Abstract

The characteristics of the whole set of tristimulus values for “all” object surfaces can be studied by applying statistical methods. This leads to the determination (a) of the relative numbers of object colors that are metameric with respect to any fixed reference color, and (b) of the distribution of the tristimulus values in one trichromatic system of all object surfaces that are metameric in another trichromatic system. Subject to certain restrictions, Gaussian distributions are found to apply. The conclusions bear on important colorimetric problems such as color-rendering and degree of metamerism.

© 1962 Optical Society of America

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References

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  1. International Commission on Illumination: Proceedings 8th Session, Cambridge, 1931 (Cambridge University Press, London, England, 1932), p. 19.
  2. E. Schrödinger, Ann. Physik 62, 603 (1920).
    [Crossref]
  3. S. Rösch, Fortschr. Mineral., Krist., Petrog. 13, 143 (1929).
  4. D. L. MacAdam, J. Opt. Soc. Am. 25, 249 (1935).
    [Crossref]
  5. This multiple integral with simple limits is valid provided it is understood that F(x1,x2⋯xM) is defined to be zero if any xi lies outside the interval 0≤xi≤1.
  6. See for example: H. Cramér, Mathematical Methods of Statistics. (Princeton University Press, Princeton, New Jersey, 1946).
  7. H. Cramér, Random variables and probability distributions. Cambridge Tracts in Mathematics and Mathematical Physics No. 36. (Cambridge University Press, London, 1937).
  8. See, for example: S. Goldman, Information Theory (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1953).
  9. G. Wyszecki, J. Opt. Soc. Am. 48, 451 (1958).
    [Crossref]
  10. G. Wyszecki, CIE Proc. Brussels 1959, Vol.  A, 146 (1960).
  11. G. Wyszecki, J. Opt. Soc. Am. 49, 811 (1959).
    [Crossref]
  12. R. Courant and D. Hilbert, Methoden der Mathematischen Physik I (Verlag Julius Springer, Berlin, 1931), p. 19.

1960 (1)

G. Wyszecki, CIE Proc. Brussels 1959, Vol.  A, 146 (1960).

1959 (1)

1958 (1)

1935 (1)

1929 (1)

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

1920 (1)

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

Courant, R.

R. Courant and D. Hilbert, Methoden der Mathematischen Physik I (Verlag Julius Springer, Berlin, 1931), p. 19.

Cramér, H.

See for example: H. Cramér, Mathematical Methods of Statistics. (Princeton University Press, Princeton, New Jersey, 1946).

H. Cramér, Random variables and probability distributions. Cambridge Tracts in Mathematics and Mathematical Physics No. 36. (Cambridge University Press, London, 1937).

Goldman, S.

See, for example: S. Goldman, Information Theory (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1953).

Hilbert, D.

R. Courant and D. Hilbert, Methoden der Mathematischen Physik I (Verlag Julius Springer, Berlin, 1931), p. 19.

MacAdam, D. L.

Rösch, S.

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

Schrödinger, E.

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

Wyszecki, G.

Ann. Physik (1)

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

CIE Proc. Brussels 1959 (1)

G. Wyszecki, CIE Proc. Brussels 1959, Vol.  A, 146 (1960).

Fortschr. Mineral., Krist., Petrog. (1)

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

J. Opt. Soc. Am. (3)

Other (6)

International Commission on Illumination: Proceedings 8th Session, Cambridge, 1931 (Cambridge University Press, London, England, 1932), p. 19.

R. Courant and D. Hilbert, Methoden der Mathematischen Physik I (Verlag Julius Springer, Berlin, 1931), p. 19.

This multiple integral with simple limits is valid provided it is understood that F(x1,x2⋯xM) is defined to be zero if any xi lies outside the interval 0≤xi≤1.

See for example: H. Cramér, Mathematical Methods of Statistics. (Princeton University Press, Princeton, New Jersey, 1946).

H. Cramér, Random variables and probability distributions. Cambridge Tracts in Mathematics and Mathematical Physics No. 36. (Cambridge University Press, London, 1937).

See, for example: S. Goldman, Information Theory (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1953).

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Figures (10)

Fig. 1
Fig. 1

Slice of the object-color space cut perpendicularly to the Y axis. Each cell of the slice contains the number of colors found in a Monte Carlo computation.

Fig. 2
Fig. 2

Cumulative relative frequency distribution of Y values of approximately 10 000 object colors obtained in a Monte Carlo computation.

Fig. 3
Fig. 3

Color-rendering ellipsoid for CIE source “A” as compared to CIE source “C”. Ellipses shown are cross sections of the ellipsoid at planes of Y′=const. Value given for each cross section indicates displacement from central section (0.0) located at Y′=25.00.

Fig. 4
Fig. 4

Color-rendering ellipsoid for Macbeth daylight (6800°K) (source “M”) as compared to CIE source “C”. Ellipses shown are cross sections of the ellipsoid at planes of Y′=const. Value given for each cross section indicates displacement from central section (0.0) located at Y′=25.00.

Fig. 5
Fig. 5

Color-mixture-function ellipsoid for Stiles’ 10° Pilot observer as compared to CIE 2° observer. Ellipses shown are cross sections of the ellipsoid at planes of Y′=const. Value given for each cross section indicates displacement from central section (0.0)located at Y′=25.00.

Fig. 6
Fig. 6

Spectral-reflectance functions ρλ which with respect to CIE source “C” and the CIE 2° standard observer give metameric grays of tristimulus values X1Y=25.00, X2X=24.51, X3Z=29.53.

Fig. 7
Fig. 7

The degree of metamerism with respect to Stiles’ 10° Pilot observer as compared to the CIE 2° standard observer. The four solid dots indicate location of four colors as seen by Stiles’ 10° Pilot observer but which are metameric grays for the CIE 2° standard observer.

Fig. 8
Fig. 8

Cross sections of a sphere of reference colors in XYZ space with center at X=49.02, Y=50.00, Z=59.05 and radius r=10. Value given for each cross section indicates displacement from central section (0.0) located at Y=50.00.

Fig. 9
Fig. 9

Distortion ellipsoid for CIE source “A” as compared to reference sphere for CIE source “C” (See Fig. 8). Ellipses shown are cross sections of the ellipsoid at planes of Y′=const. Value given for each cross section indicates displacement from central section (0.0) located at Y′=50.00.

Fig. 10
Fig. 10

Distortion ellipsoid for source “M” as compared to reference sphere for CIE source “C” (See Fig. 8). Ellipses shown (which are practically circles) are cross sections of the ellipsoid at planes of Y′=const. Value given for each cross section indicates displacement from central section (0.0) located at Y′=50.00.

Tables (3)

Tables Icon

Table I Departure from normality for a single variable. (Numerical example for a simplified case.)

Tables Icon

Table II Frequency distribution of Y values obtained from Monte Carlo calculations.

Tables Icon

Table III Values of Dβγ. (Dβγ=Dγβ).

Equations (94)

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X = k λ ρ λ E λ x ¯ λ d λ ,             Y = k λ ρ λ E λ y ¯ λ d λ ,             Z = k λ ρ λ E λ z ¯ λ d λ ,
k = 100 / λ E λ y ¯ λ d λ .
λ ρ λ ( 1 ) E λ x ¯ λ d λ = λ ρ λ ( 2 ) E λ x ¯ λ d λ λ ρ λ ( 1 ) E λ y ¯ λ d λ = λ ρ λ ( 2 ) E λ y ¯ λ d λ λ ρ λ ( 1 ) E λ z ¯ λ d λ = λ ρ λ ( 2 ) E λ z ¯ λ d λ ,
X ( 1 ) = X ( 2 ) Y ( 1 ) = Y ( 2 ) Z ( 1 ) = Z ( 2 ) ,
0 ρ λ 1 ,
X = 1 M ρ i a i             with             a i = ( i ) E λ x ¯ λ d λ ,
Y = 1 M ρ i b i             with             b i = ( i ) E λ y ¯ λ d λ ,
Z = 1 M ρ i c i             with             c i = ( i ) E λ z ¯ λ d λ .
F ( ρ 1 , ρ M ) d ρ 1 d ρ M
ρ 1 = 0 1 ρ M = 0 1 F ( ρ 1 , ρ M ) d ρ 1 d ρ M = 1 .
F ( ρ 1 , ρ M ) = F 1 ( ρ 1 ) F M ( ρ M ) .
X = i = 1 M ρ i a i
F ( ρ 1 , ρ M - 1 , X - i = 1 M - 1 a i ρ i a M ) d ρ 1 d ρ M - 1 d X a M .
Φ ( X ) = ρ M - 1 = 0 1 ρ 1 = 0 1 F ( ρ 1 , ρ M - 1 , X - i = 1 M - 1 a i ρ i a M ) × d ρ 1 d ρ M - 1 a M .
F i ( ξ i / a i ) d ξ i / a i .
ξ ¯ i = 0 a i ξ i F i ( ξ a i ) d ξ a i = a i 0 1 ρ F i ( ρ ) d ρ ,
σ i 2 = 0 a i ξ i 2 F i ( ξ i a i ) d ξ i a i - ( ξ ¯ i ) 2 , = a i 2 { 0 1 ρ 2 F i ( ρ ) d ρ - [ 0 1 ρ F i ( ρ ) d ρ ] 2 } .
Φ ( X ) d X = d X σ ( 2 π ) 1 2 exp - 1 2 σ 2 ( X - X m ) 2 ,
X m = i = 1 M ξ ¯ i ,
σ 2 = i = 1 M σ i 2 .
ξ ¯ i = a i / 2 ,             σ i 2 = a i 2 / 12             for all i ,
X m = 1 2 i = 1 M a i = 1 2 λ E λ x ¯ λ d λ
σ 2 = 1 12 i = 1 M ( ( i ) E λ x ¯ λ d λ ) 2 .
σ 2 = δ 12 λ ( E λ x ¯ λ ) 2 d λ .
σ 2 = 12 λ ( E λ x ¯ λ λ ) 2 d λ .
X m = r 2 λ E λ x ¯ λ d λ ,
r = 2 0 1 ρ F 0 ( ρ ) d ρ
σ 2 = s δ 12 λ ( E λ x ¯ λ ) 2 d λ ,
s = 12 { 0 1 ρ 2 F 0 ( ρ ) d ρ - [ 0 1 ρ F 0 ( ρ ) d ρ ] 2 } .
X k = λ E λ x ¯ k , λ d λ
Φ ( X 1 , X 2 X n ) d X 1 d X 2 d X n ,
X k = i = 1 M ξ i ,
Φ ( X 1 , X n ) d X 1 d X n = d X 1 d X n ( 2 π ) n / 2 σ k l 1 2 exp - 1 2 P n
P n = k = 1 n l = 1 n σ k l ( X k - X m , k ) ( X l - X m , l ) .
X m , k = i = 1 M a k , i 0 1 ρ F 0 ( ρ ) d ρ .
with             σ k l = s 12 i = 1 M a k , i a l , i a k , i = ( i ) E λ x ¯ k , λ d λ             and             a l , i = ( i ) E λ x ¯ l , λ d λ .
u k , λ = l = 1 n t k l ( E λ x ¯ l , λ )             ( k = 1 to n ) ,
λ u k , λ u l , λ d λ = { 1 for k = l 0 for k l .
( t 11 0 0 0 t 21 t 22 0 0 t n 1 t n 2 t n n ) .
U k = l = 1 n t k l X l             ( k = 1 to n ) .
ψ ( U 1 , U n ) d U 1 d U n = d U 1 d U n ( 2 π s δ / 12 ) n / 2 exp - 12 2 s δ k = 1 n ( U k - U m , k ) 2 ,
s = 12 { 0 1 ρ 2 F 0 ( ρ ) d ρ - [ 0 1 ρ F 0 ( ρ ) d ρ ] 2 }
ψ ( U 1 , U n ) d X 1 d X n J = d X 1 d X n ( 2 π s δ / 12 ) n / 2 J exp - 1 2 Q n ,
J = | U 1 X 1 U 1 X n U n X 1 U n X n | = | t 11 t 1 n t n 1 t n n | ,
Q n = 12 s δ [ l = 1 n j = 1 n ( k = 1 n t k l t k j ) ( X l - X m , l ) ( X j - X m , j ) ] .
U 4 = - + U n = - + ψ ( U 1 , U n ) d U 1 d U n .
Ω E ( U 1 , 0 , U 2 , 0 , U 3 , 0 ) d U 1 d U 2 d U 3 = d U 1 d U 2 d U 3 ( 2 π s δ / 12 ) 3 2 exp - 1 2 Q 3 ,
Q 3 = 12 s δ [ ( U 1 , 0 - U m , 1 ) 2 + ( U 2 , 0 - U m , 2 ) 2 + ( U 3 , 0 - U m , 3 ) 2 ] .
Ω F ( U 4 , U n ) d U 4 d U n = d U 4 d U n ( 2 π s δ / 12 ) ( n - 3 ) / 2 exp - 1 2 Q n - 3
Q n - 3 = 12 s δ [ k = 4 n ( U k - U m , k ) 2 ] .
f n - 3 ( χ 2 ) d ( χ 2 ) = ( χ 2 / 2 ) ( n - 5 ) / 2 2 Γ [ ( n - 3 ) / 2 ] e - χ 2 / 2 d ( χ 2 ) .
F ( χ 2 ) = 0 χ 2 ( t / 2 ) ( n - 5 ) / 2 2 Γ [ ( n - 3 ) / 2 ] e - t / 2 d t
k = 4 n ( U k - U m , k ) 2 = s δ 12 χ 2
l = 4 n j = 4 n [ ( k = 4 n t k l l k j ) ( X l - X l , 0 ) ( X j - X j , 0 ) ] = s δ 12 χ 2 ,
j = 4 n τ l j X j , 0 = b l             ( l = 4 to n ) ,
τ l j = k = 4 n t k l t k j
b l = k = 4 n L k t k l ,
L k = t k 1 X 1 , 0 + t k 2 X 2 , 0 + t k 3 X 3 , 0 - U m , k .
- ( X 4 , 0 - X m , 4 ) = ( 1 / t 44 ) [ t 41 ( X 1 , 0 - X m , 1 ) + t 42 ( X 2 , 0 - X m , 2 ) + t 43 ( X 3 , 0 - X m , 3 ) ] - ( X 5 , 0 - X m , 5 ) = ( 1 / t 44 t 55 ) [ ( t 51 t 44 - t 41 t 54 ) ( X 1 , 0 - X m , 1 ) + ( t 52 t 44 - t 42 t 54 ) ( X 2 , 0 - X m , 2 ) + ( t 53 t 44 - t 43 t 54 ) ( X 3 , 0 - X m , 3 ) ] - ( X 6 , 0 - X m , 6 ) = ( 1 / t 44 t 55 t 66 ) [ ( t 61 t 44 t 55 - t 51 t 44 t 65 + t 41 t 54 t 65 - t 41 t 55 t 64 ) ( X 1 , 0 - X m , 1 ) + ( t 62 t 44 t 55 - t 52 t 44 t 65 + t 42 t 54 t 65 - t 42 t 55 t 64 ) ( X 2 , 0 - X m , 2 ) + ( t 63 t 44 t 55 - t 53 t 44 t 65 + t 43 t 54 t 65 - t 43 t 55 t 64 ) ( X 3 , 0 - X m , 3 ) ] .
- 0.3 < ( Y / 2 Y m - 0.5 ) < + 0.3
1 / 2 σ 2 = 43.7.
1 / 2 σ 2 = 44.0
x ¯ 1 ( λ ) = E λ y ¯ λ , x ¯ 4 ( λ ) = F λ y ¯ λ x ¯ 2 ( λ ) = E λ x ¯ λ , x ¯ 5 ( λ ) = F λ x ¯ λ x ¯ 3 ( λ ) = E λ z ¯ λ , x ¯ 6 ( λ ) = F λ z ¯ λ .
X k = i = 1 30 ρ i a k i ,
a k i = ( i ) x ¯ k ( λ ) d λ
a k i = δ x ¯ k i ,
λ x ¯ k ( λ ) d λ = δ i = 1 30 x ¯ k i .
τ 44 ( X 4 - X 4 , 0 ) 2 + τ 55 ( X 5 - X 5 , 0 ) 2 + τ 66 ( X 6 - X 6 , 0 ) 2 + 2 τ 45 ( X 4 - X 4 , 0 ) ( X 5 - X 5 , 0 ) + 2 τ 56 ( X 5 - X 5 , 0 ) × ( X 6 - X 6 , 0 ) + 2 τ 64 ( X 6 - X 6 , 0 ) ( X 4 - X 4 , 0 ) = ( δ / 12 ) χ 2 ,
τ 44 = t 44 2 + t 54 2 + t 64 2 = 156.13 τ 55 = t 55 2 + t 65 2 = 3.20 τ 66 = t 66 2 = 16.76 τ 45 = t 54 t 55 + t 64 t 65 = 4.28 τ 56 = t 65 t 66 = - 4.02 τ 64 = t 64 t 66 = 11.15.
X 1 , 0 = 25.000 X 2 , 0 = 24.510 X 3 , 0 = 29.527 ,
X 4 , 0 = 25.00 X 5 , 0 = 27.46 X 6 , 0 = 8.89.
τ 44 = 62.80 τ 55 = 50.74 τ 66 = 31.65 τ 45 = - 45.81 τ 56 = - 3.90 τ 64 = 0.82
X 4 , 0 = 25.00 X 5 , 0 = 23.86 X 6 , 0 = 28.55
x ¯ 1 ( λ ) = E λ y ¯ λ x ¯ 4 ( λ ) = E λ y ¯ λ x ¯ 2 ( λ ) = E λ x ¯ λ x ¯ 5 ( λ ) = E λ x ¯ λ x ¯ 3 ( λ ) = E λ z ¯ λ x ¯ 6 ( λ ) = E λ z ¯ λ ,
λ s λ ( 1 ) x ¯ k , λ d λ = λ s λ ( 2 ) x ¯ k , λ d λ ,             ( k = 1 , 2 , 3 )
x ¯ 1 , λ = x ¯ 1 , λ E λ ,             x ¯ 2 , λ = x ¯ 2 , λ E λ ,             x ¯ 3 , λ = x ¯ 3 , λ E λ ,
x ¯ 4 , λ = x ¯ 4 , λ E λ ,             x ¯ n , λ = x ¯ n , λ E λ .
X 1 = 25.00 ( = Y c ) , X 2 = 24.51 ( = X c ) , X 3 = 29.53 ( = Z c ) .
D β γ = d β γ / d m             ( β , γ = 1 , 2 , 3 , 4 )
d β γ 2 = ( X 4 ( β ) - X 4 ( γ ) ) 2 + ( X 5 ( β ) - X 5 ( γ ) ) 2 + ( X 6 ( β ) - X 6 ( γ ) ) 2
F ( s 1 , s M ) d s 1 d s M ,
F i ( s i ) = F 0 ( s i ) = 1 / L for 0 s i L = 0 for s i > L .
( X k , 0 - X m k ) = l = 1 3 α l j ( X l , 0 - X m l ) ,             k = 4 , 5 , 6.
( X k - X k , c ) = l = 1 3 α l j ( X l - X l , c ) ,             k = 4 , 5 , 6.
l = 1 3 ( X l - X l , c ) 2 = r 2
γ 11 ( X 4 - X 4 , c ) 2 + γ 22 ( X 5 - X 5 , c ) 2 + γ 33 ( X 6 - X 6 , c ) 2 + 2 γ 12 ( X 4 - X 4 , c ) ( X 5 - X 5 , c ) + 2 γ 13 ( X 4 - X 4 , c ) ( X 6 - X 6 , c ) + 2 γ 23 ( X 5 - X 5 , c ) ( X 6 - X 6 , c ) = r 2 ,
γ 11 = β 11 2 + β 21 2 + β 31 2 γ 12 = β 11 β 12 + β 21 β 22 + β 31 β 32 γ 22 = β 12 2 + β 22 2 + β 32 2 γ 13 = β 11 β 13 + β 21 β 23 + β 31 β 33 γ 33 = β 13 2 + β 23 2 + β 33 2 γ 23 = β 12 β 13 + β 22 β 23 + β 32 β 33
β 11 = 1 Δ | t 42 t 43 t 44 t 52 t 53 t 54 t 62 t 63 t 64 | β 12 = - 1 Δ | t 42 t 43 0 t 52 t 53 t 55 t 62 t 63 t 65 | β 13 = t 66 Δ | t 42 t 43 t 52 t 53 | β 21 = - 1 Δ | t 41 t 43 t 44 t 51 t 53 t 54 t 61 t 63 t 64 | β 22 = 1 Δ | t 41 t 43 0 t 51 t 53 t 55 t 61 t 63 t 65 | β 23 = - t 66 Δ | t 41 t 43 t 51 t 53 | β 31 = 1 Δ | t 41 t 42 t 44 t 51 t 52 t 54 t 61 t 62 t 64 | β 32 = - 1 Δ | t 41 t 42 0 t 51 t 52 t 55 t 61 t 62 t 65 | β 33 = t 66 Δ | t 41 t 42 t 51 t 52 | Δ = - | t 41 t 42 t 43 t 51 t 52 t 53 t 61 t 62 t 63 | .
X m 1 = 50.00 = X 1 , c X m 4 = 50.00 = X 4 , c X m 2 = 49.02 = X 2 , c X m 5 = 54.91 = X 5 , c X m 3 = 59.05 = X 3 , c X m 6 = 17.77 = X 6 , c
0.0383 0 0 0 0 0 - 0.0435 0.0630 0 0 0 0 0.0141 - 0.0260 0.0256 0 0 0 - 1.1378 - 0.7064 0.1522 1.6463 0 0 - 0.3913 - 0.5970 0.1041 0.6323 0.2164 0 - 0.3369 0.0763 - 0.1622 0.3940 - 0.1420 0.5920
γ 11 = 1.8252 γ 12 = - 0.4404 γ 22 = 0.5853 γ 13 = - 0.5835 γ 33 = 13.0786 γ 23 = 0.9775
X m 1 = 50.00 = X 1 , c X m 4 = 50.00 = X 4 , c X m 2 = 49.02 = X 2 , c X m 5 = 47.73 = X 5 , c X m 3 = 59.05 = X 3 , c X m 6 = 57.08 = X 6 , c
0.0383 0 0 0 0 0 - 0.0435 0.0630 0 0 0 0 0.0141 - 0.0260 0.0256 0 0 0 - 0.6812 0.0068 0.0011 0.6660 0 0 0.9191 - 0.9945 0.0003 - 0.9324 1.0252 0 - 0.0166 0.0939 - 0.7796 0.0210 - 0.1002 0.8135
γ 11 = 0.9565 γ 12 = - 0.0256 γ 22 = 1.0826 γ 13 = 0.0037 γ 33 = 1.0892 γ 23 = - 0.0018