Analytic approximations for the CIE 1964 ten-degree-field color-matching functions*

P. M. Tannenbaum

doi:10.1364/JOSA.64.000089

Journal of the Optical Society of America
Vol. 64,
Issue 1,
pp. 89-91
(1974)
•https://doi.org/10.1364/JOSA.64.000089

Analytic approximations for the CIE 1964 ten-degree-field color-matching functions

P. M. Tannenbaum

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P. M. Tannenbaum, "Analytic approximations for the CIE 1964 ten-degree-field color-matching functions*," J. Opt. Soc. Am. 64, 89-91 (1974)

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Abstract

The similarity of the shapes and relative positions of the peaks and valleys of the first few hydrogen-atom radial probability-density functions to the CIE 1964 $\bar{x}, \bar{y}, \bar{z}$ 10°-field color-matching functions makes the former ideal candidates for analytic approximations to the latter. The hydrogen functions are, therefore, mapped into the wavelength range of the tristimulus coefficients and scaled appropriately. The two sets of curves are compared by looking at their ratios throughout the 380- to 750-nm range. Methods for curve fitting these ratios using a Chebychev-polynomial least-squares-fit procedure are described and approximate analytic expressions for $\bar{x}, \bar{y}, \bar{z}$ are given for varying degrees of fitting accuracy.

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Function	α	β	U(λ)	Mean square deviation (MSD)
${\bar{x}}_{1}$	2.370×10⁻⁵	0.101	λ-380	0.1806
${\tilde{y}}_{1}$	7.585×10⁻⁴	0.085	660-λ	0.1164
${\tilde{z}}_{1}$	4.167×10⁻¹	0.093	λ-400	0.0610
${\tilde{y}}_{2}$	6.527×10⁻⁴	0.061	716-λ	0.1060

Function	b₁	b₂	b₃	b₄	b₅	b₆	b₇	b₈	b₉	b₁₀	Function mean-square deviation
${\tilde{x}}_{1}$	0.673 24										0.1645
${\tilde{x}}_{2}$	0.676 04	−0.404 08									0.1919
${\tilde{x}}_{3}$	0.57194	−0.405 35	−0.218 49								0.1456
${\tilde{x}}_{4}$	0.570 31	−0.531	−0.216 97	−0.194 32							0.1065
${\tilde{x}}_{5}$	0.590 24	−0.530 85	−0.140 22	−0.195 61	0.120 53						0.0948
${\tilde{x}}_{6}$	0.589 56	−0.478 23	−0.138 70	−0.091 53	0.121 64	0.161 99					0.0562
${\tilde{x}}_{7}$	0.588 62	−0.478 18	−0.140 43	−0.091 63	0.118 09	0.162 08	−0.006 69				0.0565
${\tilde{x}}_{8}$	0.588 70	−0.491 03	−0.140 51	−0.100 93	0.118 10	0.130 22	−0.006 49	−0.051 49			0.0454
${\tilde{x}}_{9}$	0.582 69	−0.491 58	−0.149 62	−0.099 92	0.105 06	0.129 13	−0.046 68	−0.057 71	−0.076 95		0.0326
${\tilde{x}}_{10}$	0.582 92	−0.490 31	−0.149 92	−0.096 16	0.104 99	0.132 62	−0.046 34	−0.042 69	−0.077 46	0.031 77	0.0285
${\tilde{y}}_{1}$	0.769 56										0.1756
${\tilde{y}}_{2}$	0.767 61	0.281 05									0.1827
${\tilde{y}}_{3}$	0.575 48	0.278 70	−0.403 25								0.1075
${\tilde{y}}_{4}$	0.576 53	0.360 08	−0.404 23	0.125 84							0.0917
${\tilde{y}}_{5}$	0.590 30	0.360 17	−0.351 22	0.124 95	0.083 26						0.0778
${\tilde{y}}_{6}$	0.591 14	0.296 04	−0.353 08	−0.001 90	0.081 90	−0.197 42					0.0601
${\tilde{y}}_{7}$	0.566 90	0.297 26	−0.397 59	−0.004 47	−0.009 23	−0.195 05	−0.172 06				0.0299
${\tilde{y}}_{8}$	0.567 00	0.282 60	−0.397 68	−0.015 08	−0.009 23	−0.231 39	−0.171 83	−0.065 59			0.0364
${\tilde{y}}_{9}$	0.565 50	0.282 47	−0.399 95	−0.014 83	−0.012 48	−0.231 67	−0.181 85	−0.065 64	−0.019 17		0.0373
${\tilde{y}}_{10}$	0.565 50	0.282 47	−0.399 95	−0.014 83	−0.012 48	−0.231 67	−0.181 85	−0.065 64	−0.019 17	−0.⁺	0.0373
${\tilde{z}}_{1}$	0.368 88										0.3747
${\tilde{z}}_{2}$	0.374 00	−0.737 65									0.1297
${\tilde{z}}_{3}$	0.455 70	−0.737 65	0.171 47								0.1185
${\tilde{z}}_{4}$	0.457 44	−0.602 01	0.169 84	0.208 23							0.0756
${\tilde{z}}_{5}$	0.436 27	−0.602 16	0.088 33	0.209 60	−0.128 01						0.0782
${\tilde{z}}_{6}$	0.436 56	−0.624 71	0.087 68	0.165 00	−0.128 48	−0.069 41					0.0646
${\tilde{z}}_{7}$	0.448 24	−0.625 29	0.109 12	0.166 24	−0.084 58	−0.070 56	0.082 88				0.0592
${\tilde{z}}_{8}$	0.448 21	−0.621 26	0.109 14	0.169 16	−0.084 58	−0.060 56	0.082 82	0.018 04			0.0575
${\tilde{z}}_{9}$	0.445 41	−0.621 52	0.104 90	0.169 63	−0.090 66	−0.061 07	0.064 09	0.017 93	−0.035 87		0.0536
${\tilde{z}}_{10}$	0.445 33	−0.621 95	0.105 00	0.168 34	−0.090 64	−0.062 27	0.063 97	0.012 8	−0.035 69	−0.010 87	0.0562

Test material	Color	$Δ E - \tilde{x}, \tilde{y}, \bar{z}$	$Δ E - {\tilde{x}}_{10}, {\tilde{y}}_{10}, {\bar{z}}_{10}$
Wratten #58	Dark green	31.19	7.48
Wratten #48	Dark blue	8.39	7.35
Wratten #56	Light green	60.71	6.23
Wratten #23A	Light red	272.08	5.45
Wratten #9	Yellow	154.71	12.09
Wratten #21	Orange	208.58	5.25
Average fleshtone		119.07	13.82
Brown hair		51.32	5.78
Wratten #78	Blue	20.74	11.26
Wratten #30	Magenta	182.52	13.75
Wratten #11	Yellow green	41.60	8.13
Wratten #4	Yellow	130.82	13.72
Wratten #59	Light green	42.11	6.93
Wratten #85	Amber	140.21	14.86
Wratten #66	Light green	56.15	10.00

	Total:	1520.2	142.1
	Average:	101.3	9.47

Function	α	β	U(λ)	Mean square deviation (MSD)
${\bar{x}}_{1}$	2.370×10⁻⁵	0.101	λ-380	0.1806
${\tilde{y}}_{1}$	7.585×10⁻⁴	0.085	660-λ	0.1164
${\tilde{z}}_{1}$	4.167×10⁻¹	0.093	λ-400	0.0610
${\tilde{y}}_{2}$	6.527×10⁻⁴	0.061	716-λ	0.1060

Function	b₁	b₂	b₃	b₄	b₅	b₆	b₇	b₈	b₉	b₁₀	Function mean-square deviation
${\tilde{x}}_{1}$	0.673 24										0.1645
${\tilde{x}}_{2}$	0.676 04	−0.404 08									0.1919
${\tilde{x}}_{3}$	0.57194	−0.405 35	−0.218 49								0.1456
${\tilde{x}}_{4}$	0.570 31	−0.531	−0.216 97	−0.194 32							0.1065
${\tilde{x}}_{5}$	0.590 24	−0.530 85	−0.140 22	−0.195 61	0.120 53						0.0948
${\tilde{x}}_{6}$	0.589 56	−0.478 23	−0.138 70	−0.091 53	0.121 64	0.161 99					0.0562
${\tilde{x}}_{7}$	0.588 62	−0.478 18	−0.140 43	−0.091 63	0.118 09	0.162 08	−0.006 69				0.0565
${\tilde{x}}_{8}$	0.588 70	−0.491 03	−0.140 51	−0.100 93	0.118 10	0.130 22	−0.006 49	−0.051 49			0.0454
${\tilde{x}}_{9}$	0.582 69	−0.491 58	−0.149 62	−0.099 92	0.105 06	0.129 13	−0.046 68	−0.057 71	−0.076 95		0.0326
${\tilde{x}}_{10}$	0.582 92	−0.490 31	−0.149 92	−0.096 16	0.104 99	0.132 62	−0.046 34	−0.042 69	−0.077 46	0.031 77	0.0285
${\tilde{y}}_{1}$	0.769 56										0.1756
${\tilde{y}}_{2}$	0.767 61	0.281 05									0.1827
${\tilde{y}}_{3}$	0.575 48	0.278 70	−0.403 25								0.1075
${\tilde{y}}_{4}$	0.576 53	0.360 08	−0.404 23	0.125 84							0.0917
${\tilde{y}}_{5}$	0.590 30	0.360 17	−0.351 22	0.124 95	0.083 26						0.0778
${\tilde{y}}_{6}$	0.591 14	0.296 04	−0.353 08	−0.001 90	0.081 90	−0.197 42					0.0601
${\tilde{y}}_{7}$	0.566 90	0.297 26	−0.397 59	−0.004 47	−0.009 23	−0.195 05	−0.172 06				0.0299
${\tilde{y}}_{8}$	0.567 00	0.282 60	−0.397 68	−0.015 08	−0.009 23	−0.231 39	−0.171 83	−0.065 59			0.0364
${\tilde{y}}_{9}$	0.565 50	0.282 47	−0.399 95	−0.014 83	−0.012 48	−0.231 67	−0.181 85	−0.065 64	−0.019 17		0.0373
${\tilde{y}}_{10}$	0.565 50	0.282 47	−0.399 95	−0.014 83	−0.012 48	−0.231 67	−0.181 85	−0.065 64	−0.019 17	−0.⁺	0.0373
${\tilde{z}}_{1}$	0.368 88										0.3747
${\tilde{z}}_{2}$	0.374 00	−0.737 65									0.1297
${\tilde{z}}_{3}$	0.455 70	−0.737 65	0.171 47								0.1185
${\tilde{z}}_{4}$	0.457 44	−0.602 01	0.169 84	0.208 23							0.0756
${\tilde{z}}_{5}$	0.436 27	−0.602 16	0.088 33	0.209 60	−0.128 01						0.0782
${\tilde{z}}_{6}$	0.436 56	−0.624 71	0.087 68	0.165 00	−0.128 48	−0.069 41					0.0646
${\tilde{z}}_{7}$	0.448 24	−0.625 29	0.109 12	0.166 24	−0.084 58	−0.070 56	0.082 88				0.0592
${\tilde{z}}_{8}$	0.448 21	−0.621 26	0.109 14	0.169 16	−0.084 58	−0.060 56	0.082 82	0.018 04			0.0575
${\tilde{z}}_{9}$	0.445 41	−0.621 52	0.104 90	0.169 63	−0.090 66	−0.061 07	0.064 09	0.017 93	−0.035 87		0.0536
${\tilde{z}}_{10}$	0.445 33	−0.621 95	0.105 00	0.168 34	−0.090 64	−0.062 27	0.063 97	0.012 8	−0.035 69	−0.010 87	0.0562

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Journal of the Optical Society of America