The coefficients of the spectral reflectances β_{1}, β_{2}, and β_{3} (at 430, 560, and 660 mμ, respectively) in the formula W = aβ_{1} + bβ_{2} + cβ_{3} were determined by least squares to fit visual judgments of the whitenesses of two groups of white cement samples. Somewhat different results were obtained for the two groups, for one of which β_{2} > β_{3}, for the other set β_{2} < β_{3}. The results were combined in the formula W = 1.68β_{1} − 0.34(β_{2} + β_{3}) − q(β_{2} − β_{3}), where q = 2.28 when β_{2} > β_{3}, or q = −0.02 when β_{2} < β_{3}. This formula was tested by application to the original two sets and to two other sets of white cements which differed even more widely in dominant wavelength. The correlations between the results and the subjective judgments of whiteness were in all cases higher than for any of 15 other whiteness formulas tried.

R. Thielert and G. Schliemann J. Opt. Soc. Am. 70(4) 411-417 (1980)

References

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Standard classification of cements by decreasing visual whiteness order

To 23 observers

To a single observer

T 39

25.1

3.5

26.7

0.6

5.8

1

1

23

24.9

2.8

23.2

0.1

28.0

2

3

13

20.2

18.1

24.5

2.6

7.0

3

2

9

19.7

4.1

18.0

1.7

2.4

4

5

36

18.3

17.9

16.5

2.3

7.8

5

6

28

15.9

10.3

16.3

3.2

3.2

6

7

3

15.8

14.9

11.5

0.6

24.6

7

9

41

15.8

9.6

18.3

3.9

2.5

8

4

25

15.2

2.3

15.3

2.2

1

9

8

30

9.0

0

9.0

0

10

10

Table IV

Correlation in studied field between the dominant wavelength and the difference (β_{2} − β_{3}).

Cements

λ_{d}(mμ)

β_{2} − β_{3}(%)

T 2

575.5

−2.0

T 33

575.5

−2.0

T 39

574.5

−0.9

T 3

574

−1.0

T 42

574

−0.9

T 5

573.5

−0.7

T 12

573.5

−1.6

T 38

573

−0.6

T 6

573

−0.3

T 46

573

−0.6

T 11

573

−0.1

T 45

572.5

−0.4

T 7

572.5

−0.2

T 4

572.5

−0.4

T 21

572.0

0.1

T 9

572.0

−0.3

T 19

572.0

0

T 14

572.0

−0.1

T 41

572.0

−0.2

T 44

572.0

−0.1

T 16

571.5

0.5

T 25

571.5

0.5

T 37

571.5

−0.7

T 40

571.5

−0.3

T 20

571

−0.4

T 29

571

0.4

T 47

570.5

0.8

T 10

570.5

1.1

T 43

570

0.8

T 28

570

0.5

T 23

569.5

0.5

T 1

569.5

0.1

T 22

569

1.8

T 36

569

0.9

T 35

567.5

1.9

T 24

567.5

1.1

T 31

567.5

2.1

T 8

566.5

1.3

T 18

566.5

2.0

T 15

566

1.3

T 17

563.5

2.5

T 13

556.5

1.6

T 30

526

4.0

Table V

Correlation between the visual whiteness and the whiteness calculated from Eq. (5)W = 1.68β_{1} − 0.34(β_{2} + β_{3}) − q(β_{2} − β_{3}) where q = 2.28 if β_{2} > β_{3}, q = −0.02 if β_{2} < β_{3}.

Significance of correlation from the test of Student–Fisher.

Table VI

Correlation between the visual whiteness and the whiteness calculated from Eq. (2) of Selling15:
$W=100-{[{(\mathrm{\Delta}Y)}^{2}+3\times 3.08\times {10}^{6}{(\mathrm{\Delta}s)}^{2}]}^{\frac{1}{2}}$.^{a}

Standard classification of cements by decreasing visual whiteness order

To 23 observers

To a single observer

T 39

25.1

3.5

26.7

0.6

5.8

1

1

23

24.9

2.8

23.2

0.1

28.0

2

3

13

20.2

18.1

24.5

2.6

7.0

3

2

9

19.7

4.1

18.0

1.7

2.4

4

5

36

18.3

17.9

16.5

2.3

7.8

5

6

28

15.9

10.3

16.3

3.2

3.2

6

7

3

15.8

14.9

11.5

0.6

24.6

7

9

41

15.8

9.6

18.3

3.9

2.5

8

4

25

15.2

2.3

15.3

2.2

1

9

8

30

9.0

0

9.0

0

10

10

Table IV

Correlation in studied field between the dominant wavelength and the difference (β_{2} − β_{3}).

Cements

λ_{d}(mμ)

β_{2} − β_{3}(%)

T 2

575.5

−2.0

T 33

575.5

−2.0

T 39

574.5

−0.9

T 3

574

−1.0

T 42

574

−0.9

T 5

573.5

−0.7

T 12

573.5

−1.6

T 38

573

−0.6

T 6

573

−0.3

T 46

573

−0.6

T 11

573

−0.1

T 45

572.5

−0.4

T 7

572.5

−0.2

T 4

572.5

−0.4

T 21

572.0

0.1

T 9

572.0

−0.3

T 19

572.0

0

T 14

572.0

−0.1

T 41

572.0

−0.2

T 44

572.0

−0.1

T 16

571.5

0.5

T 25

571.5

0.5

T 37

571.5

−0.7

T 40

571.5

−0.3

T 20

571

−0.4

T 29

571

0.4

T 47

570.5

0.8

T 10

570.5

1.1

T 43

570

0.8

T 28

570

0.5

T 23

569.5

0.5

T 1

569.5

0.1

T 22

569

1.8

T 36

569

0.9

T 35

567.5

1.9

T 24

567.5

1.1

T 31

567.5

2.1

T 8

566.5

1.3

T 18

566.5

2.0

T 15

566

1.3

T 17

563.5

2.5

T 13

556.5

1.6

T 30

526

4.0

Table V

Correlation between the visual whiteness and the whiteness calculated from Eq. (5)W = 1.68β_{1} − 0.34(β_{2} + β_{3}) − q(β_{2} − β_{3}) where q = 2.28 if β_{2} > β_{3}, q = −0.02 if β_{2} < β_{3}.

Significance of correlation from the test of Student–Fisher.

Table VI

Correlation between the visual whiteness and the whiteness calculated from Eq. (2) of Selling15:
$W=100-{[{(\mathrm{\Delta}Y)}^{2}+3\times 3.08\times {10}^{6}{(\mathrm{\Delta}s)}^{2}]}^{\frac{1}{2}}$.^{a}