Whiteness formulas: a selection

Ernst Ganz

doi:10.1364/AO.18.001073

Applied Optics
Vol. 18,
Issue 7,
pp. 1073-1078
(1979)
•https://doi.org/10.1364/AO.18.001073

Whiteness formulas: a selection

Ernst Ganz

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Ernst Ganz, "Whiteness formulas: a selection," Appl. Opt. 18, 1073-1078 (1979)

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Abstract

Experience in assessing whiteness and industrial requirements is considered in the selection of whiteness formulas. The problems set by the photometry of fluorescent samples and the calorimetric prerequisites delimit the choice. A standard whiteness formula of neutral hue preference is proposed. Two supplementary whiteness formulas of green and red hue preference, respectively, are added. The feasibility of a complete characterization of white samples by whiteness, tint, and luminance factor is discussed.

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	Representative fluorescent white		Perfect diffuser

Formula	ω	ϕ	ω	ϕ	Author
W = B	556	11	638	9	(TAPPI)
W = B + G − A	603	−26	705	−30	Croes
W = 2B − A	984	−9	1343	−13	Stephansen
W = 3B + G − 3A	1319	−26	2115	−30	Berger
W = 4B − 3G	1468	11	2553	9	Taube
W = L − 3b	1964	11	2680	9	Hunter
W = L − 3b + 3a	2298	55	3384	56	Stensby

		CIE 1931 2°						CIE 1964 10°

	Whiteness formula	Representative fluorescent white			Perfect diffuser			Representative fluorescent white			Perfect diffuser
Neutral hue preference		W	∂W/∂Y	∂W/∂s	Φ	∂W/∂s	Φ	W	∂W/∂Y	∂W/∂s	Φ	∂W/∂s	Φ
1.1	W = Y − 800(x − x₀) − 1700(y − y₀)	145.6	1.000	1852	9.6	1852	9.6	144.0	1.000	1800	16.6	1800	16.6
1.2	W = 300Z/Z₀ − 200Y/Y₀	145.9	1.621	2013	11.4	1914	9.2	144.1	1.601	1934	18.2	1868	16.1
1.3	W = 3B −2G	145.9	1.621	2013	11.4	1914	9.2	144.1	1.601	1934	18.2	1868	16.1
Green hue preference
2.1	W = Y − 1700(x − x₀) − 900(y − y₀)	141.3	1.000	1709	−27.3	1709	−27.3	144.1	1.000	1804	−20.3	1804	−20.3
2.2	W = 300Z/Z₀ + 100 Y/Y₀ − 300X/X₀	140.0	1.556	1802	−25.9	1714	−30.2	143.0	1.589	1895	−18.6	1830	−23.0
2.3	W = 3B + G − 3A	151.7	1.686	2224	−25.9	2115	−30.2	155.4	1.727	2338	−18.6	2259	−23.0
2.4	W = 2.5B + 1.5G − 3A	142.4	1.582	1888	−31.3	1796	−35.5	146.4	1.626	2016	−23.9	1947	−28.4
Red hue preference
3.1	W = Y + 800(x − x₀) − 3000(y − y₀)	150.2	1.000	2006	49.7	2006	49.7	141.1	1.000	1702	56.8	1702	56.8
3.2	W = Y + 700(x − x₀) − 3100(y − y₀)	154.4	1.000	2145	47.5	2145	47.5	145.3	1.000	1843	54.6	1843	54.5
3.3	W = Y + 900(x − x₀) − 2900(y − y₀)	146.0	1.000	1867	52.1	1867	52.1	136.8	1.000	1561	59.1	1561	59.1
3.4	W = 300Z/Z₀ − 700Y/Y₀ + 500 X/X₀	155.6	1.729	2365	47.1	2249	47.9	145.9	1.621	1999	54.1	1931	54.8
3.5	W = 250Z/Z₀ − 650Y/Y₀ + 500X/X₀	146.3	1.626	2030	50.7	1930	51.6	136.9	1.521	1677	57.7	1620	58.5
3.6	W = 3B − 7G + 5A	136.1	1.512	1662	60.1	1580	61.2	125.2	1.392	1260	67.3	1217	68.2
3.7	W = 3.5B − 7.5G + 5A	145.4	1.616	1997	56.0	1899	57.1	134.3	1.492	1582	63.1	1529	63.9
3.8	W = 4B − 8G + 5A	154.7	1.719	2333	52.4	2218	53.4	143.3	1.592	1905	59.4	1840	60.2

	Tint formula	Neutral λd (nm)	T	∂f/∂T	∂f/∂T	Neutral λ_d (nm)	T	∂f/∂T	∂f/∂T
4.1	T = −950(x − x₀) + 750(y − y₀)	473.6	−2.2	0.00083	0.00083	467.3	2.2	0.00083	0.00083
4.2	T = −900(x − x₀) + 800(y − y₀)	476.2	−4.3	0.00083	0.00083	469.9	0.1	0.00083	0.00083
4.3	T = −1000(x − x₀) + 700(y − y₀)	470.2	−0.1	0.00082	0.00082	463.6	4.3	0.00082	0.00082

Sample	W_1.1
1	40
2	52
3	60
4	70
5	78
6	88
7	96
8	106
9	116
10	126
11	132
12	144

	Representative fluorescent white		Perfect diffuser

Formula	ω	ϕ	ω	ϕ	Author
W = B	556	11	638	9	(TAPPI)
W = B + G − A	603	−26	705	−30	Croes
W = 2B − A	984	−9	1343	−13	Stephansen
W = 3B + G − 3A	1319	−26	2115	−30	Berger
W = 4B − 3G	1468	11	2553	9	Taube
W = L − 3b	1964	11	2680	9	Hunter
W = L − 3b + 3a	2298	55	3384	56	Stensby

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Applied Optics