Photoelectric Tristimulus Colorimetry with Three Filters*

Richard S. Hunter

doi:10.1364/JOSA.32.000509

Journal of the Optical Society of America
Vol. 32,
Issue 9,
pp. 509-538
(1942)
•https://doi.org/10.1364/JOSA.32.000509

Photoelectric Tristimulus Colorimetry with Three Filters

Richard S. Hunter

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Richard S. Hunter, "Photoelectric Tristimulus Colorimetry with Three Filters*," J. Opt. Soc. Am. 32, 509-538 (1942)

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Abstract

The term, photoelectric colorimetry, is commonly employed to designate both photoelectric tristimulus colorimetry, used to evaluate the appearance of materials, and abridged spectrophotometry, often used to assist in chemical analyses. This paper is devoted to the first type of measurement. For a photoelectric tristimulus colorimeter, it is desired to find three or more source-filter photo-cell combinations of such spectral character that they duplicate the standard I.C.I. observer for colorimetry. With an instrument having these combinations, tristimulus values would be obtained by direct measurement. Although no one has duplicated the I.C.I. observer perfectly, several investigators have obtained source-filter photo-cell combinations suitable for the measurement of color differences between spectrally similar samples. To measure color differences as small as those which the trained inspectors of paint, textile, plastic, paper, and ceramic products can see, an instrument must have high precision. If the needed precision is available, a photoelectric tristimulus colorimeter may be used to measure: (1) I.C.I. colorimetric values, x, y, and Y, relative to those of a spectrally similar, calibrated standard; (2) relative values of α and β, components of the chromaticity departure from neutral in a new uniform-chromaticness-scale mixture diagram for representing surface colors; (3) amounts of color difference between pairs of spectrally similar samples; (4) amounts of color change accompanying fading; and (5) whiteness of white and near-white surfaces. In giving examples of the measurement of some of these different properties and in describing the errors of color measurement to which the tristimulus method is subject, reference is made to operations with the author’s recently developed multipurpose photoelectric reflectometer.

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Category	Physical	Psychophysical (physically measured, psychologically significant)	Psychological
Instrument	Spectrophotometer	Tristimulus colorimeter	Visual mechanism of the observer
Product of the instrument	Spectral apparent-reflectance curve	Designation of the surface color by a set of tristimulus values for it; X, Y and Z; A, G and B; or other.	The perceived surface color
The separate attributes of the surface color, and the perceived surface color		(1) Luminous apparent reflectance A_{θ_i,θ_υ} (chiefly designated below by Y), the lightness index L. (2) Dominant wave-length Λ, or hue angle ϕ (3) Purity p, or saturation index S	(1) Lightness (2) Hue (3) Saturation, or “strength”
	(2 and 3 combined) Chromaticity, indicated either by Λ and p, ϕ and S, or by a pair of trilinear coordinates, x and y, α and β, etc.	(2 and 3 combined) Chromaticness

Wavelength (mμ)	Spectral transmissions (Corning filters)				s Spectral response of G.E. cell No. 55 (in microamperes per milliwatt)
	T_A (Amber filter of No. 330 yellow glass and No. 978 green glass)	T_G (Green filter of No. 428 green glass and No. 330 yellow glass)	T_B (Blue filter of No. 554 blue glass and No. 038 yellow glass)	E_I Relative spectral irradiance of source at 3100°K [44]		Spectral specification of the three source-filter photo-cell combinations.(Each value of k has been chosen so that the sum of the column is 100,000.)
				E_I Relative spectral irradiance of source at 3100°K [44]		E_IT_Ask_A (amber)	E_IT_Gsk_G (green)	E_IT_Bsk_B (blue)
380		0.010	0.000	0.1188	94.5		31
90		.010	.000	.1425	101.5		39
400		.011	.000	.1688	107.8		56
10		.011	.020	.1979	113.9		70	337
20		.012	.298	.2294	119.5		92	6,109
30		.013	.443	.2634	124.7		120	10,885
40		.017	.478	.2998	129.7		185	13,901
450		.023	.482	.3383	134.2		291	16,368
60		.035	.445	.3790	138.8		514	17,511
70		.050	.338	.4215	143.0		842	15,237
80		.077	.195	.4658	147.2		1,476	10,001
90	0.004	.107	.090	.5114	151.3	96	2,315	5,208
500	.010	.138	.045	.5582	155.5	269	3,350	2,922
10	.020	.173	.015	.6061	159.1	596	4,664	1,082
20	.038	.209	.003	.6548	162.3	1,248	6,210	239
30	.063	.235	.000	.7040	164.9	2,258	7,628
40	.095	.249		.7536	167.0	3,695	8,763
550	.128	.251		.8033	168.4	5,351	9,496
60	.164	.243		.8531	168.7	7,291	9,778
70	.196	.222		.9025	167.4	9,148	9,378
80	.224	.193		.9516	162.9	10,727	8,366
90	.237	.163		1.0000	156.3	11,443	7,125
600	.238	.135		1.0476	147.5	11,363	5,833
10	.227	.109		1.0945	135.4	10,393	4,516
20	.206	.086		1.1403	117.3	8,511	3,216
30	.181	.067		1.1850	97.7	6,472	2,170
40	.153	.053		1.2284	79.1	4,594	1,440
650	.122	.040		1.2704	60.8	2,910	864
60	.099	.032		1.3111	43.7	1,752	512
70	.077	.025		1.3503	28.5	914	268
80	.057	.020	.000	1.3880	18.0	439	140
90	.043	.017	.002	1.4240	11.9	226	81	25
700	.030	.014	.003	1.4584	8.8	117	50	28
10	.022	.012	.004	1.4912	7.2	74	36	32
20	.016	.010	.004	1.5223	6.1	46	25	28
30	.011	.009	.005	1.5517	5.8	31	22	34
40	.008	.008	.004	1.5793	5.7	22	20	27
750	.005	.007	.004	1.6055	5.5	12	17	26
					Σ	99,998	99,999	100,000

Quantities computed from settings	Equations used	Uses of results
The coordinates, x, y, and Y of the I.C.I. standard system	x ≐ (0.80A +0.18B) /Σ	To find whether sample meets color requirements specified by x, y, and F.
	y ≐ G/Σ
	Σ = 0.80A+G+1.36B	To represent colors in the standard I.C.I. system.
	Y ≐ G (From Eqs. (5) and (6))	To represent colors in the standard I.C.I. system.
The chromaticity coordinates α and β of a uniform-chromatic-ness-scale system	α ≐ (A − G)/(A + 2G+B)	To designate chromaticity in a coordinate system yielding approximately uniform-chromaticness scales for surface color.
	β ≐ 0.4(G − B)/(A+2G+B) (Eqs. (8))
	β ≐ 0.4(G − B)/(A+2G+B) (Eqs. (8))	To indicate changes of chromaticness of specimens with time, exposure, change of composition, change of method of preparation, or other treatment.
The amount of color difference ΔE between two surface colors	$Δ E = f_{g} {{[7 Y^{\frac{1}{4}} {[{(Δ α)}^{2} + {(Δ β)}^{2}]}^{\frac{1}{2}} \cdot 10^{2}]}^{2} + {[k_{1} Δ (Y^{\frac{1}{2}})]}^{2}}^{\frac{1}{2}}$ (Eq. 13))	To measure the amount of color difference between two surface colors in N.B.S. units.
		To find whether a sample differs from a standard by more or less than a specified color tolerance.
		To measure the amount of color change resulting from exposure, change of composition, change of method of preparation, or other treatment of the sample.
The coordinates α′, β′, and L′ of a uniformly-spaced surface-color solid	$\begin{array}{l} α^{'} & = 700 Y^{\frac{1}{4}} α \\ β^{'} & = 700 Y^{\frac{1}{4}} β \\ L^{'} & = k_{1} Y^{\frac{1}{2}} \end{array}$ (Eqs. 14))	To designate surface color using coordinate scales which give perceptually nearly uniform measures of stimulus difference.
The hue angle ϕ, the saturation index S, and the lightness index L	$\begin{array}{l} ϕ & \equiv angle whose tangent is β / α \\ S & \equiv {[(α^{2} + β^{2}) Y^{\frac{1}{2}}]}^{\frac{1}{2}} \\ L & \equiv Y^{\frac{1}{2}} \end{array}$ (Eqs. 16))	To approximate the hue, the saturation, and the lightness of colors by separate numbers.
The hue, saturation, and lightness components ΔH′, ΔS′, and ΔL′, of a surface-color difference in N.B.S. units	$\begin{array}{l} Δ H^{'} & = 12.2 Y^{\frac{1}{4}} {(α^{2} + β^{2})}^{\frac{1}{2}} \cdot Δ ϕ & (Δ ϕ in degrees) \\ Δ S^{'} & = 700 Y^{\frac{1}{4}} \cdot Δ ({[α^{2} + β^{2}]}^{\frac{1}{2}}) \\ Δ L^{'} & = k_{1} Δ (Y^{\frac{1}{2}}) \end{array}$ (Eqs. 17))	To separate a surface-color difference into the hue, saturation, and lightness components of the difference in N.B.S. units.
The whiteness W of a white or near-white surface	$W = 1 - {{[30 {(α^{2} + β^{2})}^{\frac{1}{2}}]}^{2} + {[\frac{1 - Y}{2}]}^{2}}^{\frac{1}{2}}$ (Eqs. 19))	To designate a white or near-white surface with a number on a scale of whiteness which gives an MgO surface a value of 1.00 and a black surface a value of zero.
Yellowness	Yellowness = (A − B)/G (Eq. (20))	To give a white or near-white surface a number which indicates degree of yellowness if positive, degree of blueness if negative.

Wave-length in millimicrons	α	β
380	+0.0731	−0.3604
90	+.0725	−.3608
400	+.0718	−.3612
10	+.0705	−.3617
20	+.0679	−.3622
30	+.0613	−.3611
40	+.0484	−.3579
450	+.0265	−.3528
60	−.0095	−.3430
70	−.0729	−.3146
80	−.1876	−.2361
90	−.3373	−.0977
500	−.4440	+.0446
10	−.4651	+.1317
20	−.4166	+.1661
30	−.3387	+.1688
40	−.2605	+.1641
550	−.1793	+.1557
60	−.0924	+.1451
70	+.0001	+.1330
80	+.0952	+.1204
90	+.1879	+.1082
600	+.2688	+.0974
10	+.3320	+.0891
20	+.3754	+.0833
30	+.4035	+.0796
40	+.4229	+.0769
650	+.4352	+.0753
60	+.4423	+.0744
70	+.4458	+.0740
80	+.4483	+.0736
90	+.4501	+.0734
700 to 780 inc.	+.4506	+.0733
MgO under I.C.I. illuminant C	.0000	.0000

Conditions of observation giving equivalent visual estimates of color difference	k₁
Samples separated by a very narrow or non-existent dividing line	120
Samples separated by a contrasting, but narrow dividing line	90
Samples separated by a broad patterned area different from the areas being compared	40
Samples evaluated for whiteness without visual reference to other samples	20
(Convenient value for use when samples are separated by narrow line)	100

		For a white surface reflecting 80%	For a yellow surface reflecting 35%
If settings of the instrument should be	A	0.800	0.500
	G	.800	.350
	B	.800	.050
But instead the instrument gives	A	.801	.501
	G	.799	.349
	B	.801	.051
Changes of the following magnitudes will result	in x	+.00024	+.00049
	in y	−.00054	−.00182
	in α.	+.00062	+.00160
	in β	−.00025	−.00064
	in ${(α^{2} + β^{2})}^{\frac{1}{2}}$	.00067	.00172
	in ΔE (maxi-mum, N.B.S. units)	±.45	±.94
	in W	−.0025

Specimens compared	Size of the chromaticity difference ${[{(Δ α)}^{2} + {(Δ β)}^{2}]}^{\frac{1}{2}}$	Error in the photoelectrically measured chromaticity difference
Specimens compared		Actual	Relative to size of difference (percent)
Munsell BPB 8/2 and MgO	0.0229	0.0013	6
BG 6/4 and BG 7/4	.0150	.0018	12
Yellow Y1 and Yellow Y2	.00187	.00043	(23)
Gray G1 and Gray G2	.0160	.0061	(38)

Short cut	Errors in percent of total difference
	For 2 pairs of white samples		For 2 pairs of blue samples		For 2 pairs of orange samples
	ΔE = 1.8	ΔE = 6.6	ΔE = 1.0	ΔE = 7.6	ΔE = 1.4	ΔE = 6.0
Scale corrections omitted	0.6	1.1	0	4.1	4.2	4.8
Use of instrument values of A, G, and B having only relatively correct magnitudes	0	0	1.0	2.9	26.6	6.3
Use of 0.5 and 0.1 instead of true neutral-filter factors	—	—	3.9	3.6	7.7	1.0
Combination of above 3 short-cuts	0.6	0.3	0	8.7	27.3	4.8

Advantages		Disadvantages
1.	Small number of settings required for each sample.	1.	Errors resulting from the inaccurate spectral response of the source-filter photo-cell combinations used.
2.	Small number of computational steps required to obtain x, y, and Y, or α, β, and Y.	2.	The necessity for working standards which are spectrally similar to the samples measured.
3.	Simplicity and inexpensiveness of apparatus.	3.	The necessity for special precautions to attain the high precision required if the tristimulus method is to equal the eye in power to discriminate colors.
4.	Opportunity to convert settings rapidly to numbers giving size and character of color difference perceived between samples.	4.	Failure of tristimulus measurements to provide the physical description of a color stimulus such as is given by a spectrophotometric curve.
5.	Correspondence between values of α, β, and Y, and positions of points representing colors in the uniformly spaced surface-color solid.

Specimen	Standard	No. 1	No. 2	No. 3	No. 4
Blue settings	0.8985	0.8480	0.7900	0.7810	0.7080
Blue settings	.8955	.8450	.7880	.7790	.7070
Mean	.8970	.8465	.7890	.7800	.7075
$B = Mean \times \frac{0.890}{0.8970}$		.8398	.7828	.7739	.7020

Amber settings	.9055	.8785	.8720	.8420	.8530
Amber settings	.9030	.8770	.8680	.8400	.8520
Mean	.9042	.8778	.8700	.8410	.8525
$A = Mean \times \frac{0.906}{0.9042}$		.8795	.8717	.8427	.8542

Green settings	.9100	.8770	.8620	.8340	.8320
Green settings	.9070	.8740	.8585	.8320	.8310
Mean	.9085	.8755	.8602	.8330	.8315
$Y ≐ G = Mean \times \frac{0.909}{0.9085}$		.8760	.8607	.8335	.8320

0.18 B		.1512	.1409	.1393	.1264
1.18 B		.9910	.9237	.9132	.8284
0.80 A+0.18 B		.8548	.8383	.8135	.8098
Denom (G+0.80 A+1.36 B)		2.7218	2.6227	2.5602	2.4702
$x ≐ \frac{0.80 A + 0.18 B}{Denom}$		.3141	.3196	.3177	.3278
y ≐ G /Denom		.3218	.3282	.3256	.3368
z ≐ 1.18/Denom		.3641	.3522	.3567	.3354

Result of test		pass	fail (x)	fail (Y)	fail (all)

Specimen	Standard	Roof (4 days)	Machine 1 (43 hours)	Machine 2 (41 hours)
Blue settings	0.4995	0.300	0.685	0.597
Blue settings	.501
Mean	.5002
$B = Setting \times \frac{0.4714}{0.5002}$		.2827	.6453	.5626

Amber settings	.704	.510	.7075	.675
Amber settings	.704
Mean	.704
$A = Setting \times \frac{0.7058}{0.704}$		.5113	.7093	.6767

Green settings	.620	.424	.708	.656
Green settings	.619
Mean	.6195
$G = Setting \times \frac{0.606}{0.6195}$		.4148	.6926	.6417

A–G		+.0965	+.0167	+.0350
G–B		+.1321	+.0473	+.0791
0.4(G–B)		+.0528	+.0189	+.0316
B+A + 2G = Denom		1.6236	2.7398	2.5227
α ≐ (A–G)/Denom		+.0594	+.0061	+.0139
β ≐ 0.4(G–B)/Denom		+.0325	+.0069	+.0125

Fabric	Standard	No. 1	No. 2	No. 3	No. 4
Blue settings (with 0.427 filter)	0.1700	0.1744	0.1620	0.1690	0.2024
Blue settings (with 0.427 filter)	.1688	.1738	.1618	.1686	.2018
Mean (M)	.1694	.1741	.1619	.1688	.2021
Scale correction	−.0139	−.0138	−.0138	−.0136	−.0121
Corrected M	.1555	.1603	.1481	.1552	.1900
B = Above times 0.427	.0664	.0684	.0632	.0663	.0811

Amber settings (with 0.480 filter)	.2098	.2144	.2022	.2042	.2524
Amber settings (with 0.480 filter)	.2082	.2142	.2018	.2036	.2520
Mean (M)	.2090	.2143	.2020	.2039	.2522
Scale correction	−.0120	−.0120	−.0120	−.0120	−.0113
Corrected M	.1970	.2023	.1900	.1919	.2409
A = Above times 0.480	.0946	.0971	.0912	.0921	.1156

Green settings (with 0.479 filter)	.1916	.1946	.1848	.1882	.2300
Green settings (with 0.479 filter)	.1900	.1944	.1844	.1876	.2302
Mean (M)	.1908	.1945	.1846	.1879	.2301
Scale correction	−.0127	−.0126	−.0130	−.0129	−.0122
Corrected M	.1781	.1819	.1716	.1750	.2179

Y = G = Above times 0.479	.0853	.0871	.0822	.0838	.1044
A–G	+.0093	+.0100	+.0090	+.0083	+.0112
G–B	+.0189	+.0187	+.0190	+.0175	+.0233
0.4(G–B)	+.0076	+.0075	+.0076	+.0070	+.0093
Denom = B+A+2G	.3316	.3397	.3188	.3260	.4055
α ≐ (A–G) /Denom	+.0280	+.0294	+.0282	+.0255	+.0276
β ≐ 0.4(G–B)/Denom	+.0229	+.0221	+.0238	+.0215	+.0229

Δα = α − α_std		+.0014	+.0002	−.0025	−.0004
Δβ = β − β_std		−.0008	+.0009	−.0014	0
${[{(Δ α)}^{2} + {(Δ β)}^{2} \cdot 10^{2}]}^{\frac{1}{2}}$		.16.	.09	.29	.04
$Y^{\frac{1}{2}}$	.292	.295	.287	.289	.323
$7 (Y^{\frac{1}{4}})$ *	3.78				3.98
$Δ C = 7 (Y^{\frac{1}{4}}) {[{(Δ α)}^{2} + {(Δ β)}^{2} \cdot 10^{2}]}^{\frac{1}{2}}$		.60	.34	1.10	.16

$Δ Y^{\frac{1}{2}}$		+.003	−.005	−.003	+.031
$Δ L = k_{l} (Δ Y^{\frac{1}{2}})$		.3	.5	.3	3.1

$Δ E / f_{g} = {[{(Δ C)}^{2} + {(Δ L)}^{2}]}^{\frac{1}{2}}$		.67	.60	1.14	3.11
ΔE		0.7	0.6	1.1	3.1

Specimen	Earth Mixture	Modification No. 1	Modification No. 2
Blue settings	0.295	0.260	0.370
B = Above times 0.5	.147₅	.130	.185
Amber settings	.496	.446	.575
A = Above times 0.5	.248	.223	.287₅
Green settings	.428	.386	.508
Y = G = Above times 0.5	.214	.193	.254

$L = Y^{\frac{1}{2}} ≐ G^{\frac{1}{2}}$	.463	.439	.504
$Δ L^{'} = k_{l} Δ (Y^{\frac{1}{2}})$	—	−2.4	+4.1
A – G (note sign)	+.034	+.030	+.0335
G–B (note sign)	+.0665	+.063	+.069
0.4(G–B)	+.0266	+.0252	+.0276
Denom = B+M+2G	.8235	.739	.9805
α ≐ (A – G) /Denom	+.0413	+.0406	+.0342
β ≐ 0.4(G–B)/Denom	+.0323	+.0341	+.0281
${(α^{2} + β^{2})}^{\frac{1}{2}}$	.0525	.0528	.0443
$Δ {(α^{2} + β^{2})}^{\frac{1}{2}}$		+.0003	−.0082
$700 Y^{\frac{1}{4}}$ *		470.	487.
$Δ S^{'} = 700 Y^{\frac{1}{4}} Δ {(α^{2} + β^{2})}^{\frac{1}{2}}$		+0.14	−3.99

β/α	+.782	+.840	+.822
ϕ = angle whose tan is β/α	38.03°	40.03°	39.42°
Δϕ		+2.00°	+ 1.39°
$12.2 Y^{\frac{1}{4}}$ *		8.06	8.34
$12.2 Y^{\frac{1}{4}} {(α^{2} + β^{2})}^{\frac{1}{2}}$ *		.424	.404
$Δ H^{'} = {12.2}^{\frac{1}{4}} {(α^{2} + β^{2})}^{\frac{1}{2}} \cdot Δ ϕ$		+0.85	+0.56
Color of modification		Darker,	Lighter,
relative to that of		greener,	weaker,
actual earth, according		slightly	slightly
to the measurements		stronger	greener
$Δ E = {[{(Δ L^{'})}^{2} + {(Δ S^{'})}^{2} + {(Δ H^{'})}^{2}]}^{\frac{1}{2}}$		2.6	5.8

Specimen	Standard	New	1L	20L
Blue settings	0.898	0.837	0.845	0.765
Blue settings	.892	.850	.852	.787
Mean	.895	.8435	.8485	.776
$B = Mean \times \frac{0.890}{0.895}$		.8388	.8438	.7717
Amber settings	.9085	.892	.912	.826
Amber settings	.906	.904	.912	.836
Mean	.9072	.898	.912	.831
$A = Mean \times \frac{0.906}{0.9072}$		.8968	.9108	.8299

Green settings	.9055	.884	.900	.812
Green settings	.903	.894	.900	.824
Mean	.9042	.889	.900	.818
$G = Mean \times \frac{0.909}{0.9042}$		.8937	.9048	.8223

A–G		.0031	.0060	.0076
2.5 (A–G)		.0078	.0150	.0190
G–B		.0549	.0610	.0506
α ≐ 2.5 (A–G)/10G		.0009	.0017	.0023
β ≐ (G–B)/10G		.0061	.0067	.0062
${(α^{2} + β^{2})}^{\frac{1}{2}}$		.0062	.0069	.0066
$W_{C} = 30 {(α^{2} + β^{2})}^{\frac{1}{2}}$		.186	.207	.198
1.00−G		.106	.095	.178
W_L = (1.00−G)/2		.053	.048	.089
${({W_{C}}^{2} + {W_{L}}^{2})}^{\frac{1}{2}}$		.193	.213	.217
W = 1 − above		.807	.787	.783

Plaque No.	Standard	1	10	22	24
Blue settings	0.8955	0.7540	0.7760	0.8238	0.7804
Blue settings	.8920	.7490	.7716	.8222	.7780
Mean	.8938	.7515	.7738	.8230	.7792
$B = Mean \times \frac{0.890}{0.8938}$		.7483	.7705	.8195	.7759

Amber settings	.8945	.7276	.7236	.7990	.7980
Amber settings	.8936	.7222	.7176	.7962	.7962
Mean	.8940	.7249	.7206	.7976	.7971
$A = Mean \times \frac{0.906}{0.8940}$		.7346	.7303	.8083	.8078

Green settings	.9037	.7370	.7364	.8094	.8046
Green settings	.9030	.7316	.7324	.8074	.8030
Mean	.9034	.7343	.7344	.8084	.8038
$G = Mean \times \frac{0.909}{0.9034}$		.7389	.7390	.8134	.8088

A–B		−.0137	−.0402	−.0112	+.0319
Yellowness = (A−B)/G (Eq. 20)		−.0185	−.0544	−.0138	+.0394

Abstract

Cited By

Figures (11)

Tables (16)

Equations (72)

Journal of the Optical Society of America