3rd INTERNATIONAL TEXTILE, CLOTHING & DESIGN CONFERENCE – Magic World of Textiles
October 08th to 11th 2006, DUBROVNIK, CROATIA
COMPARISON OF KUBELKA – MUNK THEORY AND FUZZY LOGIC IN
COLOUR MATCHING
Martinia Ira GLOGAR; Đurđica PARAC-OSTERMAN & Darko GRUNDLER
Abstract: The most common approach to computer colour estimation and colour reproduction is achieved
through Kubelka–Munk theory which is being widely used in colour industry, but which was found to be
inexact in many circumstances. The systematic discrepancy between the model and the reality is a serious
indicator how theory is either too much simplified or built on wrong assumptions. In this work, application of
Kubelka–Munk theory versus fuzzy logic in colour matching was examined. The experiments were performed
on cotton knitted fabrics of two different structure. Samples were dyed with yellow direct dyestuff in five
corresponding concentrations (0.25%; 0.75%; 1.4%; 2%; 3%). After analyses of dyed samples, the recipe
correction of concentration was performed using Kubelka–Munk and fuzzy logic colour calculations. Results
obtained showed differences considering standard sample being smaller using fuzzy logic corrected
concentrations than those obtained through Kubelka–Munk theory only. Results were obtained calculating
dE values.
Keywords: Color match prediction, correction, fuzzy logic, Kubelka–Munk.
1. Introduction
One of the most important aspects of a textile material is its appearance and in textile industry the
technologists are faced with problem how to arrive at the perfect match of a colour appearance on daily
bases. The aim is to produce a sample having a spectral reflectance function as close as possible to that of
the standard and none of the commercial mathematical based colour matching system does not give the
satisfactory performance. In general, the surface – optical appearance of one sample can be defined as a
summary of visually perceived attributes of colour and structure uniquely defined by the reflectance spectrum
of the sample. Modelling it consists of setting up relationship that link the light reflectance to the optical
properties of the substrate structure. In textile industry, the most common approach is through Kubelka–
Munk theory, expressed in 1931., and nowadays is a base of all commercial colour matching system. The
Kubelka–Munk theory defines the interaction of textile material with light through the absorption coefficient K
and the scattering coefficient S, providing the K/S ratio for a given wavelength, commonly known as
"absorption coefficient", which can be written as a following equation 1:
f ( R )  f Rs ,   a c
(1)
Where “a” refers to absorption coefficient of a given dyestuff. The original expression was developed for
pigmented layers and assumed infinite layer width, diffuse illumination and reflectance, and homogeneous
colorant distribution. Theory also assumes a perfect linear relationship between the concentration (c) of the
colorant and a function of the reflectance value (K/S). However, this has resulted in several limitations in
application of theory especially due to a fact that such conditions are rarely met in textiles. Textile material
has definite structure so it cannot be assumed to be a homogeneous layer and the linearity of Kubelka–Munk
function frequently only applies at low dye concentrations. Deviation from linearity occurs, and is very
common at high dye concentration and low reflectance values 2, 3, 4. That is why numerous scientists tried
to improve the original Kubelka–Munk model and many modifications and different approaches to colour
matching have been reported in the literature. One of the modern approaches to improved colour matching is
through application of fuzzy logic system in aim of achieving greater preciseness in colour match prediction
operations 5. In the past few years a rapid growth of fuzzy logic application has been noticed to variety of
industrial processes. Fuzzy logic is especially applicable to a control processes and among them is also
colour matching that includes the colour control processes 5. The advantage of fuzzy logic usage lies in
basic characteristics which defines the simplicity and applicability of fuzzy logic system. Fuzzy logic is easy
to understand and mathematical concepts behind fuzzy reasoning are simple. It is tolerant of imprecise data
and any set of input-output data can be created as a fuzzy system. Fuzzy logic is a method that interprets
the values in the input vector and, based on some set of rules, assigns values to the output vector 6. It is a
rule – based approach that is particularly suited to complex system where accurate mathematical models
cannot give the satisfactory performance, such as Kubelka–Munk based colour matching system. The
preliminary results of fuzzy logic application in colour matching were tested in compare to Kubelka–Munk
theory in order of exposing main differences between these two approaches and illustrating possibilities of
eventual fuzzy logic use in colour match prediction operations.
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3rd INTERNATIONAL TEXTILE, CLOTHING & DESIGN CONFERENCE – Magic World of Textiles
October 08th to 11th 2006, DUBROVNIK, CROATIA
2. Approach
The analyses in this paper were carried out in a few main steps. All measurements were carried out
spectrophotometricaly on remission spectrophotometer DataColor type SF600+CT, with constant
instrument aperture (apertures “L”=2.6), using d/8 geometry. Chosen sample of cotton knitted fabric was
measured in warp directions, on front side and back side of the sample. First step in experimental work in
this paper was to provide a precise analyze of undyed samples, in aim of précising the importance of
substrate structure and its appearance characteristics influence in colour match prediction operations. For
the analyze 100% cotton knitted sample of characteristics provided in Table 1 was chosen and measured
and analyzed on both side of the specimen due to the distinguished differences in surface structure
characteristics between front and back side. The surface – optical characteristics are the basic parameter of
surface colour appearance of textile samples and the main insufficiency of Kubelka–Munk based colour
matching system is that these surface – optical characteristics are not incorporated and cannot be
considered in such system. In this work the approach are based exactly on surface – structural
characteristics of a textile samples and the aim was to model the technique which would provide the
satisfactory colour recipe correction built on these characteristics.
2.1
Characteristics of textile samples
Characteristics of chosen sample are shown in Table 1 and Figure 1.
Table 1: Characteristics of chosen textile samples
Horizontal density Dh
Vertical density Dv
Thickness
Mass per m2
Yarn fineness
Sample – left/right coated
knitting – 3+1
11.60 stitches/cm
14.00 stitches/cm
0.956 mm
283 g/m2
basally fiber – 24 tex (front side)
coating fiber – 59 tex (back side)
The surface structure of back and front side of a chosen sample is shown on Figure 1. The figure shows
microscopic scans of samples surfaces – (magnified 40x).
a.
b.
Figure 1: Microscopic scans of chosen samples: a: front side of a sample; b: back side of sample.
2.2
Analysis of dyed samples
Chosen textile samples were dyed with yellow shaded direct dyestuff suitable for cotton dyeing, in five
concentrations corresponding to calibration concentration from the colour match computer program database
(0.25%; 0.75%; 1.4%; 2%; 3%). Sample was dyed in "Linitest" apparatus in liquor ratio 1:20, following the
standard procedure recommended by the dyestuff supplier. After the dyeing, dried samples were measured
spectrophotometrically. The results obtained for each side of sample were compared calculating the dE
values. After the analyze of dyed samples, the recipe correction of concentration was performed using
Kubelka–Munk based colour matching system. Using the results of basic concentration on front side of the
sample, the correction of back side of the samples was performed. The new concentrations were calculated
for given surface, as it is done in practice in real condition of the process. The samples were dyed again in
corrected concentrations and the results were obtained and analyzed through means of colour difference
value dE.
2.3
Concentration correction using Fuzzy logic
One of the limitations of Kubelka–Munk traditional approach is one that original theory does not deal with
surface phenomena. So the main idea was to examine a different approach using fuzzy logic based
technique in order to provide the method that would include surface parameters of textile samples that are of
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3rd INTERNATIONAL TEXTILE, CLOTHING & DESIGN CONFERENCE – Magic World of Textiles
October 08th to 11th 2006, DUBROVNIK, CROATIA
vital importance in colour reproduction. As fuzzy logic is convenient way to map any input space to any
output space, the aim was to create the input space based on terms that would define the surface
characteristics of samples and through the set of rules, provide the output space which would consists the
corrected values of concentrations. So the input space was created based on two linguistic variables –
surface structure defined as roughness and the concentration. For each variable the membership function
was defined. A rule which include the two variables from input space using “AND” operator, assumes the
form: If x is A and y is B than z is C.
For the first input linguistic variable – roughness – the 7 linguistics values were used: none, very low, low,
middle – low, middle, high, very – high. For the second input linguistic variable - concentration – the 5
linguistic values were used: very low, low, middle – low, middle – high, high. For both input linguistic variable
a membership function was set. For the output linguistic variable – correction of concentration – the 5
linguistic values for correction definition were used: range 1; correction for 0.5% - 5%, range 2; correction for
4 % - 15%, range 3; correction for 12% - 25%, range 4; correction for 23% - 35%, range 5; correction for
32% - 45%. These corrections refer to decreasing of basic concentration for a given percentage due to
achieve the new concentration that would provide the nearest colour characteristics for the back side of the
sample according to standard. (The standard was front side of the sample). So 15 if – then rules were
created following the example: If roughness is middle – high and the concentration is very low the
correction of concentration is decreased in the range 4 (25% - 35%). The polynomial based curves were
used for membership functions. The membership functions for each variable are shown on Figure 2.
a.
b.
c.
Figure 2: The membership function for input and output linguistic variables: a: the membership functions for first input
linguistic variable, b: the membership functions for second input linguistic variable, c: the membership functions for
output linguistic variable.
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3rd INTERNATIONAL TEXTILE, CLOTHING & DESIGN CONFERENCE – Magic World of Textiles
October 08th to 11th 2006, DUBROVNIK, CROATIA
As the result, the corrected concentration values were provided and used for correction dyeing of sample in
order to achieve the colour of back side corresponding to colour of the standard (the standard was front side
of the sample). The results obtained were analyzed through means of colour difference value dE.
3. Results and discussion
One of the main goals for this study was to test the ability of Kubelka–Munk model to predict the results of
transition from one fabric surface appearance to another. The Kubelka–Munk theory does not deal with
surface phenomena and if no surface correction is applied, the results might be very inaccurate. Since the
subject of the analysis were two sides of one sample of definite whiteness value, the total sum of remission
which defines the amount of reflected and absorbed light arise from surface structure itself. As the remission
is the base of Kubelka–Munk calculations given the K/S values as the definition of absorbed and scattered
amount of light from given surface, total sum of K/S value were also analyzed. Results are shown in Table 2.
Table 2: Total sums of remission and K/S values for front and back side of undyed sample.
700
700
S
2444.14
2417.65
0.9346
0.9992
400
400
Front side
Back side
K
R
Certain differences obtained for the given results show the importance of surface structure characteristic
influence. It was the aim to point out the importance of difference between surface structures of both side of
specimen which can highly affect the accuracy in colour recipe calculation if it is not performed on constant
side of the sample. For back side of sample, total sum R values on average are smaller, which was expected
taking into account the roughness of the surface structure, so the certain amount of scattered and diffused
distribution of light is expected. These results are in correlation with sum of K/S results which confirm the
prior statement.
Referring to the assumption of Kubelka–Munk theory, that the correlation of optical characteristics of dyed
textile defined by the absorption coefficient and the concentration of dyestuff should be in linear relationship,
the samples were first dyed in five concentrations using the same amount of dyestuff on given concentration
for each sample. The results of obtained K/S values are showed on Figure 3.
K/S vs. concentration relationship
K/S
30
conc. 3%
25
conc. 2%
20
conc. 1,4%
15
conc. 0,75%
10
Front side
5
Back side
conc. 0,25%
0
0
0,5
1
1,5
2
2,5
3
3,5
Dyestuff concentration (%)
Figure 3: K/S vs. concentration relationship for front and back side of sample
It is immediately observed that the front side of sample (smoother surface) produce lower K/S value of the
same amount of dyestuff, as the result of the higher reflectance from the smoother surface occurred due to a
surface structure. It is in correlation with the K/S value obtained higher for the back side of sample (rougher
surface) which confirm the influence of surface structure on appearance of colour and showed that the
rougher surface, even equally coloured, will perform deeper colour. It was expected based on the preliminary
analysis performed on undyed surfaces where the higher K/S sum occurred for the rougher surface with
higher scattered and absorbed light.
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3rd INTERNATIONAL TEXTILE, CLOTHING & DESIGN CONFERENCE – Magic World of Textiles
October 08th to 11th 2006, DUBROVNIK, CROATIA
Using the CMC(l:c) equation, values of colouristic parameters of front and back side of sample were
compared and the colour difference between them for each concentration were calculated using the front
side of the sample as the standard. The difference values obtained are presented in Table 3.
Table 3: CMC(l:c) differences values for front and back side of the sample comparison.
dE – CMC(l:c)
dK/S
0.25%
1.22
0.52
0.75%
2
3.21
1.4%
3.02
4.83
2%
2.21
5.14
3%
1.75
2.72
Differences obtained significantly higher than dECMC(l:c) =1 on average, implies also the significance of
substrate surface influence. The differences obtained increase with concentration until 1.4% - 2%
concentration beyond which the differences decreasing because the influence of the structure becomes less
significant and the differences occurred are influenced by the colour itself. The dK/S values obtained higher
in general then dE values, also confirm the accented influence of the structure as for the complex surface of
sample 2 higher scattering occurs, so the spectral power distribution is inconstant across the surface which
directly influence the absorption coefficient, K/S, value.
The concentration correction for back side of the sample was performed, using front side values acquired by
the calibration concentrations stored in computer database (0.25%, 0.75%, 1.4%, 2%, 3%,) as a standard.
The new concentrations were calculated using the Kubelka–Munk based matching technique, as it is done in
practice in real condition of the process.
Table 4: Corrected concentrations for back side of the sample according to K-M based colour matching.
Calibration concentrations
Corrected concentrations for back side
Percentage of correction
0.25%
0.22%
12%
0.75%
0.68%
9.33%
1.4%
1.25%
10.7%
2%
1.53%
23%
3%
2.88%
4%
After the concentration correction, sample was dyed again according to new concentrations and aftwerword
the back side of the sample was measured and compared to front side of the sample dyed in basic
concentrations (before correction). The aim was to examine the range of correction that should be applied in
order to achieve the colour appearance characteristics on back side of the sample nearest to that of a front
side achieved after initial dyeing, due to a specific surface of a back side. As the Kubelka–Munk theory has
been developed on homogeneous layer it cannot provide sufficient preciseness in predicting the colour
characteristics of dyed material so the colour correction obtained according to K-M did not provide the
satisfactory results. The differences obtained after the dyeing according to corrected concentrations were still
out of the range of tolerance, which is for textile samples set on dE ≤ 1 in general and for yellow dyed
samples dE ≤ 2, according to ISO standard. (Table 5.).
Table 5: CMC(l:c) differences for back side of the sample after the corrected concentration using the results of front side
after the initial dyeing as the standard
dE – CMC(l:c)
0.22%
2.21
0.68%
2.43
1.25%
1.75
1.53%
2.09
2.88%
1.28
To establish the set of fuzzy rules, the number of experimental dyeing was performed in order to set the
range of concentrations that would provide the satisfactory result. Based on such experiments, the fuzzy
approach to concentration correction was established and the correction of concentration was performed.
The concentrations provided by fuzzy logic are showed in Table 6. Dyeing according to new concentration
provide the better results and the differences obtained were all in the range of the tolerance. The differences
obtained comparing the front side of the sample dyed according to calibration concentrations and the back
side of the sample dyed according to concentrations provided by fuzzy correction, are presented as the dE
value also in Table 6.
Table 6: Corrected concentrations of back side of sample according to fuzzy based program and CMC(l:c) differences
for back side of the sample after the corrected concentration using the results of front side after the initial dyeing as the
standard
Calibration concentrations
Corrected concentrations for back side
Percentage of correction
dE – CMC(l:c)
0.25%
0.165%
34%
0.73
5
0.75%
0.69%
8%
0.79
1.4%
1.21%
13.5%
1.26
2%
1.50%
25%
1.58
3%
2.60%
13.3%
0.98
3rd INTERNATIONAL TEXTILE, CLOTHING & DESIGN CONFERENCE – Magic World of Textiles
October 08th to 11th 2006, DUBROVNIK, CROATIA
As it can be seen the fuzzy correction performed more satisfactory and the results obtained could be
considered in the range of tolerance. Fuzzy model was tested by performing the recipe calculation for the
concentrations additional to those used from the database for training the fuzzy system, so the testing dyeing
was performed in concentrations 0.5%, 1%, 1.5%, 2.5%. First, the sample was dyed in order to achieve the
colour characteristics of the front side. The results of front side were used to colour matching correction for
the back side of the sample. The corrected concentrations using fuzzy method only were shown in Table 8.
Table 8: Corrected concentrations back side of sample according to fuzzy based program
Additional concentrations for front side
Corrected concentrations for back side
0.5%
0.454%
1%
0.907%
1.5%
1.36%
2.5%
1.77%
After the repeated dyeing according to corrected concentrations, the results were satisfactory providing the
difference value in the range of tolerance after the only one corrective dyeing.
Table 9: CMC(l:c) differences for back side of the sample after the corrected concentration using the results of front side
after the dyeing in additional concentrations as the standard
dE – CMC(l:c)
0.5%
0.95
1%
0.3
1.5%
1.42
2.5%
1.16
4. Conclusion
Since all commercial colour matching systems nowadays are based on original Kubelka–Munk theory which
does not concerns the surface phenomena, the efforts are made to established method based on some new
and different approaches that would provide more satisfactory performance. The results proved that the
Kubelka–Munk based technique would not consider the influence of surface structure and so would not
provide sufficient preciseness in predicting the transition from one fabric surface appearance to another.
The results of differences between two matches using K-M based technique were higher that dE=2 which is
out of the range of tolerances. As the new approach using fuzzy logic technique in this work has been
analyzed, such approach provide the matching with dE values lower than 1.5, which is highly acceptable.
The results obtained are promising and encourage further research of fuzzy logic usage in colour match
prediction operations.
References
[1] Mourad, S.; Emmel, P.; Simon, K.: (2001) “Extending Kubelka – Munks Theory with Lateral Light
Scattering”, International Conference on Digital Printing Technologies, Florida, USA
[2] Shamey, M. R.; Nobbs, J. H.: (2000) The Use of Colorimetry in the Control of Dyeing Processes, Textile
Chemists and Colourists & American Dyestuff Reporter, Vol. 32, No. 6, ISSN 0040-490X
[3] Tsoutseos, A., A., Nobbs, J. H.: (2000) Alternative Approach to Colour Appearance of Textile Materials
with Application to the Wet/Dry Reflectance Prediction, Textile Chemists and Colourists & American
Dyestuff Reporter, Vol. 32, No. 6, ISSN 0040-490X, pp. 38 – 43
[4] Joaneli, M., I.; Osterman-Parac, Đ.; Golob, D.: Textile Surface Structure Importance and Kubelka-Munk
Theory Use in Colour Match Calculations, accepted paper for publishing in annual addition of Colourage
[5] Mitra, S.: Neuro – Fuzzy Rule Generation: Survey in Soft Computing Framework, IEEE Transactions of
Neural Networks, Vol. 11, No. 3, (2000.), 748 – 768
[6] Cheetham, W; Graf, J.: Case – Based Reasonong in Color Matching, In Case – Based Reasoning
Research and Development, Leake, D., B. & Plaza, E. Eds., Berlin, Springer Verlag, (1997.)
Author(s):
Martinia Ira GLOGAR, M.Sc.
Prof. Đurđica PARAC – OSTERMAN, Ph.D.
University of Zagreb, Faculty of Textile Technology, Department of Textile Chemistry and Ecology
Prilaz baruna Filipovića 30, HR-10000 Zagreb, Croatia
Phone: +(385) (1) 4877 350
Fax: +(385) (1) 4877 355
E-mail:martina.joanelli@ttf.hr
Prof. Darko Grundler, Ph.D.
University of Zagreb, Faculty of Textile Technology, Department of basic natural and technical science
Prilaz baruna Filipovića 30, HR-10000 Zagreb, Croatia
Phone: +(385) (1) 3712 545
E-mail:darko.grundler@ttf.hr
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