Abstract

A premise was set up entailing the possibility of a synergistical combination of advantages of spectrophotometric and colorimetric matching procedures. Attempts were therefore made to test the performances of fifteen matching procedures, all based on the Kubelka–Munk theory, including two procedures utilizing the fundamental color stimulus RFCS of the spectral decomposition theory. Color differences CIE ΔE00 as well as concentration differences ΔCAVE were used to theoretically rank the fifteen color matching procedures. Results showed that procedures based on RFCS were superior in accurately predicting colors and concentrations. Additionally, the metameric black component RMB of the decomposition theory also showed promise in predicting degrees of metamerism. This preliminary study, therefore, provides evidence for the premise of this investigation.

© 2008 Optical Society of America

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    [CrossRef]
  2. P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche (An Article on Optics of Paint Layers),” Z. Tech. Phys. (Leipzig) 12, 593-601 (1931), see also www.graphics.cornell.edu/~westin/pubs/kubelka.pdf.
  3. P. Kubelka, “New contributions to the optics of intensely light scattering materials. Part I,” J. Opt. Soc. Am. 38, 448-457 (1948).
    [CrossRef] [PubMed]
  4. S. Chandrasekhar, Radiative Transfer (Oxford U. Press, 1950).
  5. D. B. Judd and G. Wyszecki, Color in Business, Science and Industry (Wiley, 1975).
  6. F. W. Billmeyer, Jr., R. L. Abrams, and J. G. Davidson, “Predicted reflectance and color of paint films by Kubelka-Munk analysis, I. Turbid medium theory,” J. Paint Technol. 45, 23-30 (1973).
  7. F. W. Billmeyer, Jr., R. L. Abrams, and J. G. Davidson, “Predicted reflectance and color of paint films by Kubelka-Munk analysis, II. Performance tests,” J. Paint Technol. 45, 30-37 (1973).
  8. R. S. Berns and M. Mohammadi, “Single-constant simplification of Kubelka-Munk turbid-media theory for paint systems--A review,” Color Res. Appl. 32, 201-207 (2007).
    [CrossRef]
  9. R. S. Berns, Billmeyer and Saltzman's Principles of Color Technology, 3rd ed. (Wiley-Interscience, 2000).
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    [CrossRef]
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  12. S. H. Amirshahi and M. T. Pailthorpe, “Applying the Kubelka-Munk equation to explain the color of blends prepared from precolored fibers,” Text. Res. J. 64, 357-364 (1994).
    [CrossRef]
  13. C. S. Haase and G. W. Meyer, “Modeling pigmented materials for realistic image synthesis,” ACM Trans. Graphics 11, 305-332 (1992).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  16. H. R. Kang, “Kubelka-Munk modeling of ink jet ink mixing,” J. Imaging Technol. 17, 76-83 (1991).
  17. D. Y. Tzeng and R. S. Berns, “Spectral-based ink selection for multiple-ink printing I. Colorant estimation of original objects,” in Proceedings of 6th IS&T/SID Color Imaging Conference, (Society for Imaging Science & Technology, 1998), pp. 106-111.
  18. D. Y. Tzeng and R. S. Berns, “Spectral-based ink selection for multiple-ink printing II. Optimal ink selection,” in Proceedings of 7th IS&T/SID Color Imaging Conference, (Society for Imaging Science & Technology, 1999), pp. 182-187.
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  22. J. C. Ragain, Jr., and W. M. Johnston, “Accuracy of Kubelka-Munk reflectance theory applied to human dentin and enamel,” J. Dent. Res. 80, 449-452 (2001).
    [CrossRef] [PubMed]
  23. E. Allen, “Basic equations used in computer color matching, II. Tristimulus match,” J. Opt. Soc. Am. 64, 991-993 (1974).
    [CrossRef]
  24. B. Sluban, “Comparison of colorimetric and spectrophotometric algorithms for computer match prediction,” Color Res. Appl. 18, 74-79 (1993).
    [CrossRef]
  25. E. Allen, “Colorant formulation and shading,” in Optical Radiation Measurements, Vol. 2, Color Measurement, F.Grum and C.J.Bartleson, eds. (Academic, 1980), pp. 289-336.
  26. A. Kumar and R. Choudhury, Modern Concepts of Color and Appearance (Science Publishers Inc., 2000).
  27. R. H. Park and E. I. Stearns, “Spectrophotometric formulation,” J. Opt. Soc. Am. 4, 112-113 (1944).
    [CrossRef]
  28. P. H. McGinnis, “Spectrophotometric color matching with the least square technique,” Col. Eng. 5, 22-27 (1967).
  29. N. Ohta and H. Urabe, “Spectral color matching by means of minimax approximation,” Appl. Phys. Lett. 11, 2551-2553 (1972).
  30. E. Walowit, C. J. McCarthy, and R. S. Berns, “Spectrophotometric color matching based on two-constant Kubelka-Munk theory,” Color Res. Appl. 13, 358-362 (1988).
    [CrossRef]
  31. D. W. Marquardet, “An algorithm for least square estimation of non linear parameters,” SIAM J. Appl. Math. 11, 431-441 (1963).
    [CrossRef]
  32. B. Sluban and O. Šauperl, “Least metameric recipe formulation,” Croat. Chem. Acta 76, 161-166 (2003).
  33. R. K. Winey, “Computer color matching with the aid of visual technique,” Color Res. Appl. 3, 165-167 (1978).
    [CrossRef]
  34. G. Wyszecki, “Valenzmetrische untersuchung des zusammen-hanges zwischen normaler und anomaler Trichromasie (Psychlogical investigation of the relation between normal and abnormal trichromatic vision),” Die Far. 2, 39-52 (1953).
  35. J. B. Cohen and W. E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszecki's metameric black,” Austral. J. Earth. Sci. 95, 537-564 (1982).
  36. J. B. Cohen and W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171-259 (1985).
    [CrossRef]
  37. H. S. Fairman, “Correction using parametric decomposition,” Color Res. Appl. 12, 261-265 (1987).
    [CrossRef]
  38. R. A. Charvat, Coloring of Plastics: Fundamentals (Wiley-Interscience, 2004).
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    [CrossRef]
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  42. M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE 2000 colour-difference formula: CIEDE 2000,” Color Res. Appl. 25, 340-350 (2001).
    [CrossRef]
  43. S. Moradian and B. Rigg, “The quantification of metamerism,” J. Soc. Dyers Colour. 103, 209-213 (1987).
    [CrossRef]
  44. F. H. Imai, M. R. Rosen, and R. S. Berns, “Comparative study of metrics for spectral match quality,” in Proceedings of CGIV 2002, C.Fernandez-Maloigne, ed. (IEEE Computer Society, 2002) pp. 492-496.
  45. M. A. López-Álvarez, J. Hernández-Andrés, E. M. Valero, and J. L. Nieves, “Colorimetric and spectral combined metric for the optimization of multispectral systems,” in Proceedings of AIC Colour 05, J.Romero, ed. (International Colour Association, 2005), pp. 1685-1688.
  46. F. Ameri, S. Moradian, M. Amani Tehran, and K. Faez, “The use of fundamental color stimulus to improve the performance of artificial neural network color match prediction systems,” Iran. J. Chem. and Chem. Eng. 24, 53-61 (2005).
  47. A. Karbasi, S. Moradian, and S. Asiaban, “The use of two constant Kubelka-Munk theory in spectrophotometric color matching,” in Proceedings of ICE2007, P.Ziegler, ed. (International Coatings Expo, 2007).
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  50. F. Ameri, S. Moradian, M. A. Tehran, and K. Faez, “Use of transformed reflectance functions for neural network color match prediction systems,” Ind. J. Fib. Tex. Res. 31, 439-443 (2006).
  51. C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Society for Industrial and Applied Mathematics, 1995).
    [CrossRef]

2007 (1)

R. S. Berns and M. Mohammadi, “Single-constant simplification of Kubelka-Munk turbid-media theory for paint systems--A review,” Color Res. Appl. 32, 201-207 (2007).
[CrossRef]

2006 (1)

F. Ameri, S. Moradian, M. A. Tehran, and K. Faez, “Use of transformed reflectance functions for neural network color match prediction systems,” Ind. J. Fib. Tex. Res. 31, 439-443 (2006).

2005 (1)

F. Ameri, S. Moradian, M. Amani Tehran, and K. Faez, “The use of fundamental color stimulus to improve the performance of artificial neural network color match prediction systems,” Iran. J. Chem. and Chem. Eng. 24, 53-61 (2005).

2004 (1)

I. Ariño, U. Kleist, and M. Rigdah, “Color of pigmented plastics--measurements and predictions,” Polym. Eng. Sci. 44, 141-152 (2004).
[CrossRef]

2003 (1)

B. Sluban and O. Šauperl, “Least metameric recipe formulation,” Croat. Chem. Acta 76, 161-166 (2003).

2001 (3)

M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE 2000 colour-difference formula: CIEDE 2000,” Color Res. Appl. 25, 340-350 (2001).
[CrossRef]

J. C. Ragain, Jr., and W. M. Johnston, “Accuracy of Kubelka-Munk reflectance theory applied to human dentin and enamel,” J. Dent. Res. 80, 449-452 (2001).
[CrossRef] [PubMed]

B. Philips-Invernizzi, D. Dupont, and C. Caze, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. (Bellingham) 40, 1082-1092 (2001).
[CrossRef]

1997 (1)

R. S. Berns, “A generic approach to color modeling,” Color Res. Appl. 22, 318-325 (1997).
[CrossRef]

1994 (1)

S. H. Amirshahi and M. T. Pailthorpe, “Applying the Kubelka-Munk equation to explain the color of blends prepared from precolored fibers,” Text. Res. J. 64, 357-364 (1994).
[CrossRef]

1993 (1)

B. Sluban, “Comparison of colorimetric and spectrophotometric algorithms for computer match prediction,” Color Res. Appl. 18, 74-79 (1993).
[CrossRef]

1992 (1)

C. S. Haase and G. W. Meyer, “Modeling pigmented materials for realistic image synthesis,” ACM Trans. Graphics 11, 305-332 (1992).
[CrossRef]

1991 (1)

H. R. Kang, “Kubelka-Munk modeling of ink jet ink mixing,” J. Imaging Technol. 17, 76-83 (1991).

1988 (1)

E. Walowit, C. J. McCarthy, and R. S. Berns, “Spectrophotometric color matching based on two-constant Kubelka-Munk theory,” Color Res. Appl. 13, 358-362 (1988).
[CrossRef]

1987 (2)

S. Moradian and B. Rigg, “The quantification of metamerism,” J. Soc. Dyers Colour. 103, 209-213 (1987).
[CrossRef]

H. S. Fairman, “Correction using parametric decomposition,” Color Res. Appl. 12, 261-265 (1987).
[CrossRef]

1985 (2)

J. B. Cohen and W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171-259 (1985).
[CrossRef]

J. H. Nobbs, “Kubelka-Munk theory and the prediction of reflectance,” Rev. Prog. Color. Relat. Top. 15, 66-75 (1985).
[CrossRef]

1982 (1)

J. B. Cohen and W. E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszecki's metameric black,” Austral. J. Earth. Sci. 95, 537-564 (1982).

1978 (1)

R. K. Winey, “Computer color matching with the aid of visual technique,” Color Res. Appl. 3, 165-167 (1978).
[CrossRef]

1974 (1)

1973 (2)

F. W. Billmeyer, Jr., R. L. Abrams, and J. G. Davidson, “Predicted reflectance and color of paint films by Kubelka-Munk analysis, I. Turbid medium theory,” J. Paint Technol. 45, 23-30 (1973).

F. W. Billmeyer, Jr., R. L. Abrams, and J. G. Davidson, “Predicted reflectance and color of paint films by Kubelka-Munk analysis, II. Performance tests,” J. Paint Technol. 45, 30-37 (1973).

1972 (1)

N. Ohta and H. Urabe, “Spectral color matching by means of minimax approximation,” Appl. Phys. Lett. 11, 2551-2553 (1972).

1967 (1)

P. H. McGinnis, “Spectrophotometric color matching with the least square technique,” Col. Eng. 5, 22-27 (1967).

1966 (1)

1963 (1)

D. W. Marquardet, “An algorithm for least square estimation of non linear parameters,” SIAM J. Appl. Math. 11, 431-441 (1963).
[CrossRef]

1953 (1)

G. Wyszecki, “Valenzmetrische untersuchung des zusammen-hanges zwischen normaler und anomaler Trichromasie (Psychlogical investigation of the relation between normal and abnormal trichromatic vision),” Die Far. 2, 39-52 (1953).

1948 (1)

1944 (1)

R. H. Park and E. I. Stearns, “Spectrophotometric formulation,” J. Opt. Soc. Am. 4, 112-113 (1944).
[CrossRef]

1942 (1)

1931 (1)

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche (An Article on Optics of Paint Layers),” Z. Tech. Phys. (Leipzig) 12, 593-601 (1931), see also www.graphics.cornell.edu/~westin/pubs/kubelka.pdf.

Abrams, R. L.

F. W. Billmeyer, Jr., R. L. Abrams, and J. G. Davidson, “Predicted reflectance and color of paint films by Kubelka-Munk analysis, II. Performance tests,” J. Paint Technol. 45, 30-37 (1973).

F. W. Billmeyer, Jr., R. L. Abrams, and J. G. Davidson, “Predicted reflectance and color of paint films by Kubelka-Munk analysis, I. Turbid medium theory,” J. Paint Technol. 45, 23-30 (1973).

Allen, E.

E. Allen, “Basic equations used in computer color matching, II. Tristimulus match,” J. Opt. Soc. Am. 64, 991-993 (1974).
[CrossRef]

E. Allen, “Basic equations used in computer color matching,” J. Opt. Soc. Am. 56, 1256-1259 (1966).
[CrossRef]

E. Allen, “Advances in colorant formulation and shading,” in Proceedings of AIC Color 77, F.W.Billmeyer, Jr., and G.Wyszecki, eds. (International Colour Association, 1978), pp 153-179.

E. Allen, “Colorant formulation and shading,” in Optical Radiation Measurements, Vol. 2, Color Measurement, F.Grum and C.J.Bartleson, eds. (Academic, 1980), pp. 289-336.

Amani Tehran, M.

F. Ameri, S. Moradian, M. Amani Tehran, and K. Faez, “The use of fundamental color stimulus to improve the performance of artificial neural network color match prediction systems,” Iran. J. Chem. and Chem. Eng. 24, 53-61 (2005).

Ameri, F.

F. Ameri, S. Moradian, M. A. Tehran, and K. Faez, “Use of transformed reflectance functions for neural network color match prediction systems,” Ind. J. Fib. Tex. Res. 31, 439-443 (2006).

F. Ameri, S. Moradian, M. Amani Tehran, and K. Faez, “The use of fundamental color stimulus to improve the performance of artificial neural network color match prediction systems,” Iran. J. Chem. and Chem. Eng. 24, 53-61 (2005).

Amirshahi, S. H.

S. H. Amirshahi and M. T. Pailthorpe, “Applying the Kubelka-Munk equation to explain the color of blends prepared from precolored fibers,” Text. Res. J. 64, 357-364 (1994).
[CrossRef]

Ariño, I.

I. Ariño, U. Kleist, and M. Rigdah, “Color of pigmented plastics--measurements and predictions,” Polym. Eng. Sci. 44, 141-152 (2004).
[CrossRef]

Asiaban, S.

A. Karbasi, S. Moradian, and S. Asiaban, “The use of two constant Kubelka-Munk theory in spectrophotometric color matching,” in Proceedings of ICE2007, P.Ziegler, ed. (International Coatings Expo, 2007).

Berns, R. S.

R. S. Berns and M. Mohammadi, “Single-constant simplification of Kubelka-Munk turbid-media theory for paint systems--A review,” Color Res. Appl. 32, 201-207 (2007).
[CrossRef]

R. S. Berns, “A generic approach to color modeling,” Color Res. Appl. 22, 318-325 (1997).
[CrossRef]

E. Walowit, C. J. McCarthy, and R. S. Berns, “Spectrophotometric color matching based on two-constant Kubelka-Munk theory,” Color Res. Appl. 13, 358-362 (1988).
[CrossRef]

F. H. Imai, M. R. Rosen, and R. S. Berns, “Comparative study of metrics for spectral match quality,” in Proceedings of CGIV 2002, C.Fernandez-Maloigne, ed. (IEEE Computer Society, 2002) pp. 492-496.

D. Y. Tzeng and R. S. Berns, “Spectral-based ink selection for multiple-ink printing I. Colorant estimation of original objects,” in Proceedings of 6th IS&T/SID Color Imaging Conference, (Society for Imaging Science & Technology, 1998), pp. 106-111.

R. S. Berns, Billmeyer and Saltzman's Principles of Color Technology, 3rd ed. (Wiley-Interscience, 2000).

D. Y. Tzeng and R. S. Berns, “Spectral-based ink selection for multiple-ink printing II. Optimal ink selection,” in Proceedings of 7th IS&T/SID Color Imaging Conference, (Society for Imaging Science & Technology, 1999), pp. 182-187.

Billmeyer, F. W.

F. W. Billmeyer, Jr., R. L. Abrams, and J. G. Davidson, “Predicted reflectance and color of paint films by Kubelka-Munk analysis, II. Performance tests,” J. Paint Technol. 45, 30-37 (1973).

F. W. Billmeyer, Jr., R. L. Abrams, and J. G. Davidson, “Predicted reflectance and color of paint films by Kubelka-Munk analysis, I. Turbid medium theory,” J. Paint Technol. 45, 23-30 (1973).

Caze, C.

B. Philips-Invernizzi, D. Dupont, and C. Caze, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. (Bellingham) 40, 1082-1092 (2001).
[CrossRef]

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, 1950).

Charvat, R. A.

R. A. Charvat, Coloring of Plastics: Fundamentals (Wiley-Interscience, 2004).

Choudhury, R.

A. Kumar and R. Choudhury, Modern Concepts of Color and Appearance (Science Publishers Inc., 2000).

Cohen, J. B.

J. B. Cohen and W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171-259 (1985).
[CrossRef]

J. B. Cohen and W. E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszecki's metameric black,” Austral. J. Earth. Sci. 95, 537-564 (1982).

Cui, G.

M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE 2000 colour-difference formula: CIEDE 2000,” Color Res. Appl. 25, 340-350 (2001).
[CrossRef]

Davidson, J. G.

F. W. Billmeyer, Jr., R. L. Abrams, and J. G. Davidson, “Predicted reflectance and color of paint films by Kubelka-Munk analysis, I. Turbid medium theory,” J. Paint Technol. 45, 23-30 (1973).

F. W. Billmeyer, Jr., R. L. Abrams, and J. G. Davidson, “Predicted reflectance and color of paint films by Kubelka-Munk analysis, II. Performance tests,” J. Paint Technol. 45, 30-37 (1973).

Dupont, D.

B. Philips-Invernizzi, D. Dupont, and C. Caze, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. (Bellingham) 40, 1082-1092 (2001).
[CrossRef]

Faez, K.

F. Ameri, S. Moradian, M. A. Tehran, and K. Faez, “Use of transformed reflectance functions for neural network color match prediction systems,” Ind. J. Fib. Tex. Res. 31, 439-443 (2006).

F. Ameri, S. Moradian, M. Amani Tehran, and K. Faez, “The use of fundamental color stimulus to improve the performance of artificial neural network color match prediction systems,” Iran. J. Chem. and Chem. Eng. 24, 53-61 (2005).

Fairman, H. S.

H. S. Fairman, “Correction using parametric decomposition,” Color Res. Appl. 12, 261-265 (1987).
[CrossRef]

Haase, C. S.

C. S. Haase and G. W. Meyer, “Modeling pigmented materials for realistic image synthesis,” ACM Trans. Graphics 11, 305-332 (1992).
[CrossRef]

Hanson, R. J.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Society for Industrial and Applied Mathematics, 1995).
[CrossRef]

Harris, R. M.

R. M. Harris, Coloring Technology for Plastics (Plastics Design Library, 1999).

Hernández-Andrés, J.

M. A. López-Álvarez, J. Hernández-Andrés, E. M. Valero, and J. L. Nieves, “Colorimetric and spectral combined metric for the optimization of multispectral systems,” in Proceedings of AIC Colour 05, J.Romero, ed. (International Colour Association, 2005), pp. 1685-1688.

Imai, F. H.

F. H. Imai, M. R. Rosen, and R. S. Berns, “Comparative study of metrics for spectral match quality,” in Proceedings of CGIV 2002, C.Fernandez-Maloigne, ed. (IEEE Computer Society, 2002) pp. 492-496.

Johnston, W. M.

J. C. Ragain, Jr., and W. M. Johnston, “Accuracy of Kubelka-Munk reflectance theory applied to human dentin and enamel,” J. Dent. Res. 80, 449-452 (2001).
[CrossRef] [PubMed]

Judd, D. B.

D. B. Judd and G. Wyszecki, Color in Business, Science and Industry (Wiley, 1975).

Kang, H. R.

H. R. Kang, “Kubelka-Munk modeling of ink jet ink mixing,” J. Imaging Technol. 17, 76-83 (1991).

Kappauf, W. E.

J. B. Cohen and W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171-259 (1985).
[CrossRef]

J. B. Cohen and W. E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszecki's metameric black,” Austral. J. Earth. Sci. 95, 537-564 (1982).

Karbasi, A.

A. Karbasi, S. Moradian, and S. Asiaban, “The use of two constant Kubelka-Munk theory in spectrophotometric color matching,” in Proceedings of ICE2007, P.Ziegler, ed. (International Coatings Expo, 2007).

Kleist, U.

I. Ariño, U. Kleist, and M. Rigdah, “Color of pigmented plastics--measurements and predictions,” Polym. Eng. Sci. 44, 141-152 (2004).
[CrossRef]

Kubelka, P.

P. Kubelka, “New contributions to the optics of intensely light scattering materials. Part I,” J. Opt. Soc. Am. 38, 448-457 (1948).
[CrossRef] [PubMed]

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche (An Article on Optics of Paint Layers),” Z. Tech. Phys. (Leipzig) 12, 593-601 (1931), see also www.graphics.cornell.edu/~westin/pubs/kubelka.pdf.

Kuehni, R.

R. Kuehni, Computer Color Formulation (Lexington Books, 1975).

Kumar, A.

A. Kumar and R. Choudhury, Modern Concepts of Color and Appearance (Science Publishers Inc., 2000).

Lawson, C. L.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Society for Industrial and Applied Mathematics, 1995).
[CrossRef]

López-Álvarez, M. A.

M. A. López-Álvarez, J. Hernández-Andrés, E. M. Valero, and J. L. Nieves, “Colorimetric and spectral combined metric for the optimization of multispectral systems,” in Proceedings of AIC Colour 05, J.Romero, ed. (International Colour Association, 2005), pp. 1685-1688.

Luo, M. R.

M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE 2000 colour-difference formula: CIEDE 2000,” Color Res. Appl. 25, 340-350 (2001).
[CrossRef]

Marquardet, D. W.

D. W. Marquardet, “An algorithm for least square estimation of non linear parameters,” SIAM J. Appl. Math. 11, 431-441 (1963).
[CrossRef]

McCarthy, C. J.

E. Walowit, C. J. McCarthy, and R. S. Berns, “Spectrophotometric color matching based on two-constant Kubelka-Munk theory,” Color Res. Appl. 13, 358-362 (1988).
[CrossRef]

McDonald, R.

R. McDonald, “Computer match prediction--dyes,” in Colour Physics for Industry, R.McDonald, ed. (Society of Dyers and Colourists, 1997), pp. 209-291.

McGinnis, P. H.

P. H. McGinnis, “Spectrophotometric color matching with the least square technique,” Col. Eng. 5, 22-27 (1967).

Meyer, G. W.

C. S. Haase and G. W. Meyer, “Modeling pigmented materials for realistic image synthesis,” ACM Trans. Graphics 11, 305-332 (1992).
[CrossRef]

Mohammadi, M.

R. S. Berns and M. Mohammadi, “Single-constant simplification of Kubelka-Munk turbid-media theory for paint systems--A review,” Color Res. Appl. 32, 201-207 (2007).
[CrossRef]

Moradian, S.

F. Ameri, S. Moradian, M. A. Tehran, and K. Faez, “Use of transformed reflectance functions for neural network color match prediction systems,” Ind. J. Fib. Tex. Res. 31, 439-443 (2006).

F. Ameri, S. Moradian, M. Amani Tehran, and K. Faez, “The use of fundamental color stimulus to improve the performance of artificial neural network color match prediction systems,” Iran. J. Chem. and Chem. Eng. 24, 53-61 (2005).

S. Moradian and B. Rigg, “The quantification of metamerism,” J. Soc. Dyers Colour. 103, 209-213 (1987).
[CrossRef]

A. Karbasi, S. Moradian, and S. Asiaban, “The use of two constant Kubelka-Munk theory in spectrophotometric color matching,” in Proceedings of ICE2007, P.Ziegler, ed. (International Coatings Expo, 2007).

Munk, F.

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche (An Article on Optics of Paint Layers),” Z. Tech. Phys. (Leipzig) 12, 593-601 (1931), see also www.graphics.cornell.edu/~westin/pubs/kubelka.pdf.

Nieves, J. L.

M. A. López-Álvarez, J. Hernández-Andrés, E. M. Valero, and J. L. Nieves, “Colorimetric and spectral combined metric for the optimization of multispectral systems,” in Proceedings of AIC Colour 05, J.Romero, ed. (International Colour Association, 2005), pp. 1685-1688.

Nobbs, J. H.

J. H. Nobbs, “Kubelka-Munk theory and the prediction of reflectance,” Rev. Prog. Color. Relat. Top. 15, 66-75 (1985).
[CrossRef]

J. H. Nobbs, “Colour-match prediction for pigmented materials,” in Colour Physics for Industry, R.McDonald, ed. (Society of Dyers and Colorists, 1997), pp. 292-372.

Ohta, N.

N. Ohta and H. Urabe, “Spectral color matching by means of minimax approximation,” Appl. Phys. Lett. 11, 2551-2553 (1972).

Pailthorpe, M. T.

S. H. Amirshahi and M. T. Pailthorpe, “Applying the Kubelka-Munk equation to explain the color of blends prepared from precolored fibers,” Text. Res. J. 64, 357-364 (1994).
[CrossRef]

Park, R. H.

R. H. Park and E. I. Stearns, “Spectrophotometric formulation,” J. Opt. Soc. Am. 4, 112-113 (1944).
[CrossRef]

Philips-Invernizzi, B.

B. Philips-Invernizzi, D. Dupont, and C. Caze, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. (Bellingham) 40, 1082-1092 (2001).
[CrossRef]

Ragain, J. C.

J. C. Ragain, Jr., and W. M. Johnston, “Accuracy of Kubelka-Munk reflectance theory applied to human dentin and enamel,” J. Dent. Res. 80, 449-452 (2001).
[CrossRef] [PubMed]

Rigdah, M.

I. Ariño, U. Kleist, and M. Rigdah, “Color of pigmented plastics--measurements and predictions,” Polym. Eng. Sci. 44, 141-152 (2004).
[CrossRef]

Rigg, B.

M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE 2000 colour-difference formula: CIEDE 2000,” Color Res. Appl. 25, 340-350 (2001).
[CrossRef]

S. Moradian and B. Rigg, “The quantification of metamerism,” J. Soc. Dyers Colour. 103, 209-213 (1987).
[CrossRef]

Rosen, M. R.

F. H. Imai, M. R. Rosen, and R. S. Berns, “Comparative study of metrics for spectral match quality,” in Proceedings of CGIV 2002, C.Fernandez-Maloigne, ed. (IEEE Computer Society, 2002) pp. 492-496.

Saunderson, J. L.

Šauperl, O.

B. Sluban and O. Šauperl, “Least metameric recipe formulation,” Croat. Chem. Acta 76, 161-166 (2003).

Schmid, H.

H. Schmid and D. Strocka, “Adaptation of computer colour matching to practical requirements,” in Proceedings of XIth FATIPEC, fatipec.com (1972), pp. 163-170.

Sluban, B.

B. Sluban and O. Šauperl, “Least metameric recipe formulation,” Croat. Chem. Acta 76, 161-166 (2003).

B. Sluban, “Comparison of colorimetric and spectrophotometric algorithms for computer match prediction,” Color Res. Appl. 18, 74-79 (1993).
[CrossRef]

Stearns, E. I.

R. H. Park and E. I. Stearns, “Spectrophotometric formulation,” J. Opt. Soc. Am. 4, 112-113 (1944).
[CrossRef]

Strocka, D.

H. Schmid and D. Strocka, “Adaptation of computer colour matching to practical requirements,” in Proceedings of XIth FATIPEC, fatipec.com (1972), pp. 163-170.

Tehran, M. A.

F. Ameri, S. Moradian, M. A. Tehran, and K. Faez, “Use of transformed reflectance functions for neural network color match prediction systems,” Ind. J. Fib. Tex. Res. 31, 439-443 (2006).

Tzeng, D. Y.

D. Y. Tzeng and R. S. Berns, “Spectral-based ink selection for multiple-ink printing I. Colorant estimation of original objects,” in Proceedings of 6th IS&T/SID Color Imaging Conference, (Society for Imaging Science & Technology, 1998), pp. 106-111.

D. Y. Tzeng and R. S. Berns, “Spectral-based ink selection for multiple-ink printing II. Optimal ink selection,” in Proceedings of 7th IS&T/SID Color Imaging Conference, (Society for Imaging Science & Technology, 1999), pp. 182-187.

Urabe, H.

N. Ohta and H. Urabe, “Spectral color matching by means of minimax approximation,” Appl. Phys. Lett. 11, 2551-2553 (1972).

Valero, E. M.

M. A. López-Álvarez, J. Hernández-Andrés, E. M. Valero, and J. L. Nieves, “Colorimetric and spectral combined metric for the optimization of multispectral systems,” in Proceedings of AIC Colour 05, J.Romero, ed. (International Colour Association, 2005), pp. 1685-1688.

Völz, H. G.

H. G. Völz, Industrial Color Testing--Fundamentals and Techniques (VCH Weinheim, 2001).
[CrossRef]

Walowit, E.

E. Walowit, C. J. McCarthy, and R. S. Berns, “Spectrophotometric color matching based on two-constant Kubelka-Munk theory,” Color Res. Appl. 13, 358-362 (1988).
[CrossRef]

Winey, R. K.

R. K. Winey, “Computer color matching with the aid of visual technique,” Color Res. Appl. 3, 165-167 (1978).
[CrossRef]

Wyszecki, G.

G. Wyszecki, “Valenzmetrische untersuchung des zusammen-hanges zwischen normaler und anomaler Trichromasie (Psychlogical investigation of the relation between normal and abnormal trichromatic vision),” Die Far. 2, 39-52 (1953).

D. B. Judd and G. Wyszecki, Color in Business, Science and Industry (Wiley, 1975).

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C. S. Haase and G. W. Meyer, “Modeling pigmented materials for realistic image synthesis,” ACM Trans. Graphics 11, 305-332 (1992).
[CrossRef]

Am. J. Psychol. (1)

J. B. Cohen and W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171-259 (1985).
[CrossRef]

Appl. Phys. Lett. (1)

N. Ohta and H. Urabe, “Spectral color matching by means of minimax approximation,” Appl. Phys. Lett. 11, 2551-2553 (1972).

Austral. J. Earth. Sci. (1)

J. B. Cohen and W. E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszecki's metameric black,” Austral. J. Earth. Sci. 95, 537-564 (1982).

Col. Eng. (1)

P. H. McGinnis, “Spectrophotometric color matching with the least square technique,” Col. Eng. 5, 22-27 (1967).

Color Res. Appl. (7)

R. K. Winey, “Computer color matching with the aid of visual technique,” Color Res. Appl. 3, 165-167 (1978).
[CrossRef]

E. Walowit, C. J. McCarthy, and R. S. Berns, “Spectrophotometric color matching based on two-constant Kubelka-Munk theory,” Color Res. Appl. 13, 358-362 (1988).
[CrossRef]

H. S. Fairman, “Correction using parametric decomposition,” Color Res. Appl. 12, 261-265 (1987).
[CrossRef]

M. R. Luo, G. Cui, and B. Rigg, “The development of the CIE 2000 colour-difference formula: CIEDE 2000,” Color Res. Appl. 25, 340-350 (2001).
[CrossRef]

R. S. Berns, “A generic approach to color modeling,” Color Res. Appl. 22, 318-325 (1997).
[CrossRef]

B. Sluban, “Comparison of colorimetric and spectrophotometric algorithms for computer match prediction,” Color Res. Appl. 18, 74-79 (1993).
[CrossRef]

R. S. Berns and M. Mohammadi, “Single-constant simplification of Kubelka-Munk turbid-media theory for paint systems--A review,” Color Res. Appl. 32, 201-207 (2007).
[CrossRef]

Croat. Chem. Acta (1)

B. Sluban and O. Šauperl, “Least metameric recipe formulation,” Croat. Chem. Acta 76, 161-166 (2003).

Die Far. (1)

G. Wyszecki, “Valenzmetrische untersuchung des zusammen-hanges zwischen normaler und anomaler Trichromasie (Psychlogical investigation of the relation between normal and abnormal trichromatic vision),” Die Far. 2, 39-52 (1953).

Ind. J. Fib. Tex. Res. (1)

F. Ameri, S. Moradian, M. A. Tehran, and K. Faez, “Use of transformed reflectance functions for neural network color match prediction systems,” Ind. J. Fib. Tex. Res. 31, 439-443 (2006).

Iran. J. Chem. and Chem. Eng. (1)

F. Ameri, S. Moradian, M. Amani Tehran, and K. Faez, “The use of fundamental color stimulus to improve the performance of artificial neural network color match prediction systems,” Iran. J. Chem. and Chem. Eng. 24, 53-61 (2005).

J. Dent. Res. (1)

J. C. Ragain, Jr., and W. M. Johnston, “Accuracy of Kubelka-Munk reflectance theory applied to human dentin and enamel,” J. Dent. Res. 80, 449-452 (2001).
[CrossRef] [PubMed]

J. Imaging Technol. (1)

H. R. Kang, “Kubelka-Munk modeling of ink jet ink mixing,” J. Imaging Technol. 17, 76-83 (1991).

J. Opt. Soc. Am. (5)

J. Paint Technol. (2)

F. W. Billmeyer, Jr., R. L. Abrams, and J. G. Davidson, “Predicted reflectance and color of paint films by Kubelka-Munk analysis, I. Turbid medium theory,” J. Paint Technol. 45, 23-30 (1973).

F. W. Billmeyer, Jr., R. L. Abrams, and J. G. Davidson, “Predicted reflectance and color of paint films by Kubelka-Munk analysis, II. Performance tests,” J. Paint Technol. 45, 30-37 (1973).

J. Soc. Dyers Colour. (1)

S. Moradian and B. Rigg, “The quantification of metamerism,” J. Soc. Dyers Colour. 103, 209-213 (1987).
[CrossRef]

Opt. Eng. (Bellingham) (1)

B. Philips-Invernizzi, D. Dupont, and C. Caze, “Bibliographical review for reflectance of diffusing media,” Opt. Eng. (Bellingham) 40, 1082-1092 (2001).
[CrossRef]

Polym. Eng. Sci. (1)

I. Ariño, U. Kleist, and M. Rigdah, “Color of pigmented plastics--measurements and predictions,” Polym. Eng. Sci. 44, 141-152 (2004).
[CrossRef]

Rev. Prog. Color. Relat. Top. (1)

J. H. Nobbs, “Kubelka-Munk theory and the prediction of reflectance,” Rev. Prog. Color. Relat. Top. 15, 66-75 (1985).
[CrossRef]

SIAM J. Appl. Math. (1)

D. W. Marquardet, “An algorithm for least square estimation of non linear parameters,” SIAM J. Appl. Math. 11, 431-441 (1963).
[CrossRef]

Text. Res. J. (1)

S. H. Amirshahi and M. T. Pailthorpe, “Applying the Kubelka-Munk equation to explain the color of blends prepared from precolored fibers,” Text. Res. J. 64, 357-364 (1994).
[CrossRef]

Z. Tech. Phys. (Leipzig) (1)

P. Kubelka and F. Munk, “Ein beitrag zur optik der farbanstriche (An Article on Optics of Paint Layers),” Z. Tech. Phys. (Leipzig) 12, 593-601 (1931), see also www.graphics.cornell.edu/~westin/pubs/kubelka.pdf.

Other (19)

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Society for Industrial and Applied Mathematics, 1995).
[CrossRef]

A. Karbasi, S. Moradian, and S. Asiaban, “The use of two constant Kubelka-Munk theory in spectrophotometric color matching,” in Proceedings of ICE2007, P.Ziegler, ed. (International Coatings Expo, 2007).

R. Kuehni, Computer Color Formulation (Lexington Books, 1975).

H. Schmid and D. Strocka, “Adaptation of computer colour matching to practical requirements,” in Proceedings of XIth FATIPEC, fatipec.com (1972), pp. 163-170.

E. Allen, “Colorant formulation and shading,” in Optical Radiation Measurements, Vol. 2, Color Measurement, F.Grum and C.J.Bartleson, eds. (Academic, 1980), pp. 289-336.

A. Kumar and R. Choudhury, Modern Concepts of Color and Appearance (Science Publishers Inc., 2000).

D. Y. Tzeng and R. S. Berns, “Spectral-based ink selection for multiple-ink printing I. Colorant estimation of original objects,” in Proceedings of 6th IS&T/SID Color Imaging Conference, (Society for Imaging Science & Technology, 1998), pp. 106-111.

D. Y. Tzeng and R. S. Berns, “Spectral-based ink selection for multiple-ink printing II. Optimal ink selection,” in Proceedings of 7th IS&T/SID Color Imaging Conference, (Society for Imaging Science & Technology, 1999), pp. 182-187.

J. H. Nobbs, “Colour-match prediction for pigmented materials,” in Colour Physics for Industry, R.McDonald, ed. (Society of Dyers and Colorists, 1997), pp. 292-372.

R. McDonald, “Computer match prediction--dyes,” in Colour Physics for Industry, R.McDonald, ed. (Society of Dyers and Colourists, 1997), pp. 209-291.

R. S. Berns, Billmeyer and Saltzman's Principles of Color Technology, 3rd ed. (Wiley-Interscience, 2000).

S. Chandrasekhar, Radiative Transfer (Oxford U. Press, 1950).

D. B. Judd and G. Wyszecki, Color in Business, Science and Industry (Wiley, 1975).

F. H. Imai, M. R. Rosen, and R. S. Berns, “Comparative study of metrics for spectral match quality,” in Proceedings of CGIV 2002, C.Fernandez-Maloigne, ed. (IEEE Computer Society, 2002) pp. 492-496.

M. A. López-Álvarez, J. Hernández-Andrés, E. M. Valero, and J. L. Nieves, “Colorimetric and spectral combined metric for the optimization of multispectral systems,” in Proceedings of AIC Colour 05, J.Romero, ed. (International Colour Association, 2005), pp. 1685-1688.

R. A. Charvat, Coloring of Plastics: Fundamentals (Wiley-Interscience, 2004).

R. M. Harris, Coloring Technology for Plastics (Plastics Design Library, 1999).

H. G. Völz, Industrial Color Testing--Fundamentals and Techniques (VCH Weinheim, 2001).
[CrossRef]

E. Allen, “Advances in colorant formulation and shading,” in Proceedings of AIC Color 77, F.W.Billmeyer, Jr., and G.Wyszecki, eds. (International Colour Association, 1978), pp 153-179.

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Figures (11)

Fig. 1
Fig. 1

Reflectance spectra of white substrate (i.e., master batch containing 0.5% Ti O 2 + PP ) together with 6 singly pigmented white PP (i.e., Y Yellow , R Red , G Green , B Blue , V Violet , K Black ).

Fig. 2
Fig. 2

Three-dimensional representation and encompassed color gamut for the 70 colored samples.

Fig. 3
Fig. 3

Schematic algorithm for a general computer matching procedure

Fig. 4
Fig. 4

Typical example of spectral reflectance curve of a target together with its corresponding fundamental color stimulus for (a) a nonmetameric match, (b) a metameric match.

Fig. 5
Fig. 5

Average color differences obtained using different matching procedures.

Fig. 6
Fig. 6

Average values of the general metameric index obtained using different matching procedures.

Fig. 7
Fig. 7

Average values of RMS obtained using different matching procedures

Fig. 8
Fig. 8

Average values of WRMS1 and WRMS2 obtained using different matching procedures

Fig. 9
Fig. 9

The number of possible recipes having Δ E less than 0.5 for different matching procedures

Fig. 10
Fig. 10

Average Δ C values obtained for comparable recipes

Fig. 11
Fig. 11

Top, linear plot of MI6 versus RMS of R MB . Center, linear plot of Δ E MAX versus RMS of R MB . Bottom, right linear plot of MI6 versus Δ E MAX .

Tables (4)

Tables Icon

Table 1 Characteristics of the Achromatic and Chromatic Pigments

Tables Icon

Table 2 Recipes of All 70 Color Centers

Tables Icon

Table 3 Different Matching Procedures

Tables Icon

Table 4 Correlation Coefficients ( r 2 ) of Various Functions Relating Potential Indices of Metamerism

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

f t f s = Φ c ,
i w i 2 [ Δ R i ] 2 min .
c = ( T E D Φ ) 1 × T E D [ f t f s ] ,
c = [ T E ( D k × Φ k + D s × Φ s ) ] 1 × T E { D k [ K t K s ] + D s [ S t S s ] } ,
Δ t = B × Δ c ,
w D 65 2 ( Δ E D 65 ) 2 + w A 2 ( Δ E A ) 2 + w F 11 2 ( Δ E F 11 ) 2 min ,
matrix R = A ( A T A ) 1 A T ,
i [ Δ R FCS ] i 2 min .
RMS = [ i ( R i t R i m ) 2 n ] 1 2 ,
WRMS = { i [ w i ( R i t R i m ) ] 2 n } 1 2 ,
Δ R FCS = matrix R × [ R i c j ] Δ c .
Δ R = [ R i c j ] × Δ c ,
R i c j = ( R K ) i K i c j + ( R S ) i S i c j ,
( R K ) i = 2 R i 2 [ S i ( 1 R i 2 ) ] ,
( R S ) i = R i ( 1 R i ) [ S i ( 1 + R i 2 ) ] ,
K i c j = K ij ,
S i c j = S ij .
[ R i c j ] = ( D k × Φ k + D s × Φ s ) ,
w × ( D k × Φ k + D s × Φ s ) × Δ c = w × Δ R .
Δ c = ( P T P ) 1 × P T × w × Δ R ,
where P = w × ( D k × Φ k + D s × Φ s ) .

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