Abstract

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 34 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.

© 1970 Optical Society of America

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References

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  1. H. Helson and W. C. Michels, J. Opt. Soc. Am. 38, 1025 (1948).
    [CrossRef] [PubMed]
  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 f(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).
    [CrossRef] [PubMed]
  14. D. V. Lindley, J. Roy. Stat. Soc. (Suppl.) 9, 218 (1947).
    [CrossRef]
  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.

1949 (1)

1948 (1)

1947 (1)

D. V. Lindley, J. Roy. Stat. Soc. (Suppl.) 9, 218 (1947).
[CrossRef]

1901 (1)

K. Pearson, Phil Mag. 2, 559 (1901).

Helson, H.

Kinney, J. A. S.

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).

J. A. S. Kinney, personal communication.

Lindley, D. V.

D. V. Lindley, J. Roy. Stat. Soc. (Suppl.) 9, 218 (1947).
[CrossRef]

Luria, S. M.

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).

MacAdam, D. L.

Michels, W. C.

Pearson, K.

K. Pearson, Phil Mag. 2, 559 (1901).

Sanders, C. L.

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.

Scarborough, J. B.

See any standard work on numerical analysis; for example, J. B. Scarborough, Numerical Mathematical Analysis, 6th ed. (Johns Hopkins Press, Baltimore, 1966), p. 201.

Stiles, W. S.

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.

Wyszecki, G.

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.

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.

J. Opt. Soc. Am. (2)

J. Roy. Stat. Soc. (Suppl.) (1)

D. V. Lindley, J. Roy. Stat. Soc. (Suppl.) 9, 218 (1947).
[CrossRef]

Phil Mag. (1)

K. Pearson, Phil Mag. 2, 559 (1901).

Other (11)

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.

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.

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.”

See, for example, C.R.C. Standard Mathematical Tables, 12th ed., edited by C. D. Hodgman (Chemical Rubber, Cleveland, 1959), p. 358.

See any standard work on numerical analysis; for example, J. B. Scarborough, Numerical Mathematical Analysis, 6th ed. (Johns Hopkins Press, Baltimore, 1966), p. 201.

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 f(r) ≡ [(a3r+ a2)r+ a1]r− 1.

See Ref. 4, p. 202.

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.

Reference 1, Fig. 2, P. 1030.

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).

J. A. S. Kinney, personal communication.

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Figures (2)

Fig. 1
Fig. 1

Adaptation ratio, r[ = (yay0)/(yby0)], as a function of the luminance ratio, Q (=La/Lb), with ρ [= (yby0)/y0] as the parameter. This is a graphical display of the family of solutions of the cubic equation (18). The partial curves for ρ = 2.7 and −0.986 delimit the range of ρ values corresponding to real background colors. The dashed curves outside these limits apply to physically unrealizable background colors.

Fig. 2
Fig. 2

Mean “white” points (×) for four observers, surrounded by one-standard-deviation scatter ellipses for the white (solid), red (dashed), and blue–green (dotted) backgrounds. Δ, Background chromaticities: W = white, R = red, BG = blue–green. □, Absolute neutral point derived from the data by the author. [Modified from a figure in an unpublished report by J. A. S. Kinney and J. C. Cooper, with the kind permission of Kinney.]

Tables (2)

Tables Icon

Table I Lines fitted to Helson and Michels data (constrained to pass through background points). Optimum intersection point: x ˆ 0 = 0.263, ŷ0 = 0.243; mean experimental absolute neutral: x0 = 0.269, y0 = 0.269.

Tables Icon

Table II Summary of Kinney and Cooper data.

Equations (47)

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R = ( X a + Y a + Z a ) / ( X b + Y b + Z b ) .
r = 1 / ( 1 + 0.21 R 2 ) .
Q = L a / L b
x 0 = 0.269 ± 0.026 y 0 = 0.269 ± 0.040 z 0 = 0.462 ± 0.057.
y = Y / ( X + Y + Z ) .
X + Y + Z = Y / y = L / y .
X b + Y b + Z b = L b / y b
X a + Y a + Z a = L a / y a .
L a = Q L b .
X a + Y a + Z a = Q L b / y a ;
R = ( X a + Y a + Z a ) / ( X b + Y b + Z b ) = Q ( y b / y a ) .
x a = x 0 + r Δ x ,             y a = y 0 + r Δ y ,             z a = z 0 + r Δ z ,
Δ x = x b - x 0 ,             Δ y = y b - y 0 ,             Δ z = z b - z 0 .
R = Q [ ( y 0 + Δ y ) / ( y 0 + r Δ y ) ] .
r = 1 / { 1 + 0.21 Q 2 [ ( y 0 + Δ y ) / ( y 0 + r Δ y ) ] 2 } ,
ρ = Δ y / y 0 .
( y 0 + Δ y ) / ( y 0 + r Δ y ) = ( 1 + ρ ) / ( 1 + r ρ ) ,
ρ 2 r 3 + ρ ( 2 - ρ ) r 2 + [ 0.21 Q 2 ( 1 + ρ ) 2 + 1 - 2 ρ ] r - 1 = 0.
f ( r ) a 3 r 3 + a 2 r 2 + a 1 r - 1 ,
a 1 = 0.21 Q 2 ( 1 + ρ ) 2 + 1 - 2 ρ ,             a 2 = ρ ( 2 - ρ ) ,             a 3 = ρ 2 .
f ( r ) 3 a 3 r 2 + 2 a 2 r + a 1 .
r n + 1 = r n - f ( r n ) / f ( r n ) .
r 0 = 1 / ( 1 + 0.21 Q 2 ) ,
f ( r ) ( r - 1 ) 3
f ( r ) 3 ( r - 1 ) 2 .
ρ = ( y b - y 0 ) / y 0 .
R = Q [ ( 1 + ρ ) / ( 1 + r ρ ) ] .
S = i = 1 n ( m i p + b i - q ) 2 m i 2 + 1 .
p = D p / D ,             q = D q / D ,
D = [ i - ( Δ y ) 2 ( Δ x ) 2 + ( Δ y ) 2 ] [ i ( Δ x ) 2 ( Δ x ) 2 + ( Δ y ) 2 ] - [ i - Δ x Δ y ( Δ x ) 2 + ( Δ y ) 2 ] 2 ,
D p = [ i Δ x Δ y ( Δ x ) 2 + ( Δ y ) 2 ] [ i Δ x ( y Δ x - x Δ y ) ( Δ x ) 2 + ( Δ y ) 2 ] - [ i ( Δ x ) 2 ( Δ x ) 2 + ( Δ y ) 2 ] [ i Δ y ( y Δ x - x Δ y ) ( Δ x ) 2 + ( Δ y ) 2 ] ,
D q = [ i ( Δ y ) 2 ( Δ x ) 2 + ( Δ y ) 2 ] [ i Δ x ( y Δ x - x Δ y ) ( Δ x ) 2 + ( Δ y ) 2 ] - [ i Δ x Δ y ( Δ x ) 2 + ( Δ y ) 2 ] [ i Δ y ( y Δ x - x Δ y ) ( Δ x ) 2 + ( Δ y ) 2 ] ,
Δ y = m Δ x ,             y = m x + b ;
background range of r red 0.25 - 0.42 white 0.18 - 0.49 blue-green 0.14 - 0.43.
Q = [ 1 + ( 2 ρ - 1 ) r - ρ ( 2 - ρ ) r 2 - ρ 2 r 3 0.21 ( 1 + ρ ) 2 r ] 1 2 ,
background range of Q red 2.3 - 3.3 white 2.5 - 5.6 blue-green 2.5 - 5.2.
r = 1 / ( 1 + 0.54 R 3 ) .
r = 1 / ( 1 + k R p ) ,
f ( r ) ( 1 + ρ r ) p ( r - 1 ) + k Q p ( 1 + ρ ) p r
f ( r ) ( 1 + ρ r ) p - 1 { 1 + ρ [ ( 1 + ρ ) r - p ] } + k Q p ( 1 + ρ ) p .
x = x - p ,             y = y - q ,
u = ( y ) 2 - ( x ) 2 2 Σ x y             ( Σ x y 0 ) ,
m ˆ = u ± ( u 2 + 1 ) 1 2 ,
b ˆ = q - m ˆ p .
orientation equation m ˆ condition vertical x = p ( y ) 2 > ( x ) 2 horizontal y = q 0 ( y ) 2 < ( x ) 2 indeterminate - - ( y ) 2 = ( x ) 2 .
x ¯ = ( 1 / n ) x ,             y ¯ = ( 1 / n ) y .
p = x ¯ ,             q = y ¯