Further Investigations of Colorant Database Development for Two-Constant Kubelka- Munk Theo

Further Investigations of Colorant Database Development for Two-Constant Kubelka-Munk Theory for Artist Acrylic and Oil PaintsYonghui ZhaoRoy S. BernsJune 2006AbstractIt is a common practice to prepare tint ladders of each colorant consisting of five or more concentrations in order to develop a colorant database for instrumental-based color matching, characterization of optical properties of a coloration system, and pigment mapping. For two-constant Kubelka-Munk theory, the unit absorption and scattering coefficients for each colorant can be solved by the linear-least-squares method using all the samples in the tint ladder. The objective of this research was to find the minimum number of samples in order to characterize each colorant. For the purpose of pigment identification, a pure pigment (masstone) and a mixture sample of unknown concentration with white paint were sufficient to describe each pigment qualitatively. On the other hand, pigment mapping, which resolves not only colorant constituents but also their concentrations, required a masstone and a mixture sample of known concentration, preferably 40% to 60% with white paint, to quantitatively characterize each colorant. For either case, only two samples were needed, which greatly simplified the process to develop a colorant database. This was tested using artist acrylic and oil paints. IntroductionIn 1931, Kubelka and Munk[1]proposed a turbid media theory that derived the relationship between reflectance factor and absorption and scattering coefficients. For single-constant Kubelka-Munk (K-M) theory, the colorant database consists of the ratio between absorption and scattering coefficients and the relationship between effective and theoretical concentrations (i.e. the scalar of the linear system and user control[2, 3]) for each colorant. Since the relationship between the linear scalar and user control is usually nonlinear, it is necessary to require five or more samples in the tint ladder for each colorant in order to build this relationship. In addition, one major assumption of single-constant K-M theory is that the scattering coefficient is essentially constant independent of the colorant formula.[4]Based on this assumption, one sample is sufficient to characterize each colorant. Reference [5] concluded that the sample near the maximal chroma was the best choice. However, in most cases where the assumption is invalid, two-constant K-M theory should be used.For two-constant Kubelka-Munk theory, the absorption and scattering coefficients for a colorant are separated as two independent variables, which, in some sense, accounts for the most nonlinear relationship between the linear scalar and user control. Moreover, two samples of the same colorant with different known concentrations with white paint are sufficient to reach a unique solution of the unit absorption and scattering coefficients for that colorant.[4] In a previous Art Spectral Imaging technical report,[5] it was concluded that two samples of paints are sufficient to determine absorption and scattering coefficients when one is masstone sample and the other is a mixture sample with white paint, near the maximum chroma. In this report, the same conclusion is drawn from a large number of pigments, and also concluded that the optimal colorant concentration for the calibration mixture is between 40% and 60%, in the neighborhood of the maximum chroma. If colorant concentration of the mixture sample used for calibration is unknown, the unit absorption and scattering coefficients of that colorant can be still estimated, but their values might be off by the same scaling factor. If those values are then used to analyze an unknown mixture sample of the same colorant with white paint, the predictedcolorant concentration is not necessarily equal to the theoretical concentration. Thus, a masstone and a mixture sample with unknown colorant concentration can be used for pigment identification but not for pigment mapping. The same conclusion still holds even if more than two mixture samples with unknown colorant concentrations are used during the calibration process. On the other hand, if the colorant concentration of the calibration mixture sample is known, the absolute values for unit absorption and scattering coefficients can be obtained, and it is then possible to achieve accurate prediction of colorant concentration for an unknown mixture sample.TheoryTwo-constant Kubelka-Munk theory establishes the relationship between a mixture sample and its constituent colorants, which is expressed in mathematical forms as Eqs. (1) and (2). It is assumed that the mixture sample is made of only one colorant at a certain concentration c with white paint. The capital symbols K and S are the absorption and scattering coefficients, while the small symbols k and s are these coefficients at unit amount. The subscript “w” means titanium white paint; “mixture” and “masstone” stand for mixture and masstone samples of the colorant, respectively. K S ! " # $ % & mixture =1'c ()k w +c k 1'c ()s w +c s (1) K S ! " # $ % & masstone =k s(2) The following equations can be readily derived from Eq. (1). c K S ! " # $ % & mixture 's (k ) * + , - . +1(c ()K S ! " # $ % & mixture 's w (k w ) * + , - . =0 (3.1) 1!c c K S " # $ % & ' mixture (s w !k w ) * + , - . +K S " # $ % & ' mixture (s !k ) * + , -. =0 (3.2) Substitute k =s !K S " # $ % & ' mastoneinto Eq. (3.1), ()01=!"#$%&'()*+,-.'+(!"#$%&)*+,-.')*+,-.w w mixture masstone mixture k s S K s S K S K c c (4.1) mixturemasstone w w mixture S K S K k s S K s !"#$%&'!"#$%&'(!"#$%&'=c c 1 (4.2) Above derivation shows mathematically the process to characterize the unit absorption and scattering coefficients of the colorant from a combination of masstone and mixture samples. The results are rewritten with a simpler format as following.k =c 0s 0K S ! " # $ % & masstone (5.1) s =c 0s 0 (5.2)c c c !=10 (5.3) s 0=K S ! " # $ % & mixture 's w (k w K S ! " # $ % & masstone (K S ! " # $ % & mixture(5.4) Given a masstone and a mixture sample of colorant with white paint, the unit k and s for this colorant can be solved as expressed in Eq. (5.1) and (5.2), respectively. It is worth noting that though the ratio between the unit k and s is uniquely determined by measurement on the masstone sample, the determination of their absolute values also requires knowledge of colorant concentration of the calibration mixture sample. If the colorant concentration is unknown, inaccuracy in estimation on its value will be manifested as the error in 0c , the same scaling factor applied to both the unit k and s . Once characterization is complete, colorant concentration of another unknown mixture sample of the same colorant with white paint can be predicted by using linear least square method, as expressed in Eq. (6.1) and (6.2), where PINV stands forpseudoinverse, and u cˆrepresents estimated colorant concentration of the unknown mixture sample. K S ! " # $ % & u is the ratio between absorption and scattering coefficients of the unknown mixture sample derived from its reflectance measurement. 1!ˆ c u ˆ c u =!PINV K S " # $ % & ' u (s w !k w ) * + , - . /K S " # $ % & ' u (s !k ) * + , - . (6.1) ˆ c u =11!PINV K S " # $ % & ' u (s w !k w ) * + , - . /K S " # $ % & ' u (s !k ) * + , - . 0 1 23 4 5 (6.2) After that, the ratio between the absorption and scattering coefficients for this mixture sample can be estimated using Eq. (6.3), where u!"#$%&'S K represents estimation in order to differentiate from u !"#$%&S K that is derived from the reflectance measurement. K S ! " # $ % & 'u =1(ˆ c u ()k w +ˆ c u k 1(ˆ c u ()s w +ˆ c us (6.3) Divide ˆ c u in both the denominator and the nominator of the fraction on the rightside, K S ! " # $ % & 'u =1(ˆ c u ˆ c u ! " # $ % & k w +k 1(ˆ c u ˆ c u ! " # $ %& s w +s (6.4) Substitute Eq. (6.1) into the above equation,s s k s S K k s S K k k k s S K k s S K S K w w w w w w +!"#$%&'()*+,-./!"#$%&'()*+,-.'+!"#$%&'()*+,-./!"#$%&'()*+,-.'=)*+,-.0u u u u u PINV PINV (6.5) Divide the unit scattering coefficient s in both denominator and the nominator,1+!"#$%&'()*+,-.!"#$%&'/()*+,-'+!"#$%&'()*+,-.!"#$%&'/()*+,-'=()*+,-0w w w w w w s s k S K k s S K s k k s k S K k s S K S K u u u u u PINV PINV (6.6) Substitute Eq. (2) into the above equation,1+!"#$%&'()*+,-'()*+,.!"#$%&-/'()*+,-'()*+,+!"#$%&'()*+,-'()*+,.!"#$%&-/'()*+,-='()*+,0w mastone w w mastone w mastone w w s S K S K k s S K S K k S K S K k s S K S K u u u u u PINV PINV (6.7) It can be readily seen from Eq. (6.7) that the estimated ratio between the absorption and scattering coefficients for the unknown mixture sample, u!"#$%&'S K , depends only on K S ! " # $ % & u and K S ! " # $ % & masstone that are derived from reflectance measurements of this unknown mixture sample and masstone respectively, and the absorption k w andscattering s w coefficients of the white paint. While u!"#$%&'S K is independent of the unit k and s of the colorant, the predicted colorant concentration of the unknown mixture sample however depends on those two parameters, as expressed in Eq. (6.2).Several conclusions can be drawn based on above derivation.1. The unit k and s for the colorant can be uniquely determined based on amasstone sample and a mixture sample of known colorant concentration with white paint, shown in Eq. (5.1) – Eq. (5.4). If the colorant concentration is unknown, the unit k or s can still be estimated but their values might be off by the same scaling factor. The colorant database is established by applying the same characterization process to all colorants of interest to obtain their unit k and s .2. The estimated ratio between the absorption and scattering coefficients for another unknown mixture sample K S ! " # $ % & 'u depends on u!"#$%&S K and K S ! " # $ % & masstone , but is independent of unit k and s for the colorant. In other word, u!"#$%&'S K is independent of the colorant concentration c of the calibration mixture sample. Therefore,goodness of fit between u!"#$%&'S K and u !"#$%&S K is the same no matter what colorant concentration is assumed for the calibration mixture.3. The predicted colorant concentration of the other unknown mixture sample depends on unit k and s of the colorant. Any error in the colorant concentration of the calibration mixture sample will affect the accuracy of prediction.In summary, the unit k and s for each colorant can be calculated based on two calibration samples – one masstone and one mixture sample. The goodness of fit for another unknown mixture sample of the same colorant is the same no matter what colorant concentration is assumed for the calibration mixture sample. However, accurate prediction of colorant concentration of the unknown mixture demands knowledge of the colorant concentration of the calibration mixture sample. These conclusions will be verified in the following section.Discussion so far assumes that the mixture samples, including the one used for characterization and those unknown samples, are made up of only one colorant and white paint. Under this assumption, pigment identification could be achieved by determining the colorant in the database that gives the best goodness of fit between u!"#$%&'S K and u !"#$%&S K . If the colorant concentration of the calibration mixture sample for this colorant is known, pigment mapping is also accomplished by accurately calculating the colorant concentration of the unknown mixture sample. Experimental ResultsSamples of Matte Fluid Golden Acrylic Artist Colors were carefully prepared by one of our group members. There are in total of 28 colorants plus white paint. All these samples were measured by an integrating-sphere spectrophotometer with spectral component included. For each colorant, there are at least six mixture samples with white paint at about 0%, 20%, 40%, 60%, 80% and 100% colorant concentrations. The unit absorption and scattering coefficients for each colorant were calculated based on white paint, masstone and a mixture sample at a certain concentration, shown as Eq. (5.1) – (5.4). After that, the concentrations for all the mixture samples with white paint were predicted using linear least squares method, shown as Eq. (6.2). Finally, the ratios between absorption and scattering coefficients for all these samples were reconstructed using the predicted concentrations, shown as Eq. (6.3). It is also worth mentioning that Saunderson equations [6] were used to transform measured reflectance into internal reflectance, which is used in two-constant K-M theory. The two parameters in Saunderson equation were 0 and 0.5986, referred to the global optimization method by the way of tint, masstone and a mixture sample with maximum chroma in an M.S. thesis concerned with characterizing the optical properties of artist acrylic paints.[7]Table I lists the spectral and colorimetric performance of the proposed approach by comparing reconstructed reflectance factors with spectrophotometer measurements. For each colorant, the mixture sample of certain colorant concentration was used as the calibration sample along with the corresponding masstone to predict reflectance factor for all the mixture samples of the same colorant. This procedure was repeated for all thecolorants and spectral and colorimetric performance metrics based on all mixture samples were calculated. For example, if all the calibrate mixture samples had 20% colorant concentration, the mean and maximal RMS errors were 0.52% and 2.92% respectively. The maximum chroma means that for each colorant, the calibration mixture sample was chosen so that it exhibited the highest chroma among all the mixture samples of the same colorant, unless it happened to be the masstone, in which case, the mixture sample with the second highest chroma was selected, instead.Both spectral and colorimetrical accuracies improved initially as the concentration of the calibration mixture sample increased, and then declined as the increase continued. It can also be seen that the performance of the calibration mixture sample with 40% ~ 60% colorant concentration is comparable to that of the sample with the maximum chroma. This simplification can facilitate buildup of the colorant database.Table I Spectral and colorimetric accuracies of reconstructed reflectance factors compared with in situ spectrophotometer measurements Calibration Mixture RMS Error (%)Color difference (∆E00, D65)Concentration (Mass, %)Mean Max90%PercentileMean Max90%Percentile200.52 2.92 1.480.34 2.280.88 400.32 1.560.840.23 1.730.59 600.34 1.940.840.26 2.590.60 800.47 3.87 1.340.36 3.490.77 Max Chroma0.36 1.940.900.25 1.530.70As discussed earlier, the goodness of fit betweenu!"#$%&'SKandu!"#$%&SKis independent of colorant concentration of the calibration mixture sample. As a result, the spectral and colorimetric accuracies of reconstructed reflectance factor should be identical no matter what colorant concentration is assumed for the calibration mixture sample. However, prediction of colorant concentration from an unknown mixture sample relies critically on knowledge of colorant concentration of the calibration mixture sample, which if unknown should be estimated as accurately as possible. One simple and relatively accurate method to estimate colorant concentration of the calibration mixture sample is based on single-constant K-M theory.[8, 9] The estimation is achieved by reflectance measurement of the white paint, masstone and mixture sample, as shown in Eq. (7.3).KS!"#$%&mixture=ks!"#$%&w+c1'cks!"#$%&masstone(7.1)ˆ c1!ˆ c=PINVks"#$%&'masstone()*+,-.KS"#$%&'mixture!ks"#$%&'w()*+,- (7.2) ˆ c =Figure 1 Predicted concentration vs. theoretical concentration based on a masstone sample and a mixture sample with 40% colorant concentration Figure 2 Predicted concentration vs. theoretical concentration based on a masstone sample and a mixture sample with colorant concentration estimated based on single-constant K-M theoryIn Figure 1, the predicted concentrations are plotted against the measured concentrations for all mixture samples. The prediction was based on calibration mixture samples with the known 40% colorant concentration. For comparison, the same calibration samples were also used for Figure 2, but their colorant concentrations wereestimated based on single-constant K-M theory using Eq. (7.3). Excellent agreement was achieved using the known colorant concentration during the calibration process. However, a nonlinear relationship was observed when colorant concentration used in calibration was estimated based on single-constant K-M theory. The inaccuracy of the estimation caused the deviation of prediction from the theoretical concentration. ConclusionsThe goal of this research was to find the minimum number of samples in order to characterize a colorant’s optical properties, which amounts to determination of its unit absorption and scattering coefficients using two-constant K-M theory. Two calibration samples, a pure pigment (masstone) and a mixture sample of unknown colorant concentration with white paint, are sufficient to obtain both unit absorption and scattering coefficients of the colorant but their absolute levels might be off by the same scaling factor. When they are used to analyze an unknown mixture sample, such uncertainty in their absolute levels will influence prediction of colorant concentration but does not affect the spectral and colorimetric accuracies of reconstructed reflectance factor. Therefore, pigment identification can still be achieved. Since the weight measurement for determination of colorant concentration is no longer necessary, preparation of calibration mixture sample becomes much easier and more convenient. On the other hand, if the colorant concentration of the calibration mixture sample is known, the absolute levels of the unit absorption and scattering coefficients of the colorant can be obtained and pigment mapping is realized since accurate prediction of colorant concentration for the unknown mixture sample becomes possible. The optimal colorant concentration was found to be 40%~60% for the calibration sample, which gave comparable performance for prediction of unknown mixture sample as that provided by the concentration that has the maximum chroma. Finally, compared with the conventional approach based on a tint ladder consisting of at least five mixture samples, this research indicates that two calibration samples are sufficient to characterize each colorant, which greatly simplifies the process to develop a colorant database. This approach is only sufficiently accurate to predict concentration for the purpose of pigment mapping, but not for precise recipe prediction. Industrial practice might still require more samples to characterize a colorant database and batch correction for the accuracy of recipe prediction.References[1] Kubelka P. and Munk F., ‘Ein Beitrag Zur Optik der Farbanstriche’, Zeitschrift fur technischePhysik, 12, pg. 593-601, (1931)[2] R. S. Berns, Billmeyer and Saltzman’s Principles of Color Technology, 3rd Ed., John Wiley& Sons, New York (2000)[3] R. S. Berns, “A generic approach to color modeling”, Color Research and Application, 22(5),pg. 318-325, (1997)[4] E. Allen, “Colorant formulation and shading”, in F. Grum and C. J. Bartleson, Eds. OpticalRadiation Measurements, Vol.2: Color Measurement, Academic Press, New York, (1980) [5] M. Mohammadi and R. S. Berns, “Verification of the Kubelka-Munk turbid media theory forartist acrylic paint”, Munsell Color Science Laboratory Report, summer, (2004)[6] J. L. Saunderson, “Calculation of the color of pigmented plastics”, J. of Opt. Soc. Am. 32, pg.727-736, (1942)[7] Y. Okumura, “Developing a spectral and colorimetric database of artist paint materials”,Rochester Institute of Technology, M.S. Thesis, September, (2005)[8] F. W. Billmeyer and R. L. Abrams, “Predicting reflectance and color of paint films byKubelka-Munk analysis, I. Turbid-medium theory,” J. of Paint Technology 45:579, 23-30 (1973)[8] R. S. Berns, “Color Forum: Single-constant simplification of Kubelka-Munk turbid-mediatheory for paint systems,” submitted Color Research & ApplicationAcknowledgmentsThe authors would like to thank the National Gallery of Art, Washington, D.C., the Museum of Modern Art, New York, the Andrew W. Mellon Foundation, and the Institute of Museum and Library Services for their financial support of the Art Spectral Imaging (Art-SI) Project.Appendix I (Gamblin Artist Oil Paints)This technique has been applied to build a colorant database for ten Gamblin artist oil paints. For each pigment, the masstone (#1) and one mixture sample (#2) were used to calculate the unit absorption and scattering coefficients, while the other mixture sample (#3) was used to evaluate the performance.From Table A1 and Figure A1, it can be seen that the concentration error is a little bit higher for cadmium red medium and venetian red, which may result from measurement inaccuracy, incomplete mixing, surface non-smoothness and limitations of the Kubelka-Munk theory.Table A1 Measured and predicted concentrations for pure paint and mixturesConcentrations (Mass, %)Golden Artist Oil Pigments #1 #2 #3 #3 Pred. DifferenceCadmium Yellow Medium 100 51.0 20.0 22.5 2.5Indian Yellow 100 52.4 18.5 22.9 4.4Cadmium Red Medium 100 51.1 79.6 68.6 11.0Venetian Red 100 52.3 78.9 89.0 10.1Quinacridone Red 100 50.0 21.4 21.4 0.0Phthalocyanine Green 100 52.1 20.5 27.4 6.9Chromium Oxide Green 100 60.5 20.4 22.1 1.7Cobalt Blue 100 60.3 20.6 21.0 0.4Phthalocyanine Blue 100 59.7 20.3 27.1 6.8Ivory Black 100 59.2 19.6 24.7 5.1 Figure A1 Measured vs. predicted concentrations for ten Gamblin artist oil paintsIn addition to prediction of concentration, the performance of spectral estimation for these mixtures (#3) was quantified using three metrics. These metrics included RMS spectral error between measured and estimated reflectances, CIEDE2000 color difference for CIE illuminant D65 and the 2-degree observer, and a metameric index from D65 to CIE illuminant A. Both the spectral and colorimetric performances were quite well.Table A2 Performance metrics for estimation accuracy of the mixture samples (#3)StatisticsSpectral% RMS ErrorColor Difference(∆E00 D65)Index of Metamerism(∆E00 D65 -> A)Average 1.1 1.0 0.1 Std. Dev. 0.8 0.6 0.1 Maximum 3.0 2.1 0.3 90% Percentile 2.4 1.9 0.3Figure A2 Left: unit absorption (red solid) and scattering (blue dashed) coefficients Right: measured (red solid lines) and predicted (blue dashed lines) reflectance factors Notes:1.Reflectance data were measured by a bidirectional spectrophotometer, calledGretagMacbeth SpectroEye.2.Two parameters in Saunderson Equations for bidirectional geometry were k1=0.03and k2=0.6, respectively.3.The unit scattering coefficient for white paint was assumed as unit.4.Unit absorption and scattering coefficients for each pigment were calculated based onone pure pigment and one pigment mixture with white paint.5.The concentrations for another mixture samples were predicted using linear-least-squares method.。