METHOD FOR CHARACTERIZING A PRINTER

Abstract

A standard printer characterization target is printed with a reference printer and measured. From the measured data, a model is derived that describes for each ink the behavior of each individual ink in combination with different combinations of the other inks. In order to reduce the complexity for obtaining a printer model of a second printer, a second target is printed that includes only a subset of the color patches in the standard printer characterization target. With the model and interpolation, it is possible to reconstruct colorimetric data for missing color patches in the reduced target.

Claims

1. A method for predicting CIE tristimulus values of a color printed with inks on a second printer, the method comprising the steps of: printing with a first printer on a first substrate a first characterization target including a first set of color patches and measuring the CIE tristimulus values of the first set of color patches to obtain a first set of measurements; and calculating from the first set of measurements a complete tone value model of the first printer, the complete tone value model including: for each primary ink of the first printer, a tone value function for the CIE tristimulus value that corresponds with a dominant absorption of the primary ink of the first printer; for each primary ink in combination with a background including remaining inks or paper white in the first characterization target, a set of three scale factors, one for each of the CIE tristimulus values, that relate the tone value function that corresponds with the dominant absorption of the primary ink with the tone value function of the primary ink printed in combination with the background; printing with the second printer on a second substrate a second characterization target including a second set of color patches that includes color patches of the primary inks of the second printer; measuring the CIE tristimulus values of the second set of color patches; calculating from the measurements of the second set of color patches for each primary ink a tone value function that describes a relationship between a tone value and the CIE tristimulus value that corresponds with the dominant absorption; calculating for a color printed with the primary inks of the second printer from the tone value functions in combination with the set of three scale factors in a look up table: a first set of CIE tristimulus values by using a tone value curve for a first primary ink on a background of a combination with the remaining inks; a second set of CIE tristimulus values by using a tone value curve for a second primary ink on a background of a combination with the remaining inks; a third set of CIE tristimulus values by using the tone value curve for a third primary ink on a background of a combination with the remaining inks; and a fourth set of CIE tristimulus values by using a tone value curve for a fourth primary ink on a background of a combination with the remaining inks; and averaging the four sets of three CIE tristimulus values to obtain a representative CIE XYZ set of tristimulus values for a selected combination of the color printed with the primary inks.

2. The method according to claim 1, further comprising the steps of: repeating the steps of calculating and averaging the four sets of three CIE tristimulus values with various combinations of inks to obtain a virtual printer model for the second printer.

3. The method according to claim 2, further comprising the steps of: inverting the virtual printer model to provide an inverted printer model; using the inverted printer model to separate a color image to provide a separated color image; printing the separated color image.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0065] FIG. 1 shows a transmissive layer through which a light beam passes.

[0066] FIG. 2 shows a layer in which light is subject to absorption and backwards scattering.

[0067] FIG. 3 shows a preferred embodiment of a reduced printer characterization target.

[0068] FIG. 4 shows a plot of a tone value curve of a cyan ink on a white paper background.

[0069] FIG. 5 shows a plot of a tone value increase curve of a cyan ink on a white paper background.

[0070] FIG. 6 shows a plot of a tone value increase curve of a cyan ink on a background with 70% magenta and 70% yellow.

[0071] FIG. 7 shows an excerpt of a printer measurement file.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Standard Printer Target

[0072] A printer characterization target contains a number of color patches of which the colorant values are known. An example is the IT8.7/3 characterization target which is a widely accepted standard target in the graphic arts industry. Detailed information on this subject is found on the website of the International Color Consortium (ICC) and by entering the search term IT8.7/3.

[0073] Printer targets are also the subject of ISO standards.

[0074] ISO 12642-1:1996 (R2001) defines an input data file, a measurement procedure and an output data format for use in characterizing any four-color print process. The technical content is identical to ANSI IT8.7/3-1993.

[0075] ISO 12642-2:2006 defines a data set of ink value combinations that are intended to be used to characterize four-color process printing. This data set is not optimized for any printing process or application area but is robust enough for all general applications. The needs of publication, commercial, and package printing with offset, gravure, flexography, and other printing processes have been considered. While it is primarily aimed at process color printing with CMYK inks, it can also be used with any combination of three chromatic colored inks and a dark ink. It is an alternate to the ISO 12642-1 data set where more robust data is required. It forms the basis for the revised 4-color process printing targetIT8.7/4

[0076] Such a printer characterization target can be printed with a specific print process and the resulting color of the patches can be measured and stored as a table in a measurement file. An excerpt of such a printer measurement file is shown in FIG. 7. A single row in this table corresponds with a single color patch in the target. The first column is a row number, the second column a row and column position indicator in the printed target, the next four columns correspond with the cyan, magenta, yellow and black colorant tone values of a color patch, the next three columns with the CIE XYZ color values of the color patch as measured on the target, and the last three columns with corresponding the CIE Lab color values. In the excerpt shown in FIG. 7 the color patches correspond with combinations of 0%, 10%, 20% 40%, 70% 100% magenta with 70% cyan, 20% yellow, 0% black colorants.

[0077] In general, a printer target comprises color patches that correspond with different amounts of a first ink of an ink set, in combination with background colors that are combinations of amounts of the remaining inks in the ink set. Such a printer target is designated as a complete printer target, and the measurement file that corresponds with it a complete measurement file.

[0078] The measurement file also comprises in its header additional meta-data such as information on the measurement geometry, the standard illuminant (usually D50) that was used, the equipment that was used for measuring the patches, the print conditions such as the raster frequency, the inks that were used, the paper etc.

[0079] A complete measurement file contains all the necessary data to calculate the color gamut of the print process and a forward printer model.

Reduced Printer Target

[0080] According to a preferred embodiment of the current invention, a reduced printer characterization is used for obtaining a printer model.

[0081] FIG. 3 shows an example of such a reduced target. In this case the reduced characterization target comprises the following colorant combinations: [0082] the primary colorants (C, M, Y, K) in different amounts (300); [0083] the secondary colorant combinations (CM, CY, MY, CK, MK, YK) in different amounts (301); [0084] the colorant combinations (CMY) in different amounts (302) that correspond with neutral or near neutral colors; [0085] a set of colorant combinations that represent the maximum tone value sum (TVS) boundary (303). In this specific example, the maximum tone value sum is 220%.

[0086] The reduced target should contain all edge points of the colorant domain, as they are necessary input to use the tone value model. Preferably also a step wedge is included of the 1-ink sub processes to model tone value of the different colorants.

[0087] The colors of a reduced printer characterization file can be measured and be stored in a reduced measurement file.

Tone Value Increase

[0088] As an example of a tone value model we now describe a model that characterizes the tone value increase of a 1-ink subprocess. The concept of tone value increase is mainly used in conventional printing processes as discussed in ISO 12647/2 till ISO 12647/6.

[0089] In remaining part of this text, we will use the symbols c, m, y and k to refer to a tone value of tints printed with respectively cyan, magenta, yellow and black inks.

[0090] According to the preferred embodiment of the current invention, tone value is not modeled by means of the Murray Davies equation which uses measured densities, but by using colorimetric data such as for example XYZ colorimetric data or spectral data. This is explained.

[0091] Consider a halftone tint printed with black ink. The digital tone value is denoted as k, whereas the measured tone value is denoted as k. The CIE Y value Y.sub.k of such a tint can be predicted as an additive mixture of a portion (1k) white paper having a CIE Y value equal to Y.sub.0 and a portion k of solid ink having a CIE Y value equal to Y.sub.1:

Y.sub.k=(1k).Math.Y.sub.0+k.Math.Y.sub.1(11)

[0092] Rearranging the above equation leads to an expression for the measured tone value k of the tint:

[00009]$\begin{array}{cc}{k}^{}=\frac{Y_k-Y_0}{Y_1-Y_0}& \left(12\right)\end{array}$

[0093] FIG. 4 shows a plot of a tone value curve in which the measured tone value k is plotted versus the digital tone value k for a series of halftone tints having tone values ranging from 0% to 100%.

[0094] In the above example the CIE Y values were used as a basis for calculating a tone value curve. In general the CIE XYZ primary should be selected that exhibits the widest contrast for a specific ink. For example, if a tone value curve is to be calculated for a cyan ink, preferably the CIE X primary will be used for that purpose. Similarly for a magenta ink the CIE Y primary will be used and for a yellow ink the CIE Z primary. More generally: for the purpose of measuring tone value, the CIE XYZ tristimulus is used that corresponds with the dominant absorption of an ink.

Mathematical Modeling of a Tone Value Curve

[0095] To enable smooth interpolation, it is advantageous to mathematically model a tone value curve. Two possible solutions for this purpose are the cubic spline interpolation model and polynomial modeling. The inventors have found that a fourth degree polynomial was perfectly suitable for modeling the tone value curves if the following additional constraints are imposed: [0096] the polynomial is required to pass through the begin (0%,0%) and end (100%,100%) points of the curve; and [0097] nowhere in the range from 0% to 100% digital tone value should the derivative of the tone value curve have a value lower than 0, because this would make the tone value curve non-monotonous. [0098] nowhere in the range from 0% to 100% digital tone value should the second derivative of the tone value curve have a value higher than 0 (this latter constraint is optional).
A tone value function that maps the digital tone value a to a measured tone value a is denoted as:

a=(a)(13)

[0099] In some instances it is advantageous not to model the measured tone value a as a function of a digital tone value a, but rather the tone value increase itself. If the tone value increase for a given a tone value is denoted as g(a), then the following expression is easily devised:

g(a)=(a)a(14)

[0100] The tone value increase can be graphically represented by means of a curve such as the one shown in FIG. 5.

Use of a Tone Value Curve to Predict CIE XYZ Tristimulus Values

[0101] If a tone value curve of a cyan ink was derived from a study of the CIE X values, then it is straightforward to calculate the CIE X tristimulus value for a given digital tone value c. The calculation involves two steps: first the digital tone value c is transformed into a measured tone value c using formula (13), and next the measured tone value c is transformed into a CIE X tristimulus value using the equation (11). This leads to the following expression in which fX.sub.c refers to the tone value function that is based on measurements of the CIE X tristimulus values of a cyan ink.

X.sub.c=(1X.sub.c(c)).Math.X.sub.0+X.sub.c(c).Math.X.sub.1(15)

[0102] So it is sufficient to know X.sub.0, X.sub.1 and the tone value curve fX.sub.c(c) to calculate the CIE X.sub.c tristimulus value for any digital tone value c.

[0103] A question that arises is if the tone value curve fX.sub.c(c) can also be used for the purpose of calculating the two remaining CIE tristimulus values Y and Z.

[0104] In this context, a first approach could be to simply assume that the non-dominant absorptions (as measured by the CIE Y or Z tristimulus values for a cyan ink, for example) would predict exactly the same increase as a function of digital tone value, and hence that the resulting tone value curves would also be identical. If this were true, the CIE Y and Z tristimulus values for a tint c could be simply calculated as:

Y.sub.c=(1X.sub.c(c)).Math.Y.sub.0+X.sub.c(c).Math.Y.sub.1(16)

and

Z.sub.c=(1X.sub.c(c)).Math.Z.sub.0+X.sub.c(c).Math.Z.sub.1(17)

[0105] Experiments, however, have demonstrated that this approach, simple and elegant as it may seem, does not yield values that sufficiently correspond with the measured observations.

[0106] For this reason, an alternative approach was developed. According to this new approach, it is assumed that the tone value curves fX.sub.c(c), fY.sub.c(c) and fZ.sub.c(c) for the different tristimulus values still have the same overall shape, but that absolute amounts of tone value increases are allowed to be different. The effect can be modeled by introducing a scaling factor for scaling the two tone value increase curves corresponding with the two non-dominant absorptions relative to the tone value increase curve corresponding to the dominant absorption.

[0107] In the above example for the case of a cyan ink, the scaling factors for the tone value increase curves for predicting the CIE Y and Z tristimulus values could be called kY.sub.c and kZ.sub.c. This leads to the following expressions (for the case of a cyan ink):

gX.sub.c(c)=X.sub.c(c)c(18)

gY.sub.c(c)=kY.sub.c.Math.gX.sub.c(c)(19)

gZ.sub.c(c)=kZ.sub.c.Math.gX.sub.c(c)(20)

Or also:

X.sub.c(c)=kX.sub.c.Math.X.sub.c(c)(21)

wherein kX.sub.c=1 and X.sub.c(c) based on measuring CIE X;

Y.sub.c(c)=kY.sub.c.Math.(X.sub.c(c)c)+c(22)

Z.sub.c(c)=kZ.sub.c.Math.(X.sub.c(c)c)+c(23)

[0108] The value of kY.sub.c is mathematically determined using a regression technique that minimizes the least mean square error over the complete range of digital tone values in the target between the Y values as measured and the values as predicted by the above formula. In a similar way a value for kZ.sub.c is determined.

[0109] In summary, iffor the case of a cyan ink: [0110] a tone value function fX.sub.c(c) is made available by measuring the CIE X tristimulus values corresponding with the dominant absorption, and; [0111] if two scale factors are kY.sub.c and kZ.sub.c are calculated corresponding with the non-dominant absorptions, then the CIE XYZ tristimulus values can be predicted as follows:

X.sub.c=(1X.sub.c(c)).Math.X.sub.0+X.sub.c(c).Math.X.sub.1(24)

Y.sub.c=(1Y.sub.c(c)).Math.Y.sub.0+Y.sub.c(c).Math.Y.sub.1(25)

Z.sub.c=(1Z.sub.c(c)).Math.Z.sub.0+Z.sub.c(c).Math.Z.sub.1(26)

wherein:

X.sub.c(c)=kX.sub.c.Math.(X.sub.c(c)c)+c(27)

wherein kX.sub.c=1.0

Y.sub.c(c)=kY.sub.c.Math.(X.sub.c(c)c)+c(28)

Z.sub.c(c)=kZ.sub.c.Math.(X.sub.c(c)c)+c(29)

[0112] Similar models can be derived for the case of magenta, yellow and black inks. For every ink a tone value function fX.sub.c(c), fY.sub.m(m), fZ.sub.y(y), fY.sub.k(k) is obtained from measurements and a set of three scale factors of which one is equal to 1.0: [0113] for the cyan ink: the scale factors are: [kX.sub.c, kY.sub.c, kZ.sub.c] in which kX.sub.c=1.0; [0114] for the magenta ink: the scale factors are: [kX.sub.m, kY.sub.m, kZ.sub.m] in which kY.sub.m=1.0; [0115] for the yellow ink: the scale factors are: [kX.sub.y, kY.sub.y, kZ.sub.y] in which kZ.sub.y=1.0; [0116] for the black ink: the scale factors are: [kX.sub.k, kY.sub.k, kZ.sub.k] in which kY.sub.k=1.0;

Use of a Tone Value Increase Curve to Predict CIE Tristimulus Values of a Colorant in the Presence of Other Inks.

[0117] In the previous paragraphs the study of tone value increase was implicitly limited to the case of printing a single colorant on white paper. The inventors have investigated how the tone value increase of a colorant behaves when it is printed on a background wherein the other colorants have a non-zero value. An example is the tone value increase of a cyan colorant when it is printed on a background of, for example, magenta (m=70%) and yellow (y=70%) inks and no black ink (k=0%).

[0118] FIG. 5 shows a tone value increase curve that was measured of cyan ink on white paper (C=0; Y=0; K=0) whereas FIG. 6 shows a tone value increase curve of cyan printed on a combination of magenta ink (m=70%), yellow ink (y=70%) and no black ink (k=0%).

[0119] It appears that the shape of both tone value increase curves is very similar, but that the actual tone value increase of the cyan ink on a background of magenta and yellow inks is lower, more precisely and in this particular case, that it is approximately 75% as that for the cyan ink printed on white paper.

[0120] A similar observation was made when the tone value increase of the cyan was analyzed in the presence of other backgrounds comprising other combinations of amounts of magenta, yellow and black inks.

[0121] More in general, the inventors have found by analyzing tone value increase curves for different colorants on different backgrounds and using different printing processes, that the following was always approximately and sufficiently true: [0122] The overall shape of the tone value increase curve of a specific colorant in a specific process was constant, independent of the background color; [0123] The tone value increases themselves, however, were not always the same, in the sense that the tone value increases are usually lower when a background contains more ink.

[0124] This has lead the inventors to come up with a generalized tone value increase model in which the tone value increase curves of the pure inks serve as reference tone value increase curves and in which the other tone value increase curves (for cases in which colorant values of the background are not zero) are derived from the reference tone value increase curve by means of scaling factors k.

[0125] The background colors for a specific ink are preferably all (available) combinations of remaining three inks that are present in the target.

[0126] For example, if for a cyan ink the tone value increase curves of the pure ink on a white background are:

gX.sub.c(c)(30)

gY.sub.c(c)(31)

gZ.sub.c(c)(32)

[0127] Then for a combination of a set of magenta (m), yellow (y) and black digital tone values that make up a background, scaling factors kX.sub.c.sup.myk, kY.sub.c.sup.myk and kZ.sub.c.sup.myk can be determined that relate the tone value increase curves as a function of the curves for the white background:

gX.sub.c.sup.myk(c)=kX.sub.c.sup.myk.Math.gX.sub.c(c)(33)

gY.sub.c.sup.myk(c)=kY.sub.c.sup.myk.Math.gY.sub.c(c)(34)

gZ.sub.c.sup.myk(c)=kZ.sub.c.sup.myk.Math.gZ.sub.c(c)(35)

[0128] The values of the scale factors kX.sub.c.sup.myk, kY.sub.c.sup.myk and kZ.sub.c.sup.myk are determined using a regression technique that minimizes the least mean square error between the CIE X, Y and Z values that are measured in the target, and the ones that are predicted by means of the following tristimulus prediction model:

X.sub.c=(1X.sub.c.sup.myk(c)).Math.X.sub.0myk+X.sub.c.sup.myk(c).Math.X.sub.1myk(36)

Y.sub.c=(1Y.sub.c.sup.myk(c)).Math.Y.sub.0myk+Y.sub.c.sup.myk(c).Math.Y.sub.1myk(37)

Z.sub.c=(1Z.sub.c.sup.myk(c)).Math.Z.sub.0myk+Z.sub.c.sup.myk(c).Math.Z.sub.1myk(38)

wherein:

X.sub.c.sup.myk(c)=kX.sub.c.sup.myk.Math.(X.sub.c(c)c)+c(39)

Y.sub.c.sup.myk(c)=kY.sub.c.sup.myk.Math.Y.sub.c(c)c)+c(40)

Z.sub.c.sup.myk(c)=kZ.sub.c.sup.myk.Math.(Z.sub.c(c)c)+c(41)

[0129] The expressions 36-41 are very similar in nature to the expressions 24-29. Both take the tone value function as based on measurements of the dominant absorption of the pure cyan ink, and use a scale factor to derive other tone value functions for predicting CIE tristimulus values.

[0130] The difference however is that the tristimulus values of the paperwite (X.sub.0, Y.sub.0 and Z.sub.0) in expressions 24-26 here are replaced by the tristimulus of the background color (X.sub.0myk, Y.sub.0myk and Z.sub.0myk). Similarly, the tristimulus values of the (pure) solid ink (X.sub.1, Y.sub.1 and Z.sub.1) have been replaced by the tristimulus values (X.sub.1myk, Y.sub.1myk, Z.sub.1myk) of the solid ink as it is printed on this background.

[0131] Obviously, the exact same approach can be used for obtaining scale factors of tone value curves for predicting: [0132] tone value functions for a magenta ink in the presence of a background comprising cyan, yellow and black inks: [0133] tone value functions for a yellow ink in the presence of a background comprising cyan, magenta and black inks: [0134] tone value functions for a black ink in the presence of a background comprising cyan, magenta and yellow inks.

Tone Value Model

[0135] According to a preferred embodiment a tone value model comprises the following information: [0136] for each ink (c, m, y, k) there is a tone value function ((fX.sub.c(c), fY.sub.m(m), fZ.sub.y(y), fY.sub.k(k)), preferably a fourth degree polynomial, that describes the tone value for the ink, based on measurements of the tristimulus value (X, Y or Z) that corresponds to the dominant absorption of the ink; [0137] there is also a look up table that comprises for each ink in combination with each background (myk, cyk, cmk, cmy combinations including paper white) a set of three scale factors (kX.sub.c.sup.myk, kY.sub.c.sup.myk, kZ.sub.c.sup.myk, kX.sub.m.sup.cyk, kY.sub.m.sup.cyk, kZ.sub.m.sup.cyk, kX.sub.y.sup.cmk, kY.sub.y.sup.cmk, kZ.sub.y.sup.cmk, kX.sub.k.sup.cmy, kY.sub.k.sup.cmy, kZ.sub.k.sup.cmy) for deriving functions corresponding with the CIE XYZ tristimulus values.

Using the Reduced Printer Characterization Target

[0138] It is now disclosed how a reduced printer target can be used in combination with the tone value increase model of a first printer to reconstruct a virtual complete printer measurement file for a second printer, without having to print, measure and process the complete printer characterization target for said second printer.

[0139] The goal is to obtain a complete virtual measurement file for the second printer.

[0140] 1) obtain the tone value model using the complete measurement file of the first printer.

[0141] 2) obtain the tone value increase curves for each one of the primary inks from the reduced measurement file that corresponds with the second printer using the method explained in paragraphs [0097] to [0099].

[0142] 3) obtain for a combination of cyan, magenta, yellow and black inks a set of CIE XYZ tristimulus values. This is achieved by consulting the tone value model and: [0143] using the tone value curve for the cyan ink on a background of magenta, yellow and black inks, yielding a first set of CIE XYZ values; [0144] using the tone value curve for the magenta ink on a background of cyan, yellow and black inks, yielding a second set of CIE XYZ values; [0145] using the tone value curve for the yellow ink on a background of cyan, magenta and black inks, yielding a third set of CIE XYZ values; [0146] using the tone value curve for the black ink on a background of cyan, magenta and yellow inks, yielding a fourth set of CIE XYZ values.

[0147] Each of the four sets of tristimulus values is obtained by using the method that was described in the paragraphs [0102] to [0114], whereby the tone value curves are used that were obtained in step 2 and whereby the scaling factors are used that were stored in the tone value model of the first printer.

[0148] 4) average the four sets of three CIE tristimulus value obtained in the previous step to obtain a representative CIE XYZ set of tristimulus value for the selected combination of cyan, magenta, yellow and black inks. The averaged CIE XYZ values are stored in the virtual complete printer measurement file for a second printer.

[0149] The steps 3) to 4) are repeated until the virtual complete printer measurement file for the second printer is completely populated.

[0150] It is obvious that we need to populate first the 1-dimensional edge processes of the colorant cube before we can calculate the color values of the 2-dimensional edge processes that we need to populate the 2-dimensional edge boundaries, before we can calculate the color values of 3-dimensional edge processes, etc. till all edge processes are defined. In a last step the internal colorant points will be predicted.

Dealing with the Tone Value Sum (TVS)

[0151] Until now in this text, it was implicitly assumed that the range of digital tone values was always in a range from 0% to 100%. This is rarely the case, because when excess ink is printed on a location, physical and optical effects start deteriorating the quality of the print. Depending on the characteristics of the printing process, the tone value sum (TVS) i.e. the sum of the digital tone values of the c, m, y and k inks should not exceed a value in the range between 180% and 340%.

[0152] This has implications in the context of the current invention.

[0153] In the context of the tone value model, we are applying the tone value curve between 0% and 100% of ink. So we need a method of extrapolation to obtain the color values from the edge points that lie outside the ink limited colorant domain. We give an example of how this can be done. Let us take an ink limitation of 240%. We can predict the 300% edge point (100% c, 100% m, 100% y, 0% k) from the edge points (0% c, 100% m, 100% y, 0% k) and (40% c, 100% m, 100% y, 0% k) by rearranging equations (36)-(38). Alternatively, we can predict the same edge point from edge points (100% c, 0% m, 100% y, 0% k) and (100% c, 40% m, 100% y, 0% k). We can also predict the same edge point from the points (100% c, 100% m, 0% y, 0% k) and (100% c, 100% m, 40% y, 0% k). The final color values for the missing edge point can be obtained by averaging these three set of values. Likewise other missing edge points of the full colorant domain can be obtained.

[0154] In the context of a model that allows for interpolation along directions in the ink limiting planes, it is not necessary to use an extrapolation technique, because we have no need of the edge points outside the ink limitations to do the model interpolation. Alternatively we can supplement our tone value model with a means of interpolation in the ink limiting plane(s). This can e.g. be a linear interpolation between the edge points in the ink limitation plane. Also this interpolation can be part of the tone value model.

Other Embodiments

[0155] The current approach to predict colors is based on 1-dimensional interpolation schemes. However, a similar approach can be developed based on multidimensional interpolation, e.g. by making use of polynomials or typical ink mixing models such as Neugebauer, Lambert-Beer, Kubelka-Munk.

[0156] In the description of the preferred embodiment, tone value functions are used that predict CIE XYZ tristimulus values as a function of tone value. As mentioned before, the method works just as well for tristimulus values that are based on filters that are a linear transformation of the CIE XYZ color matching functions. In fact, the invented method can also be used for multispectral color values that are non-linear functions of the CIE XYZ color matching functions, such as for example narrow band densitometric multispectral color values.

[0157] The method is also applicable if not tristimulus values are used, but a set of N (N>1) spectral values instead. In the latter case, scaling factors are calculated and used for each spectral component as opposed for each tristimulus value.

[0158] In the description of the preferred embodiments the term ink is used, but the method works also in combination with any colorant such as toner.