is a reduction in the

optical effect, and thus the effective concentration is less than expected assuming

linearity. This nonlinearity can be modeled empirically. For this reason, one needs to

prepare a tint ladder. Spectral scalability is required for colorant selection. Having a

12

linear relationship between actual and effective concentration or modeling any nonlinear

behavior is required for recipe prediction.

Experimental

Sample Preparation Samples for this experiment were selected from acrylic emulsion paints manufactured by

Liquitex! Artist Materials and Golden! Artist Colors. These samples were Cobalt Blue

(PB 28) and Cadmium Yellow Medium (PY 37) from Liquitex and Green Gold (PY 150,

PG 36, PY3), Phthalocyanine Green (blue shade) (PG 7), and Titanium Dioxide (PW 6)

from Golden. Acrylics were selected because of their ease of use. These particular colors

were selected to have a range of scattering and absorption properties in order to

investigate K-M theory. For each paint, a tint ladder was prepared at different

concentrations with titanium white [24,25]. The term “tint ladder” includes a drawdown

of the paint at masstone. A BYK-Gardner drawdown bar was used to apply the paint

uniformly on Lenetta opacity charts. The white and black sections of these charts enabled

the determination of opacity. In this research, concentration was expressed as the ratio of

the weight of chromatic paint to the total weight of the mixture. As a ratio, the summation

of the concentrations must equal unity, i.e., c +cw = 1. This equality was imposed as a

constraint during all optimizations. A notation for each mixture was defined in this

experiment as H/L/C, where H represents the colorant, L represents CIELAB lightness,

and C represents CIELAB chroma. A cobalt blue tint would have the notation CB/76/28.

Spectral reflectance factor was measured using a GretagMacbeth Color Eye XTH

integrating sphere spectrophotometer with the specular component included.

Measurements were collected from 360 to 750 nm in 10 nm intervals. The spectral

13

measurements of the tint ladders of each of the four colorants are shown in Figure 3. The

measured reflectance factor was converted to internal diffuse reflectance factor, equation

(8), then, in turn, to (K/S)!, equation (1). The Saunderson coefficients for these paints

were 0.03 and 0.65 for K1 and K2, respectively. These values were obtained as a result of

research by Okumura [24] for these paints. (K/S)! is a quantity to use in both single- and

two-constant K-M theory to characterize a colorant’s concentration-independent

properties.

Figure 3 Spectrophotometric curves of the tint ladder of (a)- Cobalt Blue, (b)- Cadmium

Yellow Medium, (c)- Green Gold, and (d)- Phthalocyanine Green.

The created gamut of each tint ladder of the colorants based on the measured

spectral reflectance factor is shown in Figure 4 as CIE L* versus CIE C*ab (illuminant

D65 and the 1931 Standard observer). The lightness of the masstone of the selected

14

paints varied in the range of between 25 and 84. The darkest sample was Phthalocyanine

Green and the lightest was Cadmium Yellow Medium.

Figure 4 Colorimetric plots of the tint ladder of (a)- Cobalt Blue, (b)- Cadmium Yellow

Medium, (c)- Green Gold, and (d)- Phthalocyanine Green.

Single-Constant K-M Solution In single-constant theory, it is assumed that the scattering of a mixture is attributed to the

largest scattering component. In the case of having a chromatic and white colorant in the

mixture, this component is white. For a mixture of two colorants, equation (10) was used.

In this equation c is the concentration of the chromatic colorant, defined as the ratio of the

weight to the total weight of the mixture and similarly, cw is the ratio of the weight of the

white colorant to the total weight of the mixture. The (k/s)! of the chromatic colorant for

15

each mixture was derived from equation (10), based on knowing (K/S)!,m, (k/s)!,w and the

concentration of the paints. The assumption of single-constant K-M theory, in which all

of the scattering is attributed to the white, is inapplicable for the masstone, a sample that

contains no white. Therefore, the (k/s)! of the masstone was equal to (K/S)! using

equation (1).

Recall that the linearity between actual and effective concentration and the

spectral scalability of (k/s)! were the two assumptions in the application of single-

constant K-M theory. Figure 5 shows the spectral scalability of each colorant. The

maximum absorption occurred at 600, 450, 450, 630 nm for Cobalt Blue, Cadmium

Yellow Medium, Green Gold, and Phthalocyanine Green, respectively. Samples with

high colorant concentration, especially the masstone, which are shown using bold lines,

have different spectral shapes; thus the scalability requirement has not been met for these

samples. Since the masstone, by definition, is free of white, its behavior differs from the

tint samples. Also the idea of having negligible scattering at high concentration may be

false. These are possible reasons to explain the masstone samples’ optical behavior.

Therefore, the masstone might not be a good sample to characterize a colorant. This issue

can be proven if the masstone has poor performance in predicting the other mixtures of

the corresponding ramp, to be shown below. The scalability of the other mixtures for the

four colorants is poor at short wavelengths. This might be due to measurement

imprecision or the strong absorption of white at these wavelengths obscuring each

chromatic colorant’s optical behavior.

Determining a paint’s (k/s)! requires, at minimum, a single tint, shown in equation

(11) by re-arranging equation (10):

16

k

s

!"#

$%&'

=

K

S

!"#

$%&' , tint

(k

s

!"#

$%&' ,w

ctint

cw

(11)

For the ideal case shown in Figure 1 where perfect spectral scalability was achieved, any

sample would be appropriate. For the four paints, the sample choice would affect

performance since none of the paints had spectral scalability, seen in Figure 5. This was

tested by calculating a (k/s)! for each tint via equation (11). The effective concentration

was determined for the other samples of the tint ladder by scaling the (k/s)! such that the

root-mean square spectral reflectance factor difference was minimized. That is, all the

wavelengths were used to determine effective concentration rather than a single

wavelength at maximum absorption as employed in Beer’s law and shown in Figure 3.

The advantage of this approach is that both spectral scalability and the degree of linearity

between actual and effective concentration can be evaluated simultaneously. Nonlinear

optimization was used [26]. This was performed twice, either including or omitting the

Saunderson correction for refractive index discontinuity.

17

Figure 5 The normalized (k/s)! of the tint ladders of (a)- Cobalt Blue, (b)- Cadmium

Yellow Medium, (c)- Green Gold, and (d)- Phthalocyanine Green at the maximum

absorption wavelength corresponding to each colorant. Masstone spectra are shown in

bold.

The relationship between actual and effective concentration determines whether

the tint ladder was prepared properly and also indicates whether the number of samples

can be reduced for database development. For each paint and for either including or

omitting the Saunderson correction, the best-achieved performance is plotted in Figure 6.

Without the Saunderson correction, the best cobalt tint was CB/54/48 (i.e., L*=54 and

C*ab=48) while CB/65/40 was the best cobalt tint including the Saunderson correction.

The best performing specific sample depends on the spectral model. Goodness was

quantified by two metrics, r2 and slope, both based on fitting a straight line through the

18

data points. The r2 metric is an overall indicator of regression fit and varies between 0 and

1 for poor and well-modeled data, respectively. A higher r2 indicates better spectral

scalability and that the number of tints can be reduced for colorant selection. A slope near

unity indicates that the number of tints can be reduced for recipe prediction. None of the

tints achieved both criteria. The cobalt tint, CB/65/40, was quite close. The statistics for

all the tints and for each of the paints are listed in Table 1. Because each masstone was a

poor predictor of the tints, their performance is excluded from this table. The average

results are the most important statistic as this value is the most concentration

independent. Two trends are evident. First, if both colorant selection and recipe

prediction are required, tints are required, the specific number would be paint specific.

However, if only colorant selection is required, the number of tints can be reduced,

possibly to a single tint. Second, including the Saunderson correction improved

performance, dramatically, particularly, in improving linearity between actual and

effective concentrations. This trend was found for all analyses. Therefore, the remainder

of this the publication will only describe the results including the Saunderson correction.

19

Figure 6 Effective versus actual concentration of Cobalt Blue, Cadmium Yellow Medium,

Green Gold, and Phthalocyanine Green using single-constant K-M theory and both

omitting (top) and including (bottom) the Saunderson correction. The (k/s)! for each

colorant was derived from the tint resulting in the largest r2 values, listed at top of each

individual Figures.

20

Table 1 r2

and slope values of fitting model to predict all the mixtures in the tint ladder

except the masstone using single-constant K-M theory both including and omitting the

Saunderson correction.

Cobalt Blue

Omitting surface correction Including surface correction

r2 Slope r

2 Slope

Average 0.9920 0.79 0.9989 0.92

Minimum 0.9844 0.68 0.9976 0.90

Maximum 0.9972 0.84 0.9995 0.92

Std. Dev. 0.0054 0.06 0.0006 0.01

Cadmium Yellow Medium

Average 0.9618 0.64 0.9721 0.79

Minimum 0.8889 0.60 0.9369 0.76

Maximum 0.9976 0.67 0.9842 0.83

Std. Dev. 0.0379 0.03 0.0166 0.03

Green Gold

Average 0.9934 0.79 0.9981 0.89

Minimum 0.9844 0.70 0.9955 0.84

Maximum 0.9985 0.86 0.9995 0.93

Std. Dev. 0.0065 0.14 0.0018 0.16

Phthalocyanine Green

Average 0.9836 0.78 0.9959 0.91

Minimum 0.9558 0.67 0.9876 0.85

Maximum 0.9966 0.83 0.9994 0.94

Std. Dev. 0.0169 0.06 0.0048 0.04

The performance of predicting the spectral reflectance factor for CB/71/31,

CY/88/77, GG/76/63, and PG/62/47 as a representative example of each paints is shown

in Figure 7. The effective and the actual concentrations were used in the prediction

process separately. The tint resulting in the lowest r2 was used as (k/s)". The curve shape

of the spectral reflectance factor of each colorant was predicted accordingly. The spectral

RMS% error for Cobalt Blue, Cadmium Yellow Medium, Green Gold, and

Phthalocyanine Green was 3.26, 4.47, 3.27, and 2.89, respectively. As an average for the

four colorants, the slope between the effective and actual concentration was 0.88 and it

still deviated from unity, as shown in Table 1. Therefore, single-constant K-M theory is

only recommended for colorant selection but not for recipe prediction. Since the model

21

could create the curve shape of the spectral reflectance factor of the four paints, having

just one tint of the colorant is sufficient for colorant selection purposes.

Figure 7 The spectral reflectance factor of CB/71/31, CY/88/77, GG/76/63, and

PG/62/47. The solid line represents the measured, the dashed line is for the predicted

with the actual concentration, and the dashed-dotted line predicted with the effective

concentration using single-constant K-M theory. The unit (k/s)! for each colorant derived

from mixtures obtained the smallest r2 values, respectively. The %RMS values

correspond to predictions using the actual concentrations.

The masstone performance was poor. In this case, the (k/s)" would be equal to

(K/S)". The r2 values for Cobalt Blue, Cadmium Yellow Medium, Green Gold, and

Phthalocyanine Green were 0.6294, 0.9677, 0.7108, 0.7382, respectively. The slopes of

the line between the effective and the actual concentration were in the range of 0.43-0.53.

22

This observation was expected because of the inapplicability of single-constant K-M

theory for the masstone. The reason of the discrepant behavior of Cadmium Yellow

Medium compared to the others might be related to be a very light colorant, which was

hard to differentiate with white at wavelengths longer than 550 nm and the scattering

would be attributed to this colorant. The performance of predicting the spectral

reflectance factor for representative tints using (K/S)" of the masstone as unit (k/s)" is

shown in Figure 8. The performance of predicting the mixtures using the (k/s)" of the

masstone is lower than any other (k/s)" derived from the tint samples. For Cobalt Blue

and Cadmium Yellow Medium, the curve shapes based on the masstone reasonably

approximate the tint spectra, and thus, these masstones could be used for colorant

selection. For Green Gold and Phthalocyanine Green, the spectra are quite different. It

appears as that the masstone is not appropriate for colorant selection in the general case.

23

Figure 8 The spectral reflectance factor of CB/71/31, CY/88/77, GG/76/63, and

PG/62/47. The solid line represents the measured, the dashed line is for the predicted

with the actual concentration, and the dashed-dotted line predicted with the effective

concentration using single-constant K-M theory. The unit (k/s)! for each colorant was the

(K/S)! of the masstone.

Spectral scalability was analyzed directly by calculating the spectral root mean

squared error, RMS%, comparing measured and predicted spectra using the effective

concentrations, based on the optimizations described above. Shown in Table 2, each

column lists spectral RMS% error using the (k/s)" of each of the tints in order to predict

the other samples composing the tint ladder. Based on the average results, the

performance of each (k/s)" seems very close to each other except in the case of

masstone’s (k/s)". The masstone in each case had poor performance in predicting tints.

This was expected because the masstone does not contain any white component and the

24

assumption of the single-constant K-M theory has not met in this case. There was a trend

in the maximum errors where the lowest concentration tints (e.g., CB/76/28) had the

poorest predictions for the highest concentration tints (e.g., masstone) and vice versa.

Table 2 Spectral RMS% error between the measured and the predicted reflectance factor

using (k/s)! of each of the tints to predict the others in single-constant K-M theory with

Saunderson correction.

Cobalt Blue

All

Tints CB/76/28 CB/72/32 CB/71/31 CB/65/40 CB/58/45 CB/54/48 CB/47/51 CB/33/50(Masstone)

Average 1.17 1.59 1.59 1.30 1.09 1.00 1.08 1.26 2.25

Maximum 2.28 4.58 4.30 3.91 3.49 2.85 2.48 2.17 2.66

Std. Dev. 0.74 1.54 1.44 1.29 1.07 0.84 0.79 0.73 0.41

Cadmium Yellow Medium

All

Tints CY/94/41 CY/92/55 CY/89/71 CY/88/77 CY/87/82 CY/86/86 CY/84/90 CY/82/94(Masstone)

Average 0.90 2.01 0.90 0.74 0.61 0.64 0.80 1.04 1.92

Maximum 1.61 4.19 1.76 1.55 1.22 1.03 1.39 1.86 3.14

Std. Dev. 0.54 1.44 0.63 0.50 0.38 0.37 0.47 0.66 1.01

Green Gold

All

Tints GG/86/55 GG/81/62 GG/76/63 GG/68/61 GG/50/42(Masstone)

Average 0.63 0.59 0.52 0.53 0.76 4.10

Maximum 1.80 1.75 1.82 1.85 1.79 5.86

Std. Dev. 0.67 0.68 0.74 0.75 0.66 2.41

Phthalocyanine Green

All

Tints PG/75/38 PG/70/43 PG/62/47 PG/55/47 PG/46/42 PG/26/8(Masstone)

Average 1.50 0.51 0.47 0.50 0.70 1.73 9.29

Maximum 2.58 1.34 1.35 1.36 1.36 3.06 14.55

Std. Dev. 0.92 0.47 0.50 0.49 0.46 1.14 5.49

It was also of interest to evaluate whether using all the tints to derive a single

(k/s)" would improve performance. A non-negative least square technique was used to

derive a single (k/s)" that minimized average spectral reflectance factor error for all of the

tints. The spectral performance of this (k/s)" is shown in Table 2 under the column titled

”all tints.” The average performance was almost in the range of the other cases, but the

maximum spectral RMS% error using this (k/s)" was less than the others. The maximum

error always occurred when predicting the masstone.

25

With the exception of the Green Gold and Phthalocyanine Green masstones, the

average results were very similar. A multiple comparison [27] statistical evaluation of the

means was performed to determine statistical significance [28,29]. At a 95% confidence

interval, all the results were not significantly different except the Green Gold and

Phthalocyanine Green masstones. These results support the trends described above and

indicate that a single tint of any arbitrary concentration can be used for colorant selection.

Certainly, inpainting is required for losses where the colorants are applied as

masstones, that is, without tinting with white. The performance of predicting the

masstone using different tints were not significantly difference at #=0.05 for each

colorant except for the cobalt blue in which CB/54/48 and CB/47/51 had significantly

better performance than the others. Accordingly, the masstone for each of the colorants

was predicted using CB/54/48, CY/84/90, GG/68/61, PG/46/42 to derive the (k/s)", the

results plotted in Figure 9. The performance shows that the curve shape of the masstone

can be predicted using a tint at any ratio. Again single-constant K-M theory would be a

proper model for colorant selection.

26

Figure 9 The spectral reflectance factor of the masstone of the Cobalt Blue, Cadmium

Yellow Medium, Green Gold, and Phthalocyanine Green. The solid line represents the

measured, the dashed line is for the predicted with the actual concentration, and the

dashed-dotted line predicted with the effective concentration using single-constant K-M

theory.

Relative Two-Constant K-M Solution

The results of single-constant K-M theory demonstrate that this model is only suitable for

colorant selection. The alternative method for colorant recipe prediction would be two-

constant K-M theory, in which the scattering and absorption coefficients are considered

separately. In this research, the relative two-constant K-M theory was used where the

scattering of each paint was defined relatively to the scattering of titanium white, defined

as unity across wavelength, equation (7). The (k)" and (s)" of each paint were derived

using equation (6) At least two samples from the tint ladder for each paint are needed to

determine (k)" and (s)". All possible combinations were tested to determine these

27

quantities. There were 28, 28, 10, and 15 combinations for Cobalt Blue, Cadmium

Yellow Medium, Green Gold, and Phthalocyanine Green, respectively. The measured

reflectance factor of the tint ladders was predicted using each set of derived (k)" and (s)".

The same optimization procedure was used as for the single-constant solution, the results

shown in Table 3. A low slope or poor r2 occurred when two light tints were used to

derive (k)" and (s)". The average of the slopes 0.99 shows the relative two-constant K-M

theory had better performance than single-constant K-M theory. The relationships

between actual and effective concentrations for the pair of samples resulting in the largest

r2 are shown in Figure 10. There is a marked improvement compared with single-constant

K-M theory, shown in Figure 6. These results demonstrate that relative two-constant K-M

theory can be used for both colorant selection and recipe prediction. It is also important to

note that for each paints, one of the samples was the masstone.

Table 3 r2

and slope values of fitting model to predict all the mixtures in the tint ladder

using all combination of mixtures to derive (k)! and (s)! using relative two-constant K-M

theory.

Cobalt Blue

r2 Slope

Average 0.9972 0.99

Minimum 0.9811 0.92

Maximum 0.9995 1.07

Std. Dev. 0.0047 0.04

Cadmium Yellow Medium

Average 0.9971 0.97

Minimum 0.9848 0.88

Maximum 0.9997 1.02

Std. Dev. 0.0037 0.04

Green Gold

Average 0.9942 1.00

Minimum 0.9751 0.95

Maximum 0.9995 1.07

Std. Dev. 0.0077 0.04

Phthalocyanine Green

Average 0.9842 1.01

Minimum 0.8500 0.97

Maximum 0.9999 1.12

Std. Dev. 0.0386 0.05

28

Figure 10 Effective versus actual concentration of Cobalt Blue, Cadmium Yellow

Medium, Green Gold, and Phthalocyanine Green using relative two-constant K-M theory

including the Saunderson correction. The (k)! and (s)! for each paint were derived from

the pair of samples from the tint ladder that obtained the largest r2 values.

The effective concentration was employed to predict the spectral reflectance

factor of each tint using the different sets of (k)" and (s)". The spectral RMS% error

between the predicted and the measured reflectance factor are summarized in Table 4.

The evaluation was performed on the mean of spectral %RMS error of each set to predict

all the tints. Since the prediction of the masstone was not as good as the other samples,

particularly when light tints were used to derive to (k)" and (s)", the masstones were

excluded from statistical analysis shown in Table 4. Multiple comparison analysis of the

29

means at 95% confidence intervals was performed for all the sets of (k)" and (s)". For

Cobalt Blue and Green Gold, all the combinations were not statistically different. For

Cadmium Yellow Medium, two of the sets were statistically different,

(CY/94/41,CY/92/55) and (CY/94/41,CY/82/94). Nonetheless, the mean spectral %RMS

error 1.15 and 1.46 for these two sets are still acceptable for both colorant selection and

colorant recipe prediction. For Phthalocyanine Green, one set was statistically different,

(PG/75/38, PG/70/43), with the mean of spectral %RMS error of 3.14.

Table 4 Statistical result on the mean of spectral RMS% error using all possible

combinations to predict the tint ladder excluding the masstone using relative two-

constant K-M theory.

Cobalt Blue Cadmium Yellow Medium Green Gold Phthalocyanine Green

Average 0.73 0.38 0.28 0.58

Minimum 0.34 0.18 0.11 0.20

Maximum 1.97 1.46 0.53 3.14

Std. Dev. 0.43 0.29 0.13 0.73

The performance of each pair of samples, including the masstone, to predict the

masstone was evaluated, the results listed Table 5. The poorest performance (maximum

spectral %RMS) occurred when using light tints. The best performance (minimum

spectral %RMS) occurred when one of the samples was the masstone. Therefore, the

performance of all the combinations that included the masstone was separated, the results

listed in Table 6. A T-test revealed that any tint produced results not statistically different

except in two cases: set (CB/47/51,CB/33/50) that was significantly better than the other

tints and (CY/94/41, CY/82/94) that was significantly worse than the others.

30

Table 5 Spectral RMS% error to predict the masstone using all possible combination of

mixtures to derive unit (k)! and unit (s)! using relative two-constant K-M theory.

Cobalt Blue

Samples # RMS%

Average 9.29

Minimum CB/47/51,CB/33/50 0.00

Maximum CB/72/32,CB/65/40 23.12

Std. Dev. 6.29

Cadmium Yellow Medium

Average 1.44

Minimum CY/86/86,CY/82/94 0.00

Maximum CY/94/41,CY/92/55 4.57

Std. Dev. 1.28

Green Gold

Average 6.33

Minimum GG/81/62,GG/50/42 0.00

Maximum GG/86/55,GG/81/62 18.22

Std. Dev. 7.17

Phthalocyanine Green

Average 4.96

Minimum PG/46/42,PG/26/8 0.00

Maximum PG/75/38,PG/70/43 20.15

Std. Dev. 5.88

Table 6 Spectral RMS% error to predict the masstone using all possible combination of

mixtures where one of them was masstone to derive unit (k)! and unit (s)!.

Cobalt Blue Cadmium Yellow Medium Green Gold Phthalocyanine Green

Average 2.52 0.51 0.00 0.00

Minimum 0.00 0.00 0.00 0.00

Maximum 3.22 3.55 0.00 0.00

Std. Dev. 1.14 1.34 0.00 0.00

Having determined that one of the samples should be the masstone, the (k)" and

(s)", derived from the masstone and one tint, were used to predict each of tint ladders.

Again r2and slope of the line between the effective and the actual concentrations were

considered. The range of r2

and slopes for the four paints were 0.9911-0.9999 and 0.96-

1.02, respectively. These values are very close to the desired values of unity and

compared with the ranges listed in Table 3, the masstone and a single tint produced

excellent results. The spectral %RMS errors statistics are listed Table 7. In this case, the

31

masstone is included in the analysis whereas in previous analyses (Tables 2 and 4), it was

excluded because of poor performance. Multiple comparison of the means at 95%

confidence intervals showed that the mixtures including the masstone for Cobalt Blue,

Green Gold, and Phthalocyanine Green had statistically the same performance in

predicting tints and the masstone. For Cadmium Yellow Medium, one set had statistically

poorer performance, (CY/94/41, CY/82/94) with average spectral RMS% of 1.72.

Evaluating trends in the tints revealed that the best performance occurred when the tint

had high chroma. The predicted and the measured spectral reflectance factor of each

paint’s masstone and representative tint using the sets that obtained the largest slope to

derive unit (k)" and unit (s)" are shown in Figure 11. The spectral matches are excellent

and nearly coincident.

Table 7 Statistical result on the mean of spectral RMS% error using all possible

combinations where one of them was masstone to derive (k)! and (s)! to predict the tint

ladder including the masstone sample using relative two-constant K-M theory.

Cobalt Blue Cadmium Yellow Medium Green Gold Phthalocyanine Green

Average 1.08 0.46 0.27 0.38

Minimum 0.96 0.18 0.21 0.28

Maximum 1.32 1.72 0.33 0.47

Std. Dev. 0.14 0.56 0.07 0.09

32

Figure 11 The spectral reflectance factor of a tint (left plots) and the masstone (right

plot) of the Cobalt Blue, Cadmium Yellow Medium, Green Gold, and Phthalocyanine

Green using relative two-constant K-M theory. The solid and the dashed lines represent

the measured and the predicted spectral reflectance factor. The (k)! and (s)! for each

paint were derived from the pair of samples from the tint ladder that obtained the largest

slope values

Conclusions

This research evaluated the validity of the single- and relative two- constant forms of

Kubelka-Munk (K-M) theory for colorant selection and colorant recipe prediction for

inpainting. Accounting for the refractive index discontinuity between the medium and air,

known as the “Saunderson correction,” was also evaluated. Four acrylic emulsion paints

including cobalt blue, cadmium yellow medium, Green Gold (a three-pigment mixture),

33

and phthalocyanine green were tested. Three metrics were used to evaluate performance:

r2 and the slope of the line between the effective and the actual concentrations and

spectral RMS% error based on spectral reflectance matching. r2

and spectral RMS% error

indicate whether an approach can be used for colorant selection. All three metrics

indicate whether an approach can be used for colorant selection and recipe prediction.

For both forms of K-M theory, the Saunderson correction improved performance

and its use is recommended. For colorant selection, either the single-constant or relative

two-constant form of K-M theory was appropriate for colorant selection. This supports

previous research in colorant selection for inpainting where the single-constant form of

K-M theory was used, though with a different geometry and not including the Saunderson

correction [4]. For recipe prediction, only the relative two-constant form of K-M theory

was appropriate. This agrees with industrial practice where both colorant selection and

recipe prediction are required [1].

The main question posed in this research concerned the minimum number of

samples required for either task. For colorant selection, only a single tint is required. A

tint is preferred over a masstone. For recipe prediction two samples are required, the

masstone and a single tint, preferably at high chroma. This is a very significant result. A

colorant database for inpainting can be prepared with just a few samples. A step-by-step

procedure is described in the appendix.

This research also provides guidance for creating a spectral database of historical

colorants and paints. Typically, only the masstone is formed into a paint film. One should

also form a tint. Ideally both films should be opaque.

34

Appendix I: Procedures for Developing Colorant

Database The following is a list of procedures to characterize a colorant using single- and

two-constant K-M theory. This procedure is given for retouching paints. In the future, the

authors hope to implement these procedures in Microsoft Excel that will be available

from the second listed author.

Single-Constant K-M Theory

1. Create a tint of a desired retouching paint at any ratio with titanium white and

apply uniformly on a support. The film of the paint mixture should be opaque.

(See the experimental procedure described above for an example.)

2. Record the concentration of the paint as the ratio of the weight to the total weight

of the mixture and similarly for titanium white. Label them as c and cw,

respectively. For example, if the chromatic paint was 4 g. and the white paint was

12 g., c = 4/(4+12) = 0.25 and cw = 12/(4+12) = 0.75.

3. Create an opaque film of titanium white.

4. Measure the spectral reflectance factor of each sample with any available

spectrophotometer three times with replacement. Make note of the instrument

geometry. If it is an integrating sphere spectrophotometer, measure with the

specular component included (total hemispherical). Plot the spectral reflectance as

a function of wavelength. The three curves should be nearly identical in shape.

Calculate the average of three consistent curves and label it as measured

reflectance factor, R!,m.

5. Calculate internal reflectance using Saunderson correction and label it as R!,i.

35

Assign K1 the value of 0.04 and K2 the value of 0.6, or alternatively, explicitly

calculate K1 and K2 based on the paint’s refractive index. The mathematics are

given in Reference 20. This equation applies to integrating sphere geometry,

specular component included.

R! ,i =R! ,m " K1

1" K1" K

2+ K

2R! ,m

(A1)

For bidirectional geometry, or using a integrating sphere spectrophotometer with

the specular component excluded (diffuse hemispherical), the following equation

applies:

R! ,i =R! ,m

1" K1( ) 1" K2( ) + K2

R! ,m

(A2)

6. Convert internal reflectance factor to (K/S)! :

!

K

S

"

# $

%

& ' (

=1) R( ,i( )

2

2R( ,i

(A3)

7. Calculate (k/s)":

k

s

!"#

$%&'

=

K

S

!"#

$%&' , tint

(k

s

!"#

$%&' ,w

ctint

cw

(A4)

8. The derived (k/s)" can be used for colorant selection.

Two-Constant K-M Theory

1. Create a tint of a desired retouching paint at any ratio with titanium white and

apply uniformly on a support. The film of paint mixture should be opaque. (See

36

the experimental procedure described above for an example.)

9. Apply the retouching paint uniformly on a support. This is the masstone. The film

should be opaque. (See the experimental procedure described above for an

example.)

2. Follow steps 2-6 of the single-constant procedure.

3. Create the (k)" and (s)" database for all wavelengths using non-negative least

square technique as

!

T =CX (A5)

where,

!

T =

cw1

K

S

"

# $

%

& ' 1

( kw

)

* +

,

- .

.

.

.

cwn

K

S

"

# $

%

& ' n

( kw

)

* +

,

- .

)

*

+ + + + + + + + +

,

-

.

.

.

.

.

.

.

.

. /

; C =

c1

(c1

K

S

"

# $

%

& ' 1

.

.

.

cn

(cn

K

S

"

# $

%

& ' n

)

*

+ + + + + + + +

,

-

.

.

.

.

.

.

.

. /

X =k

s

)

* + ,

- . /

Repeat the calculation for each wavelength. The quantities (K/S)1 … (K/S)n should be

calculated from the internal reflectance factor of each sample, equation (A1) or (A2).

For having just one tint and a masstone the above equations would be simplified as

!

T =cw( tint)

K

S

"

# $

%

& ' tint

( kw

)

* +

,

- .

0

)

*

+ + +

,

-

.

.

. /

; C =

ctint

(ctint

K

S

"

# $

%

& ' tint

1 (K

S

"

# $

%

& ' masstone

)

*

+ + + +

,

-

.

.

.

. /

X =k

s

)

* + ,

- . /

(A6)

where,

37

!

sw

= 1,

K

S

"

# $

%

& ' w

=k

s

"

# $ %

& ' w

= kw. (A7)

4. The derived (k)" and (s)" would be quantities to be employed for colorant selection

and colorant recipe prediction.

Suppliers

The instruments and materials employed in this research were

• Spectrophotometer: GretagMacbeth Color Eye XTH manufactured by

GretagMacbeth AG, www.gretagmacbeth.com.

• Film applicator (drawdown bar): BYK-Gardner drawdown bar with 10 mils

thickness (one mil is equal to 1/1000 of an inch, or 25.4 microns (µm)),

www.byk-gardner.de.

• Paints: Matte fluid acrylic paints produced by Golden Artist Colors, Inc.,

www.goldenpaints.com, and high viscosity acrylic artist colors by Liquitex Artist

Acrylic, www.liquitex.com.

• Paper: Leneta opacity charts, Form 3B (7-5/8 $ 11-3/8 inch), produced by The

Leneta Company, www.leneta.com.

Acknowledgments The authors would like to acknowledge their sponsors: the Andrew W. Mellon

Foundation, the National Gallery of Art, Washington, DC, the Museum of Modern Art,

New York, the Institute of Museum and Library Services, Washington, DC, and

Rochester Institute of Technology. We also thank Yoshio Okumura for the sample

preparation of two of the paints.

38

References 1. Berns, R. S., Billmeyer and Saltzman’s Principle of Color Technology, 3rd edn,

John Wiley & Sons, New York (2000).

2. Staniforth, S., ‘Retouching and colour matching: the restorer and metamersim’,

Studies in Conservation 30 (1985) 101-111.

3. Kubelka, P., and Munk, F., Ein beitrag zur optik der farbanstriche, Zeitschrift für

technische Physik 12 (1931) 593-601.

4. Berns, R. S., Krueger, J. and Swicklik, M., Multiple pigment selection for

inpainting using visible reflectance spectrophotometry, Studies in Conservation

47 (2002) 46-61.

5. Davidson, H. R., and Hemmendinger, H., Color prediction using the two-constant

turbid-media theory, Journal Of The Optical Society Of America, 56 (8) (1966)

1102-1109.

6. Billmeyer, JR. F. W., and Abrams, R. L., Predicting reflectance and color of paint

films by Kubelka-Munk analysis I. Turbid-medium Theory, Journal of Paint

Technology 45 (579) (1973) 23-30.

7. Cairns, E. L., Holtzen, D. A., Spooner, D. L., Determining absorption and

scattering constants for pigments, Color Research and Application 1 (4) (1976)

174- 180.

8. Allen, E., ‘Colorant formulation and shading’, in Optical Radiation Color

Measurements, F. Grum and C.J. Bartleson, Academic Press, New York, (1980)

289-336.

9. Johnston ,R. M., and Feller, R. L. , The use of differential spectral curve analysis

in the study of museum objects, Dyestuffs 44 (9) (1963) 1-10.

39

10. Walowit, E., McCarthy C. J., and Berns R. S., ‘An algorithm for the optimization

of Kubelka-Munk absorption and scattering coefficients’, Color Research and

Application 12 (6) (1987) 340-343.

11. Walowit, E., McCarthy, C. J., and Berns, R. S., ‘Spectrophotometric color

matching based on two-constant Kubelka-Munk theory’, Color Research and

Application 13 (6) (1988) 358-362.

12. ASTM E 284, Standard Terminology of Appearance, Seventh edition, 7th

edn,

ASTM International, West Conshohocken, PA, (2004).

13. Kortum, G., Reflectance Spectroscopy, Principle, Methods, Application, Sptinger-

Verlag (1969).

14. Kubelka, P., ‘New contribution to the optics of intensely light-scattering

materials: Part I’, Journal Of The Optical Society Of America 38 (5) (1948) 448-

457.

15. Nobbs, J. H., ‘Kubelka-Munk Theory and the prediction of reflectance’, Review.

Progress in Coloration 15 (1985) 66-75.

16. Judd, D. B., and Wyszecki, G., Color in Business, Science and Industry, 3rd

edn,

John Wiley & Sons (1975).

17. Duncan, D. R., ‘The color of pigment mixtures’, Journal of Oil Color Chemical

Association 32 (1949) 296-321.

18. Saunderson, J. L., ‘Calculation of the color of pigmented plastics’, Journal Of The

Optical Society Of America 32 (1942) 727-736.

19. Ryde, J. W., ‘The scattering of light by turbid media- part I’, Proceedings Royal

Society, A131 (1931) 451-464.

40

20. Berns R. S., and de la Rie, E. R., ‘The effect of the refractive index of a varnish

on the appearance of oil paintings’, Studies in Conservation 48 (2003) 251-262.

21. Berns, R. S., and de la Rie, E. R., ‘Exploring the optical properties of picture

varnishes using imaging techniques, Studies in Conservation 48 (2003) 73-82.

22. Berns, R. S., ‘A generic approach to color modeling’, Color Research and

Application 22 (5) (1997) 318-325.

23. Derby, R.E., ‘Applied spectrophotometry. I. Color matching with the aid of the “

‘R’ cam’, American Dyestuff Reporter 41 (1952) 550-557.

24. Okumura, Y., Developing a Spectral and Colorimetric Database of Artist Paint

Materials, M.S Thesis, Rochester Institute of Technology (2005).

25. Mohammadi, M., and Berns, R. S., ‘Verification of the Kubelka-Munk turbid

media theory for artist acrylic paint’, MCSL Technical Report, www.art-si.org

(accessed 14 July 2006).

26. www.mathworks.com/access/helpdesk/help/techdoc/ref/fminsearch.html

(accessed 14 July 2006).

27. Bowerman, B., and O’Connell, R. T., Applied Statistics, Times Mirror Higher

Education Group (1997).

28. http://radio.feld.cvut.cz/Docs4Soft/matlab/toolbox/stats/multcompare.html

(accessed 14 July 2006).

29. http://www.mathworks.com/access/helpdesk/help/toolbox/stats/anova1.html

(accessed 14 July 2006).