January 1972

JOURNAL OF THE OPTICAL SOCIETY OF AMERICA VOLUME 62, NUMBER I

Color Matching and Color-Difference Matching

GUNTER WYSZECKI

National Research Council of Canada, Ottawa, Ontario, Canada KIA OSI

(Received 21 July 1971)

The precision of color matching is commonly described by color-matching ellipses, and the precision ofmatching color differences can be described by color-difference-matching ellipses. Recently obtained setsof color-matching ellipses and color-difference-matching ellipses have been compared and some importantcorrelations are indicated. At a given location in color space, and for a given observer, the orientation andshape of a color-difference-matching ellipse are similar to those of the color-matching ellipse. The size of thecolor-difference-matching ellipse is always larger than the color-matching ellipse and increases linearly withthe perceptual size of the color differences to be matched. There is also an indication that color-matchingellipses generally cannot be used to predict the equality of color differences that are several times a just-noticeable difference. This may have important consequences on the present practice of matching such colordifferences by methods based on color-matching ellipses.INDEX HEADINGS: Color vision; Colorimetry.

In two recent papers',2 three sets of color-matchingellipses and three sets of color-difference-matchingellipses for observers GF, AR, and GW were presented.These ellipses describe, at certain locations in colorspace, the precision of visual color matching and ofcolor-difference matching, respectively. In the secondof the two papers it was noted that the orientation,size, and shape of the color-difference-matching ellipsesvary across the chromaticity diagram in a mannertypical of color-matching ellipses. This suggests thatan interesting relationship may exist between colormatching and color-difference matching. The purposeof this paper is to study this relationship by making useof the data presented in the two previous papers.

PAIRING THE DATA

In the color-difference-matching experiment2 theobserver was presented with a pair of test colors (i)and (j) of different chromaticity but equal brightness.He was then required to produce, in a visual tristimuluscolorimeter, a third color (ij) of the same brightnesssuch that he perceived the differences between the threecolors to be of equal size. When the three colors arerepresented by points in the CIE chromaticity diagram,a triangle is formed, which is perceptually equilateral.

In a single session of observation the observer madeup to 30 settings, which were used to calculate his mean

color (ij) and a color-difference-matching ellipsoid,centered at color (ij). The cdm ellipsoid defines a smallcolor gamut that contains, on the average, 95% of anyrandom set of observed colors (ij).

In the color-matching experiment,' the same observerwas presented with a single test color (p) and requiredto produce, by additive mixture of three fixed primaries(1?), (G), (B), a color that matched the given test color(p). In a single session of observation he made up to 30settings. The coordinates of his mean color were madeto agree exactly with those of the given test color (p)and a color-matching ellipsoid, centered at color (p),was calculated in the same way as for the color-difference-matching experiment.

Both the color-difference-matching and color-matching experiments were made under similar ob-serving conditions involving the same apparatus andobservers. Many of the test colors (p) used in the color-matching experiment are members of the set of testcolors (i) and (j) used in the color-difference-matchingexperiment. Other test colors (p) have chromaticitiessimilar to colors (i) and (j). Consequently, for nearlyevery color-difference-matching ellipsoid we can finda closely neighboring color-matching ellipsoid. Becauseof their close proximity in color space, we can makedirect comparisons between the two kinds of ellipsoid.

Our particular interest concerns chromaticity dis-crimination and we will confine our comparison to the

JANUARY 1972

1 17

GUNTER WYSZECKI Vol. 62

TABLE I. Chromaticity coordinates of test colors, observed and predicted mean colors, and their differences.

Mean color Mean color Mean colorPair Test color (i Test color (j) Test color (P) (i,) (ibi" (i43)' T ATNo. Obs. () x y (J) x y (f) x Y x y x Y x y (Si) (iJ) e" e'

I GF I 0.4786 0.4327 2 0.4673 0.4073

3 0.3909 0.4375

7 0.4932 0.3462

9 0.3079 0.3595

11 0.4459 0,4809

4 0.3828 0.4111

8 0.4830 0.3273

10 0.2926 0.3417

12 0.4572 0.4756

ARGW

2 GFARGW

3 GCARGW

4 GEARGW

S GPARGW

6 GFARGW

GFARGW

8 GFARGW

9 GFARGW

10 GFARGW

11 GFARGW

12 GFARGW

13 GFARGW

14 GFARGW

15 GFARGW

16 GFARGW

17 GFARGW

18 GF

ARGW

19 GFARGW

20 GFARGW

21 GF

ARGW

22 GFARGW

1 0.4786 0.4327 0.4370 0.4057 0.4364 0.4116 0.4478 0.4180 6.90.4346 0.4075 0.4561 0.4210 0.4328 0.4065 7.00.4472 0.4145 0.3952 0.4064 0.4384 0.4045 7.6

2 0.3909 0.4375 0.3656 0.4039 0.3633 0.4061 0.3481 0.4014 7.60.3550 0.4006 0.3619 0.4115 0.3668 0.4163 5.40.3651 0.4074 0.3623 0.4107 0.3621 0.4086 8.2

3 0.4932 0.3462 0.4613 0.3367 0.4222 0.3526 0.4149 0.3419 11.3

0.4560 0.3341 0.4564 0.3357 0.4638 0.3292 8.00.4659 0.3471 0.4494 0.3493 0.4287 0.3401 8.4

4 0.3079 0.3595 0.2955 0.3532 0.2960 0.3651 0.2941 0.3641 7.8

0.2951 0.3582 0.2942 0.3576 0.2972 0.3662 7.50.3031 0.3729 0.2924 0.3656 0.2941 0.3630 7.5

5 0,4459 0.4809 0.4363 0.4620 0.4299 0.4534 0.4344 0.4547 4.00.4313 0.4505 0.4386 0.4472 0.4372 0.4600 4.30.4359 0.4587 0.4430 0.4620 0.4385 0.4608 4.4

6 0.4382 0.3417 0.4361 0.3364 0.4405 0.3373 0.4376 0.3372 2.50.4313 0.3320 0.4341 0.3343 0.4375 0.3324 4.00.4325 0.3371 0.4398 0.3376 0.4381 0.3348 2.8

7 0.3822 0.3371 0.3928 0.3334 0.3977 0.3283 0.4041 0.3306 8.10.4014 0.3347 0.3951 0.3324 0.4044 0.3369 6.90.4019 0.3384 0.4061 0.3317 0.4043 0.3380 8.2

8 0.2295 0.2987 0.2410 0.2989 0.2434 0.2908 0.2479 0.2965 7.20.2478 0.3084 0.2414 0.2992 0.2409 0.2966 5.70.2422 0.3063 0.2423 0.2938 0.2413 0.2943 6.2

9 0.3310 0.2749 0.3376 0.2760 * * 0.3409 0.2724 4.60.3432 0.2766 0,3473 0.2812 0.3452 0.2765 6.50.3382 0.2754 0.3430 0.2769 0.3392 0.2765 4.3

9 0.3310 0.2749 0.3352 0.2816 0.3474 0.2799 0.3434 0.2876 4.20.3411 0.2803 0.3467 0.2828 0.3431 0.2841 5.20.3408 0.2893 0.3412 0.2842 0.3454 0.2899 4.6

9 0.3310 0.2749 0.3258 0.2713 * * 0.3317 0.2769 4.30,3256 0.2673 0.3213 0.2685 0.3277 0.2764 7.20.3274 0.2695 0.3288 0.2723 0.3313 0.2747 4.2

10 0.2766 0.2551 0.2878 0.2701 0.2872 0.2659 0.2906 0.2649 5.5

0.2913 0.2767 0.2958 0.2736 0.2902 0.2768 7.50.2935 0.2808 0.2911 0.2777 0.2934 0.2812 9.7

11 0.2054 0.2544 0.2052 0.2406 0.2080 0.2422 0.2043 0.2428 4.30.2069 0.2308 0.2070 0.2374 0.2046 0.2365 4.70.2087 0.2383 0.2052 0.2374 0.2061 0.2363 4.3

12 0.5S868 0.3360 0.5855 0.3305 0.5877 0.3329 0.5838 0.3329 1.10.5859 0.3293 0.5857 0.3318 0.5851 0.3314 1.90.5788 0.3263 0.5836 0.3334 0.5813 0.3329 1.2

12 0.5868 0.3360 0.6039 0.3259 0.6194 0.3150 0.6298 0.3202 6.40.6016 0.3229 0.6168 0.3192 0.6117 0.3218 7.60.6015 0.3205 0.6225 0.3172 0,6374 0.3233 6.5

12 0,5868 0.3360 0.5670 0.3376 0.5472 0.3394 0.5415 0.3339 5.90.5696 0.3328 0.5482 0.3392 0.5581 0.3338 7.30.5582 0.3301 0.5624 0.3364 0.5363 0.3307 5.8

13 0.3545 0.4518 0.3368 0.4242 0.3354 0.4333 0.3388 0.4289 4.20.3384 0.4289 0.3348 0.4256 0.3401 0.4307 5.70.3367 0.4307 0.3391 0.4420 0.3409 0.4337 4.2

13 0.3545 0.4518 0.3422 0.4266 0,3465 0.4264 0.3449 0.4178 5.30.3418 0.4307 * * 0.3442 0.4148 8.10.3378 0.4108 0.3308 0.4267 0.3452 0.4132 6.3

13 0,3545 0.4518 0.3493 0.4461 0.3487 0.4495 0.3543 0.4572 5.40.3471 0.4429 0.3500 0.4497 0.3537 0.4586 7.80.3552 0.4499 0.3472 0.4432 0.3519 0.4570 5.6

13 0.4382 0.3417 14 0.4475 0.3387

15 0.3756 0.3182 16 0.3822 0.3371

17 0.2282 0.2815 18 0.2295 0.2987

19 0.3310 0.2749 20 0.3451 0.2886

36 0.3402 0.2727 19 0.3310 0.2749

20 0.3451 0.2886 36 0.3402 0.2727

21 0.2861 0.2531 22 0.2766 0.2551

23 0.2054 0.2544 24 0.2156 0,2523

25 0.5868 0.3360 26 0.5943 0.3333

27 0.5741 0.3184 25 0.5868 0.3360

26 0.5943 0.3333 27 0.5741 0.3184

29 0.3545 0,4518 30 0.3477 0,4240

28 0.3355 0.4285 29 0.3545 0.4518

30 0.3477 0.4240 28 0.3355 0.4285

31 0.2492 0.2769 32 0.2617 0.2912

32 0.2617 0.2912 33 0.2594 0.2746

33 0.2594 0.2746 31 0.2492 0.2769

5.05.27.2

5.25.2SA5

7.0 1.7 1.36.2 1.8 1.58,5 2.6 0.9

14 0.2617 0.2912 0.2530 0.2710 0.2623 0.2776 0.2534 0.26930.2598 0.2709 0.2571 0.2726 0.2562 0.27170.2603 0.2716 0.2597 0.2726 0.2529 0.2630

14 0.2617 0.2912 0.2522 0.2740 0.2485 0.2772 0.2506 0.27310.2487 0.2674 0.2550 0.2794 0.2508 0.27630.2520 0.2761 0.2540 0.2838 0.2539 0.2775

14 0.2617 0.2912 0.2607 0.2866 0.2648 0.2871 0.2660 0,30060.2625 0.2870 0.2722 0.2977 0.2630 0.29500.2623 0.2993 0.2592 0.2954 0.2632 0.3024

2.72.51.1

3.62.22.6

2.12.01.7

3.23.62.3

0.91.91.7

1.62.01.9

2.92.61.6

2.82.62.3

*

2.81.8

2.31.63.1

*

2.41.9

2,62.33,7

2.61.71.8

1.81.31.5

1.51.9

1.4

1.81.32.4

1.22.21.22.5

*

2.2

2.12.22.8

I.62.62.9

1.13.22.6

0.72.73.9

0.50.90.9

3.40.41.8

1.30.12.0

0.71.41.2

1.00.61.3

1.10.92.3

1.31.12.0

0.80.8

2.11.20.7

1.10.4

0.72.10.4

0.51.00.7

0.60.92.1

2.21,62.0

1.42.70.9

1.50.51.50.8

1.9

0.40.91.2

2.00.70.2

2.21.60.9

1.10.22.5

2.11.20.7

3.41.65.3

1.41.01.9

0.70.70.5

0.61.11.0

1.5

0.40.3

1.21.31.9

0.40.9

1.51.40.6

1.80.6

1.20.30.3

0.31.20.6

0.60.72.0

2.41.03.8

2.31.51.7

0.60.31.0

1.5*

1.1

1.12.10.9

0.50.9

1.5

0.40.90.5

5.01.8

0.5

January 1972 COLOR MATCHING AND COLOR-DIFFERENCE MATCHING 119

TABLE I (continued)

Mean color Mean color Mean colorPair Test color (i) Test color (j) Test color (p) (i0) (ij)" (ilJ)' T ATNo. Obs. () x y (3) x y (P) x Y x y x y x y (ij) (ij) e" e'

23 GF 34 0.3101 0.3163 35 0.3251 0.3325 15 0.3101 0.3163 0.3257 0.3221 0.3220 0.3178 0.3217 0.3126 7.1 4.4 0.4 0.8

35 0.3251 0.3325 34 0.3101 0.3163 16 0.3251 0.3325

34 0.3101 0.3163 37 0.3061 0,2984 17 0.2959 0.3309

ARGW

24 CFARGW

25 GFARGW

26 CFARGW

27 GFARGW

28 GFARGW

29 GFARGW

30 GFARGW

31 GFARGW

32 GFARGW

33 GFARGW

34 GFARGW

35 GFARGW

36 GF

ARGW

37 GF

ARGW

38 GFARGW

54 0.5561 0.3999

45 0.2670 0.4240

46 0.3753 0.3867

58 0.5321 0.3115

61 0.2848 0.3665

62 0.4735 0.3096

64 0.4146 0.2885

15 0.3101 0.3163

16 0.3251 0.3325

1 7 0.2959 0.3309

19 0.3425 0.5280

20 0.3022 0.4725

2 1 0.4873 0.3994

55 0.5649 0,3962 22 0.5428 0.3776

56 0.2800 0.4195 23 0.2670 0.4240

57 0.3861 0,3830 24 0.3753 0.3867

59 0.5138 0.2974 25 0.5242 0.3140

60 0.2699 0,3481 26 0.2730 0.3701

63 0.4563 0.2954 27 0.4650 0.3122

65 0.3991 0.2751 28 0.4058 0.2909

0.3292 0.32680.3209 0.3150

0.3181 0.33140.3150 0.33500.3117 0.3269

0.2986 0.30540.2969 0.29870.2973 0.3038

0.3232 0.32460.3211 0.31400.3295 0.3292

0.3206 0.34160.3135 0.32750.3194 0,3422

0.2926 0.29890.2956 0.30270.2986 0.3062

0.3388 0.50680.3383 0.49620.3324 0.4756

0.2941 0.43840.2975 0.44670.2954 0.4328

0.4600 0.40130.4731 0.40040.4690 0.4070

0.5554 0.38130.5554 0.37240.5179 0.3740

0.2725 0.40340.2734 0.39850.2791 0.4024

0.3616 0.37000.3686 0.37110.3644 0.3687

0.5115 0.29930.4978 0.29450.4962 0.3063

0.2705 0.36220.2768 0.3677

0.4502 0.30370.4476 0.29450.4497 0.3074

0.3955 0.28180.3916 0.27200.3962 0.2830

0.3243 0.31690.3241 0.3089

0.3159 0.33260.3098 0.32910.3104 0.3309

0.2989 0.30280.2965 0.30240.2990 0.3001

0.3172 0.31130.3256 0.31580.3196 0.3160

0.3128 0.33440.3160 0.33100.3134 0.3315

0.2985 0.30370.2938 0.29990.2974 0.3049

0.3433 0.49950.3377 0.49750.3389 0.4796

0.2986 0.42950.3011 0.42330.2940 0.4317

0.4608 0.39860.4610 0.40410.4588 0.4042

0.5533 0.3936* *

* *

0.2690 0.37770.2649 0.39260.2753 0.4041

0.3718 0.36920.3673 0.36880.3704 0.3716

* *

* *

0.4963 0.3093

0.2704 0.36720.2711 0.3675

0.4411 0.3080* 01

0.4455 0.3110

* *

* *

* *

0.3217 0.31120.3189 0,3112

0.3095 0.34120.3117 0.33660.3128 0.3310

0.2964 0.30750.2979 0.30540.29.54 0.2982

0.3198 0,31160.3184 0.30940.3193 0.3140

0.3079 0.33050.3110 0.33110.3163 0.3333

0.3006 0.30530.2990 0.30230.2938 0.2982

0.3389 0.49390.3408 0.48250.3344 0.4682

0.2909 0.42550.2994 0.41020.2870 0.4121

0.4537 0.41400.4661 0.41040.4394 0.3965

0.5523 0.39430.5535 0.39350.5506 0.3934

0.2662 0.39900.2689 0.38700.2688 0.3944

0.3645 0.36430.3700 0.36940.3605 0.3680

0.4791 0.31750.5041 0.31090.4994 0.3096

0.2711 0.36620.2742 0.3751

0.4348 0.31490.4499 0.31020.4443 0.3062

0.3881 0.28650.4016 0.28780.3994 0.291t

9.1 2.4 2.0 3.27.5 2.1 1.4 0.7

11.8 3.8 1.0 5.68.9 2.1 1.5 1.46.9 1.8 1.3 1.1

7.7 4.4 0.4 0.96.2 2.0 0.8 1. t5.9 2.5 1.1 0.9

6.4 3.1 1.5 1.47.3 1.8 0.9 1.06.8 3.8 1.3 1.5

7.3 4.7 0.9 5.16.8 1.4 1.1 2.17.3 2.4 2.0 1.6

6.5 4.4 0.7 0.96.3 1.8 0.5 1.06.6 2.1 0.3 1.7

4.1 2.1 0.9 0.96.8 3.1 0.2 1.44.9 2.3 0.8 0.6

37 0.3061 0.2984 34 0.3101 0,3163

34 0.3101 0.3163 39 0.2996 0.3191

39 0.2996 0.3191 34 0.3101 0.3163

41 0.3425 0.5280 51 0.3565 0.5216

52 0.2848 0.4474 42 0.3022 0,4725

53 0.4992 0.4240 43 0.4873 0.3994

1.82.20.3

0.51.80.8

2.2*

2.01.80.7

2.60.30.7

**

0.5

1.01.3

1.4

0.8

*

*

1.53.22.0

2.72.21.8

2.3

*

1.61.22.5

1.70.40.5*

*

0.6

0.71.7

2.9*

0.6

*

*

*

6.79.26.9

10.57.77,3

2.12.41.7

5.16.44.8

5.55.02.3

6.64.74.0

6.16.7

6.65.14.0

6,33.74.0

cross sections through the centers of the ellipsoids at aconstant luminance of Po= 12 cdnm-2 .

The CIE 1931 (x,y) chromaticity coordinates of thecenter [test color (p)] of the cross sections of each cmellipsoid and those of the center Cmean color (ij)] ofthe cross section of its neighboring cdm ellipsoid aregiven in Table I. Each such pair of cross sections is givena serial number. Table I also lists other quantities thatwe will refer to later.

Table II gives the ellipse parameters for each cm andcdm ellipse observed by the three observers. Theorientation of each ellipse is denoted by 0, which is theangle of inclination of the major axis of the ellipse with

respect to the x axis of the chromaticity diagram. Thelengths of the major and minor semiaxes are denotedby a and b. The shape of each ellipse is defined by theratio a/b and the logarithm of the size is given bylogioA, where A =rab. Table II also gives the orienta-tion and shape of (cdm)' ellipses, which are ellipsespredicted by a method to be described later.

Figure 1 shows an example, drawn from Tables Iand II, of a pair of neighboring cm and cdm ellipses andtheir location with respect to the perceptually equi-lateral triangle formed by colors (i), (j), and (i,j). Thepair of ellipses shown in Fig. 1 refers to Pair No. 2obtained by observer &W.

1.42.21.9

3.42.22.3

1.9

*

1.31.8

2.0

1.72.31.5*

1.8

1.72.1

2.1

1.3

*

*

GUNTER WYSZECKI Vol. 62

TABLE II. Color-matching and color-difference-matching ellipses for observers GF, AR, and GW.

Pair GF AR GWNo. Ellipse O(deg) a(.103) b(103) a/b logioA G(deg) a(.103) b(.103) a/b logioA O(deg) a(.103) b( 103) a/b logioA

I cmcdm

(cdm)'2 cm

cdm(cdm)'

3 cmcdm

(cdm)'

4 cmcdm(cdm)'

5 cmcdm

(cdm)'

6 cmcdm

(cdm)'

7 cmcdm

(cdm)'8 cm

cdm(cdm)'

9 cmcdm

(cdm) '

10 cmcdm

(cdm)'

ll cmcdm

(cdm)'

12 cmcdm

(cdm)'13 cm

cdm(cdm)'

14 cmcdm

(cdm)'

15 cmcdm

(cdm)'

16 cmcdm

(cdm)'

17 cmcdm

(cdm)'

18 cmcdm

(cdm)'

19 cmcdm

(cdm)'

20 cmcdm

(cdm)'

21 cmcdm

(cMm)'

22 cmcdm

(cdm)'

34 5.0 2.9 1.71 -4.33125 19.1 8.2 2.32 -3.30732 2.83

39 7.2 2.2 3.22 -4.29151 19.6 6.0 3.25 -3.43055 3.96

177 7.5 1.7 4.35 -4.394168 14.6 3.8 3.79 -3.756

14 1.90

65 3.2 1.9 1.73 -4.72666 10.8 5.4 1.99 -3.73548 3.90

53 8.8 3.0 2.87 -4.07547 16.9 5.4 3.14 -3.54241 2.20

177 4.6 1.9 2.46 -4.572166 6.5 2.6 2.49 -4.27S

3 2.28

21 3.8 2.1 1.81 -4.60430 8.9 6.1 1.47 -3.77151 2.25

34 3.5 2.1 1.70 -4.64350 7.0 6.2 1.13 -3.86261 2.16

55 4.5 2.1 2.13 -4.525* * * * *

47 5.00

55 4.5 2.1 2.13 -4.52518 9.3 3.0 3.12 -4.056

135 1.35

55 4.5 2.1 2.13 -4.525* * * # *

43 2.63

41 3.3 1.5 2.17 -4.81355 7.5 4.5 1.69 -3.97765 2.19

38 3.2 2.1 1.54 -4.66875 7.9 6.2 1.27 -3.81360 1.60

176 8.9 2.9 3.04 -4.085166 13.4 4.5 2.94 -3.721

7 1.75

176 8.9 2.9 3.04 -4.085167 11.1 4.8 2.34 -3.779

16 1.90

176 8.9 2.9 3.04 -4.085169 14.1 5.8 2.42 -3.588

4 1.41

68 7.6 2.4 3.12 -4.23963 8.7 4.3 2.02 -3.92764 3.56

68 7.6 2.4 3.12 -4.23959 15.0 4.9 3.06 -3.63556 5.03

68 7.6 2.4 3.12 -4.23974 11.0 5.1 2.17 -3.75968 1.90

62 4.7 1.4 3.24 -4.67249 6.0 3.4 1.77 -4.18960 3.70

62 4.7 1.4 3.24 -4.67257 4.2 2.3 1.86 -4.52968 3.20

62 4.7 1.4 3.24 -4.672

37 4.4 2.0 2.20 -4.55545 1.59

28 7.5 2.7 2.80 -4.19452 10.3 6.2 1.67 -3.69728 2.69

52 6.1 2.9 2.10 -4.25146 16.3 6.8 2.41 -3.46045 3.54

31 4.2 1.6 2.55 -4.66812 9.8 4.5 2.20 -3.86114 2.32

67 3.6 1.7 2.16 -4.72151 11.3 5.1 2.24 -3.74553 2.34

49 6.6 2.8 2.32 -4.23267 16.9 5.6 3.04 -3.52952 3.19

27 2.9 2.1 1.38 -4.71116 7.1 3.3 2.13 -4.12625 2.42

32 4.8 2.0 2.37 -4.50945 9.4 4.8 1.93 -3.84733 2.13

60 3.7 1.9 1.90 -4.64859 10.0 4.6 2.17 -3.83552 3.18

42 3.0 1.6 1.91 -4.82436 9.1 3.2 2.88 -4.04744 2.34

42 3.0 1.6 1.91 -4.82429 5.3 2.1 2.52 -4.46020 1.63

42 3.0 1.6 1.91 -4.82439 10.4 3.5 3.01 -3.94748 3.50

68 3.8 1.2 3.10 -4.82951 8.4 2.6 3.26 -4.16665 3.04

66 4.0 2.2 1.85 -4.55677 6.8 3.8 1.80 -4.09381 2.53

1 4.9 2.4 2.02 -4.427176 6.5 2.9 2.21 -4.22016 1.44

1 4.9 2.4 2.02 -4.427174 10.4 4.6 2.26 -3.820

13 1.58

1 4.9 2.4 2.02 -4.427169 8.4 3.6 2.36 -4.02520 1.81

70 5.2 1.7 3.16 -4.56261 13.2 4.6 2.88 -3.72066 3.09

70 5.2 1.7 3.16 -4.562* * * * *

53 8.44

70 5.2 1.7 3.16 -4.56270 8.4 3.8 2.22 -4.00573 1.56

62 4.1 1.6 2.49 -4.67862 10.4 4.4 2.38 -3.84870 1.79

62 4.1 1.6 2.49 -4.67876 10.7 3.3 3.19 -3.95163 4,02

62 4.1 1.6 2.49 -4.678

48 9.6 2.7 3.57 -4.09442 1.84

351235

484753

179165159

637168

506048

S168

4

342038

595661

544448

545069

544950

656664

736986

17917529

179171

5

179179

8

727864

72

5965

727358

706569

708267

7076

70

6.5 2.2 2.99 -4.35614.0 3.4 4.07 -3.824

1.97

4.9 1.8 2.74 -4.55712.4 4.9 2.52 -3.715

3.20

8.2 2.3 3.57 -4.2319.5 4.2 2.24 -3.896

1.33

3.3 1.9 1.76 -4.7187.8 5.1 1.54 -3.904

3.36

6.0 2.8 2.17 -4.2808.0 4.8 1.67 -3.916

2.37

3.9 2.4 1.59 -4.5336.9 2.6 2.64 -4.247

3.35

4.2 1.7 2.50 -4.6616.7 3.4 2.00 -4.147

2.60

3.2 2.0 1.57 -4.7007.2 5.0 1.43 -3.946

3.38

5.0 1.9 2.59 -4.5268.3 3.4 2.48 -4.057

3.89

5.0 1.9 2.59 -4.5269.0 5.7 1.57 -3.791

2.01

5.0 1.9 2.59 -4.52610.3 3.9 2.65 -3.900

3.15

3.7 1.0 3.80 -4.93811.2 3.6 3.12 -3.900

3.60

4.4 2.4 1.79 -4.4768.1 4.3 1.88 -3.960

1.68

10.0 2.8 3.63 -4.06113.1 3.6 3.69 -3.836

2.73

10.0 2.8 3.63 -4.06110.7 4,2 2.56 -3.856

1.43

10.0 2.8 3.63 -4.06113.0 7.8 1.67 -3.497

2.43

6.9 2.3 2.97 -4.3017.8 4.0 1.98 -4.014

2.47

6.9 2.3 2.97 -4.30111.1 5.1 2.17 -3.747

4.47

6.9 2.3 2.97 -4.30110.4 6.5 1.59 -3.671

2.47

3.9 1.2 3.21 -4.8248.3 4.3 1.82 -3.928

1.71

3.9 1.2 3.21 -4.8248.6 4.0 2.17 -3.966

2.66

3.9 1.2 3.21 -4.8247.0 3.1 2.23 -4.164

2.76

January1972 COLOR MATCHING AND COLOR-DIFFERENCE MATCHING

TARLT II (continued)

Pair GF AR GWNo, Ellipse O(deg) a( 10') b(.10') /b logtaA O(deg) oa(101) b(-l0) a/b logioA O(deg) a( 10') 6(-10s) a/b logIDA

59 3.3 1.7 1.9623 cmcdm

(cd n)'

24 cmcdm

(cdm)'

25 cmcdm

(cdm) '

26 cmcdm

(cdm)'

27 cmcdm

(cdm)'

28 cmcdm

(cdm)'

29 cmcdm

(cdm)'

30 cmcdm

(cdmY'

31 cmccinm

(cdrn)'

32 cm

cdm(cdm)'

33 cmcdm

(cdm)'

34 cmcdm

(cdm)'

35 cmcdm

(cdm)'

36 cmcdm

(cdm)'

37 cmodin

(cdm)'

38 cmcdm

(cdm)'

so 13.542

8l 2.253 7.651

67 2.465 11.667

59 3.366 9.858

81 2.26t 14.155

67 2.460 12.4

53

73 9.380 15.355

75 6.981 8.969

6 4.434 14.930

2 S.95 t1.0

75

72 5.685 13.597

50 5.456 6.683

169 8.0*' *

34

56 3.958 6.650

169 5.7

5 10.225

9 4.4

30

- -

COMPARING COLOR-DIFFERENCE-MATCHING of the three observers GF, AR, and GW. Figures 2-4ELLIPSES WITH COLOR-MATCHING ELLIPSES are the resulting correlation diagrams.

tprecision The correlation between the orientation, 0, of corre-The first comparison is made between the ciin sponding cdm and cm ellipses would be perfect if all the

of color-difference matching and that of color matching. points plotted would fall on the dashed 45 line shown inFollowing this, an attempt will be made to relate the Figs. 2-4. Similarly, perfect correlations in the (logoA)precision of color matching to the perceptually equi- diagrams and (a/b) diagrams would be indicated if, inlateral triangles in the chromaticity diagram. each case, all the points would fall on the dashed 450

The precision of color-difference matching and color line. Evidently, we do not have perfect correlations andmatching is characterized by the cdm ellipses and cm did not expect it because of the inherent uncertaintiesellipses, respectively. A graphical method'" has been in the obsen'ational data, which we had already notedchosen to show the correlation between the values of 6 in the earlier papers.' 2

(orientation), log28A (log of size), and a/b (shape) of Nevertheless, we consider the correlations shown inthe cdm ellipses and corresponding cm ellipses for each the 6 diagrams quite good and those shown in the (a/h)

121

-4.861

-4.0415.4 2.492.67

1.5 1.482.5 3.00

4.70

1.7 1.444.8 2.43

1.96

1.7 1.964.2 2.33

4.24

L.5 1.487.0 2.02

3.25

1.7 1.447.3 1.69

2.56

3.7 2.497.6 2.01

1.S9

1.9 3.583.8 2.S3

2.94

2.3 1.905.2 2.85

2.56

2.7 2.235.7 1.94

2.12

2.7 2.053.5 3.82

1.67

2.0 2.753.5 1.87

3.07

2.7 2.90* *

9.71

2.0 1.943.6 1.83

2.45

2.8 2.034.9 2.11

2.58

2.2 1.98

3.03

-4.762 64 2.7 1.4 1.86-3.638 49 5.8 3.0 1.91

38 3.55

-4.991 62 2.6 1.6 1.67-4.21,9 44 5.4 2.4 2.19

57 2.37

-4.897 66 3.1 1.7 1.82-3,760 59 6.8 3.3 2.03

60 2.81

-4.762 64 2.7 1.4 1.86-3.890 44 6.6 3.1 2.14

55 2.15

-4,991 62 2.6 1.6 1.67-3.510 42 4.2 2.9 2.18

33 1.81

-4.897 66 3.1 1.7 1.82-3.545 54 7.0 2.9 2.38

53 2.02

-3.965 80 7.4 2.2 3.38-3.438 67 16.0 7.2 2.23

69 2.31

-4 374 90 6.6 2.1 3.20-3.979 88 I1.5 4.8 2.41

62 2.63

-4,488 27 4.6 2.7 1.68-3 611 28 12.6 5.5 2.29

47 3.19

-4.305 4 4.8 2.6 1.85-3.707 * * * *

80 3.23

-4.326 84 6.4 2.1 3.05-3.826 74 10.3 3.9 2.54

96 2.12

-4.477 St 4.3 2.2 1.96-4.142 45 11.4 4.9 2.33

41 2.10

-4. 165 1 1 5.9 3.3 1.79* * * * *

27 7.47

-4.608 72 4.3 2.0 2.16-4.125 39 7.3 3.8 1.92

58 3.36

-4.293 18 4,8 3.1 1.57- 3.807 * * * *

32 7.65

-4.617 39 S.S 2.6 2.14* 3 * * *

37 14.00

-4.920 60 3.3 1.4 2.42-4.255 72 6.7 4.2 1.59

55 2.34

-4.886 so 3.2 1.4 2.34-4.3S4 47 5.9 3.0 1.97

45 3.22

-4.779 57 4.0 1.6 2,56-4.150 65 9.1 3.0 3.05

62 1,96

-4.920 60 3.3 1.4 2.42-4.198 59 13.2 4.8 2.73

52 4.44

-4.886 SO 3.2 1.4 2.34-4.600 S 6.3 3.4 1.87

58 3.17

-4.779 57 4.0 1.6 2.56-4.196 51 5.3 3.1 1,70

50 1.51

-4,288 76 14.0 3.1 4.52-3.445 78 25.2 7.1 3.57

72 4.00

-4,369 77 8.6 1.8 4,71-3 761 74 11.4 3.9 2.92

75 2.65

-4,407 21 9.0 2.7 3.36-3.663 25 13.8 6,0 2.31

36 2.00

-4.398 8 8.4 3.7 2.30* 4' * 4' *

29 7.49

-4,367 82 6.8 2.8 2.41-3.910 109 9.1 5.0 1.83

117 2.36

-4.535 36 6.1 2.0 3,t0-3.753 44 11.4 6.1 1.87

40 2.69

-4.210 8 7.8 3.7 2.10* 5 15.9 6.6 2.41

10 2.79

-4,575 .. ... ... ...

-4.054 ...

-4.335 14 7.3 3.4 2.15* 3 9.5 5.8 1.64

29 1.89

-4.348 40 5.1 3.5 1.46

30 2.70

-4.855-4.257

-4.699-4.074

-4.861-3.699

-4.855-4.173

-4.699-4.283

-3.866-3.252

-4.302-3.858

-4.119-3.588

-4.014*

-4.220-3.849

-4.429-3.665

-4.042-3.482

-4.111-3.761

-4.255*

GUNTER WYSZECKI

(i )=(3)(p) = (2)

', r(p)

0.450

y

0.400

-

1001

W4)

Z

a<b

80

60

40

20

R(ij) 0

1601I I I I I I

0.350X

-Obs. GF

0

* I

/ 8I

1600.400

S

0/* /

- /

/g 0

l I I I

0 20 40 60B(c m ellipse)

FIG. 1. Portion of CIE 1931 (xy) chromaticity diagram showingcolor-matching ellipse for test color (p) = (2), color-difference-matching ellipse centered at (ij)= (3,4), and perceptually equi-lateral triangle (3)-(4)-(3,4). The observer is GW. Note thattest colors (p) and (i) coincide in this case. Data are taken fromTables I and II for pair No. 2.

diagrams not negligibly low. However, we note animportant systematic deviation between the sizes ofcorresponding ellipses. A best-fitting straight linethrough the points of each of the (logioA) diagramswould roughly run parallel to the dashed 45° line,displaced upward by about 0.6 log units. This meansthat, on the average, the size of a cdm ellipse is aboutfour times as large as the neighboring cm ellipse.

From an earlier investigation,3 we deduce that thesize of a cdm ellipse is a function of the chromaticitydifference between the test colors (i) and (j). The sizeincreases with increasing difference between (i) and (j)and decreases when (i) and (j) move closer together.In the limit, when (i) coincides with (j), we expect thecdm ellipse to coincide exactly with the cm ellipse.

Each cdm ellipse of the present study relates to apair of test colors (i) and (j) whose CIE color differenceis the same for every pair and approximately equal to6.7 CIE color-difference units.2 If we assume that theCIE 1964 color-difference formula makes accuratepredictions of the perceptual differences between testcolors (i) and (j), we expect the sizes of our cdm ellipsesto vary only as a function of their location in thechromaticity diagram. The (logioA) correlation dia-gramns of Figs. 2-4 are first viewed with this assumptionconsidered to be valid.

However, there is an indication that the CIE 1964color-difference formula does not predict accurately theperceptual difference between the colors (i) and (j) ofevery pair. Such an indication can be extracted fromthe data given in Tables I and II; the following analysismav be of interest.

4)

a.

a)

0

4CW

0)

.- -4.0k

-5.0 k

0* 0

SC' S

S S

* //

* /o /

/

//

//

/

/

-5.0 -4.0log1OA(cm ellipse)

7;

43

a

5 r

41-

31-

2

S

-,

0* /o

00 0

00. 0SkO#°

'S4 *

//

//

//

S/

I l l I I

1 2 3 4 5a/b(c m ellipse)

FIG. 2. Diagram showing for observer GF the correlations be-tween the orientation 6, the logarithm of the size A, and the shapea/D of the observed cdm ellipses with the corresponding cmellipses. The points are derived from data given in Table II. Opencircles represent average data for those pairs of correspondingellipses that involve the same cm ellipse. For example, pairsNo. 9, No. 10, and No. 11 of Table II involve cm ellipse (p=9)in each case.

Obs. GW

(i, I)=5( )=(4)

*0p /

'I-,

0

r-

80 100

122 Vol. 62

v

-3.0r-

Januaryl972 COLOR MATCHING AND COLOR-DIFFERENCE MATCHING

Obs. AR* / .

* /0 0

//

/

/ *.W

/

0

SO

cL

-aU

qZ

I I I I I I

160 0 20 40 609(c m ellipse)

80 100

Obs.

601-

40H

201-

HI-

1601L160

-3.01

S

0*o- 0

Jr)bs 0

S6S

//

//

/

07

0.

0-c

0

0

0

/

0

W

E

0

C

100

80

I I

0 20 40 601(cm ellipse)

7

-4.0 -

S0

*0 4

/

//

-5.01-

-5.0 -4.0logloA(c m ellipse)

-5.0 -4.0loglo A(cm ellipse)

41-

3[

2

0/- .0 /

- *&/ 0Kh/0 0

/

/1 1 1

//

//

/

In;0.-

E

U

.0

a

1 2 3 4 5a/b (c m ellipse)

FIG. 3. Diagram showing for observer AR the correlations be-tween the orientation 0, the logarithm of the size A, and the shapea/b of the observed cdm ellipses with the corresponding cmellipses. The points are derived from data given in Table II. Opencircles represent average data for those pairs of correspondingellipses that involve the same cm ellipse. For example, pairsNo. 9, No. 10, and No. 11 of Table II involve cm ellipse (p=9)in each case.

3

2

e/

.

// 00

.

0

I I I I I1 2 3 4 5

a/b (c m ellipse)

FIG. 4. Diagram showing for observer GW the correlations be-tween the orientation 0, the logarithm of the size A, and the shapea/b of the observed cdm ellipses with the corresponding cmellipses. The points are derived from data given in Table II. Opencircles represent average data for those pairs of correspondingellipses that involve the same cm ellipse. For example, pairs No. 9,No. 10, and No. 11 of Table II involve cm ellipse (p=9) in eachcase.

100 l

80H

60l

40H

/0

/ 0/ 0

123

.

/

20

0

160 /

- 3 .0 r-

0

0.-CD

I I

80 100

E

0

0a)0

-4.0

-5.01-

/

I

I

00 4%

0 0

0 &

5 F-

I

. W //

o W//i/

*/.

.4 00

0/0

t.2

GUNTER WYSZECKI

4-

.002 0

1 .1 1

2 4L I

color differences At that can be fitted along the straightline connecting (i) and (j). If the distance in AP be-tween (i) and (j) is denoted by D(i,j) and the distanceAt(i,j) corresponds to a just-noticeable color differencein the same direction, the number T(i,j) of just-noticeable color differences between (i) and (j) isgiven by

6 8 10 12T ( iXj)

FIG. 5. Relationship between the just-noticeable color-differenceincrement AT (i,j) and the number T(i,j) of just-noticeable colordifferences between test colors (i) and (j). Points representaverage data of the three observers GF, AR, and GW.

Consider a small domain AP of the (x,y,Y) colorspace, such as that shown in Fig. 1. The x and y coordi-nates of this domain are restricted to the range0.350<x<0.400 and 0.400<y<0.450, respectively, andthe Y coordinate is constant and equal to 12 cd.m-2 .Thus AP is a small portion of a plane in the (x,y,Y)color space whose points represent color stimuli withchromaticity coordinates x and y and a constantluminance of 12 cd.m72 .

When the color stimuli, represented by points in theplane AP, are viewed under the given observing condi-tions, the corresponding perceived colors can be mappedonto a plane AP* of constant brightness. The pointsof AP are transformed linearly into the points of AP*.

The observed cm ellipse, centered at (p) in AP, de-fines a locus of chromaticity points whose correspondingcolor stimuli are assumed to be perceptually equidistantfrom the color stimulus represented by (p). Morespecifically, it is assumed that the length r of the radiusvector of the cm ellipse is a measure of a just-noticeablecolor difference At, and we may set

AIl= air. (1)

The constant ai is independent of the direction of theradius vector and the location of (p). It is difficult todetermine its value accurately. However, for the presentstudy we propose to set a, equal to unity. We recall thatthe radii of our cm ellipses are (7.81)i = 2.79 times thestandard deviation of our color-matching data,' andMacAdam4 estimated that a just-noticeable colordifference corresponds to approximately three timesthe standard deviation of color matching.

In the AP* plane, the observed cm ellipse transformsinto a circle of radius r, and with a= 1; this circle definesa locus of colors that are perceived to be just-noticeablydifferent from the central color (p).

The observed cm ellipse in AP, or its correspondingcircle in AP*, applies to every point within the smalldomain AP (or AP*); that is, the orientation, shape,and size of the cm ellipse are the same everywhere inAP. With this assumption, we can readily determinethe color difference between the given test colors (i)and (j) in AP in terms of the number of just-noticeable

T(i,j) =D(ij)/A1(ij). (2)

For the example illustrated in Fig. 1 we find T(3,4) = 8.2.If all pairs of test colors (i) and (j) used in our

experiment would have the same perceptual colordifference, as predicted by the CIE 1964 color-differenceformula, we would expect the corresponding values ofT(i,j) to be the same for every pair. However, as shownin Table I, for each of the three observers T(i,j) variesconsiderably from pair to pair and the spread aroundthe mean value, approximately 6.0, cannot entirely beexplained by the uncertainties we know to exist in theobserved cm ellipses. Instead, we must conclude that forat least some pairs of colors (i) and (j) the CIE 1964color-difference formula fails to predict accurately theperceptual color difference.

We now consider the observed cdm ellipse in thesmall domain AP. This ellipse defines a locus of chroma-ticity points whose corresponding color stimuli areassumed to be perceptually equidistant from the colorstimulus represented by (i,j). In AP*, the cdm ellipsetransforms into a circle whose radius R is a measureof the just-noticeable color-difference increment ATwhich, expressed in terms of just-noticeable colordifferences At, may be represented by

AT= a2R/A t. (3)

The constant a2 is set equal to a,, i.e., unity.This just-noticeable color-difference increment AT

applies to the given difference T(i,j) between colors(i) and (j). To calculate AT(i,j) from the data givenin the domain AP, we determine R(i,j) in the directionspecified by the straight line between (i) and (j) anddivide it by At(i,j) corresponding to the same direction.The results of such calculations for the three observersand the 38 pairs of colors are given in Table I.

The values of AT(i,j) can be compared with thecorresponding values of T(i,j). Figure 5 illustrates thecorrelation between these two quantities for the averageof the three observers. Correlation diagrams for theindividual observers (not shown here) are similar to theaverage shown in Fig. 5, but the scatter of the points issomewhat greater, particularly for observer GF.

As expected, the quantity AT(i,j) is not independentof T(i,j), but rather increases with T(i,j) over therange of 1-10 just-noticeable color differences. Theincrease appears to follow the linear relation

AT(ij) =aT(i,j)+1, (4)

where the constant a has an average value of approxi-

Vol. 62124

January1972 COLOR MATCHING AND COLOR-DIFFERENCE MATCHING

mately 0.2±40.1. Note that the straight line describedby Eq. (4) passes through the point T(i,j) =0,A T(i,j) = 1. This case represents color matching, and,as expected earlier, the transition from color-differencematching to color matching-or vice versa-is smoothand appears to be linear, to the extent that the precisionof the visual observations permits judgment.

The relationship between AT(i,j) and T(i,j) givenby Eq. (4) may be used to adjust the sizes of the ob-served cdm ellipses so that they correspond to the casewhere the difference T(i,j) between the test colors (i)and (j) is always the same.

Suppose we choose the difference T(i, j) to be T= 6.0,which is about the average difference of all pairs. Thenthe size of an ellipse observed with T(i,j) <6.0 must beincreased, and similarly, the size of an ellipse observedwith T(i,j)>6.0 must be decreased. The factor, f(i,j),required to make the adjustment is given by

f(i,j) =[2.2/AT(iJ)]2 , (5)

where the value 2.2 is obtained from Eq. (4) witha=0.2 and T(ij)=T=6.0; the value of AT(i,j) is alsofound from Eq. (4) with T(i,j) set equal to the par-ticular value given in Table I appropriate for the giventest colors (i) and (j) and observer.

When the adjustment to the sizes of the observedcdm ellipses is made, new correlation diagrams of 1og1sAcan be drawn. These diagrams are very similar to thosealready shown in Figs. 2-4, corresponding to the un-adjusted sizes of cdm ellipses. In the new diagrams(not reproduced in this paper), the scatter of the pointsis generally somewhat smaller, but again the distribu-tion of points is roughly along a straight line parallel tothe dashed 450 line, displaced upward by about 0.6log units.

PERCEPTUALLY EQUILATERAL TRIANGLE ANDCOLOR-MATCHING ELLIPSE

In the previous section, we found a close relationshipbetween cm ellipses and cdm ellipses. In this section wewill relate the cm ellipses with the perceptually equi-lateral triangles formed by colors (i), (j), and (i,j).

Again, let us consider the small domain AP of the(x,y,Y) color space, such as that shown in Fig. 1, andits corresponding domain AP* in color-perception space.In transforming AP to AP*, we expect both the cmellipse and the cdm ellipse to become circles and thetriangle (i), (j), (i,j) to become equilateral.

The transformation from AP to AP* is linear andcorresponds to a distortion of AP such that in AP*the x,y axes form an oblique coordinate system withdifferent scale units in its two directions. Figure 6 showsan example. On the left-hand side, the domain AP is thesame as that already shown in Fig. 1. On the right-handside is the linear transform of AP, the domain AP*.In AP* the two coordinate axes x and y intersect at anangle w, and the ratio of the scale unit in the x direction

0440

0.420

y

0.400

0.360

I)

0.380 0.400

AP,

0.380 as0340 0360 0380 0400

X

FIG. 6. Example of a small domain aP in (x,y, Y) space(Y= const) and its linear transform AP* in color-perceptionspace (brightness=constant). Under ideal conditions it is hy-pothesized that the observed triangle (i), (j), (i j), the cm ellipse[centered at (i)], and the cdm ellipse [centered at (ij)], shownin AP, transform into an equilateral triangle, and circles, respec-tively, in AP*. The x,y coordinate system in AP transforms intoan oblique coordinate system whose axes have different scaleunits and intersect at an angle ca.

to that in the y direction is s0 /Sy where s. and s, areconstants.

There are three independent but equivalent methodsof determining w and s.7s, from the data embedded inAP. One method is to make use of the cm ellipse

gli(x-xp) 2±+2gl 2 (x-x 5 )(y--yp)+g22(y-yP) = 7.81, (6)

where xp,yp are the coordinates of the center (p) andgs1, g12, g22 are constants derived from the observationaldata. It has been shown elsewhere5 that the followingrelations exist between the gik's and cosw and sx/s 0,

cosW =g12/(g1ug22)'

s0 /sy= (gs1/g 22)A. (7)The second method makes use of the cdm ellipse

Gii(x-x Xj) 2+2Gl2 (x-Xii) (y -Yij)+G22(y-yij)2 =7.81, (8)

where xjj, yij are the coordinates of the center (i,j)and Gil, G12, G22 are constants derived from the ob-servational data. The relations between the Gik's andcosw and s./s, are identical to those given in Eqs. (7)except that now the Gik's take the place of the gik's.

The use of w and s0 /s5 derived from the gik's makesthe cm ellipse an exact circle in AP*. The cdm ellipsewill become an exact circle in the same AP* only if itscoefficients Gik are proportional to the correspondinggik's. The triangle (i), (j), (i,j) will be exactly equi-lateral in the same AP* if the gik's are proportional to athird set of corresponding coefficients rik defining anellipse that, when centered at any one of the threecorners of the triangle, will go through the remainingtwo. The coefficients rik are the solutions of the set ofsimultaneous linear equations

r 11(xi-xj)-+2r 12(xi - xi) (yi-yj)+r 25 (yi-y.)

2= T2

r11(xj-xij)1+2r12(XJ-Xij) (Yi-Yij)+Ir22(Yi -yij)2 = T2

rT1 (x 1 - x,)2+2±22Fi2 (x i) - (xY)Y(yi 2-y.)+±P 2 2 (y-yi)

2= T

2. (9)

125

GUNTER WYSZECKIVo.6

801- Obs. GF

40 V-

20 F-

160 -

80 I-

I

140 160

/ Sc(

0

0 //

0/

0// *

/

/

/

// *

3

20 40 60(C di m V ellipsel

Obs. AR

60?-

40 [-

20 1--

80 500

//

o 0/

(SI /

A

/ '0

140IS

0 208[(c di m

601-

40F-

I 140 60ellipse]

091g

.8

PA *.S / 0

1 20 I140 160

I I I I0 20 40 60

8[(c di m )' ellipse]

S

555*a

0

data, the results of the previous section have indicatedthat the cm ellipses, defined by the giks, correlate wellwith the corresponding cdm ellipses, defined by theGik's. What remains to be explored is the correlationbetween the perceptually equilateral triangles, withwhich the rik'5 are associated, and the cm ellipses.

The ellipse defined by the coefficients Fik may beinterpreted as a color-difference-matching ellipse pre-dicted from the perceptually equilateral triangle (i),(j), (i~j). We will call this ellipse a (cdm)' ellipse, butnote that only its orientation (0) and shape (a/b) canbe determined, whereas its size (A) cannot. Table IIgives the values of 0 and a/b for each case and eachobserver. Figures 7 and 8 show 6 and (a/b) correlationdiagrams for the (cdm)' ellipses and the corresponding

0

a.L

0

-J80

Ft11.

/A

0

3S

.0

2

80 100 120

The. 7. Diagrams showing f or observers OF, AR, and OW thecorrelations between the orientation 0 of the observed cm ellipseswith the corresponding (cdm)' ellipses predicted from the observedperceptually equilateral triangles. The points are derived fromdata given in Table IL. Open circles represent average data forthose pairs of corresponding ellipses that involve the same cmellipse. Points marked with an asterisk (*) denote cases for whichno observed cdm ellipses are available and the observed data arein some doubt.

For the purpose of this investigation, TV can be anypositive and finite number. If we substitute the rik's forthe gis.'s in Eqs. (7), we obtain new values of cosw ands./s, that are identical to the previous ones only if the

rksare proportional to the corresponding gtk'S.Exact proportionality botveen the gjk-'s and either

the corresponding Coefficients Gia or Fai does not occurfor any of the 38 domains AP. However, if allowance ismade for the inherent uncertainties of the observational

-5

= 4

0

C,

C.

Obs. G F A/

/

/

a /_

0 * 34w* 5/-

W0

00. ' 0/, 0 a

/

I a 3 4 5 6 7a/b [(C d M V ellipse]

8 9 10

5F .A R

*

/ 0/ / 0B (

0

/9,66M (~

I 2 3 4 5 6 7 a 9 10a/b [(c d m V ellipse]

-Obs. GW

0/

0 * 0/0

- .53 4 5 6 7 8 9 1

a/b [cr d nm)' eIlipsej

FnG. s. Diagrams showing for observers OF, AR, and GW thecorrelations between the shapes (a/b) of the observed cm ellipsesand of the corresponding (cdm)' ellipses predicted from the ob-served perceptually equilateral triangles. The points are derivedfrom data given in Table II. Open circles represenit average datafor those pairs of corresponding ellipses that involve the same cmellipse. Points marked with an asterisk (*) denote cases for whichno observed cdm ellipses are available and the observed data a-rein some doubt.

0/ 0O0

01

1001

aol- Obs. GW

C.L

0Z

20

0

Vol. 62

roo �-

I

January1972 COLOR MATCHING AND COLOR DIFFERENCE MATCHING

cm ellipses. These diagrams can be compared withthose given6 in Figs. 2 to 4.

In comparing the 0-correlation diagrams, our firstimpression is of a general over-all agreement for each ofthe three pairs of diagrams. However, closer inspectionindicates some slight systematic deviations. In the caseof observer GF (Fig. 7), several (cdm)' ellipses arerotated counterclockwise in AP relative to the corre-sponding cm ellipses; such a deviation is not noticeablebetween the cdm ellipses and the cm ellipses (Fig. 2).This is indicated particularly for the ellipses with lowvalues of 9, i.e., ellipses in the chromaticity region ofsaturated purples and reds. Systematic deviations of asimilar kind are observed for the other two observers,but they are not as pronounced.

A comparison of the (a/b) correlation diagrams ismore difficult because the majority of points in everydiagram distribute within a nearly circular area so thattrends of deviations between the distributions of Fig. 8and the corresponding distributions of Figs. 2-4 cannotbe extracted easily or their significance established.However, some extreme deviations are noted. The(a/b) ratios of some of the (cdm)' ellipses are extremelyhigh, compared to those of the corresponding cmellipses. These (cdm)' ellipses were all derived fromtriangles for which the observer had great difficultiesfinding an acceptable mean color (i,j), indicated by thefact that he was unable to produce a set of repeatmeasurements of (ij) suitable for calculating a cdmellipse. We recall that these cases are identified inTable II by asterisks placed in the appropriate entriesfor the ellipse parameters. In Fig. 8, the same cases arealso marked with an asterisk. Because of the extremeobservational uncertainties of these data, we are inclinedto ignore them in this comparison, except to note thatall of them have (a/b) ratios that are considerablygreater than those of the corresponding cm ellipses.

The (a/b) ratios of the remaining majority of (cdm)'ellipses lead to a distribution of points in Fig. 8 re-sembling closely those in Figs. 2-4. However, inspectionof the actual (a/b) ratios given in Table II shows thatall of them are different from those of the corresponding(observed) cdm ellipses. Some of these deviationsappear to be not negligible, but when we try to correlate,for example, the increased ratios (of which there areabout as many as decreased ratios) with the location ofthe corresponding triangles in the (xy) chromaticitydiagram, no clear pattern seems to emerge that mighthint at some visual significance of these deviations otherthan merely large observational scatter.

Another method of correlating the observed triangleswith the observed cm ellipses and cdm ellipses is thefollowing. From an observed cm ellipse, given byEq. (6), we can predict the location of the mean color(i,j) that we expect to be observed by matching thedifferences between colors (i), (j), and (i,j). The pre-dicted mean color (i,j), which we will denote by (i,j)',can be compared with the actually observed mean

0.350- Obs. GW

y

0.300

(26)

(27)

I I I I I I I0.550 0.600

X

FIG. 9. Portion of CIE 1931 (xy) chromaticity diagram showingcm ellipse for test color (p)= (12), cdm ellipse centered at(ij)= (26,27), and observed perceptually equilateral triangle(26), (27), (26,27). Also shown are the colors (26,27)' and (26,27)" predicted from the cm ellipse and cdm ellipse, respectively.These predicted colors deviate from the observed color (26,27),and the points R' and R" on the cdm ellipse are used to calculatethe size of the deviations.

color (i,j) and any discrepancy between them can beestimated as to its significance by relating it to theobserved cdm ellipse centered at color (i,j).

Similarly, we can use the observed cdm ellipse, givenby Eq. (8), to predict the location of the mean color(i,j) and compare it (denoted by (i,j)") with theactually observed mean color.

In Table II, the chromaticity coordinates are givenfor both the predicted color (i,j)' and the predictedcolor (i,j)". Figure 9 illustrates the case of triangle (26),(27), (26,27) observed by GW with predicted colors(26,27)' and (26,27)". The lines drawn from the ob-served color (26,27) through (26,27)' and (26,27)"intersect the observed cdm ellipse at R' and R",respectively.

The ratio e' of the distance between (26,27)' and(26,27) over the distance between R' and (26,27) can beused as a measure of the significance of the discrepancybetween the predicted color (26,27)' and the observedcolor (26,27). In this particular case, we find e'= 1.7,which means that the predicted color (26,27)' is welloutside the observed cdm ellipse and thus its differencefrom the observed color (26,27) is considered significant.

The ratio e" of the distance between (26,27)" and(26,27) over the distance between R" and (26,27) ise"=0.9, which means that the predicted color (26,27)"which falls just inside the observed cdm ellipse is con-sidered insignificantly different from the observed color(26,27).

Table I gives, for each case and observer, the calcu-lated ratios e' and e". For each observer roughly 50%of the ratios e' are greater than unity. This means thatabout 50% of the predicted colors (i,j)' are significantlydifferent from the observed color (i,j). A very similarresult is found for the predicted colors (i,j)".

127

GUNTER WYSZECKI

The average ratios e' and e" are, for observer GE,ed=1.6, e" =1.3; for observer AR, e'=1,3, e"= 12; forobserver GW, e'=1.3, e"=1.2. The average ratios of e'are slightly higher than those of e".

Every chromaticity difference between the predictedcolors (ij)' or (i,j)" and the observed color (ij) is adifference vector with a certain direction in the chroma-ticity diagram. An attempt was made to discover somesystematic pattern in the difference-vector field, butnothing of significance emerged. The slight systematicdeviations noted earlier in the 0-correlation diagramsof Fig. 7, which referred to saturated purples and reds,could not be reconfirmed in the difference-vector field.

CONCLUSIONS

The main points that may be made on the basis ofthis investigation are:

(1) The precision of color matching, characterizedby cm ellipses (cross sections of cm ellipsoids), appearsclosely related to the precision of color-differencematching, characterized by cdm ellipses (cross sectionsof cdm ellipsoids). In particular, the orientation (0) ofcm ellipses correlates well with the orientation of cdmellipses observed in the same area of the chromaticitydiagram. Also the shapes (a/b) and sizes (A) of corre-sponding ellipses show significant correlations.

(2) The precision of color-difference matching is alinear function of the perceptual size of the differencebetween the given test colors (i) and (j). The precisionincreases (cdm ellipse decreases) with decreasing differ-ence between (i) and (j). In the limit, when the differ-ence between (i) and (j) tends to zero, the precisionapproaches that of color matching.

(3) The prediction of perceptually equilateral tri-angles from the shape and orientation of cm ellipses orcdm ellipses is not always in agreement with observa-tion. About 50% of the cases studied indicate significantdiscrepancies between prediction and observation.However, an attempt has failed to discover a systematic

pattern for these discrepancies with respect to theircorresponding locations in the chromaticity diagram.On the other hand, our failure to find a systematicpattern does not mean that such a pattern does notexist,

(4) The possibility that cm ellipses generally cannotbe used to predict the equality of color differences of thesize of several (2-10) just-noticeable differences mayhave important consequences on the present practice ofevaluating color differences by methods based on cmellipses. Some graphical methods, such as the one onwhich the Simon-Goodwin charts7 are based, and somenumerical methods, such as the one given by theFriele-MacAdam-Chickering formula,8 are based onthe cm ellipses of MacAdam's observer P. G. Nutting,9

which are similar to our cm ellipses.1(5) Unfortunately, our color-matching and color-

difference matching data exhibit large inherent in-consistencies," 2 characteristic of this type of visualexperimentation, which makes it difficult to extractfrom them systematic trends that can be documentedwith certainty.

More experimental data seem required, possibly ob-tained by different techniques, to check on the con-clusions given above, particularly those in paragraphs(3) and (4).

REFERENCES

1 G. Wyszecki and G. H. Fielder, J. Opt. Soc. Am. 61, 1135(971).

2 G. Wyszecki and G. H. Fielder, J. Opt. Soc. Am. 61, 1501(1971).

a G. Wyszecki, J. Opt. Soc. Am. 55, 1319 (1965).4 D. L. MacAdam, J. Opt. Soc. Am. 32, 247 (1942).' See, for example, D. L. MacAdam, J. Opt. Soc. Am. 33, 18

(1943).eIn making the comparison, note that in Figs. 2-4 the data of

the cm ellipses are plotted as the abscissa, whereas in Figs. 7 and 8the same data are plotted as the ordinate.

F. T. Simon and W. J. Goodwin, Am. Dyestuff Reptr. 47, (4),105 (1958).8 K. D. Chickering, J. Opt. Soc. Am. 57, 537 (1967).

9 D. L. MacAdam, J. Opt. Soc. Am. 32, 247 (1942).

128 Vol. .62