Abstract

Color discrimination for the general case of combined chromaticity and luminance differences has been investigated by analysis of the errors of visual trichromatic colorimetry. Formulas based on those devised by Silberstein have been used to determine the coefficients and axes of ellipsoids in color space from the data. Methods for transforming the results to other coordinate systems are presented and used to give the results in terms of both the ICI tristimulus X, Y, Z values and the ICI x, y, chromaticity values. Most of the results are nearly symmetrical above and below the chromaticity (constant luminance) cross section of each ellipsoid. Those cross sections are comparable with results previously published, and the agreement is satisfactory. The influence on color discrimination of changing the level of luminance has been studied for three colors. The ratio between the standard of deviation of each primary and its amount seems to be a function of only the luminance contributed by that primary to the color mixture.

© 1949 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. D. Wright, Proc. Phys. Soc. 53, 93–112 (1941);D. L. MacAdam, J. Opt. Soc. Am. 32247–274 (1942).
    [Crossref]
  2. L. Silberstein and D. L. MacAdam, J. Opt Soc. Am. 3532–39 (1945).
    [Crossref]
  3. L. Silberstein, Phil. Mag., Ser. 7,  37, 126–144 (1946).
  4. D. L. MacAdam, J. Opt. Soc. Am. 3318–26 (1943).
    [Crossref]
  5. Hecht, Peskin, and Patt, J. Gen. Physiol. 22, 7–19 (1938).
  6. D. L. MacAdam, J. Opt. Soc. Am. 32, 247–274 (1942).
    [Crossref]
  7. A. W. Wundheiler, J. Opt. Soc. Am. 36, 288–291 (1946).
    [Crossref]
  8. L. Silberstein, J. Opt. Soc. Am. 36, 464–468 (1946).
    [Crossref] [PubMed]
  9. L. Tuttle and J. Satterly, The Theory of Measurements (Longmans Green and Company, London, 1925).

1946 (3)

1945 (1)

L. Silberstein and D. L. MacAdam, J. Opt Soc. Am. 3532–39 (1945).
[Crossref]

1943 (1)

1942 (1)

1941 (1)

W. D. Wright, Proc. Phys. Soc. 53, 93–112 (1941);D. L. MacAdam, J. Opt. Soc. Am. 32247–274 (1942).
[Crossref]

1938 (1)

Hecht, Peskin, and Patt, J. Gen. Physiol. 22, 7–19 (1938).

Hecht,

Hecht, Peskin, and Patt, J. Gen. Physiol. 22, 7–19 (1938).

MacAdam, D. L.

Patt,

Hecht, Peskin, and Patt, J. Gen. Physiol. 22, 7–19 (1938).

Peskin,

Hecht, Peskin, and Patt, J. Gen. Physiol. 22, 7–19 (1938).

Satterly, J.

L. Tuttle and J. Satterly, The Theory of Measurements (Longmans Green and Company, London, 1925).

Silberstein, L.

L. Silberstein, J. Opt. Soc. Am. 36, 464–468 (1946).
[Crossref] [PubMed]

L. Silberstein, Phil. Mag., Ser. 7,  37, 126–144 (1946).

L. Silberstein and D. L. MacAdam, J. Opt Soc. Am. 3532–39 (1945).
[Crossref]

Tuttle, L.

L. Tuttle and J. Satterly, The Theory of Measurements (Longmans Green and Company, London, 1925).

Wright, W. D.

W. D. Wright, Proc. Phys. Soc. 53, 93–112 (1941);D. L. MacAdam, J. Opt. Soc. Am. 32247–274 (1942).
[Crossref]

Wundheiler, A. W.

J. Gen. Physiol. (1)

Hecht, Peskin, and Patt, J. Gen. Physiol. 22, 7–19 (1938).

J. Opt Soc. Am. (1)

L. Silberstein and D. L. MacAdam, J. Opt Soc. Am. 3532–39 (1945).
[Crossref]

J. Opt. Soc. Am. (4)

Phil. Mag. (1)

L. Silberstein, Phil. Mag., Ser. 7,  37, 126–144 (1946).

Proc. Phys. Soc. (1)

W. D. Wright, Proc. Phys. Soc. 53, 93–112 (1941);D. L. MacAdam, J. Opt. Soc. Am. 32247–274 (1942).
[Crossref]

Other (1)

L. Tuttle and J. Satterly, The Theory of Measurements (Longmans Green and Company, London, 1925).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (16)

F. 1
F. 1

Standard deviation ellipsoid, representing combined chromaticity and luminance discrimination, with constant-luminance cross sections showing displacement of center and of point of maximum luminance.

F. 2
F. 2

Front view of colorimeter used to determine visual sensitivities for combined chromaticity and luminance differences.

F. 3
F. 3

End view of colorimeter.

F. 4
F. 4

Top view of colorimeter.

F. 5
F. 5

Calibration curves for various assumed values of the optical distance from the scale index to the diffuser.

F. 6
F. 6

Slopes of calibration curves for various assumed values of optical distances from the scale indices to the diffuser.

F. 7
F. 7

Chromaticity diagram showing primaries used in colorimeter, filters used in calibration, coordinate system based on one calibration, and colors around which discrimination was investigated.

F. 8
F. 8

Constant-luminance cross sections of discrimination ellipsoids for WRJB (solid), DLM (broken), PGN (dotted), and displacements of points of maximum luminance, for colors 1 to 3 in left column, 4 to 6 in right column.

F. 9
F. 9

Constant-luminance cross sections of discrimination ellipsoids and displacements of points of maximum luminance for colors 7 to 9 (left), 11 to 13 (right).

F. 10
F. 10

Constant-luminance cross sections of discrimination ellipsoids and displacements of points of maximum luminance for colors 14 to 19.

F. 11
F. 11

Constant-luminance cross sections of discrimination ellipsoids, and displacements of points of maximum luminance for colors 20 to 25.

F. 12
F. 12

Constant-luminance cross sections of discrimination ellipsoids and displacements of points of maximum luminance for colors 26 to 31.

F. 13
F. 13

Constant-luminance cross sections of discrimination ellipsoids, and displacements of points of maximum luminance for colors 32 to 36.

F. 14
F. 14

Constant-luminance cross sections of discrimination ellipsoids and displacements of points of maximum luminance for colors 37, 38 and 39.

F. 15
F. 15

Cross sections at constant luminance of discrimination ellipsoids for WRJB, enlarged ten times on chromaticity diagram.

F. 16
F. 16

Fractional errors of colorimeter primaries as functions of luminance contributed by each primary to match: +=Values for red primary●●●=Values for green primary×××=Values for blue primary

Tables (9)

Tables Icon

Table I Experimental means (in millimeters).

Tables Icon

Table II Standard deviations and correlations of instrument primaries.

Tables Icon

Table III Discrimination coefficients for instrument primaries.

Tables Icon

Table IV Discrimination coefficients for ICI primaries: X, Y, Z.

Tables Icon

Table V Coefficients (×10−4) of discrimination in x, y, and principal axes in section for Δl=0.

Tables Icon

Table VI Determinants (×10−12) and minors (×10−8) of discrimination ellipsoids in x, y, l and coordinates of point having maximum l.

Tables Icon

Table VII Lengths and orientations of principal axes of discrimination ellipsoids in x, y, 1 5 coordinates.

Tables Icon

Table VIII Influence of luminance variation on precision of color-matching.

Tables Icon

Table IX Luminance of components and fractional error of color-matching

Equations (69)

Equations on this page are rendered with MathJax. Learn more.

P = A exp ( c 11 x 1 2 + c 12 x 1 x 2 + c 13 x 1 x 3 + c 21 x 2 x 1 + c 22 x 2 2 + c 23 x 2 x 3 + c 31 x 3 x 1 + c 32 x 3 x 2 + c 33 x 3 2 ) · d x 1 d x 2 d x 3 ,
P = A exp ( c i k x i x k ) · d x 1 d x 2 d x 3 .
f ( x 1 , x 2 , x 3 ) = A exp ( c i k x i x k ) .
1 = A + exp ( c i k x i x k ) · d x 1 d x 2 d x 3 .
A = ( D / π 3 ) 1 2 ,
D = | c 11 c 12 c 13 c 21 c 22 c 23 c 31 c 32 c 33 | = | c i k |
f ( x 1 , x 2 , x 3 ) = ( | c i k | / π 3 ) 1 2 exp ( c i k x i x k ) .
j = 1 n log f i ,
( 1 / n ) ( log f j ) / c i k = ( 1 2 log | c i k | c i k x i x k / n ) / c i k = 0 = C i k / 2 | c i k | x i x k / n = 0 .
( x 1 2 / n ) 1 2 , ( x 2 2 / n ) 1 2 , and ( x 3 2 / n ) 1 2
x i x k x k x i ,
f ( x 1 , x 2 , x 3 ) = ( | c i k | / π 3 ) 1 2 exp ( ( c 11 x 1 2 + 2 c 12 x 1 x 2 + c 22 x 2 2 + 2 c 23 x 2 x 3 + c 33 x 3 2 + 2 c 13 x 1 x 3 ) ) .
x i = Δ X i = 2 X i Δ L i / L i = 2 X i Δ S i / L i ,
x i x k / n = ( 4 X i X k / L i L k ) ( Δ S i Δ S k / n )
c i k = ( 8 X i X k / L i L k ) ( Δ S i Δ S k / n ) ,
( c 11 ) 1 2 = 2 σ r , ( c 22 ) 1 2 = 2 σ g , and ( c 33 ) 1 2 = 2 σ b .
ρ i k = 1 ρ r g ρ r b ρ r g 1 ρ g b ρ r b ρ g b 1 .
ρ r g = 1 n Δ S r · Δ S g / ( Δ S r 2 ) 1 2 · ( Δ S g 2 ) 1 2 .
σ r = 2 [ R ¯ / ( S ¯ r + 455 ) ] · [ Δ S r 2 / n ] 1 2 ,
c i k = ρ i k / 2 σ i σ k .
Δ S 2 = c 11 Δ R 2 + 2 c 12 Δ R Δ G + c 22 Δ G 2 + 2 c 23 Δ G Δ B + c 33 Δ B 2 + 2 c 13 Δ B Δ R = G 11 Δ X 2 + 2 G 12 Δ X Δ Y + G 22 Δ Y 2 + 2 G 23 Δ Y Δ Z + G 33 Δ Z 2 + 2 G 13 Δ X Δ Z = g 11 Δ x 2 + 2 g 12 Δ x Δ y + g 22 Δ y 2 + 2 g 23 Δ y Δ Y + g 33 Δ Y 2 + 2 g 13 Δ x Δ Y .
G j l = ( x i / x j ) ( x k / x l ) · c i k ,
x i , x k = R , G , B x j , x l = X , Y , Z .
x i = a i j x j ,
R = a 11 X + a 12 Y + a 13 Z G = a 21 X + a 22 Y + a 23 Z B = a 31 X + a 32 Y + a 33 Z .
x 1 / x 1 = a 11 , x 2 / x 1 = a 21 , x 3 / x 1 = a 31 .
( x 1 / x 1 ) ( x 2 / x 1 ) = a 11 a 21 ( x 2 / x 1 ) ( x 3 / x 1 ) = a 21 a 31
( x 1 / x 1 ) ( x 3 / x 1 ) = a 11 a 31 .
G 11 = ( x 1 / x 1 ) ( x 1 / x 1 ) c 11 + ( x 1 / x 1 ) ( x 2 / x 1 ) c 12 + ( x 1 / x 1 ) ( x 3 / x 1 ) c 13 + ( x 2 / x 1 ) ( x 1 / x 1 ) c 21 + ( x 2 / x 1 ) ( x 2 / x 1 ) c 22 + ( x 2 / x 1 ) ( x 3 / x 1 ) c 23 + ( x 3 / x 1 ) ( x 1 / x 1 ) c 31 + ( x 3 / x 1 ) ( x 2 / x 1 ) c 32 + ( x 3 / x 1 ) ( x 3 / x 1 ) c 33 ,
G 11 = a 11 2 c 11 + 2 a 11 a 21 c 12 + a 21 2 c 22 + 2 a 21 a 31 c 23 + a 31 2 c 33 + 2 a 11 a 31 c 13 .
G 12 = a 11 a 12 c 11 + ( a 11 a 22 + a 21 a 12 ) c 12 + a 21 a 22 c 22 + ( a 21 a 32 + a 31 a 22 ) c 23 + a 31 a 32 c 33 + ( a 11 a 32 + a 31 a 12 ) c 13 .
g j l = ( x i / x j ) ( x k / x l ) G i k .
x 1 = x 1 x 3 / x 2 , i.e. , X = x Y / y x 2 = x 3 , i.e. , Y = Y x 3 = ( 1 x 1 x 2 ) x 3 / x 2 , i.e. , Z = ( 1 x y ) Y / y .
g 11 = S 2 ( G 11 + G 33 2 G 13 ) g 12 = S 2 ( A G 11 + B G 33 + { A B } G 13 ) g 22 = S 2 ( A 2 G 11 + B 2 G 33 + 2 A B G 13 ) g 23 = S ( A 2 G 11 A G 12 B G 23 B C G 33 A { B + C } G 13 ) g 33 = A 2 G 11 + 2 A G 12 + G 22 + 2 C G 23 + C 2 G 33 + 2 A C G 13 g 13 = S ( A G 11 + G 12 G 23 C G 33 + { C A } G 13 )
A = x / y , B = ( 1 x ) / y , C = ( 1 x y ) / y
S = X + Y + Z = Y / y .
Δ S 2 = g 11 Δ x 2 + 2 g 12 Δ x Δ y + g 22 Δ y 2 + ( 2 g 23 k Y ) Δ y ( Δ Y / k Y ) + ( g 33 k 2 Y 2 ) ( Δ Y / k Y ) 2 + ( 2 g 13 k Y ) Δ x ( Δ Y / k Y ) .
1 = g 11 Δ x 2 + 2 g 12 Δ x Δ y + g 22 Δ y 2 + 2 k Y g 23 Δ y Δ l + ( k Y ) 2 g 33 Δ l 2 + 2 k Y g 13 Δ x Δ l .
g 11 = g 11 , g 12 = g 12 , g 22 = g 22 , g 23 = k Y g 23 , g 33 = ( k Y ) 2 g 33 , g 13 = k Y g 13 .
1 = g 11 Δ x 2 + 2 g 12 Δ x Δ y + g 22 Δ y 2 + 2 g 23 Δ y Δ l + g 33 Δ l 2 + 2 g 13 Δ x Δ l .
g 11 = S 2 [ D 1 2 c 11 + 2 D 1 D 2 c 12 + D 2 2 c 22 + 2 D 2 D 3 c 23 + D 3 2 c 33 + 2 D 1 D 3 c 13 ] g 12 = ( S 2 / y ) [ D 1 E 1 c 11 + ( D 1 E 2 + D 2 E 1 ) c 12 + D 2 E 2 c 22 + ( D 2 E 3 + D 3 E 2 ) c 23 + D 3 E 3 c 33 + ( D 1 E 3 + D 3 E 1 ) c 13 ] g 22 = ( S 2 / y 2 ) [ E 1 2 c 11 + 2 E 1 E 2 c 12 + E 2 2 c 22 + 2 E 2 E 3 c 23 + E 3 2 c 33 + 2 E 1 E 3 c 13 ] g 23 = ( k S / y ) [ E 1 R c 11 + ( E 1 G + E 2 R ) c 12 + E 2 G c 22 + ( E 2 B + E 3 G ) c 23 + E 3 B c 33 + ( E 1 B + E 3 R ) c 13 ] g 33 = k 2 [ R 2 c 11 + 2 R G c 12 + G 2 c 22 + 2 G B c 23 + B 2 c 33 + 2 R B c 13 ] g 13 = k S [ D 1 R c 11 + ( D 1 G + D 2 R ) c 12 + D 2 G c 22 + ( D 2 B + D 3 G ) c 23 + D 3 B c 33 + ( D 3 R + D 1 B ) c 13 ] ,
D 1 = a 11 a 13 , D 2 = a 21 a 23 , D 3 = a 31 a 33
E 1 = D 1 x + a 13 , E 2 = D 2 x + a 23 , E 3 = D 3 x + a 33 .
1 = g 33 Δ l 2 = g 33 ( Δ Y / 11.53 Y ) 2 .
g 33 = 1 / ( Δ Y / 11.53 Y ) 2 ( 3 × 11.53 ) 2 / W 2 1200 / W 2 ,
Δ x = Δ y = 0 .
g 11 , g 12 , g 22 , g 23 , g 23 and g 13 . 8
σ 3 g 2 σ 2 + g 1 σ g 0 = 0
a 1 = σ 1 1 2 , a 2 = σ 2 1 2 , and a 3 = σ 3 1 2 .
c 1 : c 2 : c 3 = [ ( g 22 σ 1 ) ( g 33 σ 1 ) g 23 2 ] : [ g 13 g 23 g 12 ( g 33 σ 1 ) ] : [ g 12 g 23 g 13 ( g 22 σ 1 ) ] .
Δ x m = G 13 / Δ l m · g 0 and Δ y m = G 23 / Δ l m · g 0 .
[ 1 ( Δ Y / Δ Y m ) 2 ] 1 2 .
log B = log B 0 + log T = log B 0 D ,
log E + log k 2 = log B 0 D .
R = 10 6 T r / ( S r + C r ) 2 , G = 10 6 T g / ( S g + C g ) 2 , B = 10 6 T b / ( S b + C b ) 2 .
R = a 11 X + a 12 Y + a 13 Z G = a 21 X + a 22 Y + a 23 Z B = a 31 X + a 32 Y + a 33 Z ,
{ R , G , B } = a i k ( X , Y , Z ) .
X = A 11 R + A 12 G + A 13 B Y = A 21 R + A 22 G + A 23 B Z = A 31 R + A 32 G + A 33 B .
r = R / ( R + G + B ) and g = G / ( R + G + B ) .
ρ i k = Δ S i Δ S k / ( Δ S i 2 · Δ S k 2 ) 1 2 .
σ r = ( Δ R 2 / n ) 1 2 , σ g = ( Δ G 2 / n ) 1 2 ,
σ b = ( Δ B 2 / n ) 1 2 .
ρ b g = Δ G Δ B / n σ g σ b , ρ r g = Δ R Δ B / n σ r σ b , ρ r g = Δ R Δ G / n σ r σ g .
R = 0.28 X 0.157 Y 0.019 Z G = 0.36 X + 0.810 Y 0.046 Z B = 0.043 X 0.061 Y + 0.490 Z .
G 11 = 0.0775 c 11 0.1988 c 12 + 0.1271 c 22 0.0305 c 23 + 0.0019 c 33 + 0.0238 c 13 G 12 = 0.0437 c 11 + 0.2820 c 12 0.2890 c 22 + 0.0564 c 23 0.0027 c 33 0.0239 c 13 G 22 = 0.0244 c 11 0.2533 c 12 + 0.6565 c 22 0.0991 c 23 + 0.0038 c 33 + 0.0191 c 13 G 23 = 0.0030 c 11 0.0080 c 12 0.0375 c 22 + 0.3990 c 23 0.0300 c 33 0.0755 c 13 G 33 = 0.0003 c 11 + 0.0017 c 12 + 0.0022 c 22 0.0454 c 23 + 0.2390 c 33 0.0184 c 13 G 13 = 0.0053 c 11 0.0063 c 12 + 0.0166 c 22 0.1768 c 23 + 0.0210 c 33 + 0.1360 c 13 .
R = 0.0780 X 0.0386 Y 0.0097 Z G = 0.2552 X + 0.5961 Y 0.0401 Z B = 0.0192 X 0.0415 Y 0.3604 Z .
G 11 = 0.0061 c 11 0.0398 c 12 + 0.0650 c 22 0.0098 c 23 + 0.0004 c 33 + 0.0030 c 13 G 12 = 0.0030 c 11 + 0.0564 c 12 0.1522 c 22 + 0.0220 c 23 0.0008 c 33 0.0040 c 13 G 22 = 0.0015 c 11 0.0460 c 12 + 0.03550 c 22 0.0494 c 23 + 0.0017 c 33 + 0.0032 c 13 G 23 = 0.0004 c 11 0.0042 c 12 0.0239 c 22 + 02160 c 23 0.0150 c 33 0.0135 c 13 G 33 = 0.0001 c 11 + 0.0008 c 12 + 0.0016 c 22 0.0290 c 23 + 0.1300 c 33 0.0070 c 13 G 13 = 0.0008 c 11 0.0007 c 12 + 0.0102 c 22 0.0929 c 23 + 0.0069 c 33 + 0.0279 c 13 .
R = 0.07000 X 0.01557 Y 0.01282 Z G = 0.47550 X + 1.12596 Y 0.06750 Z B = 0.00745 X 0.01828 Y 0.27440 Z .
G 11 = 0.0049 c 11 0.0666 c 12 + 0.2261 c 22 0.0071 c 23 + 0.0001 c 33 + 0.0010 c 13 G 12 = 0.0011 c 11 + 0.0862 c 12 0.5354 c 22 + 0.0171 c 23 0.0001 c 33 0.0014 c 13 G 22 = 0.00024 c 11 0.0351 c 12 + 1.2678 c 22 0.0412 c 23 + 0.0003 c 33 0.0006 c 13 G 23 = 0.00020 c 11 0.0134 c 12 0.0760 c 22 + 0.3102 c 23 0.0050 c 33 0.0040 c 13 G 33 = 0.00016 c 11 + 0.0017 c 12 + 0.0046 c 22 0.0370 c 23 + 0.0753 c 33 0.0070 c 13 G 13 = 0.0009 c 11 + 0.0014 c 12 + 0.0321 c 22 0.1310 c 23 + 0.0020 c 33 + 0.0191 c 13 .