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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 60, Iss. 7 — Jul. 1, 1970
  • pp: 951–958

Achromatic-Point Prediction

GERALD L. HOWETT  »View Author Affiliations

JOSA, Vol. 60, Issue 7, pp. 951-958 (1970)

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This paper constitutes a mathematical elaboration, in a form permitting direct predictions of data, of the empirical discoveries of Helson and Michels concerning the “effect of chromatic adaptation on achromaticity.” An equation is developed permitting the chromaticity of a test spot that appears achromatic against an extended chromatic background to be predicted from the background chromaticity and the luminance ratio of spot to background. The technique for numerical solution of the equation is explained, and a family of curves allowing approximate solution by interpolation is presented. If the luminance of the achromatic spot is no greater than that of the background, the achromatic chromaticity always lies more than ¾ of the way along the line directed from the absolute (black background) neutral point to the background point, regardless of the color of the background. As part of a discussion of additional numerical methods useful in applying the model to data, general least-squares formulas are presented for the coordinates of the point best representing the common intersection of any set of given lines; and for the line, passing through a specified point, that best fits a set of given points by the criterion of perpendicular deviations.

GERALD L. HOWETT, "Achromatic-Point Prediction," J. Opt. Soc. Am. 60, 951-958 (1970)

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  1. H. Helson and W. C. Michels, J. Opt. Soc. Am. 38, 1025 (1948).
  2. The notation to be used here is consistent with that of Ref. 1, except that the subscript a (for "achromatic") replaces Helson and Michel's subscript s (for "spot"). It is too easy to think of s as standing for "surround."
  3. See, for example, C.R.C. Standard Mathematical Tables, 12th ed., edited by C. D. Hodgman (Chemical Rubber, Cleveland, 1959), p. 358.
  4. See any standard work on numerical analysis; for example, J. B. Scarborough, Numerical Mathematical Analysis, 6th ed. (Johns Hopkins Press, Baltimore, 1966), p. 201.
  5. A polynomial of degree p requires no more than p multiplications to evaluate. For example, the cubic in Eq. (21) is evaluated with 3 multiplications by using the form ƒ(r)≡[(a3r+a2)r+a1]r-1.
  6. See Ref. 4, p. 202.
  7. Based on the smoothed interpolations for 1-nm intervals of the 1931 CIE color-matching functions, adopted by the CIE in 1967. See G. Wyszecki and W. S. Stiles, Color Science: Concepts and Methods, Quantitative Data and Formulas (Wiley, New York, 1967), p. 242.
  8. Reference 1, Fig. 2, P. 1030.
  9. J. A. S. Kinney and S. M. Luria, in Proceedings of the Fourth Symposium on Underwater Physiology, edited by C. J. Lambertsen (Academic, New York, 1970).
  10. J. A. S. Kinney, personal communication.
  11. K. Pearson, Phil Mag. 2, 559 (1901).
  12. C. L. Sanders and G. Wyszecki, in CIE Proceedings Vienna 1963 (CIE, Paris, 1964), Vol. B (Lighting Technique), p. 221. See entry in their Table I, p. 225, for -492/8.
  13. D. L. MacAdam, J. Opt. Soc. Am. 39, 454 (1949).
  14. D. V. Lindley, J. Roy. Stat. Soc. (Suppl.) 9, 218 (1947).
  15. See Ref. 14, p. 236. Note that Lindley asserts incorrectly that in his analog of Eq. (43) [his Eq. (70)], the plus sign is always the correct choice for minimizing the criterion sum. This error has, been perpetuated in some later texts, such as Statistical Methods in Research and Production, 3rd ed., edited by O. L. Davies (Oliver and Boyd, London and Edinburgh, 1957), p. 174. Other texts, however, present the equation in a form that yields the correct sign. See, for example, M. G. Kendall and A. Stuart, The Advanced Theory of Statistics (Charles Griffin, London, 1961), Vol. 2, p. 381.

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