Abstract

Three new sets of color-matching experiments have been made by three observers binocularly viewing a bipartite field (each half subtending 3°) of a three-primary colorimeter. Each set of color-matching data refers to 28 test colors scattered over the chromaticity gamut provided by the colorimeter. The luminance of each test color was 12 cd · m−2; the white surround (subtending 40°) was maintained at 6 cd · m−2. The elliptical cross sections (for Y = const) of the observed color-matching ellipsoids are compared with results previously published by MacAdam, and Brown and MacAdam. In view of the inherent experimental uncertainties of data of this kind, the new color-matching ellipses correlate well with those obtained by Brown, and Brown and MacAdam, but show significant deviations from those obtained by MacAdam’s observer P. G. Nutting, Jr. The discrepancies are puzzling. A note of caution is added with regard to the usefulness of color-matching ellipses in testing line elements. A set of color-matching ellipses can reveal only little of the visual mechanism that governs the precision of color matching and its assumed direct correlation with judging small color differences. It appears that fundamentally different line elements can reproduce almost equally well a given set of color-matching ellipses.

© 1971 Optical Society of America

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References

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  1. D. L. MacAdam, J. Opt. Soc. Am. 32, 247 (1942).
    [Crossref]
  2. W. R. J. Brown and D. L. MacAdam, J. Opt. Soc. Am. 39, 808 (1949).
    [Crossref] [PubMed]
  3. A review of empirical and inductive line elements can be found in G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967), pp. 511–560.
  4. G. Wyszecki, J. Opt. Soc. Am. 58, 290 (1968).
  5. L. F. C. Friele, Die Farbe 10, 193 (1961).
  6. L. F. C. Friele, J. Opt. Soc. Am. 55, 1314 (1965).
    [Crossref]
  7. D. L. MacAdam, J. Opt. Soc. Am. 55, 91 (1965).
    [Crossref]
  8. D. L. MacAdam, Acta Chromatica 1 (4), 1 (1965).
  9. K. D. Chickering, J. Opt. Soc. Am. 57, 537 (1967).
    [Crossref] [PubMed]
  10. G. Wyszecki, J. Opt. Soc. Am. 55, 1319 (1965).
    [Crossref] [PubMed]
  11. C. L. Sanders and W. Gaw, Appl. Opt. 6, 1639 (1967).
    [Crossref] [PubMed]
  12. See, for example, Ref. 3, pp. 536–541.
  13. There are discrepancies between the coordinates (x0,y0,l0) of the test color and the corresponding coordinates (x¯0=1n∑ixi,y¯0=1n∑iyi,l¯0=1n∑ili) of the mean color match calculated from the n observations. However, these discrepancies appear to follow the pattern we would expect from the statistical variations that characterize the observational data, superimposed on small unavoidable errors in the calibration of the instrumental scales of the two colorimetric fields. For approximately 90% of all of the color-matching ellipsoids [with (ds)2= 7.81] we have studied, the point (x¯0,y¯0,l¯0) falls well within the ellipsoid centered at the point (x0, y0, l0). For convenience of intercomparing different sets of color matches made by the same or by different observers on the same test color, the individual points (xi, yi, li) of each set of observations have been shifted so as to make their respective mean point (x¯0,y¯0,l¯0) coincide exactly with (x0, y0, l0). This small translation of the observed points is of no apparent consequence to the conclusions drawn from the experiment as a whole.
  14. To illustrate the observer’s precision of chromaticity matching, it might be more appropriate to use the (x,y) projection rather than the cross sections of the color-matching ellipsoid. However, in most cases we found only negligible differences between the two ellipses, that is, differences of the kind and magnitude shown in Fig. 3. This indicates that our color-matching ellipsoids are either only slightly tilted against the (x,y) plane, or their projections in the (x,l) and (y,l) planes are ellipses with little or only moderate eccentricities. The directions and magnitudes of the tilts are of interest in developing color-difference formulas or line elements, but we will not discuss this aspect of color-matching ellipsoids in the present paper.
  15. W. R. J. Brown, J. Opt. Soc. Am. 47, 137 (1957).
    [Crossref] [PubMed]
  16. Note that the g coefficients and parameters of the elliptical cross section of the mean ellipsoid are not the arithmetic means of the corresponding quantities of the ellipsoids derived from the data of the individual sessions.
  17. W. S. Stiles, Proc. Phys. Soc. (London) 58, 41 (1946).
    [Crossref]
  18. If the areas of two ellipses differ by a factor F, the chromaticity differences between the center color and a color on the ellipse differ by a factor √F.
  19. The calculation of the FMC ellipses were made with the constant (ds)2 set equal to 7.81, that is, identical to the one used in calculating the GW ellipses [see Eq. (1)].
  20. D. L. MacAdam, J. Opt. Soc. Am. 54, 1161 (1964).
    [Crossref]
  21. To calculate Δxi, Δyi (see Fig. 19)(1)Calculate ∊1 (in degrees) fromtan∊1=-(a/b) tanθ.(2)Set up a series of angles ∊i at constant intervals Δ∊ to give the desired number m of points on the semicircle∊i=∊1+(i-1)Δ∊         i=1,2,…mwith m=180/Δ∊.(3)Calculate the corresponding angles (ψi− θ) fromtan(ψi-θ)=(b/a) tan∊iand record ψi.(4)CalculateΔxi=ri cosψi         Δyi=ri sinψi,whereri=c(G11 cos2ψi+2G12 sinψi cosψi+G22 sin2ψi)-12.Note: For Δ∊= 30°, that is m= 6, MacAdam’s20 and Chickering’s9 case is obtained.
  22. When we square our computed value 〈e2〉12=0.175 for α= 1.0, we obtain 0.03 (approx.) which agrees with the value Chickering9 reported.

1968 (1)

1967 (2)

1965 (4)

1964 (1)

1961 (1)

L. F. C. Friele, Die Farbe 10, 193 (1961).

1957 (1)

1949 (1)

1946 (1)

W. S. Stiles, Proc. Phys. Soc. (London) 58, 41 (1946).
[Crossref]

1942 (1)

Brown, W. R. J.

Chickering, K. D.

Friele, L. F. C.

L. F. C. Friele, J. Opt. Soc. Am. 55, 1314 (1965).
[Crossref]

L. F. C. Friele, Die Farbe 10, 193 (1961).

Gaw, W.

MacAdam, D. L.

Sanders, C. L.

Stiles, W. S.

W. S. Stiles, Proc. Phys. Soc. (London) 58, 41 (1946).
[Crossref]

A review of empirical and inductive line elements can be found in G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967), pp. 511–560.

Wyszecki, G.

G. Wyszecki, J. Opt. Soc. Am. 58, 290 (1968).

G. Wyszecki, J. Opt. Soc. Am. 55, 1319 (1965).
[Crossref] [PubMed]

A review of empirical and inductive line elements can be found in G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967), pp. 511–560.

Acta Chromatica (1)

D. L. MacAdam, Acta Chromatica 1 (4), 1 (1965).

Appl. Opt. (1)

Die Farbe (1)

L. F. C. Friele, Die Farbe 10, 193 (1961).

J. Opt. Soc. Am. (9)

Proc. Phys. Soc. (London) (1)

W. S. Stiles, Proc. Phys. Soc. (London) 58, 41 (1946).
[Crossref]

Other (9)

If the areas of two ellipses differ by a factor F, the chromaticity differences between the center color and a color on the ellipse differ by a factor √F.

The calculation of the FMC ellipses were made with the constant (ds)2 set equal to 7.81, that is, identical to the one used in calculating the GW ellipses [see Eq. (1)].

To calculate Δxi, Δyi (see Fig. 19)(1)Calculate ∊1 (in degrees) fromtan∊1=-(a/b) tanθ.(2)Set up a series of angles ∊i at constant intervals Δ∊ to give the desired number m of points on the semicircle∊i=∊1+(i-1)Δ∊         i=1,2,…mwith m=180/Δ∊.(3)Calculate the corresponding angles (ψi− θ) fromtan(ψi-θ)=(b/a) tan∊iand record ψi.(4)CalculateΔxi=ri cosψi         Δyi=ri sinψi,whereri=c(G11 cos2ψi+2G12 sinψi cosψi+G22 sin2ψi)-12.Note: For Δ∊= 30°, that is m= 6, MacAdam’s20 and Chickering’s9 case is obtained.

When we square our computed value 〈e2〉12=0.175 for α= 1.0, we obtain 0.03 (approx.) which agrees with the value Chickering9 reported.

Note that the g coefficients and parameters of the elliptical cross section of the mean ellipsoid are not the arithmetic means of the corresponding quantities of the ellipsoids derived from the data of the individual sessions.

A review of empirical and inductive line elements can be found in G. Wyszecki and W. S. Stiles, Color Science (Wiley, New York, 1967), pp. 511–560.

See, for example, Ref. 3, pp. 536–541.

There are discrepancies between the coordinates (x0,y0,l0) of the test color and the corresponding coordinates (x¯0=1n∑ixi,y¯0=1n∑iyi,l¯0=1n∑ili) of the mean color match calculated from the n observations. However, these discrepancies appear to follow the pattern we would expect from the statistical variations that characterize the observational data, superimposed on small unavoidable errors in the calibration of the instrumental scales of the two colorimetric fields. For approximately 90% of all of the color-matching ellipsoids [with (ds)2= 7.81] we have studied, the point (x¯0,y¯0,l¯0) falls well within the ellipsoid centered at the point (x0, y0, l0). For convenience of intercomparing different sets of color matches made by the same or by different observers on the same test color, the individual points (xi, yi, li) of each set of observations have been shifted so as to make their respective mean point (x¯0,y¯0,l¯0) coincide exactly with (x0, y0, l0). This small translation of the observed points is of no apparent consequence to the conclusions drawn from the experiment as a whole.

To illustrate the observer’s precision of chromaticity matching, it might be more appropriate to use the (x,y) projection rather than the cross sections of the color-matching ellipsoid. However, in most cases we found only negligible differences between the two ellipses, that is, differences of the kind and magnitude shown in Fig. 3. This indicates that our color-matching ellipsoids are either only slightly tilted against the (x,y) plane, or their projections in the (x,l) and (y,l) planes are ellipses with little or only moderate eccentricities. The directions and magnitudes of the tilts are of interest in developing color-difference formulas or line elements, but we will not discuss this aspect of color-matching ellipsoids in the present paper.

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Figures (19)

Fig. 1
Fig. 1

Configuration of the visual field. Hatched area is the white surround (not to scale).

Fig. 2
Fig. 2

CIE 1931 chromaticity diagram showing chromaticity points of the primaries R, G, B of the colorimeter, the surround S, and the 28 test colors. The triangle (R), (G), (B) represents the chromaticity gamut provided by the colorimeter.

Fig. 3
Fig. 3

Portion of (x,y,l) space, in its three main projections, showing the distribution of points (solid dots) corresponding to color matches made by observer GW on test color 16 (open circle). The ellipses (solid lines) are projections of the corresponding color-matching ellipsoid [with (ds)2 = 7.81] into the respective coordinate planes. The ellipse (dashed line) in the (x,y) plane is the cross section through the center of the ellipsoid with l = const = 0.2158.

Fig. 4
Fig. 4

Portion of the CIE 1931 chromaticity diagram showing cross sections (l = const = 0.2158) of color-matching ellipsoids obtained in (x,y,l) space for observer GF. Points R, G, B represent the chromaticities of the instrumental primaries and S the surround. The axes of the ellipses are five times actual size.

Fig. 5
Fig. 5

Portion of the CIE 1931 chromaticity diagram showing cross sections (l = const = 0.2158) of color-matching ellipsoids obtained in (x,y,l) space for observer AR. Points R, G, B represent the chromaticities of the instrumental primaries and S the surround. The axes of the ellipses are five times actual size.

Fig. 6
Fig. 6

Portion of the CIE 1931 chromaticity diagram showing cross sections (l = const = 0.2158) of color-matching ellipsoids obtained in (x,y,l) space for observer GW. Points R, G, B represent the chromaticities of the instrumental primaries and S the surround. The axes of the ellipses are five times actual size.

Fig. 7
Fig. 7

Portion of the CIE 1931 chromaticity diagram showing cross sections (l = const = 0.2158) of color-matching ellipsoids obtained in (x,y,l) space for the same observer (GF) producing sets of color matches at different occasions but otherwise identical observing conditions.

Fig. 8
Fig. 8

Portion of the CIE 1931 chromaticity diagram showing cross sections (l = const = 0.2158) of color-matching ellipsoids obtained in (x,y,l) space for the same observer (AR) producing sets of color matches at different occasions but otherwise identical observing conditions.

Fig. 9
Fig. 9

Portion of the CIE 1931 chromaticity diagram showing cross sections (l = const = 0.2158) of color-matching ellipsoids obtained in (x,y,l) space for the same observer (GW) producing sets of color matches at different occasions but otherwise identical observing conditions.

Fig. 10
Fig. 10

CIE 1931 chromaticity diagram showing color-matching ellipses (10 times enlarged) obtained by observer P. G. Nutting, Jr. (after MacAdam1).

Fig. 11
Fig. 11

CIE 1931 chromaticity diagram showing color-matching ellipses (three times enlarged) predicted by the line element of Stiles (after Stiles17).

Fig. 12
Fig. 12

CIE 1931 chromaticity diagram showing cross sections of color-matching ellipsoids (10 times enlarged) obtained by observer W. R. J. Brown (after Brown and MacAdam2).

Fig. 13
Fig. 13

CIE 1931 chromaticity diagram showing cross sections of color-matching ellipsoids (10 times enlarged) obtained by observer D. L. MacAdam (after Brown and MacAdam2).

Fig. 14
Fig. 14

CIE 1931 chromaticity diagram, showing cross sections (10 times enlarged) of mean color-matching ellipsoids obtained by 12 observers (after Brown15).

Fig. 15
Fig. 15

Diagrams showing the correlation of the orientations (θ), sizes (log10A), and shapes (a/b) between corresponding color-matching ellipses of observers GF, GW, and observers AR, GW.

Fig. 16
Fig. 16

Diagrams showing the correlation of the orientations (θ), sizes (log10A), and shapes (a/b) between corresponding color-matching ellipses of observers PGN, WSS, and observers WRB, DLM.

Fig. 17
Fig. 17

Diagrams showing the correlation of the orientations (θ), sizes (log10A), and shapes (a/b) between corresponding color-matching ellipses of observers FMC, PGN, and observers FMC, GW.

Fig. 18
Fig. 18

Diagrams showing the correlation of the orientations (θ), sizes (log10A), and shapes (a/b) between corresponding color-matching ellipses of observers FMC, WRB, and observers FMC, 12B.

Fig. 19
Fig. 19

Selection of points (1–18), uniformly spaced (10°) on a circular projection of a color-matching ellipse with center x0, y0, semi-axes a, b, and orientation θ.

Tables (6)

Tables Icon

Table I Color-matching ellipsoids and their cross sections of observers GF, AR, and GW.

Tables Icon

Table II Results of repeat observations.a

Tables Icon

Table III Color-matching ellipses of observers PGN, WSS, and FMC.a

Tables Icon

Table IV Color-matching ellipses of observers WRB and DLM.a

Tables Icon

Table V Color-matching ellipses of observer 12B.a

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Table VI Numerical correlations between sets of color-matching ellipses of different observers.a

Equations (17)

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g 11 ( x - x 0 ) 2 + g 22 ( y - y 0 ) 2 + g 33 ( l - l 0 ) 2 + 2 g 12 ( x - x 0 ) ( y - y 0 ) + 2 g 13 ( x - x 0 ) ( l - l 0 ) + 2 g 23 ( y - y 0 ) ( l - l 0 ) = ( d s ) 2 .
σ 2 ( a ) = a 2 / [ 2 ( n - 1 ) ] σ 2 ( b ) = b 2 / [ 2 ( n - 1 ) ] σ 2 ( θ ) = ( a / b ) 2 / { ( n - 1 ) [ a / b ) 2 - 1 ] 2 } ,
θ is in radians .
G 11 ( Δ x ) 2 + 2 G 12 Δ x Δ y + G 22 ( Δ y ) 2 = ( d s ) 2 ,
g 11 ( Δ x ) 2 + 2 g 12 Δ x Δ y + g 22 ( Δ y ) 2 = ( d s ) 2 .
e 2 = 1 m n j = 1 n i = 1 m ( α ( d s ) i j - c c ) 2
α = α 1 = c / 1 m n j = 1 n i = 1 m ( d s ) i j ,
α = α 2 = c j = 1 n i = 1 m ( d s ) i j / j = 1 n i = 1 m ( d s ) i j 2 ,
e 1 2 = e 2 2 1 - e 2 2 .
d = 1 n m j = 1 n i = 1 m | α ( d s ) i j c - 1 | ,
d = ω e 2 1 / 2 ,
(x¯0=1nixi,y¯0=1niyi,l¯0=1nili)
tan1=-(a/b)tanθ.
i=1+(i-1)Δ         i=1,2,mwithm=180/Δ.
tan(ψi-θ)=(b/a)tani
Δxi=ricosψi         Δyi=risinψi,
ri=c(G11cos2ψi+2G12sinψicosψi+G22sin2ψi)-12.