Abstract

This paper describes a fundamental procedure for deriving “within” and “between” variances and covariances in the spectral tristimulus values. The within variances are based on the replications of color-mixture data by an observer. The between variances are based on differences among the color-mixture data of individual observers. A statistical model for the system in which the experimental data are obtained is given, expected values (means), variances, and covariances are developed, and the method by which these variances and covariances may be distributed among the sources of uncertainties in the experimental data is presented. There is developed a procedure for determining the uncertainties in the constants of a linear transformation to a system analogous to the present CIE system. The formulas for variances and covariances after linear transformation are given for an arbitrary choice of transformation constants. The complete set of variances and covariances derived from an arbitrary transformation is listed for 20 mμ intervals. This set, together with the mean spectral tristimulus values, forms a complete standard observer system. The between-observer variabilities are found to be about 10% of the averages of the color-mixture data and the average ratio of the between-observer variabilities to the within-observer variabilities is found to be about 5.7.

© 1962 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. I. Nimeroff, J. Opt. Soc. Am. 47, 697 (1957).
    [CrossRef]
  2. W. S. Stiles and J. M. Burch, Optica Acta 6, 1 (1959).
    [CrossRef]
  3. N. I. Speranskaya, CIE Committee Report, and in part, NPL Symposium 8 (Her Majesty’s Stationery Office, London) 1, 317 (1958); also Optics and Spectroscopy VII, 429 (1959), (OSA translation).
  4. I. Nimeroff, J. R. Rosenblatt, and M. C. Dannemiller, J. Research Natl. Bur. Standards 65A, 475 (1961).
  5. D. B. Judd, Detailed Report of Progress (September, 1957, to April, 1959), CIE Working Committee W-1.3.1, Colorimetry.
  6. I. Nimeroff, J. R. Rosenblatt, and M. C. Dannemiller, J. Opt. Soc. Am. 49, 1137 (1959).
  7. H. Hotelling, Ann. Math. Stat. 2, 360 (1931); J. E. Jackson, J. Opt. Soc. Am. 49, 585 (1959).
    [CrossRef]
  8. D. B. Judd, J. Research Natl. Bur. Standards 43, 277 (1949).
    [CrossRef]
  9. G. Wyszecki, National Research Council of Canada Rept. APOI-907, June, 1959.
  10. G. Wyszecki, J. Opt. Soc. Am. 49, 389 (1959).
    [CrossRef]

1961 (1)

I. Nimeroff, J. R. Rosenblatt, and M. C. Dannemiller, J. Research Natl. Bur. Standards 65A, 475 (1961).

1959 (3)

I. Nimeroff, J. R. Rosenblatt, and M. C. Dannemiller, J. Opt. Soc. Am. 49, 1137 (1959).

W. S. Stiles and J. M. Burch, Optica Acta 6, 1 (1959).
[CrossRef]

G. Wyszecki, J. Opt. Soc. Am. 49, 389 (1959).
[CrossRef]

1958 (1)

N. I. Speranskaya, CIE Committee Report, and in part, NPL Symposium 8 (Her Majesty’s Stationery Office, London) 1, 317 (1958); also Optics and Spectroscopy VII, 429 (1959), (OSA translation).

1957 (1)

1949 (1)

D. B. Judd, J. Research Natl. Bur. Standards 43, 277 (1949).
[CrossRef]

1931 (1)

H. Hotelling, Ann. Math. Stat. 2, 360 (1931); J. E. Jackson, J. Opt. Soc. Am. 49, 585 (1959).
[CrossRef]

Burch, J. M.

W. S. Stiles and J. M. Burch, Optica Acta 6, 1 (1959).
[CrossRef]

Dannemiller, M. C.

I. Nimeroff, J. R. Rosenblatt, and M. C. Dannemiller, J. Research Natl. Bur. Standards 65A, 475 (1961).

I. Nimeroff, J. R. Rosenblatt, and M. C. Dannemiller, J. Opt. Soc. Am. 49, 1137 (1959).

Hotelling, H.

H. Hotelling, Ann. Math. Stat. 2, 360 (1931); J. E. Jackson, J. Opt. Soc. Am. 49, 585 (1959).
[CrossRef]

Judd, D. B.

D. B. Judd, J. Research Natl. Bur. Standards 43, 277 (1949).
[CrossRef]

D. B. Judd, Detailed Report of Progress (September, 1957, to April, 1959), CIE Working Committee W-1.3.1, Colorimetry.

Nimeroff, I.

I. Nimeroff, J. R. Rosenblatt, and M. C. Dannemiller, J. Research Natl. Bur. Standards 65A, 475 (1961).

I. Nimeroff, J. R. Rosenblatt, and M. C. Dannemiller, J. Opt. Soc. Am. 49, 1137 (1959).

I. Nimeroff, J. Opt. Soc. Am. 47, 697 (1957).
[CrossRef]

Rosenblatt, J. R.

I. Nimeroff, J. R. Rosenblatt, and M. C. Dannemiller, J. Research Natl. Bur. Standards 65A, 475 (1961).

I. Nimeroff, J. R. Rosenblatt, and M. C. Dannemiller, J. Opt. Soc. Am. 49, 1137 (1959).

Speranskaya, N. I.

N. I. Speranskaya, CIE Committee Report, and in part, NPL Symposium 8 (Her Majesty’s Stationery Office, London) 1, 317 (1958); also Optics and Spectroscopy VII, 429 (1959), (OSA translation).

Stiles, W. S.

W. S. Stiles and J. M. Burch, Optica Acta 6, 1 (1959).
[CrossRef]

Wyszecki, G.

G. Wyszecki, J. Opt. Soc. Am. 49, 389 (1959).
[CrossRef]

G. Wyszecki, National Research Council of Canada Rept. APOI-907, June, 1959.

Ann. Math. Stat. (1)

H. Hotelling, Ann. Math. Stat. 2, 360 (1931); J. E. Jackson, J. Opt. Soc. Am. 49, 585 (1959).
[CrossRef]

CIE Committee Report, and in part, NPL Symposium 8 (Her Majesty’s Stationery Office, London) (1)

N. I. Speranskaya, CIE Committee Report, and in part, NPL Symposium 8 (Her Majesty’s Stationery Office, London) 1, 317 (1958); also Optics and Spectroscopy VII, 429 (1959), (OSA translation).

J. Opt. Soc. Am. (3)

I. Nimeroff, J. R. Rosenblatt, and M. C. Dannemiller, J. Opt. Soc. Am. 49, 1137 (1959).

G. Wyszecki, J. Opt. Soc. Am. 49, 389 (1959).
[CrossRef]

I. Nimeroff, J. Opt. Soc. Am. 47, 697 (1957).
[CrossRef]

J. Research Natl. Bur. Standards (2)

D. B. Judd, J. Research Natl. Bur. Standards 43, 277 (1949).
[CrossRef]

I. Nimeroff, J. R. Rosenblatt, and M. C. Dannemiller, J. Research Natl. Bur. Standards 65A, 475 (1961).

Optica Acta (1)

W. S. Stiles and J. M. Burch, Optica Acta 6, 1 (1959).
[CrossRef]

Other (2)

D. B. Judd, Detailed Report of Progress (September, 1957, to April, 1959), CIE Working Committee W-1.3.1, Colorimetry.

G. Wyszecki, National Research Council of Canada Rept. APOI-907, June, 1959.

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 (6)

Fig. 1
Fig. 1

Values of the T2 statistic for evaluating differences between Stiles’ data obtained under condition I and under condition II. Critical values of T2 at three probability levels are indicated, showing that even at the 2.5% level the results obtained under the two conditions are for the most part significantly different.

Fig. 2
Fig. 2

Means and variability of the blue-primary data. Means ○; standard deviation between observers (bσb): Stiles Δ, Speranskaya ×; standard deviation within observers (wσb), ●.

Fig. 3
Fig. 3

Mean standard deviation between observers for the Stiles-Burch and the Speranskaya data.

Fig. 4
Fig. 4

Ratio of the between- to within- standard deviations, for the three primaries r ¯ ( ) , g ¯ ( × ), and b ¯ ( ), showing over-all average ratio at 5.7.

Fig. 5
Fig. 5

Correlation coefficients ρij for ij equal to r ¯ g ¯ ( ) , r ¯ b ¯ ( × ) , g ¯ b ¯ ( ). Dashed lines define the region within which these coefficients are not (statistically) significantly different from zero at the 5% level.

Fig. 6
Fig. 6

Plot of R500 against G500 for Stiles’ individual observers, indicating no bimodality and very little skewness: condition I (●), condition II (○), and mean (■). Straight lines join outlying values obtained under the two conditions.

Tables (1)

Tables Icon

Table I Between-observer variances V, within-observer variances υ, and covariances C.

Equations (28)

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

R i = R + b R i + R i , G i = G + b G i + G i , B i = B + b B i + B i ,
E i [ R i ] = R + b R i , E i [ G i ] = G + b G i , E i [ B i ] = B + b B i ,
V i ( R i ) = E i [ ( R i E i [ R i ] ) 2 ] = E i [ 2 R i ] .
σ R 2 = b σ R 2 + w σ R 2 . σ G 2 = b σ G 2 + w σ G 2 . σ B 2 = b σ B 2 + w σ B 2 .
σ R G = E i [ ( R i E i [ R i ] ) ( G i E i [ G i ] ) ] = E i [ R i G i ] .
C ( R , G ) = b σ R G + w σ R G , C ( R , B ) = b σ R B + w σ R B , C ( G , B ) = b σ G B + w σ G B .
C ( R , G ) = ρ R G σ R σ G .
σ ˆ R 2 = i = 1 n ( R i R ¯ ) 2 ( n 1 ) ,
x ¯ λ = f x ( k 1 r ¯ λ + k 2 g ¯ λ + k 3 b ¯ λ ) = f x x ¯ λ , y ¯ λ = f y ( k 4 r ¯ λ + k 5 g ¯ λ + k 6 b ¯ λ ) = f y y ¯ λ , z ¯ λ = f z ( k 7 r ¯ λ + k 8 g ¯ λ + k 9 b ¯ λ ) = f z z ¯ λ .
| k 1 k 2 k 3 k 4 k 5 k 6 k 7 k 8 k 9 | 0.
V λ = y ¯ λ = L r r ¯ λ + L g g ¯ λ + L b b ¯ λ = k 4 r ¯ λ + k 5 g ¯ λ + k 6 b ¯ λ ,
x ¯ λ = k 1 r ¯ λ + k 2 g ¯ λ + k 3 b ¯ λ ,
x ¯ λ = r ¯ λ + k 2 g ¯ λ + k 3 b ¯ λ .
x ¯ 1 = r ¯ 1 + k 2 g ¯ 1 + k 3 b ¯ 1 ,
x ¯ 2 = r ¯ 2 + k 2 g ¯ 2 + k 3 b ¯ 2 ,
k 2 = [ ( x ¯ 1 r ¯ 1 ) b ¯ 2 ( x ¯ 2 r ¯ 2 ) b ¯ 1 ] / ( g ¯ 1 b ¯ 2 g ¯ 2 b ¯ 1 ) ,
k 3 = [ ( x ¯ 2 r ¯ 2 ) g ¯ 1 ( x ¯ 1 r ¯ 1 ) g ¯ 2 ] / ( g ¯ 1 b ¯ 2 g ¯ 2 b ¯ 1 ) .
k 7 = [ ( z ¯ 3 b ¯ 3 ) g ¯ 4 ( z ¯ 4 b ¯ 4 ) g ¯ 3 ] / ( r ¯ 3 g ¯ 4 r ¯ 4 g ¯ 3 ) , k 8 = [ ( z ¯ 4 b ¯ 4 ) r ¯ 3 ( z ¯ 3 b ¯ 3 ) r ¯ 4 ] / ( r ¯ 3 g ¯ 4 r ¯ 4 g ¯ 3 ) , k 9 = 1.
σ U 2 = n = 1 k ( U υ n ) 2 σ υ n 2 + m n ( U υ n ) ( U υ m ) σ υ m υ n ,
σ U V = n = 1 k ( U υ n ) ( V υ n ) σ υ n 2 + m n ( U υ m ) ( V υ n ) σ υ m υ n ,
V ( k i ) = C ( k i , k j ) = C ( k i , υ i ) = 0 ,
x ¯ λ = k 1 r ¯ λ + k 2 g ¯ λ + k 3 b ¯ λ , y ¯ λ = k 4 r ¯ λ + k 5 g ¯ λ + k 6 b ¯ λ , z ¯ λ = k 7 r ¯ λ + k 8 g ¯ λ + k 9 b ¯ λ ,
V ( x ¯ ) = k 1 2 V ( r ¯ ) + k 2 2 V ( g ¯ ) + k 3 2 V ( b ¯ ) + 2 [ k 1 k 2 C ( r ¯ , g ¯ ) + k 1 k 3 C ( r ¯ , b ¯ ) + k 2 k 3 C ( g ¯ , b ¯ ) ] , V ( y ¯ ) = k 4 2 V ( r ¯ ) + k 5 2 V ( g ¯ ) + k 6 2 V ( b ¯ ) + 2 [ k 4 k 5 C ( r ¯ , g ¯ ) + k 4 k 6 C ( r ¯ , b ¯ ) + k 5 k 6 C ( g ¯ , b ¯ ) ] , V ( z ¯ ) = k 7 2 V ( r ¯ ) + k 8 2 V ( g ¯ ) + k 9 2 V ( b ¯ ) + 2 [ k 7 k 8 C ( r ¯ , g ¯ ) + k 7 k 9 C ( r ¯ , b ¯ ) + k 8 k 9 C ( g ¯ , b ¯ ) ] ,
C ( x ¯ , y ¯ ) = k 1 k 4 V ( r ¯ ) + k 2 k 5 V ( g ¯ ) + k 3 k 6 V ( b ¯ ) + ( k 1 k 5 + k 2 k 4 ) C ( r ¯ , g ¯ ) + ( k 1 k 6 + k 3 k 4 ) C ( r ¯ , b ¯ ) + ( k 2 k 6 + k 3 k 5 ) C ( g ¯ , b ¯ ) , C ( x ¯ , z ¯ ) = k 1 k 7 V ( r ¯ ) + k 2 k 8 V ( g ¯ ) + k 3 k 9 V ( b ¯ ) + ( k 1 k 8 + k 2 k 7 ) C ( r ¯ , g ¯ ) + ( k 1 k 9 + k 3 k 7 ) C ( r ¯ , b ¯ ) + ( k 2 k 9 + k 3 k 8 ) C ( g ¯ , b ¯ ) , C ( y ¯ , z ¯ ) = k 4 k 7 V ( r ¯ ) + k 5 k 8 V ( g ¯ ) + k 6 k 9 V ( b ¯ ) + ( k 4 k 8 + k 5 k 7 ) C ( r ¯ , g ¯ ) + ( k 4 k 9 + k 6 k 7 ) C ( r ¯ , b ¯ ) + ( k 5 k 9 + k 6 k 8 ) C ( g ¯ , b ¯ ) .
x ¯ 10 = 0.341080 r ¯ 10 + 0.189145 g ¯ 10 + 0.387529 b ¯ 10 , y ¯ 10 = 0.139058 r ¯ 10 + 0.837460 g ¯ 10 + 0.073316 b ¯ 10 , z ¯ 10 = 0.000000 r ¯ 10 + 0.039553 g ¯ 10 + 2.026200 b ¯ 10 .
T 2 = N 1 N 2 N 1 + N 2 i , j = 1 3 S i j ( p ¯ 1 i p ¯ 2 i ) ( p ¯ 1 j p ¯ 2 j ) ,
S i j = 1 N 1 + N 2 2 [ k = 1 N 1 ( p ¯ 1 i k p ¯ 1 i ) ( p ¯ 1 j k p ¯ 1 j ) + k = 1 N 2 ( p ¯ 2 i k p ¯ 2 i ) ( p ¯ 2 j k p ¯ 2 j ) ] .
ρ i j = σ i j / σ i σ j .