Colour matching for ink-jet prints on paper

Colour Matching for Ink-Jet Printson Paper

C. de M. Bezerra,1 C. J. Hawkyard,2,*H. M. Kulube,3 S. Reyner2

1CNPq, Brasilia-DF, Brazil

2Department of Textiles, UMIST, Manchester, United Kingdom

3Department of Applied Chemistry, National University of Science & Technology, Bulawayo, Zimbabwe

Received 21 September 1996; accepted 11 August 1997

The image’s quality depends on such factors as theAbstract: Predicting the colour of trichromatic prints bymeasuring the colour of the primaries and their areas resolution/speed defined in dots per inch (dpi), the shape

and uniformity of dots, the amount of overlap, the solventhas been shown to be possible using large area textileprints. This approach has been extended to paper printing used and its drying time, and the nature of the substrate’s

surface (for paper there are several different types: matt,with much smaller coloured areas, in a predeterminedarray of squares. As before, the light reflected from the brilliant, polished, coated, etc.) .

A wide gamut of colours can be obtained through ancoloured areas mixed additively within the integratingsphere of a spectrophotometer. Cyan, magenta, and yel- ink-jet printer, using a matrix of dots to produce each

pixel ( the smallest picture element3 Fig. 1) , or by jettinglow were used, with the addition of red, green, blue, andblack, to simulate the typical situation likely to arise in coloured primary inks in such a way that they can mix

on the substrate. It is also possible to apply the ink incommercial paper ink-jet printers. Partitive colour-mix-ing theory was used to predict the colours, and the results different densities by controlling the amount of ink used.

In a previous study,4 it was established that ink-jet print-compared well with those obtained by measurement.However, unwanted overlaps of the printed coloured ing involves both partitive (additive) and subtractive mix-

ing, due to the overlaps. For jet-printing technology thesquares caused all the measured results to be darker andof higher chroma. The reverse process, whereby the area subtractive primary colours cyan, magenta, and yellow

(CMY) are usually employed, together with black (K) forof each colour was calculated for a given target colour,specified by its tristimulus values, was also carried out. dark colours. However, there has been a recent upsurge in

interest in increasing the number of colours to seven orThe results were generally in agreement with the actualnumber of squares used to produce the target, within one eight5–7 in order to achieve a wider gamut.square out of 16. q 1998 John Wiley & Sons, Inc. Col Res Appl,

23, 18–26, 1998MAXWELL MODEL

Key words: jet printing; CMYK inks; paper; colour mix-Maxwell’s equations can be summarized as:ing; partitive; prediction; areas; spectrophotometer

Xmix Å Sai Xi (1)JET PRINTING

Ymix Å SaiYi (2)Jet printing is defined as a nonimpact printing technology,

Zmix Å SaiZi , (3)which projects inks onto the substrate in a controlledseries of drops.1,2

where Xmix , Ymix , Zmix are the tristimulus values of themixture; Xi , Yi , and Zi are the tristimulus values for thesingle colours, and ai are the fractional areas of the col-* Correspondence to: Dr. C. J. Hawkyard, UMIST, Department ofours, i.e., Sai Å 1.Textiles, P.O. Box 88, Manchester M60 1QD, United Kingdom

q 1998 John Wiley & Sons, Inc. These equations were originally developed by Maxwell

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Green: cy(1 0 m) Å fg (9)

Blue: cm(1 0 y) Å fb (10)

Black: cmy Å fk . (11)

Neugebauer’s Eqs. (12) – (14) for the tristimulus val-ues of the total area are obtained from these relationshipsby weighting each of the eight fractional areas by theirtristimulus values and summing.

R * Å (1 0 c)(1 0 m)(1 0 y)RW

/ c(1 0 m)(1 0 y)RC / m(1 0 c)(1 0 y)RMFIG. 1. Schematic representation of the samples printed

/ y(1 0 m)(1 0 c)RY / my(1 0 c)RRwith the HP ink-jet printer.

/ cy(1 0 m)RG / cm(1 0 y)RB / cmyRK . (12)

G * Å (1 0 c)(1 0 m)(1 0 y)GW

for his spinning disc experiments, 8 but have since been/ c(1 0 m)(1 0 y)GC / m(1 0 c)(1 0 y)GMapplied to a diversity of colour-matching situations.

/ y(1 0 m)(1 0 c)GY / my(1 0 c)GR

NEUGEBAUER MODEL / cy(1 0 m)GG / cm(1 0 y)GB / cmyGK (13)

When three inks cyan (C), magenta (M), and yellow B * Å (1 0 c)(1 0 m)(1 0 y)BW(Y) are printed on a white (W) substrate such that the

/ c(1 0 m)(1 0 y)BC / m(1 0 c)(1 0 y)BMdots overlap, then four additional colours will result:namely, red (R), green (G), blue (B), and black (K). / y(1 0 m)(1 0 c)BY / my(1 0 c)BRIf the fractional dot areas of the separate cyan, magenta,

/ cy(1 0 m)BG / cm(1 0 y)BB / cmyBK (14)and yellow colours are represented by c , m , y , respec-tively, then the fractional areas occupied by the eight

where:colours (including the white substrate) can be estimatedby the Demichel’s equations as follows9,10 :

a) C Å Cyan, M Å Magenta, Y Å Yellow, R Å Red, GWhite: (1 0 c)(1 0 m)(1 0 y) Å fw (4) Å Green, B Å Blue, W Å White, and K Å Black.

b) c , m , y are the percentage areas covered by cyan,Cyan: c(1 0 m)(1 0 y) Å fc (5)magenta, or yellow ink, respectively.

Magenta: m(1 0 c)(1 0 y) Å fm (6) c) RW , GW , BW are tristimulus values for white.d) RC , GC , BC , RM , GM , BM , RY , GY , BY . . . . etc. are

Yellow: y(1 0 c)(1 0 m) Å fy (7)the tristimulus values for cyan, magenta, yellow . . . .etc.Red: my(1 0 c) Å fr (8)

FIG. 2. Examples of printed samples: (a) 2-colour mixture 11Y5C, (b) 3-colour mixture 5Y6C5M, and (c) 8-colour mixture3Y2C2M2R3G1B1K2W. (The numbers represent the number of each color in the 4 1 4 matrix—see text.)

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TABLE I. Values for samples printed by the HP be placed into any of 16 pixels in a 4 1 4 array. Eachink-jet printer with single colours (D65, 107 observer). printed pixel was set to be slightly under 1 mm2 and the

4 1 4 array was replicated a further eight times in hori-Chromaticity zontal and vertical directions producing a square sample,Tristimulus values coordinates

Fig. 1. Each repeat, therefore, contains 16 1-mm2 squaresSamples X Y Z x y as is illustrated in Fig. 2, which shows samples of 2-, 3-,

and 8-colour mixtures. The samples were printed at fullYellow 67.25 69.61 12.40 0.4506 0.4663intensity on A4 commercial copier paper (Copier 80Cyan 20.37 29.00 64.24 0.1793 0.2552

Magenta 26.15 16.79 26.13 0.3787 0.2430 g/m2). The number of samples generated using the soft-Red 25.68 17.60 9.01 0.4912 0.3365 ware was 7 printed with a single colour, 45 with 2-colourGreen 12.76 21.07 11.09 0.2840 0.4691

mixtures, 21 with 3-colour mixtures, and 22 with 8-colourBlue 11.99 11.40 32.33 0.2151 0.2046Black 3.04 3.17 3.27 0.3206 0.3342 mixtures. The printed samples were subsequently mea-White 82.28 87.00 95.26 0.3110 0.3289 sured on a Spectraflash 600 spectrophotometer, using the

mean value of five measurements in different positions,with specular component excluded, large area of viewand ultraviolet excluded. The calculated and measuredtristimulus values X, Y, Z and chromaticities coordinatese) R *, G *, B * are the tristimulus values for the overallx, y are shown in Tables I–VII.image when the inks cover the areas c , m and y .10

Examination of the red, green, and blue areas under amicroscope showed that these colours consisted of dotsThe Neugebauer model shows that the average frac-of two of the primary colours within a matrix. Thus, fortional area of dot overlap is proportional to the productexample, green was a mixture of cyan and yellow, withof the fractional dot sizes of the separate dot patterns.9no magenta or black ink. The black consisted only ofThese equations are, therefore, very useful in predictingblack ink, not a mixture containing cyan, magenta, andthe colour gamut achievable with a given trichromat ofyellow. However, in the 2-colour mixtures it was apparentinks.that the printed squares in the matrix did not fit togetherprecisely, but there was some overlap. As a result, all

EXPERIMENTAL the prints were darker than anticipated. It was, therefore,decided to further investigate the extent of the overlap.Printing with a Commercial Paper Ink-Jet Printer

Software was written in C// in a Windows environ-Image Analysis

ment to control a Hewlett–Packard Deskjet 660C printer,which allowed for a selection from a pallette of eight In order to verify and quantify the existence of overlap-

ping areas in the printed samples, image analysis wascolours (yellow, magenta, cyan, red, green, blue, black,and white) over a range of intensities (except white) to carried out. The sample’s image was captured using a

TABLE II. Two-colour mixtures (cyan-magenta) printed by the HP ink-jet printer—tristimulus values and chroma-ticity coordinates (D65, 107 Observer) for the measured and calculated samples, and DE (CIELAB).

Tristimulus values Chromaticity coordinates Number of pixels

Measured Calculated Measured Calculated Used Calculated

Sample X Y Z X Y Z x y x y C M C M DE

1C15M 25.24 16.76 27.76 25.79 17.55 28.51 0.3618 0.2403 0.3590 0.2442 1 15 0.9 15.3 2.232C14M 24.59 17.18 29.70 25.43 18.31 30.89 0.3440 0.2404 0.3407 0.2454 2 14 1.9 14.4 2.833C13M 23.79 17.56 31.52 25.07 19.08 33.28 0.3265 0.2410 0.3238 0.2464 3 13 2.8 13.2 3.014C12M 23.05 17.95 33.47 24.71 19.84 35.66 0.3095 0.2411 0.3081 0.2474 4 12 3.8 12.2 3.295C11M 22.92 18.89 36.31 24.35 20.60 38.04 0.2934 0.2418 0.2934 0.2483 5 11 4.9 11.4 3.106C10M 22.26 19.32 38.18 23.98 21.37 40.42 0.2791 0.2422 0.2796 0.2491 6 10 5.8 10.4 3.347C9M 21.89 20.06 40.42 23.62 22.13 42.80 0.2657 0.2436 0.2668 0.2499 7 9 6.8 9.4 3.118C8M 21.59 20.89 42.88 23.26 22.89 45.19 0.2530 0.2447 0.2547 0.2506 8 8 7.8 8.4 2.859C7M 21.33 21.71 45.49 22.90 23.66 47.57 0.2409 0.2452 0.2433 0.2513 9 7 8.8 7.5 2.7710C6M 20.90 22.40 47.78 22.54 24.42 49.95 0.2294 0.2459 0.2326 0.2520 10 6 9.8 6.5 2.6811C5M 20.50 23.09 49.99 22.18 25.18 52.33 0.2190 0.2468 0.2225 0.2526 11 5 10.8 5.5 2.6212C4M 20.16 23.84 52.28 21.82 25.94 54.72 0.2094 0.2476 0.2129 0.2532 12 4 11.7 4.5 2.5113C3M 20.22 25.19 55.61 21.45 26.71 57.10 0.2002 0.2494 0.2038 0.2537 13 3 12.9 3.4 1.8214C2M 20.05 26.10 58.05 21.09 27.47 59.48 0.1924 0.2505 0.1952 0.2543 14 2 13.9 2.4 1.6115C1M 19.84 27.05 60.70 20.73 28.23 61.86 0.1844 0.2514 0.1871 0.2548 15 1 14.9 1.4 1.39

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TABLE III. Two-colour mixtures (yellow-magenta) printed by the HP ink-jet printer—tristimulus values and chro-maticity coordinates (D65, 107 Observer) for the measured and calculated samples, and DE (CIELAB).



Sample X Y Z X Y Z x y x y Y M Y M DE

1Y15M 27.85 18.84 24.65 28.72 20.09 25.27 0.3903 0.2641 0.3877 0.2712 1 15 0.9 15.6 3.422Y14M 29.84 21.35 23.72 31.29 23.39 24.41 0.3983 0.2850 0.3956 0.2957 2 14 1.8 14.9 5.043Y13M 31.93 24.03 22.63 33.86 26.69 23.56 0.4063 0.3057 0.4026 0.3173 3 13 2.7 14.2 5.704Y12M 33.91 26.71 21.34 36.43 29.99 22.70 0.4137 0.3259 0.4088 0.3365 4 12 3.7 13.2 5.705Y11M 36.46 29.91 20.78 39.00 33.29 21.84 0.4183 0.3432 0.4143 0.3537 5 11 4.6 12.3 5.626Y10M 38.64 32.76 19.79 41.56 36.59 20.98 0.4237 0.3593 0.4192 0.3691 6 10 5.5 11.4 5.667Y9M 41.20 36.03 18.85 44.13 39.90 20.12 0.4288 0.3750 0.4237 0.3830 7 9 6.5 10.5 5.168Y8M 43.57 39.15 17.83 46.70 43.20 19.27 0.4333 0.3894 0.4278 0.3957 8 8 7.5 9.5 4.689Y7M 46.28 42.56 17.22 49.27 46.50 18.41 0.4364 0.4013 0.4315 0.4072 9 7 8.5 8.5 4.4510Y6M 48.68 45.65 16.35 51.84 49.80 17.55 0.4398 0.4125 0.4349 0.4178 10 6 9.4 7.5 4.3411Y5M 51.49 49.27 15.30 54.41 53.10 16.69 0.4437 0.4245 0.4381 0.4275 11 5 10.6 6.5 3.4812Y4M 54.29 52.86 14.31 56.98 56.40 15.83 0.4469 0.4352 0.4409 0.4365 12 4 11.7 5.4 2.9213Y3M 57.34 56.79 13.77 59.54 59.70 14.98 0.4483 0.4440 0.4436 0.4448 13 3 12.7 4.1 2.2914Y2M 60.34 60.62 12.98 62.11 63.00 14.12 0.4505 0.4526 0.4461 0.4525 14 2 13.8 3.0 1.9315Y1M 63.69 64.92 12.30 64.68 66.31 13.26 0.4520 0.4607 0.4484 0.4597 15 1 15.0 1.7 1.64

Ave. 4.14

MaxVision video camera, colour-separated using Visilog measuring a sample with a spectrophotometer with anintegrating sphere results in a reflected beam in which4.1 software, and the areas measured. By this means it

was established that there was 6.04% area overlap when the individual component colours have mixed homoge-neously. The purpose of this work was to show that thistwo pixels were printed side by side. This can be attrib-

uted to the spread of the ink and the absorbancy of the assumption is also tenable for the more complex patternsprinted by the Hewlett–Packard Deskjet 660C printer.paper.

The samples were printed using a single colour, andmixtures of two, three, or eight. Each sample was desig-

RESULTS AND DISCUSSIONnated according to the colours and number of pixels usedto print its array. Thus, the sample labeled 5C11M wasThe assumption employed in the analysis of the results

was derived from earlier work, which used two different printed with 5 pixels of cyan and 11 of magenta, and5Y1C10M was printed with 5 pixels of yellow, 1 of cyan,colours placed side by side,4 where it was shown that

TABLE IV. Two-colour mixtures (yellow-cyan) printed by the HP ink-jet printer—tristimulus values and chroma-ticity coordinates (D65, 107) for the measured and calculated samples, and DE (CIELAB).



Sample X Y Z X Y Z x y x y Y C Y C DE

1Y15C 21.74 30.00 59.24 23.30 31.54 61.00 0.1959 0.2703 0.2011 0.2722 1 15 1.1 15.0 2.092Y14C 23.58 31.74 55.25 26.23 34.07 57.76 0.2133 0.2871 0.2222 0.2886 2 14 2.4 14.5 3.823Y13C 26.04 33.96 51.69 29.16 36.61 54.52 0.2331 0.3041 0.2424 0.3044 3 13 3.5 13.7 4.264Y12C 28.29 35.96 47.86 32.09 39.15 51.28 0.2524 0.3207 0.2619 0.3195 4 12 4.6 12.9 4.755Y11C 30.96 38.35 44.49 35.02 41.69 48.04 0.2720 0.3370 0.2807 0.3342 5 11 5.8 12.1 4.836Y10C 33.29 40.42 40.80 37.95 44.23 44.80 0.2907 0.3530 0.2989 0.3483 6 10 6.9 11.3 5.207Y9C 36.13 42.81 37.48 40.88 46.76 41.56 0.3104 0.3677 0.3164 0.3619 7 9 7.8 10.3 4.768Y8C 38.92 45.22 34.16 43.81 49.30 38.32 0.3290 0.3822 0.3333 0.3751 8 8 8.8 9.3 4.629Y7C 41.88 47.89 31.27 46.74 51.84 35.08 0.3460 0.3957 0.3497 0.3878 9 7 9.9 8.4 4.5410Y6C 44.67 50.28 27.68 49.67 54.38 31.84 0.3642 0.4100 0.3655 0.4002 10 6 10.9 7.4 4.6611Y5C 48.14 53.29 25.04 52.60 56.92 28.60 0.3806 0.4214 0.3808 0.4121 11 5 11.8 6.3 4.1512Y4C 51.66 56.35 22.38 55.53 59.45 25.36 0.3962 0.4321 0.3957 0.4236 12 4 12.7 5.2 3.6713Y3C 54.93 59.09 19.25 58.46 61.99 22.12 0.4122 0.4434 0.4100 0.4348 13 3 13.6 4.0 3.6714Y2C 58.64 62.29 16.43 61.39 64.53 18.88 0.4269 0.4535 0.4240 0.4456 14 2 14.6 2.8 3.6115Y1C 62.86 65.79 13.95 64.32 67.07 15.64 0.4408 0.4614 0.4375 0.4562 15 1 15.3 1.3 2.95

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TABLE V. Tristimulus values and chromaticity coordinates assuming 6% overlap.

Tristimulus values Chromaticity coordinates

Measured Calculated/recalculated Measured Calculated/recalc.

Sample X Y Z X Y Z x y x y DE

Y8C8 38.92 45.22 34.16 43.81 49.30 38.32 0.3290 0.3822 0.3333 0.3751 4.628Y8C-recal 38.92 45.22 34.16 41.95 47.61 36.69 0.3290 0.3822 0.3323 0.3771 3.20C8M8 21.59 20.89 42.88 23.26 22.89 45.19 0.2530 0.2447 0.2547 0.2506 2.858C8M-rcal. 21.59 20.89 42.88 22.58 22.20 44.42 0.2530 0.2447 0.2532 0.2489 2.18Y8M8 43.57 39.15 17.83 46.70 43.20 19.27 0.4333 0.3894 0.4278 0.3957 4.688Y8M-recal. 43.57 39.15 17.83 45.44 41.66 18.65 0.4333 0.3894 0.4297 0.3939 3.16

and 10 of magenta. Similarly, the sample designatedXmixture Å

116

Xy /2

16Xc /

216

Xm /1

16Xr3Y2C1M4R1G1B3K1W was printed using 3 pixels of

yellow, 2 of cyan, 1 of magenta, 4 of red, 1 of green, 1of blue, 3 of black, and 1 of white (no ink). / 4

16Xg /

116

Xb /416

Xk /116

Xw , (15)The tristimulus values and chromaticity coordinates of

the samples printed with a single colour are presented inand similar expressions for Ymixt and Zmixt .Table I. The green, red, and blue colours contained two

inks.In order to simulate the samples with 2-, 3-, and 8-

Two-Colour Mixturescolour mixtures, Eqs. (1) – (3) were used together withthe tristimulus values in Table I. Thus, for example, an The combinations cyan-magenta, yellow-magenta, and

yellow-cyan were used, as the three colours constitute the8-colour mixture whose component colours have the areasyellow (y) 1, cyan (c) 2, magenta (m) 2, red (r) 1, green trichromat used for ink-jet printing. Since there is a good

degree of agreement between the measured and calculated(g) 4, blue (b) 1, black (k) 4, and white (w) 1 mm2

would have its tristimulus values calculated as follows: tristimulus values—the average colour difference, DE

TABLE VI. Three-colour mixtures (yellow-cyan-magenta) printed by the HP ink-jet printer—tristimulus valuesand chromaticity coordinates for the measured and calculated samples and DE (D65, 107 Observer), togetherwith the number of pixels used and calculated.



Sample X Y Z X Y Z x y x y Y C M Y C M DE

2Y2C12M 28.78 22.76 27.87 30.57 24.92 29.18 0.3624 0.2866 0.3611 0.2943 2 2 12 1.6 1.7 12.2 3.772Y4C10M 27.34 23.57 31.68 29.84 26.44 33.94 0.3310 0.2853 0.3308 0.2931 2 4 10 1.5 3.5 10.1 4.022Y6C8M 26.34 24.95 35.97 29.12 27.97 38.71 0.3018 0.2859 0.3040 0.2920 2 6 8 1.6 5.5 7.8 3.362Y8C6M 25.37 26.30 40.59 28.40 29.49 43.47 0.2750 0.2850 0.2802 0.2910 2 8 6 1.5 7.5 5.7 3.342Y10C4M 24.59 27.78 45.18 27.68 31.02 48.24 0.2521 0.2848 0.2588 0.2901 2 10 4 1.6 9.5 3.7 3.302Y12C2M 24.40 30.18 50.68 26.95 32.55 53.00 0.2318 0.2867 0.2396 0.2893 2 12 2 1.7 11.7 1.4 3.124Y2C10M 32.46 27.68 25.76 35.70 31.52 27.46 0.3779 0.3223 0.3771 0.3329 4 2 10 3.2 1.6 10.4 5.514Y4C8M 30.86 28.80 29.17 34.98 33.04 32.23 0.3474 0.3242 0.3489 0.3296 4 4 8 3.3 3.5 7.6 4.114Y6C6M 29.55 29.81 33.31 34.26 34.57 36.99 0.3188 0.3217 0.3237 0.3267 4 6 6 3.3 5.5 5.4 4.384Y8C4M 28.87 31.65 37.81 33.54 36.10 41.76 0.2936 0.3219 0.3011 0.3241 4 8 4 3.4 7.5 3.0 4.274Y10C2M 28.16 33.37 42.53 32.81 37.62 46.52 0.2706 0.3207 0.2806 0.3217 4 10 2 3.5 9.6 0.8 4.826Y2C8M 37.31 34.03 23.93 40.84 38.12 25.75 0.3916 0.3572 0.3901 0.3641 6 2 8 5.2 1.6 8.3 4.526Y4C6M 35.05 34.29 27.33 40.12 39.65 30.51 0.3626 0.3547 0.3638 0.3595 6 4 6 5.0 3.5 5.8 4.666Y6C4M 33.47 35.56 30.76 39.40 41.17 35.28 0.3354 0.3563 0.3401 0.3554 6 6 4 5.2 5.5 2.8 4.676Y8C2M 33.62 38.13 35.84 38.67 42.70 40.04 0.3125 0.3544 0.3185 0.3517 6 8 2 5.4 7.6 0.7 4.418Y2C6M 42.23 40.43 22.02 45.98 44.72 24.03 0.4034 0.3862 0.4007 0.3898 8 2 6 7.1 1.5 6.3 3.748Y4C4M 40.47 41.55 25.52 45.26 46.25 28.79 0.3763 0.3864 0.3762 0.3845 8 4 4 7.3 3.6 3.3 3.258Y6C2M 39.64 43.27 29.64 44.53 47.78 33.56 0.3522 0.3844 0.3538 0.3796 8 6 2 7.4 5.6 0.8 3.6310Y2C4M 47.49 47.42 20.18 51.12 51.33 22.31 0.4126 0.4120 0.4097 0.4114 10 2 4 9.3 1.6 4.0 2.4910Y4C2M 46.02 48.96 23.96 50.39 52.85 27.08 0.3869 0.4116 0.3867 0.4055 10 4 2 9.5 3.8 0.9 3.1812Y2C2M 53.17 54.84 18.41 56.25 57.93 20.60 0.4206 0.4338 0.4174 0.4298 12 2 2 11.5 1.7 1.6 2.15

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TABLE VII. Eight-colour mixtures (yellow-cyan-magenta-red-green-blue-black-white) printed by the HP ink-jetprinter—tristimulus values and chromaticity coordinates for the measured and calculated samples, and DE(D65, 107 Observer).

Tristimulus values Tristimulus values Chromat. coord. Chromat. coord.measured calculated measured calculated

Sample X Y Z X Y Z x y x y DE

1Y1C1M2R2G3B3K3W 25.85 26.74 28.17 30.16 31.09 33.47 0.3201 0.3311 0.3184 0.3282 4.001Y1C2M2R1G4B1K4W 31.75 31.88 37.87 36.51 36.57 41.98 0.3128 0.3141 0.3173 0.3179 4.021Y1C2M2R3G3B3K1W 18.87 19.33 21.01 22.31 22.58 23.89 0.3187 0.3264 0.3243 0.3283 3.971Y1C3M1R3G2B1K4W 32.15 33.06 35.74 36.64 37.73 40.39 0.3185 0.3275 0.3192 0.3288 3.641Y2C2M1R4G1B4K1W 18.11 19.98 20.53 21.46 23.38 24.20 0.3090 0.3408 0.3109 0.3386 3.801Y2C2M3R3G3B1K1W 20.93 21.58 23.88 24.81 25.10 28.06 0.3153 0.3250 0.3182 0.3219 4.201Y3C1M3R2G1B4K1W 18.83 19.89 21.50 22.72 23.71 26.32 0.3127 0.3303 0.3123 0.3259 4.411Y3C2M1R4G1B1K3W 27.79 31.11 34.02 32.45 35.47 39.51 0.2991 0.3348 0.3021 0.3302 4.381Y4C1M1R2G2B1K4W 31.91 35.44 43.85 36.39 39.75 48.48 0.2869 0.3187 0.2920 0.3190 3.691Y4C1M1R3G1B3K2W 22.99 26.45 31.09 26.53 29.88 35.65 0.2855 0.3284 0.2882 0.3246 3.741Y4C1M4R1G1B2K2W 25.30 26.15 30.15 29.56 30.35 35.75 0.3101 0.3205 0.3090 0.3172 4.042Y1C4M1R1G3B1K3W 31.71 30.77 33.77 36.49 35.77 37.48 0.3294 0.3197 0.3325 0.3260 4.632Y1C4M1R4G1B1K2W 28.32 29.20 26.35 32.24 32.86 29.57 0.3376 0.3481 0.3405 0.3471 3.442Y2C1M4R1G4B1K1W 23.80 23.48 24.86 28.13 27.58 28.40 0.3299 0.3254 0.3345 0.3279 4.272Y2C2M2R2G2B2K2W 25.81 26.63 26.34 31.19 31.95 31.72 0.3276 0.3381 0.3288 0.3368 4.772Y2C3M3R3G1B1K1W 24.62 24.72 22.83 29.15 29.07 26.43 0.3411 0.3425 0.3443 0.3434 4.292Y3C3M3R2G1B1K1W 25.80 25.88 26.43 29.62 29.57 29.75 0.3303 0.3313 0.3331 0.3324 3.533Y1C3M2R1G4B1K1W 26.77 26.07 24.49 31.12 30.01 27.30 0.3462 0.3372 0.3519 0.3394 4.213Y2C1M4R1G1B3K1W 25.13 25.15 19.58 30.47 30.18 23.52 0.3597 0.3600 0.3620 0.3586 4.904Y1C1M2R2G1B4K1W 25.61 26.43 16.82 31.18 32.04 20.05 0.3719 0.3838 0.3744 0.3848 5.364Y1C1M3R1G3B2K1W 28.30 28.14 20.35 33.10 32.85 23.56 0.3685 0.3664 0.3698 0.3670 4.244Y1C4M1R1G2B2K1W 28.91 27.91 21.67 34.05 33.08 25.31 0.3683 0.3556 0.3683 0.3579 4.64

Aver. 4.19

(CIELAB) for the cyan-magenta combination was 2.61Zmix Å

47100

∗12.40 / 47100

∗64.24 / 6100

∗11.09.(Table II) , yellow-magenta was 4.14 (Table III) , and foryellow-cyan was 4.11 (Table IV) —the premise estab-

Assuming the same 6% overlap for the samples 8C8Mlished for the mixing of light reflected from a pair ofand 8Y8M, and using the same procedure used for thedifferently coloured samples measured simultaneouslysample 8Y8C, the recalculated values for those sampleshas been borne out, even though the samples in this case

are much smaller. The corollary of that statement wouldbe that, were the squares as small as the dots producedby a commercial ink-jet printer, typically 300–400 dpi,then predicting the colour required for a particular printwould be rendered feasible.

The chromaticities for 2-colour mixtures, Tables II–IV, bear further testimony to the small difference betweenthe measured and calculated values.

Figure 3 was drawn from the measured and calculatedchromaticity coordinates and, as expected, the mixturesfall in order on the respective lines linking the relevantpair of single colours. Ideally, primaries should be at thecorners of triangles whose sides pass through the mixturepoints, but in this case they do not, mainly because theareas overlap, as shown in Fig. 4.

Determination of the overlap by image analysis wascarried out on sample 8Y8C in Table IV. If the 6% overlapis allowed for, using the tristimulus values for green inTable I, the recalculated tristimulus values are as follows:

Xmix Å47100

∗67.25 / 47100

∗20.37 / 6100

∗12.76,FIG. 3. Chromaticity coordinates for the measured and cal-culated samples, printed with 2-colour mixtures (yellow-cyan, yellow-magenta, and cyan-magenta, D65 107 Ob-Ymix Å

47100

∗69.61 / 47100

∗29.00 / 6100

∗21.07,server) .


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FIG. 4. Yellow and cyan stripe print showing overlap.

are placed in Table V. And as can be seen, the recalculatedFIG. 6. Chromaticity coordinates for the measured and cal-

values are closer to the measured ones. culated samples printed with 3-colour mixtures (yellow-In order to test this hypothesis further, the program cyan-magenta, D65, 107 Observer) .

was amended so that the pixel size was 110th the previous

size. The resulting mixtures showed a great deal moreoverlap, as shown in Fig. 5, and the mixture colours were,therefore, darker and more saturated. The data in TableV indicate that the level of overlap should have been of The average DE for the 3-colour mixtures in Table VI isthe order of 15%, rather than 6% for the larger pixel size. 3.84, smaller than the average DE for the YM and YCThe explanation for the difference is probably that there 2-colour mixtures (Tables III and IV), and larger thanis a certain amount of splashing of one ink into adjacent the average DE for the CM colour mixtures (Table II) .areas, and this can be seen in Fig. 5. The premise promulgated in the simple case where two

differently coloured samples were placed in the port of aspectrophotometer has, therefore, been proven to hold forThree-Colour Mixturesthe trichromatic situation. The purpose of using the HP

The observations made for 2-colour mixture are like- ink-jet printer for paper to simulate mixing of the 1 mm2

wise valid for 3-colour mixtures, even though the arrange- squares of cyan, magenta, and yellow was to imitate morement of the 1 mm2 coloured squares is more intricate. closely the results of the ink-jet printing of textile sub-

strates. Thus far, it has been shown that the equationsadapted from Maxwell can aptly be applied to the circum-stances where the printed dots are juxtaposed withoutoverlap, that is, when partitive mixing takes place. Green,red, and blue were included, because these represent thesituation where there is partial overlap. Microscopic ex-amination, plus observations of a print that did not havethe yellow ink, indicated that only two inks were presentin these areas. Partitive theory also applies in areas wherethere is a partial overlap, as long as the colours and areasare measurable.

The differences in the measured and calculated chroma-ticities of the 3-colour mixtures are again small, as evi-denced by Table VI. From the measured and calculatedchromaticities, Fig. 6 was drawn, and it can be seen thatthe mixtures form a gamut that falls within the trianglejoining the chromaticity points of cyan, magenta, andyellow. The gamut is a small one, showing how overlapto give red, green, and blue is essential to provide an

FIG. 5. Pixel size reduces to 110th size (approx. 0.1 mm). adequate range of colours.


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Calculation of Fractional Areas 216ths of each of the eight colours was used. The total

(3-colour mixtures—YCM) fractional area of the cyan print is assumed to be as fol-lows:Using the same mathematical model [Eqs. (1) – (3)] ,

the XYZ measured tristimulus values for yellow, cyan,c Å 2

16(c) / 2

16(in blue) / 2

16(in green)and magenta (Table I) , together with the Xmix , Ymix , and

Zmix measured tristimulus values for the samples printedwith the CM, YM, YC 2-colour mixtures (Tables II, III,

/ 216

(in black) Å 816Å 1

2,and IV), and YCM 3-colour mixture (Table VI), the

fractional area of each colour (number of pixels per array)to print the 2-colour or 3-colour sample could be calcu- and similarly for magenta and yellow. Although blacklated using an inverse matrix method and compared with results from the mixture of cyan, magenta, and yellow, itthe fractional area really used. As can be seen in Tables II, is also included as a separate ink. Therefore,III, IV, and VI, the calculated number of pixels required to

c Å m Å y Å 12match the measured tristimulus values are almost the

same as those used. The calculated number of pixels forand (1 0 c) Å (1 0 m) Å (1 0 y) Å 1

2.cyan and yellow (CM and YM colour mixtures, TablesII and III) was always smaller than the number used,

Substituting in Eq. (12) – (14) and taking the modifica-while the number for the magenta colour was larger. Fortions cited above, the calculations for the Neugebauerthe YC colour mixtures, Table IV, both the Y and Ctristimulus values gives:calculated values were higher than the ones used. In the

3-colour mixtures, Table VII, the calculated number ofX Å 0.125Xw / 0.125Xc / 0.125Xm / 0.125Xypixels for Y and C colours were smaller than the ones

used, while for C colours it varied from smaller to larger / 0.125Xr / 0.125Xg / 0.125Xb / 0.125Xkthan the number used.

Y Å 0.125Yw / 0.125Yc / 0.125Ym / 0.125Yy

Eight-Colour Mixtures / 0.125Yr / 0.125Yg / 0.125Yb / 0.125Yk

Experiments with 8-colour mixtures were carried out Z Å 0.125Zw / 0.125Zc / 0.125Zm / 0.125Zyto test the applicability of Maxwell’s Eqs. (1) – (3) for

/ 0.125Zr / 0.125Zg / 0.125Zb / 0.125Zkmixtures under such extreme conditions.The assumption was that if the equations worked for

X Å 31.19, Y Å 31.95, Z Å 31.728-colour mixtures then they would work for any numberof colours. An average DE of 4.19 was obtained for the x Å 0.3288, y Å 0.3368.8-colour mixtures, Table VII, a value close to that of the2- and 3-colour mixtures. All the calculated values for The X ,Y ,Z values for the single colours were takenthe calculated samples were higher, a result that can from Table I. The tristimulus values obtained with thelargely be attributed to the amount of overlap areas, when Neugebauer equations are the same as those calculatedthe large number of colours was used. with Maxwell’s equations, since 2

16 Å 0.125, and higherNeugebauer reworked Maxwell’s equations incorporat- than the measured, see sample Y2C2M2R2G2B2K2W in

ing the Demichel’s approximations in an effort to address Table VII. Thus, for overlapping areas, the Neugebauerthe issue of dot areas that only overlap fractionally. The model could be used as the method for predicting theNeugebauer’s Eqs. (12) – (14) are derived from the eight colour or, more precisely, the tristimulus values of thecolours that are present when cyan, yellow, and magenta resultant mixture with a significant degree of accuracy.inks overlap, the secondary colours being red, green, and The fact that the DE’s average value is around 4, andblue-violet, and the tertiary, black, unprinted areas being there is good agreement between measured and calculatedwhite. The equations were formulated for the calculation chromaticities, Table VII, further support the validity ofof fractional area of dot overlap. Where cyan, magenta, the Neugebauer method. Lightness, Y , is poor due to theand yellow overlap, it was recognized that black and sub- presence of overlap areas.tractive colours similar to the additive primaries would The tristimulus values in the chromaticity calcula-be formed thus: tions were derived from Maxwell’s equations. The mea-

sured chromaticities for the mixture with an equal area216

b Å (m / c) ;216

r Å m / y ; for each colour likewise compare well with those calcu-lated from the Neugebauer tristimulus values ( i.e., x Å0.3288, y Å 0.3368) . However, as expected, the Neu-2

16g Å c / y ;

216

k Å m / c / y . gebauer and Maxwell chromaticities are identical, sincetheir tristimulus values were the same, see sample2Y2C2M2R2G2B2K2W in Table VII. From the mea-In order to test Neugebauer’s equations, the print with


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sured and calculated chromaticities, Fig. 7 was drawn.It shows the mixtures clustered in the center of thepolygon, which forms the boundaries of the 8-colourgamut. The range is small, because the inclusion ofblack and white in each 8-colour mixture desaturatedthe colours.

CIELAB Colour Space of the Partitive Gamut

The entire CIELAB colour space formed by the 2-, 3-,and 8-colour mixtures together with the eight individualprimaries printed is shown in Fig. 8. The black and whiteare situated at the center on this two-dimensional diagram,while on the three-dimensional version they would be atopposite ends. By and large, the mixtures retain the shapethat was apparent on the chromaticity diagrams.

CONCLUSIONS FIG. 8. CIELAB colour space for YC, CM, and YM 2-colourmixture; YCM 3-colour mixture; YCMGRBKW 8-colour mix-

The use of partitive colour theory has proved valid for ture, and the single colours of Y, C, M, R, G, B, K, and W.ink-jet prints on paper, as long as the areas of each ofthe colours and their respective tristimulus values areknown. The use of Demichel’s approximation means thatthe areas of cyan, magenta, and yellow only are required,but tristimulus values of the overlap colours are again necessary. It may be possible to postulate a theory for

approximating these from the tristimulus values of theprimary inks.

The inverse matrix method has proved useful whenonly 2 or 3 colours are present, but cannot be used for 8colours.

1. T. L. Dawson, Jet printing. Rev. Prog. Coloration 22, 22–31(1992).

2. J. P. Stefanini, Jet printing for the textile industry. Textile Chem.Colour. 28(9), 19–23 (1996).

3. C. J. Hawkyard, Colour in visual displays. Rev. Prog. Coloration21, 43–55 (1991).

4. H. M. Kulube and C. J. Hawkyard, Predicting the colour of trichro-matic prints. Color Res. Appl. 20, 55–61 (1995).

5. L. W. MacDonald, J. M. Deane, and D. N. Rughani, Extending thecolour gamut of printed images. J. Photographic Sci. 42, 97–99(1994).

6. T. Kohler and R. S. Berns, Reducing metamerism and increasinggamut using 5 or more coloured inks, in Proc. IS&T Tech. Symp.On Prepress, Proofing & Printing, 24–28, Chicago, Nov. 1993.

7. V. Ostromoukhov, Chromaticity gamut by heptatone multi-colorprinting, Device-Independent Colour Imaging and Imaging SystemIntegration, J. Motta and H. A. Berberian, Ed., Proc. SPIE 1909,139 (1993).

8. J. C. Maxwell, Philosophical Magazine 14 (1857).9. F. J. Pollack, The relationship between the densities and dot sizes

of multi-colour halftone images. J. Photographic Science 3, 112–FIG. 7. Chromaticity coordinates for the measured and cal-116 (1955).culated samples printed with 8-colours mixture (yellow-

10. H. E. J. Neugebauer, Theory of masking for colour correction. J.cyan-magenta-red-green-blue-black-white, D65, 107 Ob-Opt. Soc. of America 42, 740–748 (1952).server) .


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Colour matching for ink-jet prints on paper