A Printer Model for Color Printing
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IS&T©s 50th Annual Conference IS&T©s 50th Annual Conference Copyright 1997, IS&T A Printer Model for Color Printing Huanzhao Zeng, and Bob Chin Encad, Inc., San Diego, CA Abstract 2. Neugebauer Equation and Its Yule-Nielsen Modification A new model to predict color for dot-on-dot color printing is presented. The Neugebauer narrow-band color mixing model The simplest tone reproduction model, Murray-Davies was applied with modifications. The Yule-Nielsen factor n is model, can be expressed by the following equation ∆ ∆ optimized by minimizing E*L*a*b* or E*94. Dot area at each wavelength was calculated by the Balasubramanian’s R = (1 - a) R0 + a Ri , (2.1) cellular model with eighty-one primaries. Neugebauer colorimetric quality factor (CQF) was applied as a weighting where R is the total reflectance, R 0 is the reflectance of the function for the optimization of dot areas. The application of base which is usually paper, Ri is the reflectance of ink, and the CQF decreases average color difference significantly. We a is the relative dot area of the ink. also analyzed the difference of optimizing the Yule-Nielsen The narrow-band equation based on this model can be ∆ ∆ n-value by minimizing E* L*a*b* and by minimizing E*94. written as λ λ λ There is almost no further improvement in the optimization R( ) = (1 - a) R0 ( ) + a R i ( ), (2.2) ∆ ∆ of the n-value by using E*94 instead of E* L*a*b* with the λ λ λ data set we used. where R( ) is the total reflectance at wavelength , R0 ( ) is λ λ the reflectance of paper at wavelength , and R i ( ) is the 1. Introduction reflectance of ink at wavelength λ. In the case of a four-color printing process, a color is In 1936, Murray and Davies1 published a model to predict the combination of four colorants, cyan (C), magenta (M), the binary tone reproduction printing process. This model yellow (Y), black (K), and white paper (W). The Neugebauer was extended by Neugebauer2 in 1937 for color halftoning. equation for four-color binary printing process is given by However, variation from this linear optical model is often significant. Yule and Nielsen3 modified the tone reproduction 16 RaR()λλ= (), (2.3) model in which they incorporated an empirical factor n to fit CMYK∑ i i =1 the non-linear relationship between total reflectance and dot i where R(λ) is the total spectral reflectance in a given area. Viggiano4,5 further improved the model by applying CMYK wavelength λ, i is the i-th Neugebauer primary, R (λ) is the Yule-Nielsen factor n into the Neugebauer narrow-band i spectral reflectance of the i-th primary in the given model. This model and its variations have been widely used wavelength λ, and a is the relative area of the i-th primary. to model binary color printers.6-9 Arney’s model 10,11 reserves i The a s are given by the linearity of the Murray-Davies model, and explains the i reflectances of paper and inks as functions of ink area. a = (1-c)(1-m)(1-y)(1-k) Balasubramanian12 extended the Neugebauer model into a 0 a = c(1-m)(1-y)(1-k) cellular model in which the primary colors are not limited to 1 a = m(1-c)(1-y)(1-k) sixteen. Due to the increase number of primaries in the 2 a = y(1-c)(1-m)(1-k) cellular model, it yields more accurate result from a finer 3 a = k(1-c)(1-m)(1-y) interpolation. 4 a = cm(1-y)(1-k) In this paper, the Neugebauer colorimetric quality 5 a = cy(1-m)(1-k) factor13,14 (CQF) is applied to optimize the dot areas of 6 a = ck(1-m)(1-y) colorants. The Neugebauer 16-primary model and the 7 a = my(1-c)(1-k) (2.4) Balasubramanian’s cellular model are used to predict 8 a = mk(1-c)(1-y) colorimetric values. Yule-Nielsen factor n is optimized by 9 a = yk(1-c)(1-m) minimizing ∆E* or ∆E* 15,16 and the differences 10 L*a*b* 94, a = cmy(1-k) between using both color difference equations are analyzed. 11 a = cmk(1-y) This model was tested on a Novajet Pro wide-format ink 12 a = cyk(1-m) jet color printer which was based on a four-color dot-on-dot 13 a = myk(1-c) printing process. 14 a15 = cmyk 2842 IS&T©s 50th Annual Conference IS&T©s 50th Annual Conference Copyright 1997, IS&T where c, m, y, and k are dot areas of cyan, magenta, 700 =λλλ yellow, and black colorants, respectively. awadii∫()() , (3.2) The Murray-Davies equation is valid only for zero λ= 400 λ λ optical gain behavior. Yule and Nielsen improved the where w( ) is a weighting function. w( ) is determined Murray-Davies equation by incorporating an n factor. Thus, based on the characteristics that the human visual sensitivity equation (2.1) becomes is a function of wavelength. Neugebauer’s CQF is used as the weighting function 1 / n 1 / n 1 / n w(λ). Figure 1 shows the CQF function. Since human eyes R = ( 1 - a ) R 0 + a R i (2.5) are almost non-sensitive to the wavelengths below 400nm This is the Yule-Nielsen equation. Viggiano extended and wavelengths above 700nm, the weights in these areas the Yule-Nielsen effect to the Neugebauer equation to predict are set to zero. Three peeks approximately correspond to the the color of halftone printing by equation most sensitive wavelengths of three cone types of the human eye.17 16 1 /n ()λλ1/n = () 3.2 The Determination of Primaries RaRCMYK ∑ ii . (2.6) i=1 Balasubramanian’s cellular framework model was used The dot area ai can be obtained by minimizing the color to determine primaries. The eighty-one Neugebauer difference or by minimizing the difference between the primaries were implemented. In the 16-primary model (the predictive reflectance and the corresponding measured Neugebauer model), primaries are the combinations of C, reflectance. M, Y, and K being 0 and 1. In the 81-primary model, the primaries are the combinations of C, M, Y, and K being 0, 3. A New Model 0.5, and 1 (the medium value is not necessary to be exactly 0.5). Primaries to predict an unknown color are still sixteen The Neugebauer color mixing equation (2.6) will be used for in the 81-primary model. The difference between the 16- printer color formation with some modifications, and the primary model and the 81-primary model is that the sixteen Neugebauer colorimetric quality factor (CQF) will be applied primaries in the 16-primary model are unchanged, while the to optimize dot areas. In the Neugebauer model, sixteen sixteen primaries in the 81-primary model are changed for primaries are always the same for any CMYK values. With different colors. For example, if cyan dot area c ≤ 0.5, cyan the application of the cellular model, sixteen primaries are dot areas to compose primaries are 0 and 0.5, and c is re- determined based on CMYK values. The processes are scaled to cc'=⋅2 (see Figure 2-a); if cyan dot area c > 0.5, described in following sections. cyan dot areas to compose primaries are 0.5 and 1, and c is re-scaled to cc'(.)=⋅205 − (see Figure 2-b). 3.1 The Determination of Dot Areas When equation (2.6) is used to calculate dot areas for 3.3 The Determination of Yule-Nielsen Factor n each wavelength, dot area becomes a function of wavelength. The Yule-Nielsen factor n can be obtained by Thus a can be treated as a function of λ, ∆ i minimizing the average color difference, such as E*L*a*b* or 16 1 /n 1 ∆E* , between the measured colorimetric values and the R(λλλ)/n = a ()() R (3.1) 94 CMYK ∑ ii corresponding predictive values from a set of colors which =1 i are almost uniformly distributed in the whole color gamut. The wavelength independent dot area coverage ai is then obtained by different techniques. The simplest technique is 4. Experimental Results λ by averaging ai( ) in the visual spectrum, or by using a wide-band color mixing model. This kind of algorithm does An X-Rite 938 spectrophotometer was used to measure not give very good result. Since the human vision spectral reflectances and CIE L*a*b* values. The measuring sensitivities in different wavelengths are different, errors at geometry is 0/45. The validation of the predictive model was different wavelengths should not be treated evenly. Another tested on an Encad Novojet Pro wide-format ink jet color technique to optimize a is by minimizing the color differ- i printer. ence between the measured colorimetric value and the The 4 by 12 cyan, magenta, yellow, and black primary corresponding predictive colorimetric value in a uniform ramps (CMYK digital values from 0 to 255) were printed color space, such as CIE L*a*b* color space. This and the spectral reflectances were measured. The dot areas algorithm gives good results but takes a lot of computation. were calculated by equations (3.1) and (3.2). Four one- A new technique is to calculate a (λ) in the visual i dimensional look-up-tables for looking the CMYK dot areas spectrum (we used wavelengths from 400nm to 700nm) by from CMYK digital counts were obtained by linear equation (3.1), and a is then optimized by equation i interpolation.