Analytical Approximations for Color Metric Coefficients. II. Friele Approximation

David L. MacAdam

doi:10.1364/JOSA.54.000249

Journal of the Optical Society of America
Vol. 54,
Issue 2,
pp. 249-256
(1964)
•https://doi.org/10.1364/JOSA.54.000249

Analytical Approximations for Color Metric Coefficients. II. Friele Approximation

David L. MacAdam

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David L. MacAdam, "Analytical Approximations for Color Metric Coefficients. II. Friele Approximation," J. Opt. Soc. Am. 54, 249-256 (1964)

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Abstract

Friele has published a set of formulas based on an analysis of experimental color metric coefficients published by Brown and MacAdam. These formulas permit the direct computation of color metric coefficients that may be satisfactory for many uses. The results are here compared with the original experimental data. Contour diagrams showing the dependence of the Friele metric coefficients, g₁₁, g₁₂, and g₂₂, on chromaticity are presented for comparison with the corresponding diagrams published by MacAdam in 1943, which are used for many applications.

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		Coefficients						Ellipse constants
Color	Observer	g₁₁	g₁₂	g₂₂	g₂₃	g₃₃	g₃₁	θ	a	b
1	B	6	2	21	− 4	32	− 1	− 9°	4.3	2.2
	M	8	6	20	− 8	17	− 5	− 21°	4.3	2.1
	F	27	11	56	13	66	15	− 19°	2.1	1.3

2	B	17	− 13	21	− 6	42	5	40°	4.0	1.8
	M	9	− 7	21	− 6	30	0	25°	4.1	2.0
	F	16	− 6	17	8	68	6	42°	3.1	2.1

3	B	34	− 11	13	− 2	51	− 6	67°	3.4	1.6
	M	12	− 4	9	− 3	18	1	58°	3.9	2.6
	F	12	− 0.2	5	3	61	3	88°	4.3	2.9

4	B	21	− 20	35	− 2	39	3	35°	3.8	1.4
	M	14	− 17	45	5	15	− 2	24°	3.9	1.4
	V	28	− 24	35	7	68	6	41°	3.8	1.3

5	B	21	− 4	25	− 3	47	2	32°	2.4	1.9
	M	14	− 2	20	− 5	30	0	18°	2.8	1.4
	F	13	0.1	11	2	60	4	93°	3.0	2.8

6	B	72	− 21	13	− 1	62	1	72°	4.0	1.1
	M	45	− 24	26	12	54	6	56°	3.2	1.3
	F	14	0.1	7	8	69	5	91°	3.8	2.6

7	B	31	− 11	28	− 8	44	3	49°	2.3	1.6
	M	18	− 1	35	−20	103	5	3°	2.4	1.4
	F	13	− 0.3	10	4	62	4	85°	3.1	2.7

8	B	23	0	10	− 6	62	11	90°	3.2	2.1
	F	7	0.6	3	2	60	1	98°	6.2	3.7

9	B	45	35	107	−14	22	2	− 24°	1.8	0.9
	M	28	32	88	− 8	10	− 3	− 30°	1.8	1.1
	F	134	68	273	9	60	18	− 22°	1.0	0.6

11	B	19	− 1	77	−18	30	4	1°	2.3	1.1
	M	18	5	66	− 1	17	− 3	− 6°	2.4	1.2
	F	57	2	216	26	68	20	− 1°	1.3	0.7

12	B	31	− 31	44	− 5	48	0	39°	4.0	1.2
	M	35	− 38	69	−25	58	20	32°	3.5	1.0
	F	44	− 34	39	8	68	5	47°	3.5	1.1

13	B	79	− 47	77	7	52	0	46°	1.8	0.9
	M	156	−119	150	9	67	14	46°	1.7	0.6
	F	101	− 30	23	11	67	3	71°	2.9	0.9

14	B	16	8	41	−13	30	6	− 16°	2.7	1.5
	M	17	14	57	6	22	− 3	− 18°	2.8	1.3
	F	78	33	137	0	60	12	− 24°	1.3	0.8

15	B	15	− 17	36	− 3	24	− 1	29°	4.3	1.5
	M	7	− 4	23	− 8	26	2	15°	3.8	2.1
	F	19	− 18	42	7	68	8	29°	3.4	1.4

16	B	49	− 35	66	4	105	−11	17°	3.8	1.6
	M	39	− 24	89	−60	117	−14	22°	1.8	1.0
	F	15	− 4	17	8	68	7	37°	2.9	2.2

17	B	11	0	22	1	36	− 3	0°	3.0	2.1
	M	11	8	46	−11	41	0	− 12°	3.3	1.4
	F	17	− 5	23	2	63	8	29°	2.7	2.0

18	B	10	− 9	38	−11	26	2	17°	3.8	1.6
	M	15	− 2	31	− 8	43	−13	16°	3.7	1.5
	F	20	− 4	47	16	68	10	7°	2.2	1.4

19	B	43	− 20	18	1	32	2	61°	3.5	1.4
	M	58	− 33	32	5	39	− 4	56°	3.2	1.1
	F	59	− 20	16	9	69	2	69°	3.5	1.2

20	B	31	− 5	6	− 2	48	5	79°	4.4	1.8
	M	29	− 13	19	− 8	38	7	56°	3.0	1.6
	F	19	− 2	6	7	68	3	80°	4.2	2.2

21	B	45	− 13	11	−10	52	12	66°	4.3	1.3
	M	47	− 19	27	−18	66	10	59°	2.5	1.3
	F	21	− 4	7	7	68	3	77°	4.1	2.2

22	B	31	− 14	14	−10	80	21	61°	4.0	1.6
	M	48	− 34	40	−12	81	5	51°	2.8	1.2
	F	30	− 12	13	7	68	4	63°	3.8	1.7

23	B	57	− 18	15	− 2	54	13	69°	3.6	1.3
	M	49	− 23	22	− 5	37	11	60°	3.4	1.3
	F	34	− 8	9	8	68	3	73°	3.9	1.7

24	B	35	0	15	−10	61	1	90°	2.6	1.7
	M	22	− 2	13	− 8	57	4	78°	2.8	2.1
	F	14	− 1	7	− 4	61	3	82°	3.9	2.7

25	B	24	1	5	− 2	46	3	93°	4.5	2.0
	M	24	− 2	9	3	105	− 6	83°	3.4	2.0
	F	8	0.7	3	2	61	1	98°	5.7	3.5

26	B	23	− 14	21	− 7	98	1	43°	3.8	1.6
	M	40	− 29	42	−18	134	26	43°	3.0	1.2
	F	20	− 6	11	8	68	5	64°	3.5	2.1

27	B	22	− 19	29	−11	32	3	40°	2.5	1.5
	M	12	− 10	19	− 5	28	3	35°	4.5	2.0
	F	23	− 22	38	6	68	6	36°	3.8	1.4

28	B	16	3	59	−19	43	4	− 4°	2.5	1.3
	M	27	20	39	−28	65	−22	− 37°	2.9	1.4
	F	37	16	101	20	69	19	− 14°	1.7	1.0

29	B	25	− 13	44	−10	72	−10	27°	2.3	1.4
	M	12	− 8	34	8	97	− 6	18°	3.3	1.6
	F	15	− 7	25	8	69	8	28°	3.0	1.9

30	B	40	− 34	47	−12	59	6	42°	3.3	1.1
	M	21	− 22	37	−11	81	11	35°	4.2	1.4
	F	27	− 20	28	7	68	5	45°	3.7	1.4

31	B	31	1	7	− 5	117	5	92°	3.8	1.8
	M	43	− 11	14	−13	174	31	71°	3.1	1.5
	F	11	1	6	5	64	3	100°	4.3	3.0

32	B	40	− 6	12	−12	122	9	78°	3.0	1.6
	M	59	− 27	25	−35	166	15	61°	3.1	1.2
	F	18	− 2	7	7	68	4	78°	4.0	2.3

33	B	17	− 6	4	− 3	72	4	69°	7.8	2.3
	M	30	− 10	8	− 4	83	− 2	69°	4.9	1.7
	F	18	− 3	7	7	70	4	78°	4.0	2.3

34	B	52	− 21	24	−16	88	17	62°	2.8	1.3
	M	23	− 4	25	− 7	119	6	38°	2.2	1.9
	F	16	− 3	13	8	67	7	61°	3.0	2.4

35	B	10	1	2	− 1	40	0	97°	7.2	3.2
	M	28	0	2	− 1	58	2	90°	7.1	1.9
	F	8	3	4	6	67	3	118°	6.3	3.3

36	B	15	2	1	4	107	22	98°	11.8	2.6
	M	10	− 1	2	− 4	77	7	83°	7.4	3.2
	F	5	1	2	2	60	0	109°	8.8	4.3

37	B	43	12	7	12	142	42	107°	5.5	1.5
	M	17	4	4	1	53	3	106°	5.9	2.3
	F	10	2	4	6	69	3	107°	5.4	3.0

38	B	167	5	9	12	82	2	92°	3.3	0.8
	M	238	− 4	9	4	23	6	89°	3.3	0.6
	F	43	4	8	9	68	1	97°	3.7	1.5

39	B	48	− 3	4	0	83	− 1	86°	5.1	1.4
	M	37	5	10	4	56	16	100°	3.3	1.6
	F	15	2	4	7	69	2	98°	5.0	2.5

		Coefficients						Ellipse constants
Color	Observer	g₁₁	g₁₂	g₂₂	g₂₃	g₃₃	g₃₁	θ	a	b
1	B	88	−27	34	− 3	11	− 2	67°	2.1	1.1
	F	48	−12	34	0	22	0	60°	1.9	1.3

2	B	41	− 1	7	0	17	5	88°	3.7	1.6
	F	20	5	13	0	21	0	118°	3.0	2.1

3	B	38	−20	41	2	25	0	43°	2.3	1.3
	F	41	−38	108	3	22	− 3	24°	2.1	0.9

4	B	13	7	46	4	15	− 2	169°	2.9	1.4
	F	30	12	35	1	22	− 1	141°	2.2	1.5

5	B	28	− 6	105	− 3	31	− 9	4°	1.9	1.0
	F	34	18	58	0	21	− 1	151°	2.0	1.2

6	B	23	5	5	1	21	4	104°	5.4	2.0
	F	10	3	5	0	22	0	114°	5.5	3.0

7	B	87	23	40	− 3	55	12	112°	1.8	1.0
	F	19	9	18	0	21	1	133°	3.3	1.9

8	B	38	32	194	8	17	1	169°	1.8	0.7
	F	96	− 3	192	8	22	− 5	2°	1.0	0.7

9	B	177	−37	27	0	33	− 8	77°	2.3	0.7
	F	158	−31	63	− 1	22	1	74°	1.4	0.8

10	B	74	−26	31	5	32	− 7	65°	2.3	1.1
	F	60	− 5	105	2	23	− 2	33°	1.9	0.8

11	B	22	− 1	9	2	22	− 1	86°	3.4	2.1
	F	22	3	16	0	21	0	113°	2.6	2.1

12	B	39	− 9	37	− 6	31	0	47°	1.8	1.5
	F	28	4	20	0	21	1	111°	2.3	1.8

13	B	32	16	88	−16	45	0	165°	1.9	1.0
	F	32	1	34	0	21	0	160°	1.8	1.7

14	B	18	10	124	6	17	− 2	175°	2.4	0.9
	F	63	− 9	153	7	23	− 4	6°	1.3	0.8

15	B	10	3	8	− 1	26	− 1	128°	4.3	2.9
	F	17	2	9	0	21	0	103°	3.4	2.4

16	B	17	9	12	− 2	29	1	127°	4.5	2.0
	F	13	5	8	0	21	1	122°	4.4	2.5

17	B	36	− 9	43	− 1	32	0	35°	1.8	1.4
	F	45	− 9	31	0	22	0	64°	2.0	1.4

18	B	30	−26	129	1	28	− 7	14°	2.0	0.9
	F	55	−56	245	8	22	− 5	15°	1.6	0.6

19	B	11	7	8	1	55	1	128°	6.0	2.4
	F	10	4	6	0	21	1	122°	5.7	2.9

20	B	18	− 3	46	− 3	56	− 2	6°	2.4	1.5
	F	30	7	19	0	21	1	115°	2.5	1.7

21	B	15	11	37	− 3	47	3	158°	3.0	1.6
	F	18	9	15	0	22	1	131°	3.8	2.0

22	B	48	−36	62	19	56	−16	40°	2.8	1.0
	F	96	−32	47	0	21	0	64°	1.8	1.0

	J	K	L	M	N
For Ref. 4	1111			1333	1600
G<R		4/R⁴
G>R		4/G⁴
G>2.5B			6.25B²/G²
G<2.5B			1
For Ref. 5	400			800	1600
G<R		16/R⁴
G>R²		16/G⁴
G>2B and G>3			16B²/G²
G<2B and B>1.5			4
G<3 and B<1.5			16B²/9

		Coefficients						Ellipse constants
Color	Observer	g₁₁	g₁₂	g₂₂	g₂₃	g₃₃	g₃₁	θ	a	b
1	B	6	2	21	− 4	32	− 1	− 9°	4.3	2.2
	M	8	6	20	− 8	17	− 5	− 21°	4.3	2.1
	F	27	11	56	13	66	15	− 19°	2.1	1.3

2	B	17	− 13	21	− 6	42	5	40°	4.0	1.8
	M	9	− 7	21	− 6	30	0	25°	4.1	2.0
	F	16	− 6	17	8	68	6	42°	3.1	2.1

3	B	34	− 11	13	− 2	51	− 6	67°	3.4	1.6
	M	12	− 4	9	− 3	18	1	58°	3.9	2.6
	F	12	− 0.2	5	3	61	3	88°	4.3	2.9

4	B	21	− 20	35	− 2	39	3	35°	3.8	1.4
	M	14	− 17	45	5	15	− 2	24°	3.9	1.4
	V	28	− 24	35	7	68	6	41°	3.8	1.3

5	B	21	− 4	25	− 3	47	2	32°	2.4	1.9
	M	14	− 2	20	− 5	30	0	18°	2.8	1.4
	F	13	0.1	11	2	60	4	93°	3.0	2.8

6	B	72	− 21	13	− 1	62	1	72°	4.0	1.1
	M	45	− 24	26	12	54	6	56°	3.2	1.3
	F	14	0.1	7	8	69	5	91°	3.8	2.6

7	B	31	− 11	28	− 8	44	3	49°	2.3	1.6
	M	18	− 1	35	−20	103	5	3°	2.4	1.4
	F	13	− 0.3	10	4	62	4	85°	3.1	2.7

8	B	23	0	10	− 6	62	11	90°	3.2	2.1
	F	7	0.6	3	2	60	1	98°	6.2	3.7

9	B	45	35	107	−14	22	2	− 24°	1.8	0.9
	M	28	32	88	− 8	10	− 3	− 30°	1.8	1.1
	F	134	68	273	9	60	18	− 22°	1.0	0.6

11	B	19	− 1	77	−18	30	4	1°	2.3	1.1
	M	18	5	66	− 1	17	− 3	− 6°	2.4	1.2
	F	57	2	216	26	68	20	− 1°	1.3	0.7

12	B	31	− 31	44	− 5	48	0	39°	4.0	1.2
	M	35	− 38	69	−25	58	20	32°	3.5	1.0
	F	44	− 34	39	8	68	5	47°	3.5	1.1

13	B	79	− 47	77	7	52	0	46°	1.8	0.9
	M	156	−119	150	9	67	14	46°	1.7	0.6
	F	101	− 30	23	11	67	3	71°	2.9	0.9

14	B	16	8	41	−13	30	6	− 16°	2.7	1.5
	M	17	14	57	6	22	− 3	− 18°	2.8	1.3
	F	78	33	137	0	60	12	− 24°	1.3	0.8

15	B	15	− 17	36	− 3	24	− 1	29°	4.3	1.5
	M	7	− 4	23	− 8	26	2	15°	3.8	2.1
	F	19	− 18	42	7	68	8	29°	3.4	1.4

16	B	49	− 35	66	4	105	−11	17°	3.8	1.6
	M	39	− 24	89	−60	117	−14	22°	1.8	1.0
	F	15	− 4	17	8	68	7	37°	2.9	2.2

17	B	11	0	22	1	36	− 3	0°	3.0	2.1
	M	11	8	46	−11	41	0	− 12°	3.3	1.4
	F	17	− 5	23	2	63	8	29°	2.7	2.0

18	B	10	− 9	38	−11	26	2	17°	3.8	1.6
	M	15	− 2	31	− 8	43	−13	16°	3.7	1.5
	F	20	− 4	47	16	68	10	7°	2.2	1.4

19	B	43	− 20	18	1	32	2	61°	3.5	1.4
	M	58	− 33	32	5	39	− 4	56°	3.2	1.1
	F	59	− 20	16	9	69	2	69°	3.5	1.2

20	B	31	− 5	6	− 2	48	5	79°	4.4	1.8
	M	29	− 13	19	− 8	38	7	56°	3.0	1.6
	F	19	− 2	6	7	68	3	80°	4.2	2.2

21	B	45	− 13	11	−10	52	12	66°	4.3	1.3
	M	47	− 19	27	−18	66	10	59°	2.5	1.3
	F	21	− 4	7	7	68	3	77°	4.1	2.2

22	B	31	− 14	14	−10	80	21	61°	4.0	1.6
	M	48	− 34	40	−12	81	5	51°	2.8	1.2
	F	30	− 12	13	7	68	4	63°	3.8	1.7

23	B	57	− 18	15	− 2	54	13	69°	3.6	1.3
	M	49	− 23	22	− 5	37	11	60°	3.4	1.3
	F	34	− 8	9	8	68	3	73°	3.9	1.7

24	B	35	0	15	−10	61	1	90°	2.6	1.7
	M	22	− 2	13	− 8	57	4	78°	2.8	2.1
	F	14	− 1	7	− 4	61	3	82°	3.9	2.7

25	B	24	1	5	− 2	46	3	93°	4.5	2.0
	M	24	− 2	9	3	105	− 6	83°	3.4	2.0
	F	8	0.7	3	2	61	1	98°	5.7	3.5

26	B	23	− 14	21	− 7	98	1	43°	3.8	1.6
	M	40	− 29	42	−18	134	26	43°	3.0	1.2
	F	20	− 6	11	8	68	5	64°	3.5	2.1

27	B	22	− 19	29	−11	32	3	40°	2.5	1.5
	M	12	− 10	19	− 5	28	3	35°	4.5	2.0
	F	23	− 22	38	6	68	6	36°	3.8	1.4

28	B	16	3	59	−19	43	4	− 4°	2.5	1.3
	M	27	20	39	−28	65	−22	− 37°	2.9	1.4
	F	37	16	101	20	69	19	− 14°	1.7	1.0

29	B	25	− 13	44	−10	72	−10	27°	2.3	1.4
	M	12	− 8	34	8	97	− 6	18°	3.3	1.6
	F	15	− 7	25	8	69	8	28°	3.0	1.9

30	B	40	− 34	47	−12	59	6	42°	3.3	1.1
	M	21	− 22	37	−11	81	11	35°	4.2	1.4
	F	27	− 20	28	7	68	5	45°	3.7	1.4

31	B	31	1	7	− 5	117	5	92°	3.8	1.8
	M	43	− 11	14	−13	174	31	71°	3.1	1.5
	F	11	1	6	5	64	3	100°	4.3	3.0

32	B	40	− 6	12	−12	122	9	78°	3.0	1.6
	M	59	− 27	25	−35	166	15	61°	3.1	1.2
	F	18	− 2	7	7	68	4	78°	4.0	2.3

33	B	17	− 6	4	− 3	72	4	69°	7.8	2.3
	M	30	− 10	8	− 4	83	− 2	69°	4.9	1.7
	F	18	− 3	7	7	70	4	78°	4.0	2.3

34	B	52	− 21	24	−16	88	17	62°	2.8	1.3
	M	23	− 4	25	− 7	119	6	38°	2.2	1.9
	F	16	− 3	13	8	67	7	61°	3.0	2.4

35	B	10	1	2	− 1	40	0	97°	7.2	3.2
	M	28	0	2	− 1	58	2	90°	7.1	1.9
	F	8	3	4	6	67	3	118°	6.3	3.3

36	B	15	2	1	4	107	22	98°	11.8	2.6
	M	10	− 1	2	− 4	77	7	83°	7.4	3.2
	F	5	1	2	2	60	0	109°	8.8	4.3

37	B	43	12	7	12	142	42	107°	5.5	1.5
	M	17	4	4	1	53	3	106°	5.9	2.3
	F	10	2	4	6	69	3	107°	5.4	3.0

38	B	167	5	9	12	82	2	92°	3.3	0.8
	M	238	− 4	9	4	23	6	89°	3.3	0.6
	F	43	4	8	9	68	1	97°	3.7	1.5

39	B	48	− 3	4	0	83	− 1	86°	5.1	1.4
	M	37	5	10	4	56	16	100°	3.3	1.6
	F	15	2	4	7	69	2	98°	5.0	2.5

		Coefficients						Ellipse constants
Color	Observer	g₁₁	g₁₂	g₂₂	g₂₃	g₃₃	g₃₁	θ	a	b
1	B	88	−27	34	− 3	11	− 2	67°	2.1	1.1
	F	48	−12	34	0	22	0	60°	1.9	1.3

2	B	41	− 1	7	0	17	5	88°	3.7	1.6
	F	20	5	13	0	21	0	118°	3.0	2.1

3	B	38	−20	41	2	25	0	43°	2.3	1.3
	F	41	−38	108	3	22	− 3	24°	2.1	0.9

4	B	13	7	46	4	15	− 2	169°	2.9	1.4
	F	30	12	35	1	22	− 1	141°	2.2	1.5

5	B	28	− 6	105	− 3	31	− 9	4°	1.9	1.0
	F	34	18	58	0	21	− 1	151°	2.0	1.2

6	B	23	5	5	1	21	4	104°	5.4	2.0
	F	10	3	5	0	22	0	114°	5.5	3.0

7	B	87	23	40	− 3	55	12	112°	1.8	1.0
	F	19	9	18	0	21	1	133°	3.3	1.9

8	B	38	32	194	8	17	1	169°	1.8	0.7
	F	96	− 3	192	8	22	− 5	2°	1.0	0.7

9	B	177	−37	27	0	33	− 8	77°	2.3	0.7
	F	158	−31	63	− 1	22	1	74°	1.4	0.8

10	B	74	−26	31	5	32	− 7	65°	2.3	1.1
	F	60	− 5	105	2	23	− 2	33°	1.9	0.8

11	B	22	− 1	9	2	22	− 1	86°	3.4	2.1
	F	22	3	16	0	21	0	113°	2.6	2.1

12	B	39	− 9	37	− 6	31	0	47°	1.8	1.5
	F	28	4	20	0	21	1	111°	2.3	1.8

13	B	32	16	88	−16	45	0	165°	1.9	1.0
	F	32	1	34	0	21	0	160°	1.8	1.7

14	B	18	10	124	6	17	− 2	175°	2.4	0.9
	F	63	− 9	153	7	23	− 4	6°	1.3	0.8

15	B	10	3	8	− 1	26	− 1	128°	4.3	2.9
	F	17	2	9	0	21	0	103°	3.4	2.4

16	B	17	9	12	− 2	29	1	127°	4.5	2.0
	F	13	5	8	0	21	1	122°	4.4	2.5

17	B	36	− 9	43	− 1	32	0	35°	1.8	1.4
	F	45	− 9	31	0	22	0	64°	2.0	1.4

18	B	30	−26	129	1	28	− 7	14°	2.0	0.9
	F	55	−56	245	8	22	− 5	15°	1.6	0.6

19	B	11	7	8	1	55	1	128°	6.0	2.4
	F	10	4	6	0	21	1	122°	5.7	2.9

20	B	18	− 3	46	− 3	56	− 2	6°	2.4	1.5
	F	30	7	19	0	21	1	115°	2.5	1.7

21	B	15	11	37	− 3	47	3	158°	3.0	1.6
	F	18	9	15	0	22	1	131°	3.8	2.0

22	B	48	−36	62	19	56	−16	40°	2.8	1.0
	F	96	−32	47	0	21	0	64°	1.8	1.0

Abstract

Cited By

Figures (7)

Tables (3)

Equations (32)

Journal of the Optical Society of America