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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 22, Iss. 5 — May. 1, 2005
  • pp: 810–819

Metamer sets

Graham D. Finlayson and Peter Morovic  »View Author Affiliations

JOSA A, Vol. 22, Issue 5, pp. 810-819 (2005)

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If two different surfaces look the same when viewed under a particular light source, then they are called metamers. We show mathematically how one can solve for the whole set of physically realizable natural surface reflectances that relate to the same tristimulus, the metamer set. Our analysis is based on very general linear models of reflectances, coupled with constraints that reflectances should adhere to (e.g., positivity and boundedness). We show that we can recover metamer sets for linear models of an arbitrary high dimension. To illustrate our new algorithm, we provide an example of calculating the metamer set and its manifestation as a mismatch region. Given a single X Y Z observed under illuminant D65, we can examine the set of X Y Z s that would be possible under illuminant A.

© 2005 Optical Society of America

OCIS Codes
(330.0330) Vision, color, and visual optics : Vision, color, and visual optics
(330.1710) Vision, color, and visual optics : Color, measurement
(330.1720) Vision, color, and visual optics : Color vision
(330.6180) Vision, color, and visual optics : Spectral discrimination
(330.7310) Vision, color, and visual optics : Vision

Original Manuscript: January 29, 2004
Revised Manuscript: July 26, 2004
Manuscript Accepted: September 21, 2004
Published: May 1, 2005

Graham D. Finlayson and Peter Morovic, "Metamer sets," J. Opt. Soc. Am. A 22, 810-819 (2005)

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  1. G. Wyszecki, W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulae, 2nd ed. (Wiley, New York, 1982).
  2. G. Wyszecki, “Evaluation of metameric colors,” J. Opt. Soc. Am. 48, 451–454 (1958). [CrossRef]
  3. J. B. Cohen, W. E. Kappauf, “Metameric color stimuli, fundamental metamers, and Wyszecki’s metameric blacks,” Am. J. Psychol. 95, 537–564 (1982). [CrossRef] [PubMed]
  4. E. H. Land, “The retinex theory of color vision,” Sci. Am. 237, 108–128 (1977). [CrossRef] [PubMed]
  5. B. Smith, C. Spiekermann, R. Sember, “Numerical methods for colorimetric calculations: sampling density requirements,” Color Res. Appl. 17, 394–401 (1992). [CrossRef]
  6. E. L. Krinov, “Spectral reflectance properties of natural formations,” Tech. Transl. TT-439 (National Research Council of Canada, Ottawa, 1947).
  7. J. B. Cohen, “Dependency of the spectral reflectance curves of the Munsell color chips,” Psychon. Sci. 1, 369–370 (1964). [CrossRef]
  8. L. T. Maloney, “Evaluation of linear models of surface spectral reflectance with small numbers of parameters,” J. Opt. Soc. Am. A 3, 1673–1683 (1986). [CrossRef] [PubMed]
  9. J. P. S. Parkkinen, J. Hallikanen, T. Jaaskelainen, “Characteristic spectra of Munsell colors,” J. Opt. Soc. Am. A 6, 318–322 (1989). [CrossRef]
  10. S. Westland, J. Shaw, H. Owens, “Color statistics of natural and man-made surfaces,” Remote Sens. Rev. 20, 50–55 (2000).
  11. L. T. Maloney, B. A. Wandell, “Color constancy: a method for recovering surface spectral reflectance,” J. Opt. Soc. Am. A 3, 29–33 (1986). [CrossRef] [PubMed]
  12. J. B. Cohen, W. E. Kappauf, “Color mixture and fundamental metamers: theory, algebra, geometry, application,” Am. J. Psychol. 98, 171–259 (1985). [CrossRef]
  13. J. B. Fraleigh, R. A. Beauregard, Linear Algebra (Addison-Wesley, Reading, Mass., 1990).
  14. S. A. Burns, J. B. Cohen, E. N. Kuznetsov, “Multiple metamers: preserving color matches under diverse illuminants,” Color Res. Appl. 14, 16–22 (1989). [CrossRef]
  15. F. J. M. Schmitt, “A method for the treatment of metamerism in colorimetry,” J. Opt. Soc. Am. 66, 601–608 (1976). [CrossRef] [PubMed]
  16. K. Takahama, Y. Nayatani, “New method for generating metameric stimuli of object colors,” J. Opt. Soc. Am. 62, 1516–1520 (1972). [CrossRef]
  17. N. Ohta, “Generating metameric object colors,” J. Opt. Soc. Am. 65, 1081–1082 (1975). [CrossRef]
  18. R. S. Berns, F. W. Billmeyer, R. S. Sacher, “Methods for generating spectral reflectance functions leading to color-constant properties,” Color Res. Appl. 10, 73–83 (1985). [CrossRef]
  19. J. M. Zoido, F. Carreño, E. Bernabeu, “Improved linear programming method to generate metameric spectral distributions,” Appl. Opt. 34, 1938–1943 (1995). [CrossRef] [PubMed]
  20. J. M. Speigle, D. H. Brainard, “Luminosity thresholds: effects of test chromaticity and ambient illumination,” J. Opt. Soc. Am. A 13, 436–451 (1996). [CrossRef]
  21. H. J. Trussell, “Applications of set-theoretic methods to color systems,” Color Res. Appl. 16, 31–41 (1991). [CrossRef]
  22. D. H. Marimont, B. A. Wandell, “Linear models of surface and illuminant spectra,” J. Opt. Soc. Am. A 9, 1905–1913 (1992). [CrossRef] [PubMed]
  23. T. Katriňák, M. Galvanec, E. Gedeonová, J. Smítal, Algebra a Teoretická Aritmetika (Alfa, Bratislava, Czechoslovakia, 1985).
  24. P. Morovic, “Metamer Sets,” Ph.D. thesis (University of East Anglia, Norwich, UK, 2002).
  25. G. H. Golub, C. F. van Loan, Matrix Computations, 3rd ed. (Johns Hopkins U. Press, Baltimore, Md., 1996).
  26. F. P. Preparata, M. I. Shamos, Computational Geometry—An Introduction, 2nd ed. (Springer-Verlag, New York, 1988).
  27. C. B. Barber, D. D. Dobkin, H. Huhdanpaa, “The Quickhull algorithm for convex hulls,” ACM Trans. Math. Softw. 22, 469–483 (1996). [CrossRef]
  28. V. Chvátal, Linear Programing (Freeman, New York, 1983).
  29. N. Ohta, G. Wyszecki, “Theoretical chromaticity mismatch limits of metamers viewed under different illuminants,” J. Opt. Soc. Am. 65, 327–333 (1975). [CrossRef]
  30. J. A. Worthey, “Calculation of metameric reflectances,” Color Res. Appl. 13, 76–84 (1988). [CrossRef]
  31. W. A. Thornton, “Matching lights, metamers, and human visual response,” J. Color Appearance 2, 23–29 (1973).
  32. N. Ohta, G. Wyszecki, “Location of the nodes of metameric color stimuli,” Color Res. Appl. 2, 183–186 (1977). [CrossRef]
  33. N. Ohta, “Intersections of spectral curves of metameric colors,” Color Res. Appl. 12, 85–87 (1987). [CrossRef]
  34. G. D. Finlayson, P. Morovic, “Metamer crossovers of infinite metamer sets,” in Proceedings of the IS&T/SID Eighth Color Imaging Conference (The Society of Imaging Science and Technology, Springfield, Va. 2000), pp. 13–17..
  35. W. S. Stiles, G. Wyszecki, “Counting metameric object colors,” J. Opt. Soc. Am. 52, 313–328 (1962). [CrossRef]
  36. G. D. Finlayson, P. Morovic, “Metamer constrained colour correction,” J. Imaging Sci. Technol. 44, 295–300 (2000).

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