Abstract

Using a large number of specially selected imaginary object colors which are metameric with respect to one set of color-mixture functions, the spatial distribution of these colors with respect to the other set of color-mixture functions provides an illustrative means of measuring the total difference of the two sets of color-mixture functions. The spatial distribution follows a normal trivariate distribution law which allows the computation of an ellipsoid that is expected to contain 95% of all theoretically and practically possible object colors of the same class used to calculate that ellipsoid. A numerical example involving the color-mixture functions of the 1931 CIE standard observer and the color-mixture functions derived from the Stiles 10° pilot data demonstrates the theory.

© 1959 Optical Society of America

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References

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  1. E. Schrödinger, Ann. Physik 62, 603 (1920).
    [CrossRef]
  2. S. Rösch, Fortschr. Mineral. Krist. Petrog. 13, 143 (1929).
  3. D. L. MacAdam, J. Opt. Soc. Am. 25, 249 (1935).
    [CrossRef]
  4. D. B. Judd, J. Opt. Soc. Am. 23, 359 (1933). Also see reference a, Table I.
    [CrossRef]
  5. G. Wyszecki, J. Opt. Soc. Am. 48, 451 (1958).
    [CrossRef]
  6. W. S. Stiles, Optica Acta (Paris) 2, 168 (1955).
    [CrossRef]
  7. Tables provided by D. B. Judd, National Bureau of Standards, Washington, D. C.

1958 (1)

1955 (1)

W. S. Stiles, Optica Acta (Paris) 2, 168 (1955).
[CrossRef]

1935 (1)

1933 (1)

1929 (1)

S. Rösch, Fortschr. Mineral. Krist. Petrog. 13, 143 (1929).

1920 (1)

E. Schrödinger, Ann. Physik 62, 603 (1920).
[CrossRef]

Judd, D. B.

D. B. Judd, J. Opt. Soc. Am. 23, 359 (1933). Also see reference a, Table I.
[CrossRef]

Tables provided by D. B. Judd, National Bureau of Standards, Washington, D. C.

MacAdam, D. L.

Rösch, S.

S. Rösch, Fortschr. Mineral. Krist. Petrog. 13, 143 (1929).

Schrödinger, E.

E. Schrödinger, Ann. Physik 62, 603 (1920).
[CrossRef]

Stiles, W. S.

W. S. Stiles, Optica Acta (Paris) 2, 168 (1955).
[CrossRef]

Wyszecki, G.

Ann. Physik (1)

E. Schrödinger, Ann. Physik 62, 603 (1920).
[CrossRef]

Fortschr. Mineral. Krist. Petrog. (1)

S. Rösch, Fortschr. Mineral. Krist. Petrog. 13, 143 (1929).

J. Opt. Soc. Am. (3)

Optica Acta (Paris) (1)

W. S. Stiles, Optica Acta (Paris) 2, 168 (1955).
[CrossRef]

Other (1)

Tables provided by D. B. Judd, National Bureau of Standards, Washington, D. C.

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Figures (1)

F. 1
F. 1

Portion of the x10,y10-chromaticity diagram showing a number of contour lines of the c.m.f. ellipsoid of the Stiles 10 pilot data as compared with the CIE 2° data. The central contour line is indicated by ±0.0 and lies in the plane Y = 25.0(%). The other contour lines, above and underneath the central contour line, are indicated by ±0.3, ±0.6, ±0.9, respectively, and lie in planes Y = const which are separated from the central plane by just these amounts. The corresponding centers of the contour lines are shown by small circles connected by a straight line. Point “C” indicates the location of CIE standard source “C” and all 108 metameric grays on the 1931 CIE x,y-chromaticity diagram.

Tables (1)

Tables Icon

Table I Classification of the difference between two sets of color-mixture functions (c.m.f.).a

Equations (20)

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U i = k i λ s λ l r ( λ ) H ( λ ) ū i ( λ ) d λ , ( i = 1 , 2 , 3 ) .
ū i ( λ ) = j a i j ū j ( λ ) , ( i , j = 1 , 2 , 3 ) ,
i ( λ ) = t ( λ ) ū i ( λ ) , ( i = 1 , 2 , 3 ) ,
V i = k i λ s λ l r ( λ ) H ( λ ) t ( λ ) ū i ( λ ) d λ .
U i = U i ( α ) = k i λ s λ l r ( α ) ( λ ) H ( λ ) ū i ( λ ) d λ . ( α = 1 , 2 , 3 , 4 , ) .
V i ( α ) = k i λ s λ l r ( α ) ( λ ) H ( λ ) i ( λ ) d λ ,
V i V i ( α ) .
u i = U i / i U i , ( i = 1 , 2 , 3 )
υ i ( λ ) = k i t ( λ ) ū i ( λ ) i k i t ( λ ) ū i ( λ ) = k i ū i ( λ ) / i k i ū i ( λ ) .
w ¯ i ( λ ) = σ i ( λ ) ū i ( λ ) ,
W i ( α ) = λ s λ l r ( α ) ( λ ) H ( λ ) w ¯ i ( λ ) d λ .
r ( α ) ( λ ) = r ¯ ( λ ) + s α ρ ( α ) ( λ ) .
0 r ( α ) ( λ ) 1.0 .
X 1 = W 1 ( α ) W ¯ 1 , X 2 = W 2 ( α ) W ¯ 2 , X 3 = W 3 ( α ) W ¯ 3 ,
W ¯ 1 = 1 m α = 1 m W i ( α ) .
Q i j = 1 m 1 α = 1 m X i X j ( i , j = 1 , 2 , 3 ) .
i , j = 1 3 Q i j X i X j
i , j = 1 3 Q i j X i X j = 7.81 .
i ( λ ) = t ( λ ) ū i ( λ )
w ¯ i ( λ ) = σ i ( λ ) ū i ( λ )