Opponent-Process Solutions for Uniform Munsell Spacing

Whitman Richards

doi:10.1364/JOSA.56.001110

Journal of the Optical Society of America
Vol. 56,
Issue 8,
pp. 1110-1120
(1966)
•https://doi.org/10.1364/JOSA.56.001110

Opponent-Process Solutions for Uniform Munsell Spacing

Whitman Richards

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Whitman Richards, "Opponent-Process Solutions for Uniform Munsell Spacing," J. Opt. Soc. Am. 56, 1110-1120 (1966)

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Abstract

An iterative method is used to transform the chromaticity coordinates of the Munsell samples into another coordinate system such that the transformed values are spaced in accordance with the perceptual spacing of the colors. Acceptable transformations are restricted to those having an opponent-process form; brightness information is assumed to be conveyed by an independent channel. Under these conditions, the optimal transformation based on two chromatic processes is similar to one stage of the Müller–Judd formulation. By changing the constraints imposed on acceptable transformations, however, support can also be found for the Hering model. Therefore, even though many quantitative transformations can already be excluded, more data are needed before this method can be applied as a decisive test for models of color vision.

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Solution type	Function	Angle (deg)	Difference (D)	Scale coeff.	Calculated constants
Solution type	Function	Angle (deg)	Difference (D)	Scale coeff.	a_i₁	a_i₂	a_i₃	b_i₁	b_i₂	b_i₃
2D-10	LN	67.5	0.31	1.0	0.73	0.21	0.06	−0.24	1.08	0.24
(4 hues)				1.0	−0.23	1.18	0.07	0.26	0.41	0.48
		22.5	0.33	1.0	0.71	0.30	−0.05	0.35	0.36	0.40
				1.0	−0.47	1.40	0.10	0.73	0.08	0.25
	SQRT	56	0.47	1.0	0.85	0.04	0.12	0.34	0.33	0.48
				1.0	0.77	−0.08	0.42	−0.18	1.01	0.24
		0	0.60	1.0	0.01	0.87	0.16	0.07	0.46	C.68
				1.0	−0.42	1.33	0.14	0.92	−0.01	0.11
	LN	45	1.8	1.0	0.70	0.22	0.08	0.28	0.47	0.25
	LN			0.4	−0.46	1.37	0.09	0.44	0.13	0.42
	SQRT	67.5	1.8	1.0	0.36	0.32	0.44	0.08	0.73	0.27
	SQRT			0.4	0.98	−0.06	0.09	−0.04	0.80	0.34
2D-10	LN	92	1.5	0.79	−0.13	0.84	0.42	0.71	0.28	−0.02
(5 hues)				1.06	−0.43	1.38	0.08	0.24	0.51	0.36
		83	1.5	0.80	−0.19	0.96	0.34	0.73	0.25	0
				0.94	−0.43	1.37	0.08	0.16	0.53	0.44
		74	0.81	0.90	−0.25	1.08	0.26	0.74	0.24	0.01
				0.97	−0.40	1.35	0.08	0.13	0.54	0.47
		65	0.87	0.94	−0.30	1.16	0.20	0.73	0.24	0.03
				0.90	−0.36	1.31	0.07	0.05	0.57	0.54
		55	1.6	1.41	−0.30	1.14	0.25	0.50	0.39	0.14
				0.81	−0.31	1.27	0.06	−0.11	0.66	0.67
		47	1.8	0.90	−0.34	1.21	0.19	0.85	0.04	0.13
				1.02	0.30	0.68	0.02	0.37	0.32	0.45
		38	1.7	0.91	−0.45	1.40	0.09	0.68	0.24	0.09
				0.66	0.26	0.76	−0.03	0.10	0.46	0.63
		30	1.75	0.92	−0.44	1.39	0.09	0.67	0.20	0.17
				0.55	0.59	0.44	−0.07	0.11	0.42	0.69
		20	2.7	0.98	−0.32	1.27	0.07	0.77	0.08	0.19
				0.47	1.03	0.03	−0.13	0.10	0.42	0.70
		12	2.9	0.98	−0.45	1.40	0.09	0.65	0.25	0.10
				0.68	0.80	0.20	0	0.10	0.50	0.60
	SQRT	72	1.2	3.1	−0.32	1.15	0.24	1.00	−0.01	−0.03
				3.9	−0.34	1.24	0.15	0.19	0.43	0.55
		64	1.1	3.6	−0.34	1.20	0.20	0.98	−0.01	0.01
				3.7	−0.24	1.14	0.16	0.13	0.44	0.63
		57	0.85	3.3	−0.24	1.21	0.05	1.09	−0.08	−0.05
				2.7	−0.25	1.20	0.08	0.23	0.26	0.75
		46	1.2	3.8	−0.35	1.24	0.17	0.90	0.00	0.11
				3.7	0.02	0.87	0.15	0.06	0.47	0.69
		34	1.35	3.9	−0.35	1.25	0.16	0.84	0.02	0.18
				3.5	0.16	0.73	0.15	0.03	0.50	0.69
		29	2.0	4.7	−0.21	1.06	0.23	0.80	−0.02	0.30
				4.0	0.76	0.11	0.16	0.33	0.25	0.60
		11	1.75	4.4	−0.20	1.06	0.21	0.70	0.01	0.40
				4.0	0.88	0.03	0.11	0.32	0.33	0.51
		3	1.6	4.2	−0.22	1.09	0.20	0.63	0.04	0.46
				4.2	0.94	−0.03	0.09	0.25	0.44	0.45
		−6	1.3	3.9	−0.26	1.13	0.19	0.54	0.08	0.54
				4.7	0.95	−0.05	0.12	0.17	0.56	0.38
		−14	1.25	3.6	−0.22	1.09	0.19	0.48	0.09	0.61
				5.3	0.96	−0.07	0.11	0.19	0.59	0.31
		−25	1.3	2.8	−0.25	1.13	0.18	0.30	0.15	0.78
				5.2	0.98	−0.08	0.11	0.19	0.62	0.25
3D-15	LN	92	1.55	0.6	0.24	0.30	0.67	0.10	0.71	0.27
(5 hues)				1.0	−0.26	1.12	0.20	0.66	0.33	−0.01
				1.1	−0.48	1.40	0.12	−0.14	0.84	0.44
		74	0.74	0.6	0.22	0.40	0.55	0.09	0.67	0.35
				1.0	−0.29	1.17	0.17	0.64	0.33	0.01
				1.1	−0.48	1.41	0.11	−0.17	0.84	0.48
		57	0.88	0.5	0.26	0.46	0.39	−0.01	0.70	0.44
				1.0	−0.31	1.21	0.14	0.65	0.32	0.01
				1.0	−0.47	1.40	0.10	−0.18	0.83	0.51
		47	1.15	0.7	0.32	0.42	0.37	0.08	0.56	0.51
				1.1	−0.28	1.17	0.16	0.61	0.33	0.06
				1.1	−0.42	1.33	0.13	−0.20	0.84	0.53
	SQRT	72	1.15	3.4	0.22	0.40	0.55	0.25	0.47	0.40
				4.5	−0.25	1.12	0.18	0.74	0.22	0.04
				4.3	−0.45	1.36	0.13	−0.18	0.82	0.52
		66	0.96	3.5	0.21	0.43	0.51	0.26	0.44	0.43
				4.6	−0.25	1.13	0.17	0.72	0.22	0.06
				4.3	−0.45	1.36	0.13	−0.19	0.82	0.54
		57	1.15	3.5	0.22	0.46	0.45	0.24	0.42	0.49
				4.8	−0.24	1.12	0.17	0.73	0.22	0.05
				4.4	−0.42	1.33	0.14	−0.19	0.82	0.55
		48	1.0	3.7	0.25	0.45	0.42	0.22	0.40	0.55
				4.9	−0.24	1.13	0.16	0.72	0.21	0.08
				4.2	−0.40	1.30	0.15	−0.20	0.81	0.57
		37	1.0	2.9	0.32	0.46	0.31	0.14	0.41	0.65
				5.0	−0.26	1.17	0.14	0.70	0.22	0.10
				3.9	−0.34	1.23	0.17	−0.21	0.82	0.57
		30	1.2	3.1	0.41	0.39	0.29	0.13	0.42	0.67
				5.1	−0.27	1.18	0.13	0.66	0.23	0.13
				3.8	−0.30	1.18	0.18	−0.22	0.84	0.56
2D-6	LN	120	2.3	1.5	0.58	0.46	−0.08	0.00	0.98	0.02
(5 hues)				1.2	0.46	0.32	0.30	0.00	0.98	0.02
		100	1.3	1.6	0.67	0.32	−0.02	0.00	0.98	0.02
				0.9	0.17	0.47	0.50	0.00	0.98	0.02
		85	1.15	1.6	0.65	0.32	0.02	0.00	0.98	0.02
				0.9	−0.07	0.72	0.50	0.00	0.98	0.02
		70	1.65	1.6	0.60	0.33	0.07	0.00	0.98	0.02
				1.1	−0.28	1.03	0.38	0.00	0.98	0.02
		57	2.5	1.7	0.54	0.37	0.12	0.00	0.98	0.02
				1.4	−0.38	1.21	0.26	0.00	0.98	0.02
3D-9	LN	120	1.85	1.0	0.18	0.39	0.62	0.27	0.48	0.35
				1.5	0.00	0.96	0.04	0.66	0.30	0.02
				0.8	0.00	0.96	0.04	0.03	0.63	0.48
		102	1.7	1.0	0.19	0.43	0.54	0.28	0.43	0.40
				1.6	0.00	0.96	0.04	0.67	0.31	0.01
				0.9	0.00	0.96	0.04	−0.03	0.66	0.53
		84	1.35	1.0	0.26	0.43	0.44	0.22	0.43	0.50
				1.6	0.00	0.96	0.04	0.64	0.32	0.04
				1.0	0.00	0.96	0.04	−0.06	0.72	0.50
		66	1.45	1.0	0.50	0.28	0.32	0.05	0.65	0.43
				1.4	0.00	0.96	0.04	0.68	0.26	0.06
				0.9	0.00	0.96	0.04	−0.08	0.68	0.51
		48	2.0	0.6	0.82	0.11	0.06	−0.10	0.75	0.50
				1.7	0.00	0.96	0.04	0.51	0.35	0.20
				0.1	0.00	0.96	0.04	−0.03	0.75	0.45

Abstract

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Equations (10)

Journal of the Optical Society of America