INTRODUCTION
One of the main efforts of the industries dealing with colorants is to find
the correct proportion of the colorants required to achieve an exact color match.
This process, called color match prediction consists in generating, usually
by means of trial and error techniques, a recipe to match a desired or target
shade and is often performed by a trained colorist. Computer Color Matching
(CCM) is a widely known technology for the color match prediction (Zhang
and Li, 2008). This method overcomes the lacks of the experimental color
matching approach thus resulting more convenient, accurate and time saving (Agahian,
2008). Spectrophotometric and colorimetric color matching are two methods
that are conventionally used. Colorimetric algorithms aim to minimize the color
differences (usually expressed in tristimulus values) between a target and a
color sample. Spectrophotometric color matching is a technique able to achieve
a sample with spectrophotometric curve similar to target reflectance curve.
Referring to spectrophotometric approaches, the most commonly adopted are based
on the KubelkaMunk theory (Kubelka, 1954) that is widely
used for describing the colour properties of a fabric.
The KubelkaMunk (KM) theory is generally used for the analysis of diffuse reflectance spectra obtained from weakly absorbing samples. It provides a correlation between reflectance and concentration.
As widely known (Burlone, 1990), KM establishes that
internal reflectance of a colorant composing a shade, p(λ, α_{1}),
depends on absorption, K_{λ} and scattering, S_{λ}
according to the following equation:
where, λ is the wavelength in the visible range (300700 nm). p(λ) is the spectral response (reflectance) of a generic colorant composing the reference shade. (K/S)_{λ} is the ratio between the absorption and the scattering coefficients for a given wavelength.
Equation 1 is valid for a single wavelength (or monochromatic
light). The values of K_{λ} and S_{λ} need to be computed
from measurements of the reflectance of the mixture composing the shade. More
in detail the (K/S) ratio of a mixture is an additive combination of each colorant’s
unit absorptivity, K_{λ} and unit scattering S_{λ},
scaled by effective concentration, c, plus the absorption and scattering of
the substrate (notated by subscript t) as described in the following Equation:
where, (K/S)_{λ,mix} is the (K/S) ratio of a mixture, α_{i} is the proportion of the ith component composing the mixture. K_{λ, i} and S_{λ, i} are, respectively, the absorptivity and the scattering coefficient of the ith component composing the mixture.
For each component in the mixture, both the absorption and scattering properties
need to be known. For opaque materials, where the colorants do not scatter in
comparison to the substrate, the mixing equation may be simplified as follows
(Westland et al., 2000):
Stearns and Noechel worked with fine wool fibers blended in the form of slubbing
with the help of a draw frame. In their approach, called SN approach, they
assumed that the reflectance value of the blend lies between the reflectance
values of the constitutive fibers and is different from the mean of the primary
reflectances weighted by the relative mass percentages. Several studies, related
to the tristimulusmatching algorithm based on the approach firstly proposed
by Stearns and Noechel (1944) and its implementations
(Thompson and Hammersley, 1978; Kazmi
et al., 1996), allows a reliable prediction of the formula for matching
a given colour standard. The SN based approach estimates the spectrum of a
blend (i.e., the term f(R(λ)) in Eq. 4) obtained by mixing
differently colored once known the empiric constant b, according to the following
relationship:
where, R(λ) is the reflectance and b is a dimensionless constant that
can be determined by means of several experimental tests (Banyard
et al., 2006; Rong and Feng, 2006). F(R(λ))
represents the mixture function (Westland et al.,
2002).
Artificial Neural Networks (ANNs) are able to provide alternative mappings
between colorant concentrations and spectral reflectances (Furferi
and Carfagni, 2010; Bishop et al., 1991)
and, more generally, determine nonlinear transforms between colour spaces.
The FFBP ANNs are known to be suitable for applications in the pattern classification
field, especially where the limits of classification are not exactly defined
(Furferi and Governi, 2008). A properly trained FFBP
ANN is capable of generalizing the shape of a spectrophotometric response on
the basis of the information acquired during the training phase. In order to
properly teach the network to respect the classification made by the picker,
a proper target set is required.
In all the previously mentioned cases, computer recipe prediction systems require a mathematical model able to obtain an accurate color matching. In other words such a model, called color recipe mapping, is required to process colorant concentrations and spectral responses, in input, so as to provide, as output, the spectral response of a desired shade obtained by mixing the colorants.
Starting from well known relationships acknowledged at the state of the art, the main objective of the work is to provide three different formulations for the assessment of the color recipe mapping. These formulations, based respectively on KM, Stearns Noechel and ANN techniques, may be adopted by researchers and practitioners in order to assess the exact color matching once they know the concentrations and the spectral response of the colorants composing a particular recipe.
MATERIALS AND METHODS
Definitions: Let p_{i}(λ) be the spectral response in the visible wavelength of the ith of the n colorants composing the desired shade (called reference): I = 1,2,…,n. The colour spectrum R(λ, α_{i}) obtained by a linear combination of the spectra of each component, called Weighted Average Spectrum (WAS), can be stated as follows:
As λ indicates the wavelength, in the range (400700 nm), the size of vectors p_{i}(λ) and R(λ, α_{i}) is 1 x 31.
Of course, the following equation must be satisfied:
The WAS may be related to the real spectral response of the reference R_{S}(λ) measured by means of a spectrophotometer.
In other words, the color recipe mapping id defined by a transfer function F that state the functional connection between R(λ, α_{i}) and R_{S}(λ):
Once known the transfer function Φ(λ, α_{i}), it is possible to give a reliable estimation of the reference spectral response starting from the spectral response and the proportion of each colorant i.e., it is possible to assess the exact color matching.
As previously mentioned, the present work aims to state three different formulations for mathematically defining the mapping function F:
• 
StearnsNoechel based formulation 
These formulations are based on the assumption that a function Φ(λ, α_{i}) exists such that:
In other words if R_{Sj} (λ) is the jth element of R_{S} (λ), R_{j} (λ, α_{i}) the jth element of R (λ, α_{i}) and φj (λ, α_{i}) is the jth element of Φ(λ, α_{i}), with j = 1, 2, …, 31, Eq. 8 can be written, element by element, as follows:
This transfer function can be considered applicable for any variation of the parameters α_{i} as long as the Eq. 2 is respected:
KM based formulation: A first mathematical definition of the transfer function φ_{j}(λ, α_{i}) may be afforded by means of the KM formulation. Combining Eq. 1 with Eq. 9 it is possible to write:
Thus obtaining:
Finally, solving for φ_{j} (λ, λ_{i}):
This equation reassumes the studies conducted by Allen
(1966) and cited by McDonald (1997) and Berns
(2000).
Stearnsnoechel based formulation: According to Allen
(1966), if the degree of metamerism between the target and the prediction
is not too great it can be written the following equation (PhilipsInvernizzi
et al., 2002):
By combining Eq. 13 and 9 it can be demonstrated
that:
According to Eq. 4 the term in
Eq. 13 can be rewritten as follows:
Finally, combining Eq. 15 with Eq. 14 it is possible to state that:
ANN based formulation: Let suppose that:
• 
Three layers: input, hidden and output layer 
• 
Hidden layer made of logistic neurons followed by an output
layer of logistic neurons again 
• 
31 input, h hidden and 31 output units 
• 
The ANN has been trained using, as input, the spectral responses
of the colorants composing the shade and, as output, the spectral factors
of the reference 
• 
The training was performed using a backpropagation algorithm
(Allen, 1980) until the MSE reach a value equal to,
at least, 0.01 
Accordingly it is possible to define the following terms (Ghwanmeh
et al., 2006):
• 
W_{1} (size n x 2n) and W_{2} (size 2n x 1),
respectively, the weight matrices of the hidden and of the output layer
of the trained ANN 
• 
b_{1} (size 1 x 2n) and b_{2} (size 1 x 1),
respectively, the bias vector of the hidden layer and the scalar bias value
of the output layer of the trained ANN 
• 
H, vector containing the 2 x n hidden neurons of the hidden
layer of the ANN 
• 
F and G, transfer functions between, respectively, input and
hidden layer and hidden and output layer (Bouzenada
et al., 2007) 
• 
O (size 1 x 31), the output vector of the ANN 
• 
ε (size 1 x 31), the output error vector of the net 
As a result, for each wavelength in the range 400700 nm it is possible to write the following equations:
Finally it is possible to evaluate the spectral response R_{S}(λ) given the n reflectance factors R(λ,α_{i}) and the proportions α_{i}:
From Eq. 17 it is possible to derive the expression for the transfer function φ_{j} (λ, α_{i}):
Equation 18 demonstrates that the color recipe mapping depends
upon the transfer functions F and G. If, for instance, both function may be
considered linear and the term ε is neglected (being the network error,
in a proper training this term tends to zero) it can be written:
If G is linear and F is a logsigmoid function (these functions are, typically, adopted for mapping functions with ANNs) and the term ε is neglected, Eq. 18 becomes:
RESULTS
The three formulations expressed by Eq. 12, 16
and 20 provide a mathematical assessment of the color recipe
mapping on the basis of three different, well recognized, methods. On the basis
of the results provided in scientific literature, each of the three formulas
may be adopted under specific restriction and in a different field of application.
In order to understand the differences between the proposed approaches, thus
analyzing their performances an experimental test may be assessed.
In detail, 80 differently colored fabrics were collected during an extensive experimental campaign conducted in 2009 by the colourists working in the company New Mill S.p.A. of Prato, Italy. Each blend is composed by a certain number of raw materials each one characterized by a different color (e.g., by a different spectrophotometric response). In Table 1 three examples of the 30 fabrics chosen for validating the approach are listed.
The validation was carried out according to the following tasks:
• 
Spectrophotometric measurement of the colorants. 
• 
Evaluation of the transfer function by using equations 12,
16 and 20 respectively 
• 
Prediction of the transformed spectral reflectance factors
of the blend 
• 
Measurement of CIE L*a*b* colour distance (DE CIEL*a*b*) between
the predicted spectrophotometric response and the measured one (Mridula
et al., 2008) 
The results of the whole validation, depicted in the last row of Table
2, shows that the color prediction, in terms of CIE L*a*b* colour distance,
is less than 0.8 for all the three approaches. The KM and the SN based formulations
prediction error, in terms of color distance, increase when the number of colorants
is greater than 8. This can be easily viewed in Table 2: the
mean values of the three approaches are quite similar for fabrics with Id.
Table 1: 
Three examples of the 30 fabrics chosen for validating the
approach 

Table 2: 
Results of the validation test. For each Fabric Id. The CIE
L*a*b* colorimetric distance between the predicted values and the measured
one is provided 

WP and CWP, whose shade is composed by 4 or 6 colorants.
In detail the KM approach provides a prediction with a color distance averagely
equal to 0.39 for 4 colorants and of 0.45 for 6 colorants. The SN approach,
considering a constant b equal to 0.15, provides a colour distance equal to
0.38 and to 0.44 for, respectively, 4 and 6 colorants. For fabrics composed
by 8 materials (i.e., 8 colorants, having every material a different color)
the color distances evaluated for the KM and the SN approaches are higher
than the ANN based one. In detail the KM and SN based approaches provides
value, respectively, of 0.75 and 0.74 while the ANN based approach give an average
distance equal to 0.57. On the basis of this validation task, it is evident
that the KM and the SN based approaches, proposed in the present work, may
be adopted when the number of colorants is less than 6. Otherwise, the ANN based
formulation may be considered more affordable.
DISCUSSION
Starting from the literature, the present paper provided in asashortaspossible
manner, three different mathematical formulations for performing the color matching.
The provided equations, useful for establishing a functional mapping between
colorant concentrations and spectrophotometric response of the reference shade,
are, in particular, based on three widely known techniques: KM, SN and ANNs.
On the basis of validation task and of literature analysis it may be affirmed
that the KM based equation (Eq. 12) is suitable for assessing
the color recipe mapping of textiles, papers, wood, paints, inks and plastics
once the absorption and scattering coefficients are evaluated. Accordingly,
it requires a database to compute both K_{λ} and S_{λ}.
Moreover, for materials such as textiles (where the colorants do not scatter
in comparison to the substrate) the mixing equation is simplified (Eq.
3). The performance of this method is well established in literature since
1980 as demonstrated by Allen (1980),
Nobbs (1997), Berns (2000) and Zhao
and Berns (2009). Such researchers showed that the color prediction, in
terms of CIE L*a*b* colour distance, is less than 0.8 and the maximum allowable
number of colorants composing the desired shape is 6. This is in accordance
with the validation task proposed in the present study.
The SN based Eq. 16 provides good results for color recipe
prediction of textile and woven fabrics. This mathematical formulation requires
the determination of constant b (Yang and Miklaveic, 2005).
Once properly defined such a constant, the SN based equation provides results,
in terms of CIE L*a*b* colour distance less than 0.8 with a maximum number of
colorants equal to 56. The method has been developed for a higher number of
components by Rong and Feng (2006) with good results:
the maximum color difference was 4.48 CIE L*a*b* units and the average color
difference was 1.02 CIE L*a*b* units for fourcomponents fiber blends under
D65 illuminant.
These results are in accordance with the one proposed in the present study. Anyway for different materials composing a fabric (e.g., a carded cloth) the constant value is hard to be computed; as a consequence this method is useful especially for textiles composed by a single, differently colored, material.
The ANN based approach (Eq. 19) that authors propose in
this work may be, probably, suitable for assessing the color recipe mapping
of papers, wood, paints, inks and plastics whose shade is composed by any number
of colorants, accordingly to Darko et al. (2008).
Moreover the approach is suitable also for shades composed by different materials
(e.g., textiles composed by a mixture of wool, cotton, polyester etc.). Although,
this approach does not need the assessment of constants or coefficients, it
requires a training phase by means of a database of spectral responses. Once
properly trained the ANNbased approach allows results, in terms of CIE L*a*b*
colour distance, less than 0.8. Finally, while the training phase may be computationally
expensive, the simulation phase, i.e., the determination of the transfer function,
may be performed in few seconds. The authors aimed to provide such techniques
into a mathematical form in order to help researchers and practitioners (colorists),
to easily evaluate the color of a shade given the spectral factors of the colorants.
Therefore, the authors want to encourage other researchers working in the field
of colorimetry and spectrophotometry to provide a large number of results of
their experimentations using the provided equations.
ACKNOWLEDGMENTS
The proposed approach is part of a FIT Project financed by the Italian Ministry of Economic Development. The project was conducted by the Department of Mechanical and Industrial Engineering of University of Florence (Italy) during the period 20072008. The devised method was applied in an important textile Company, New Mill S.p.A., working in Prato (Italy).