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 calorimeter. 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<sup>-2</sup>; the white surround (subtending 40°) was maintained at 6 cd · m<sup>-2</sup>. The elliptical cross sections (for <i>Y</i>= 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 e llipses 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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  1. D. L. MacAdam, J. Opt. Soc. Am. 32, 247 (1942).
  2. W. R. J. Brown and D. L. MacAdam, J. Opt. Soc. Am. 39, 808 (1949).
  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).
  7. D. L. MacAdam, J. Opt. Soc. Am. 55, 91 (1965).
  8. D. L. MacAdam, Acta Chromatica 1 (4), 1 (1965).
  9. K. D. Chickering, J. Opt. Soc. Am. 57, 537 (1967).
  10. G. Wyszecki, J. Opt. Soc. Am. 55, 1319 (1965).
  11. C. L. Sanders and W. Gaw, Appl. Opt. 6, 1639 (1967).
  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 [Equation] 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 calorimetric 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, l0) falls well within the ellipsoid centered at the point (x0, y0, I0). 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, l0) 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).
  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).
  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).
  21. To calculate Δxi, Δyi (see Fig. 19) (1) Calculate ε1 (in degrees) from tanε1 = - (a/b) tanθ. (2) Set up a series of angles εi at constant intervals Δε to give the desired number in of points on the semicircle εi = ε1+(i-1)Δε i=1, 2, … m with m=180/Δε. (3) Calculate the corresponding angles (ψi-θ) from tan(ψi-θ) = (b/a) tanεi and record ψi. (4) Calculate Δxi=ri cosψi Δyi =ri sinψi, where ri=c(G11 cos2ψi+2G12sinψi cosψi+G22sin2ψi). Note: For Δε=30°, that is m=6, MacAdam's20 Chickering’s 9 case is obtained.
  22. When we square our computed value 〈e2½=0.175 for α=1.0, we obtain 0.03 (approx.) which agrees with the value Chickering9 reported.

Brown, W. R. J.

W. R. J. Brown, J. Opt. Soc. Am. 47, 137 (1957).

W. R. J. Brown and D. L. MacAdam, J. Opt. Soc. Am. 39, 808 (1949).

Chickering, K. D.

K. D. Chickering, J. Opt. Soc. Am. 57, 537 (1967).

Friele, L. F. C.

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

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

Gaw, W.

C. L. Sanders and W. Gaw, Appl. Opt. 6, 1639 (1967).

MacAdam, D. L.

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

W. R. J. Brown and D. L. MacAdam, J. Opt. Soc. Am. 39, 808 (1949).

D. L. MacAdam, J. Opt. Soc. Am. 55, 91 (1965).

D. L. MacAdam, J. Opt. Soc. Am. 54, 1161 (1964).

D. L. MacAdam, J. Opt. Soc. Am. 32, 247 (1942).

Sanders, C. L.

C. L. Sanders and W. Gaw, Appl. Opt. 6, 1639 (1967).

Stiles, W. S.

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

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.

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.

G. Wyszecki, J. Opt. Soc. Am. 55, 1319 (1965).

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

Other

D. L. MacAdam, J. Opt. Soc. Am. 32, 247 (1942).

W. R. J. Brown and D. L. MacAdam, J. Opt. Soc. Am. 39, 808 (1949).

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.

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

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

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

D. L. MacAdam, J. Opt. Soc. Am. 55, 91 (1965).

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

K. D. Chickering, J. Opt. Soc. Am. 57, 537 (1967).

G. Wyszecki, J. Opt. Soc. Am. 55, 1319 (1965).

C. L. Sanders and W. Gaw, Appl. Opt. 6, 1639 (1967).

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 [Equation] 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 calorimetric 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, l0) falls well within the ellipsoid centered at the point (x0, y0, I0). 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, l0) 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.

W. R. J. Brown, J. Opt. Soc. Am. 47, 137 (1957).

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.

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

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)].

D. L. MacAdam, J. Opt. Soc. Am. 54, 1161 (1964).

To calculate Δxi, Δyi (see Fig. 19) (1) Calculate ε1 (in degrees) from tanε1 = - (a/b) tanθ. (2) Set up a series of angles εi at constant intervals Δε to give the desired number in of points on the semicircle εi = ε1+(i-1)Δε i=1, 2, … m with m=180/Δε. (3) Calculate the corresponding angles (ψi-θ) from tan(ψi-θ) = (b/a) tanεi and record ψi. (4) Calculate Δxi=ri cosψi Δyi =ri sinψi, where ri=c(G11 cos2ψi+2G12sinψi cosψi+G22sin2ψi). Note: For Δε=30°, that is m=6, MacAdam's20 Chickering’s 9 case is obtained.

When we square our computed value 〈e2½=0.175 for α=1.0, we obtain 0.03 (approx.) which agrees with the value Chickering9 reported.

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