METHOD OF TRANSFORMING COLOR SPACES, A PRINTING MACHINE AND A MEMORY MEDIUM CONTAINING CODE FOR PERFORMING THE METHOD

Abstract

A method transforms color values of a first device-dependent color space into the color values of a second device-dependent color space so that the visual impression of the color represented in the two-color spaces is essentially identical. The transformation of the color values combines at least one absolute rendering intent and at least one relative rendering intent.

Claims

1. A method of transforming color values of a first device-dependent color space into color values of a second device-dependent color space so that a visual impression of colors represented in both of the first and second device-dependent color spaces is generally identical, which comprises the steps of: performing a transformation of the color values of the first device-dependent color space into the color values of the second device-dependent color space using at least one rendering intent which is used to describe the color values, wherein the transformation of the color values of the first device-dependent color space into the color values of the second device-dependent color space combines at least one absolute rendering intent which describes absolute color values with at least one relative rendering intent which describes relative color values, and wherein the transformation of the color values of the first device-dependent color space into the color values of the second device-dependent color space is made via the color values of an intermediate color space.

2. The method according to in claim 1, wherein the at least one absolute rendering intent and the at least one relative rendering intent are colorimetric rendering intents.

3. The method according to claim 1, wherein the intermediate color space is a device-independent color space.

4. The method according to claim 1, wherein the color values of the intermediate color space correspond to a D50 2° standard observer.

5. The method according to claim 1, which further comprises transforming the color values of the first device-dependent color space into the color values of the intermediate color space and the color values of the intermediate color space are transformed into the color values of the second device-dependent color space.

6. The method according to claim 5, wherein the transformation of the color values of the first device-dependent color space into the color values of the intermediate color space combines the at least one absolute rendering intent with the at least one relative rendering intent.

7. The method according to claim 5, wherein the transformation of the color values of the intermediate color space into the color values of the second device-dependent color space combines the at least one absolute rendering intent with the at least one relative rendering intent.

8. The method according to claim 1, which further comprises transforming the color values of the first device-dependent color space into the color values of the intermediate color space via at least one profile which has at least one conversion table, wherein the color values of the intermediate color space are transformed into the color values of the second device-dependent color space via at least one profile which contains at least one conversion table, and wherein the at least one absolute rendering intent and the at least one relative rendering intent use a same conversion table.

9. The method according to claim 8, wherein the at least one profile is at least one International Color Consortium (ICC) profile.

10. The method according to claim 1, wherein a device to which the first device-dependent color spaces refers is a different device than a device to which the second device-dependent color space refers.

11. The method according to claim 1, which further comprises performing the transformation of the color values of the first device-dependent color space into the color values of the second device-dependent color space during a printing operation in which a digital print image is printed onto a printing substrate by a printing machine and wherein the first device-dependent color space is the color space of a screen and the second device-dependent color space is the color space of the printing machine.

12. The method according to claim 11, which further comprises selecting the printing operation from the group consisting of: relief printing, flat-bed printing, gravure printing, offset printing, inkjet printing, flexographic printing, screen printing, and rotogravure.

13. The method according to claim 11, wherein the transformation of the color values of the first device-dependent color space into the color values of the second device-dependent color space does not include any simulation of color values of the printing substrate being an unprinted printing substrate.

14. The method according to claim 13, wherein the color values of the unprinted printing substrate are paper white.

15. The method according to claim 1, wherein the transformation of the color values of the first device-dependent color space into the color values of the second device-dependent color space transforms a lightest point of the first device-dependent color space into a lightest point of the second device-dependent color space on an output without print dots.

16. The method according to claim 1, wherein the at least one rendering intent contains black point compensation.

17. A printing machine configured for implementing the method according to claim 1.

18. A non-transitory memory medium storing computer executable instructions which when processed by a processor of a printing machine perform the method according to claim 1.

Description

BRIEF DESCRIPTION OF THE FIGURES

[0052] FIG. 1 is a schematic illustration of the color transformation principle for a printing process adaptation in accordance with the prior art; and

[0053] FIG. 2 is a flow chart showing a method of the invention.

DETAILED DESCRIPTION OF THE INVENTION

[0054] In a preferred embodiment of the invention, original artwork data for a substrate with a white point W.sub.1 are to be output on a substrate which has a very different white point W.sub.2. ICC profiles that describe the processes are available for both processes. The two colorimetric rendering intents of “absolute rendering intent” and “relative rendering intent” are known.

[0055] Absolute rendering intent: exact reproduction of the measured color values (XYZ, L*a*b*) wherever the destination process permits.

[00001]$\begin{array}{cc}\left(\begin{array}{c}{x}_2\\ y_2\\ z_2\end{array}\right)=\left(\begin{array}{c}{x}_1\\ y_1\\ z_1\end{array}\right)& \left(4\right)\end{array}$

[0056] The paper white of the input process is simulated, i.e. in some cases (destination substrate lighter than initial paper or of different color), white areas will be printed, too. Lighter colors are cut off if the destination paper is too dark.

[0057] Relative rendering intent: reproduction of the measured color values (XYZ, L*a*b*) in relation to the white point of the respective process.

[0058] Color coordinates (x,y,z) are scaled by the ratio of the coordinates of the white points.

[00002]$\begin{array}{cc}\left(\begin{array}{c}{x}_2\\ y_2\\ z_2\end{array}\right)=\left(\begin{array}{c}{x}_1.Math.\frac{x_{\mathrm{white},2}}{x_{\mathrm{white},1}}\\ y_1.Math.\frac{y_{\mathrm{white},2}}{y_{\mathrm{white},1}}\\ z_1.Math.\frac{z_{\mathrm{white},2}}{z_{\mathrm{white},1}}\end{array}\right)& \left(5\right)\end{array}$

[0059] The color coordinates (x,y,z) may, for instance, be XYZ or a coordinate system K that is adapted to the sensitivity of the human eye. In this case, the color coordinates of input value and paper white need to be transformed into the coordinate system K as a first step and the scaled coordinates need to be transformed back. Paper white will remain unprinted in the destination process in any case. If the destination paper is dark, color hues will be darkened in a corresponding way.

[0060] What is desired, however, is often a behavior where the hues may be accurately reproduced when an area is intensely printed on (for instance a solid cyan area or a red composed of 100% magenta and 100% yellow) as it is the case with the absolute rendering intent. Towards paper white, the behavior is to correspond to the relative rendering intent where paper remains unprinted and shadow detail is preserved even in the lights.

[0061] Without ICC-based color management, such a behavior is attainable on an offset printing press by using the plates of the input paper, feeding the paper with the different white point W.sub.2, and trying to set solid tone and dot gain of the standard for the original paper as well as possible. This process is only possible as long as the original plates may be used. It cannot work if the destination process uses more or fewer process colors than the source process, if spot colors in the source process are simulated by process colors in the destination process, if the two processes differ not just in terms of paper white but also in terms of process characteristics (location of the solid tones, overprinting behavior and the like) or if the destination process is a different printing technology.

[0062] The approach of the invention uses a method

(x.sub.2,y.sub.2,z.sub.2)=F((x.sub.1,y.sub.1,z.sub.1))  (6)

of transforming the color coordinates that is a combination of (4) and (5) and exhibits the desired behavior. The color coordinates x, y, and z may again be XYZ or a coordinate system adapted to the sensitivity of the human eye.

[0063] Two limit cases for the transformation are possible:

i) relative behavior at paper white:

[00003]$\begin{array}{cc}F\left(\left(x_{white,1},y_{white,1},z_{white,1}\right)\right)=\left(x_{whi\mathrm{te},2},y_{whi\mathrm{te},2},z_{whi\mathrm{te},2}\right)=\left(\begin{array}{c}{x}_{white,1}.Math.{\left(\frac{x_{whi\mathrm{te},2}}{x_{wh\mathrm{ite},1}}\right)}^1\\ y_{white,1}.Math.{\left(\frac{y_{whi\mathrm{te},2}}{y_{wh\mathrm{ite},1}}\right)}^1\\ z_{white,1}.Math.{\left(\frac{z_{whi\mathrm{te},2}}{z_{wh\mathrm{ite},1}}\right)}^1\end{array}\right)& \left(7\right)\end{array}$

ii) absolute behavior if at least one of the color coordinates is small:

[00004]$\begin{array}{cc}{\lim }_{x_1\mapsto 0}F\left(\left(x_1,y_1,z_1\right)\right)=\left(x_1,y_1,z_1\right)=\left(\begin{array}{c}{x}_1.Math.{\left(\frac{x_{\mathrm{white},2}}{x_{\mathrm{white},1}}\right)}^0\\ y_1.Math.{\left(\frac{y_{\mathrm{white},2}}{y_{\mathrm{white},1}}\right)}^0\\ z_1.Math.{\left(\frac{z_{\mathrm{white},2}}{z_{\mathrm{white},1}}\right)}^0\end{array}\right)& \left(8\right)\end{array}$$\begin{array}{cc}\underset{y_1\mapsto 0}{\lim }\mathrm{und}\underset{z_1\mapsto 0}{\lim }:\mathrm{in}\mathrm{an}\mathrm{analogous}\mathrm{way}& \left(9\right)\end{array}$

[0064] Combining equations (7) and (8), the transformation may be expressed as follows:

[00005]$\begin{array}{cc}F\left(\left(x_1,y_1,z_1\right)\right)=\left(\begin{array}{c}{x}_1.Math.{\left(\frac{x_{\mathrm{white},2}}{x_{\mathrm{white},1}}\right)}^{{\alpha }_x}\\ y_1.Math.{\left(\frac{y_{\mathrm{white},2}}{y_{\mathrm{white},1}}\right)}^{{\alpha }_y}\\ z_1.Math.{\left(\frac{z_{\mathrm{white},2}}{z_{\mathrm{white},1}}\right)}^{{\alpha }_z}\end{array}\right)& \left(10\right)\end{array}$

[0065] The exponents α are a function of the color location, and for paper white α=1 needs to apply and α0 needs to apply when a color coordinate tends to zero. In theory, exponents α.sub.x, α.sub.y, α.sub.z may be different. In the following, the first approach is limited to a case in which the exponents are identical for all components, α.sub.x=α.sub.y=α.sub.z=α. Thus, the transformation equation (10) becomes

[00006]$\begin{array}{cc}F\left(\left(x_1,y_1,z_1\right)\right)=\left(\begin{array}{c}{x}_1.Math.{\left(\frac{x_{\mathrm{white},2}}{x_{\mathrm{white},1}}\right)}^{\alpha }\\ y_1.Math.{\left(\frac{y_{\mathrm{white},2}}{y_{\mathrm{white},1}}\right)}^{\alpha }\\ z_1.Math.{\left(\frac{z_{\mathrm{white},2}}{z_{\mathrm{white},1}}\right)}^{\alpha }\end{array}\right)& \left(11\right)\end{array}$

[0066] Now the challenge is to determine the exponent α as a function of the color coordinates x.sub.1, y.sub.1, z.sub.1 and x.sub.white,1, y.sub.white,1, z.sub.white,1. In addition to the conditions resulting from (7) and (9) further conditions need to be fulfilled:

α=1 for xyz.sub.1=xyz.sub.white,1
lim.sub.x.sub.0α=0 and lim.sub.y.sub.0α=0 and lim.sub.z.sub.0α=0
α needs to be between 0 and 1 for all values of xyz between 0 and xyzw.sub.hite.
α needs to be strictly increasing for every component of xyz, and
α needs to be continuous.

[0067] For the transformations W.sub.1W.sub.2 and W.sub.2W.sub.1 to behave in a precisely inverted way relative to one another, it needs to be irrelevant whether α is determined for the original coordinates or for the transformed coordinates. This means α(xyz.sub.1,xyz.sub.white,1)=α(xyz.sub.2,xyz.sub.white,2).

[0068] Here the abbreviation xyz=(x,y,z) was introduced for the entire coordinate vector.

[0069] To be able to guarantee condition 6 mentioned above, i.e. independence of the direction of the transformation, an intermediate color space that is independent of the white point and has the coordinates xyz.sub.m=(x.sub.m,y.sub.m,z.sub.m) is introduced as an initial step. If every input white point W.sub.1 is at least formally mapped to the intermediate color space and from there to the coordinates of the destination white point W.sub.2, independence of the direction is definitely ensured if the exponent α is determined in this intermediate color space.

[0070] For reasons of simplicity, the following paragraph will initially only deal with one component x of the color space and the mapping into the intermediate space in accordance with x=x.sub.1 or x.sub.2 is described as

x=ƒ(x.sub.m,x.sub.white):=ƒ(x.sub.m)  (12)

[0071] written without the exponent α. For the function ƒ, the following conditions need to apply:

[00007]$$\begin{gathered} \begin{array}{cc}1.f\left(0\right)=0 \\ 2.\frac{df}{d{x}_m}{|}_{x_m=0}=1 \\ 3.f\left(1\right)=x_{white} \\ 4.\frac{df}{d{x}_m}>0\forall 0\le x_m\le 1& \end{array} \end{gathered}$$

[0072] To make sure that these conditions apply, the following approach is used for a differential equation to determine ƒ:

[00008]$\begin{array}{cc}\frac{df\left(x_m\right)}{d{x}_m}=\frac{x_{white}-f\left(x_m\right)}{x_{white}}.Math.\frac{1}{\left(1-x_m\right)}& \left(13\right)\end{array}$

[0073] The general solution to this equation is

[00009]$\begin{array}{cc}f\left(x_m\right)=c_1.Math.{\left(x_m-1\right)}^{\frac{1}{x_{\mathrm{white}}}}+x_{white}& \left(14\right)\end{array}$

[0074] Substituting the boundary conditions results in

[00010]$\begin{array}{cc}x=f\left(x_m\right)=x_{white}.Math.\left(1-{\left(1-x_m\right)}^{\frac{1}{x_{\mathrm{white}}}}\right)& \left(15\right)\end{array}$

[0075] Equation (15) may now be rearranged for the direction from absolute color coordinates on paper white W.sub.1 and W.sub.2, respectively, into the intermediate color space:

[00011]$\begin{array}{cc}{x}_m=1-{\left(1-\frac{x}{x_{\mathrm{white}}}\right)}^{x_{\mathrm{white}}}& \left(16\right)\end{array}$

[0076] Substituting equation (16) for x=x.sub.1 in equation (15) for x=x.sub.2 results in transformation equation from x.sub.1 with x.sub.white,1 to x.sub.2 with x.sub.white,2:

[00012]$\begin{array}{cc}{x}_2=x_{wh\mathrm{ite},2}.Math.\left[1-{\left(1-\frac{x_1}{x_{wh\mathrm{ite},1}}\right)}^{\frac{x_{wh\mathrm{ite},1}}{x_{wh\mathrm{ite},2}}}\right]& \left(17\right)\end{array}$

[0077] In order to be able to write equation (17) in the form

[00013]$x_2=x_1.Math.{\left(\frac{x_{wh\mathrm{ite},2}}{x_{wh\mathrm{ite},1}}\right)}^{\alpha }$

we have

[00014]$\begin{array}{cc}{x}_2=x_1.Math.{\left(\frac{x_{wh\mathrm{ite},2}}{x_{wh\mathrm{ite},1}}\right)}^{\alpha }\mathrm{with}\alpha =\frac{\log \left\{\frac{x_{wh\mathrm{ite},2}}{x_1}\left[1-{\left(1\frac{x_1}{x_{wh\mathrm{ite},1}}\right)}^{\frac{x_{wh\mathrm{ite},1}}{x_{wh\mathrm{ite},2}}}\right]\right\}}{\log \left\{\frac{x_{wh\mathrm{ite},2}}{x_{wh\mathrm{ite},1}}\right\}}& \left(18\right)\end{array}$

[0078] Since the relation between x.sub.1 and x.sub.2 in equation (17) is independent of the direction of the transformation due to the use of the intermediate color space and since equation (18) is just a different way of expressing this relation, it follows that equation (18) is likewise independent of the direction of the transformation.

[0079] Equation (18) may now be applied to the three color space coordinates x, y, and z separately. This results in a relative colorimetric behavior towards white and in an absolute colorimetric behavior towards black. However, this method has two particularities:

[0080] The exponents α differ for x, y, and z, which may cause undesired color shifts.

[0081] As soon as the value of one of the coordinates x, y, or z is significantly different from 0, this coordinate will no longer be transformed in a manner similar to an absolute colorimetric way.

[0082] Thus, a representative coordinate q is introduced as having the following properties:

i) q is transformed like x, y, and z, i.e.

[00015]$\begin{array}{cc}{q}_2=q_1.Math.{\left(\frac{q_{wh\mathrm{ite},2}}{q_{wh\mathrm{ite},1}}\right)}^{\alpha }& \left(19\right)\end{array}$

[0083] ii) q tends towards 0 as soon as at least one of coordinates x, y or z tends towards 0.

[0084] A coordinate that fulfills these conditions is the geometric mean of coordinates x, y and z

[00016]$\begin{array}{cc}q=\sqrt[3]{x.Math.y.Math.z}& \left(20\right)\end{array}$$\begin{array}{cc}{q}_{\mathrm{white}}=\sqrt[3]{x_{white}.Math.y_{white}.Math.z_{white}}& \left(21\right)\end{array}$

etc.

[0085] Thus, a common exponent α for all coordinates may be determined and in summarized form, the final transformation equation is

[00017]$\begin{array}{cc}{x}_2=x_1.Math.{\left(\frac{x_{\mathrm{white},2}}{x_{\mathrm{white},1}}\right)}^{\alpha },& \left(22\right)\end{array}$$y_2=y_1.Math.{\left(\frac{y_{\mathrm{white},2}}{y_{\mathrm{white},1}}\right)}^{\alpha },$$z_2=z_1.Math.{\left(\frac{z_{\mathrm{white},2}}{z_{\mathrm{white},1}}\right)}^{\alpha }$$\begin{array}{cc}\mathrm{with}\alpha =\frac{\log \left\{\frac{q_{\mathrm{white},2}}{q_1}.Math.\left[1-{\left(1-\frac{q_1}{q_{\mathrm{white},1}}\right)}^{\frac{q_{\mathrm{white},1}}{q_{\mathrm{white},2}}}\right]\right\}}{\log \left\{\frac{q_{\mathrm{white},2}}{q_{\mathrm{white},1}}\right\}}& \left(23\right)\end{array}$$\mathrm{and}q=\sqrt[3]{x.Math.y.Math.z}$

[0086] What needs to be factored in in this context is that coordinates xyz do not necessarily correspond to XYZ. If required, an initial step needs to be to apply a linear transformation to the XYZ coordinates such as the Bradford matrix known in principle in the prior art and the inverse Bradford matrix after the transformation. It goes without saying that in this case, the coordinates of the white points need to be dealt with in an analogous way.

[0087] In addition, one must bear in mind that the utilized color coordinates xyz are always to be understood as absolute, i.e. they are not scaled using the coordinates of the white point. Since internally most calculations with ICC profiles mostly use color coordinates in units of the white point, this scaling must be kept in mind. Since it is a simple multiplication and the white points are known at all times, however, this does not result in any limitations to the applicability.

[0088] For a later examination of borderline cases it makes sense to represent equations (22) and (23) in centered color space coordinates, i.e. with color coordinates that have been scaled with the coordinates of the white point.

[00018]$\begin{array}{cc}\tilde{x}=\frac{x}{x_{white}}& \left(24\right)\end{array}$$\tilde{y}=\frac{y}{y_{white}}$$\tilde{z}=\frac{z}{z_{white}}$$\tilde{q}=\frac{q}{q_{white}}$

[0089] Substituting (24) in (22) we get

[00019]$\begin{array}{cc}{\tilde{x}}_2={\tilde{x}}_1.Math.{\left(\frac{x_{wh\mathrm{ite},2}}{x_{wh\mathrm{ite},1}}\right)}^{\alpha -1}& \left(25\right)\end{array}$$\mathrm{and}$$\tilde{y},\tilde{z}\mathrm{in}\mathrm{an}\mathrm{analogous}\mathrm{way}.$

[0090] Substituting (24) in (23), for the exponent α we get

[00020]$\begin{array}{cc}\begin{array}{c}\alpha =\frac{\log \left\{\frac{1}{{\tilde{q}}_1}.Math.\left(\frac{q_{w\mathrm{hite},2}}{q_{w\mathrm{hite},1}}\right).Math.\left[1-{\left(1-{\tilde{q}}_1\right)}^{\frac{q_{w\mathrm{hite},1}}{q_{w\mathrm{hite},2}}}\right]\right\}}{\log \left\{\frac{q_{white2}}{q_{white1}}\right\}}\\ =1+\frac{\log \left\{\frac{1}{{\tilde{q}}_1}.Math.\left[1-{\left(1-{\tilde{q}}_1\right)}^{\frac{q_{w\mathrm{hite},1}}{q_{w\mathrm{hite},2}}}\right]\right\}}{\log \left\{\frac{q_{w\mathrm{hite},2}}{q_{w\mathrm{hite},1}}\right\}}.\end{array}& \left(26\right)\end{array}$

[0091] If an {tilde over (α)} for the centered color space coordinates is defined as

[00021]$\begin{array}{cc}\tilde{\alpha }=\alpha -1=\frac{\log \left\{\frac{1}{{\tilde{q}}_1}.Math.\left[1-{\left(1-{\tilde{q}}_1\right)}^{\frac{q_{w\mathrm{hite},1}}{q_{w\mathrm{hite},2}}}\right]\right\}}{\log \left\{\frac{q_{\mathrm{white},2}}{q_{\mathrm{white},1}}\right\}}& \left(27\right)\end{array}$

equation (25) becomes

[00022]$\begin{array}{cc}{\tilde{x}}_2={\tilde{x}}_1.Math.{\left(\frac{x_{\mathrm{white},2}}{x_{w\mathrm{hite},1}}\right)}^{\tilde{\alpha }}& \left(28\right)\end{array}$$\mathrm{and}\tilde{y},\tilde{z}\mathrm{in}\mathrm{an}\mathrm{analogous}\mathrm{way}.$

[0092] Substituting {tilde over (α)} from equation (27) in (28) and introducing new abbreviations, this may be simplified as

[00023]$\begin{array}{cc}\begin{array}{c}{\tilde{x}}_2={\tilde{x}}_1.Math.\exp \left(\underset{:={\beta }_x}{\underbrace{\frac{\log \left\{\frac{x_{w\mathrm{hite},1}}{x_{w\mathrm{hite},2}}\right\}}{\log \left\{\frac{q_{w\mathrm{hite},1}}{q_{w\mathrm{hite},2}}\right\}}}}.Math.\log \underset{:={\tilde{f}}_q}{\left\{\underbrace{\frac{1}{{\tilde{q}}_1}.Math.\left[1-{\left(1-{\tilde{q}}_1\right)}^{\frac{q_{w\mathrm{hite},1}}{q_{w\mathrm{hite},2}}}\right]}\right\}}\right)\\ ={\tilde{x}}_1.Math.{\tilde{f}}_q^{{\beta }_x}\end{array}& \left(29\right)\end{array}$

[0093] Thus, the transformation of the centered coordinates may be written in a compact way as

[00024]$\begin{array}{cc}{\tilde{x}}_2={\tilde{x}}_1.Math.{\tilde{f}}_q^{{\beta }_X}& \left(30\right)\end{array}$${\tilde{y}}_2={\tilde{y}}_1.Math.{\tilde{f}}_q^{{\beta }_y}$${\tilde{z}}_2={\tilde{z}}_1.Math.{\tilde{f}}_q^{{\beta }_z}$$\begin{array}{cc}\mathrm{with}{\beta }_x=\frac{\log \left\{\frac{x_{w\mathrm{hite},1}}{x_{w\mathrm{hite},2}}\right\}}{\log \left\{\frac{q_{w\mathrm{hite},1}}{q_{w\mathrm{hite},2}}\right\}}& \left(31\right)\end{array}$${\beta }_y=\frac{\log \left\{\frac{y_{w\mathrm{hite},1}}{y_{w\mathrm{hite},2}}\right\}}{\log \left\{\frac{q_{w\mathrm{hite},1}}{q_{w\mathrm{hite},2}}\right\}}$${\beta }_Z=\frac{\log \left\{\frac{z_{w\mathrm{hite},1}}{z_{w\mathrm{hite},2}}\right\}}{\log \left\{\frac{q_{w\mathrm{hite},1}}{q_{w\mathrm{hite},2}}\right\}}$$\begin{array}{cc}\mathrm{and}{\tilde{f}}_q=\frac{1}{{\tilde{q}}_1}.Math.\left[1-{\left(1-{\tilde{q}}_1\right)}^{\frac{q_{w\mathrm{hite},1}}{q_{w\mathrm{hite},2}}}\right]& \left(32\right)\end{array}$

[0094] Factors β in equation (31) may be calculated already as soon as the transformation is stopped because they only depend on the color space coordinates of the white points.

[0095] Although a characteristic of transformation F described by equations (30), (31), (32) is that the transformation of white point W.sub.1 into white point W.sub.2 and from white point W.sub.2 to white point W.sub.1 behave in a precisely inverted way relative to one another, i.e.

F.sub.W.sub.2.sub.W.sub.1(F.sub.W.sub.1.sub.W.sub.2(xyz))=xyz.  (33)

[0096] Yet the result of the transformation is not independent of whether the transformation from white point W.sub.1 to white point W.sub.3 is initially made via white point W.sub.2 or whether the direct transformation is calculated, i.e. in general,

F.sub.W.sub.2.sub.W.sub.3(F.sub.W.sub.1.sub.W.sub.2(xyz))≠F.sub.W.sub.1.sub.W.sub.3(xyz)  (34)

applies.

[0097] To achieve this, the transformation may be modified in such a way that no direct transformation is made from source white point to destination white point, but instead the white point of illumination and observation condition of D50 illumination type and 2° observer (D50O02) defined in the ICC specification is selected as intermediate point. We get this transformation G on the basis of transformation F described in equations (30), (31), (32) as

G.sub.W.sub.1.sub.W.sub.2(xyz)=F.sub.W.sub.D50O02.sub.W.sub.2(F.sub.W.sub.1.sub.W.sub.D50O02(xyz))  (35)

[0098] In general, all centered color space coordinates {tilde over (x)}, {tilde over (y)}, and {tilde over (z)} ought to be between 0 and 1. Thus equation (32) is only valid for this interval. However, in some cases, negative values or values above 1 may occur. This may happen, for instance, when phosphorescent colors are examined. Another reason for irregular centered color space coordinates may be a linear approximation for a Bradford transformation to the color space coordinates if extremely chromatic colors are being transformed. To continue the transformation for these values, colors with at least one non-positive color space coordinate ought to be treated in an absolute colorimetric way and colors with a color space coordinate product above 1 ought to be treated in a relative colorimetric way.

[0099] Thus, transformation equation (30) may be converted into

[00025]$\begin{array}{cc}{\tilde{x}}_2=\{\begin{array}{cccc}{\tilde{x}}_1.Math.\frac{x_{w\mathrm{hite},1}}{x_{w\mathrm{hite},2}}& :& \tilde{x}\le 0\vee \tilde{y}\le \vee \tilde{z}\le 0& \mathrm{absolute}\\ {\tilde{x}}_1& :& \tilde{x}.Math.\tilde{y}.Math.\tilde{z}\ge 1& \mathrm{relative}\\ {\tilde{x}}_1.Math.{\tilde{f}}_q^{{\beta }_x}& :& \mathrm{others}& \mathrm{mixed}\end{array}& \left(36\right)\end{array}$

{tilde over (y)},{tilde over (z)}: in an analogous way.

[0100] The method presented herein is based on determining a factor {tilde over (ƒ)}.sub.q for scaling the geometric means q of the color space coordinates with the aid of equation (32) and dividing it among the color space coordinates on the basis of exponents β.sub.x, β.sub.x, β.sub.z. Initially, this method will fail precisely when truly different white points W.sub.1 and W.sub.2 have the same geometric mean, i.e.

x.sub.white,1.Math.y.sub.white,1.Math.z.sub.white,1=x.sub.white,2.Math.y.sub.white,2.Math.z.sub.white,2  (37)

and

(x.sub.white,1≠x.sub.white,2∨y.sub.white,1≠y.sub.white,2∨z.sub.white,1≠z.sub.white,2)  (38)

[0101] This may occur if the ratios

[00026]$\frac{x_{\mathrm{white},2}}{x_{w\mathrm{hite},1}},$$\frac{y_{\mathrm{white},2}}{y_{w\mathrm{hite},1}},$$\frac{z_{\mathrm{white},2}}{z_{w\mathrm{hite},1}}$

have both values above 1 and values below 1. Therefore, the logarithmic algebraic signs are defined as

[00027]$\begin{array}{cc}\begin{array}{ccc}{.Math.}_x=.Math.\left(\frac{x_{\mathrm{white},2}}{x_{\mathrm{white},1}}\right),& {.Math.}_y=.Math.\left(\frac{y_{\mathrm{white},2}}{y_{\mathrm{white},1}}\right)& {.Math.}_Z=.Math.\left(\frac{z_{\mathrm{white},2}}{z_{\mathrm{white},1}}\right)\end{array}& \left(39\right)\end{array}$$\begin{array}{cc}\mathrm{with}.Math.\left(r\right)=\{\begin{array}{ccc}-1& :& r<1\\ 0& :& r=1\\ 1& :& r>1\end{array}& \left(40\right)\end{array}$

as well as the dominant logarithmic algebraic sign as

[00028]$\begin{array}{cc}\stackrel{—}{.Math.}=\{\begin{array}{ccc}-1& :& {.Math.}_x+{.Math.}_y+{.Math.}_z<1\\ 1& :& \mathrm{other}\end{array}& \left(41\right)\end{array}$

[0102] Thus the ratio of the geometric means of the white points may be substituted in equations (31) and (32)

[00029]$\begin{array}{cc}\frac{q_{w\mathrm{hite},1}}{q_{w\mathrm{hite},2}}\sqrt[3]{{\left(\frac{x_{w\mathrm{hite},1}}{x_{w\mathrm{hite},2}}\right)}^{{\sigma }_x}.Math.{\left(\frac{y_{wh\mathrm{ite},1}}{y_{w\mathrm{hite},2}}\right)}^{{\sigma }_y}.Math.{\left(\frac{z_{w\mathrm{hite},1}}{z_{w\mathrm{hite},2}}\right)}^{{\sigma }_z}}& \left(42\right)\end{array}$

[0103] Here weights σ=−1, 1 which determine which ratios are applied inversely and which ratios are applied directly remain to be determined.

[0104] For the direct transformation from white point W.sub.1 to W.sub.2, the component whose logarithmic sign deviates from the dominant sign is to be applied inversely.

[0105] This means

[00030]$\begin{array}{cc}{\sigma }_x=\{\begin{array}{ccc}-1& :& {.Math.}_x\ne \stackrel{—}{.Math.}\\ 1& :& \mathrm{other}\end{array}& \left(43\right)\end{array}$

and σ.sub.y,σ.sub.z in an analogous manner

[0106] Thus the ratio in equation (42) may only become 1 if the two white points W.sub.1 and W.sub.2 are identical. In this case the entire transformation is an identity and unnecessary.

[0107] For the transformation with the intermediate step via D50O02, one may in general assume that the color space coordinates of the white points are below those of D50O02. Thus, in the transformation to D50O02, the ratios which have a positive logarithmic sign Σ are inverted and in the transformation from D50O02 the ratios with a negative logarithmic sign Σ are inverted:

[00031]$\begin{array}{cc}\frac{q_{\mathrm{white},1}}{q_{D50O02}}\mapsto \sqrt[3]{{\left(\frac{x_{\mathrm{white},1}}{x_{D50O02}}\right)}^{-{\Sigma }_{1,x}}.Math.{\left(\frac{y_{\mathrm{white},1}}{y_{D50O02}}\right)}^{-{\Sigma }_{1,y}}.Math.{\left(\frac{z_{\mathrm{white},1}}{z_{D50O02}}\right)}^{-{\Sigma }_{1,Z}}}& \left(44\right)\end{array}$$\mathrm{with}$$\begin{array}{cc}\begin{array}{ccc}{.Math.}_{1,x}=.Math.\left(\frac{x_{\mathrm{white},1}}{x_{D50O02}}\right)& {.Math.}_{1,y}=.Math.\left(\frac{y_{\mathrm{white},1}}{y_{D50O02}}\right)& {.Math.}_{1,z}=.Math.\left(\frac{z_{\mathrm{white},1}}{z_{D50O02}}\right)\end{array}& \left(45\right)\end{array}$$\begin{array}{cc}\frac{q_{D50O02}}{q_{\mathrm{white},2}}\mapsto \sqrt[3]{{\left(\frac{x_{D50O02}}{x_{\mathrm{white},2}}\right)}^{-{.Math.}_{2,x}}.Math.{\left(\frac{y_{D50O02}}{y_{\mathrm{white},2}}\right)}^{-{.Math.}_{2,y}}.Math.{\left(\frac{z_{D50O02}}{z_{\mathrm{white},2}}\right)}^{-{.Math.}_{2,z}}}& \left(46\right)\end{array}$$\mathrm{with}$$\begin{array}{cc}\begin{array}{ccc}{.Math.}_{2,x}=.Math.\left(\frac{x_{D50O02}}{x_{\mathrm{white},2}}\right)& {.Math.}_{2,y}=.Math.\left(\frac{y_{D50O02}}{y_{\mathrm{white},2}}\right)& {.Math.}_{2,z}=.Math.\left(\frac{z_{D50O02}}{z_{\mathrm{white},2}}\right)\end{array}& \left(47\right)\end{array}$

[0108] These ratios may likewise only be 1 if the respective white point W.sub.1 [W.sub.2] precisely corresponds to the white point of the illuminant condition. In this case, the transformation from W.sub.1 to D50O02 [and from D50O02 to W.sub.2, respectively] is the identical representation and unnecessary.

[0109] FIG. 1 is a schematic illustration of the color transformation principle for a printing process adaptation in accordance with the prior art. A first transformation 1 transforming the color values [C1, M1, Y1, K1] of the first printing process into XYZ color values and a second color transformation 2 transforming the XYZ color values into the color values [C2, M2, Y2, K2] of the second printing process are carried out successively. The two-color transformations 1 and 2 may be combined to form an equivalent color transformation 3 which directly assigns color values [C1, M1, Y1, K1] and color values [C2, M2, Y2, K2] to one another. Since color values [C1, M1, Y1, K1] and [C2, M2, Y2, K2] which are allocated to one another via the device-independent XYZ intermediate color space have the same XYZ color values, the associated process colors of the two printing processes are essentially perceived as visually identical within the process color gamut.

[0110] FIG. 2 schematically shows the method of the invention. In step 10, the color values of a first device-dependent color space are transformed into an intermediate color space. The transformation into the intermediate color space is achieved using a combination of absolute and relative rendering intent, steps 20 and 30. The color values that have been transformed to the intermediate color space are transformed into the color values of the second device-dependent color space. The transformation from the intermediate color space is achieved using a combination of absolute and relative rendering intent, steps 40 and 50.

[0111] The following is a summary list of reference numerals and the corresponding structure used in the above description of the invention: [0112] 1 first color transformation [0113] 2 second color transformation [0114] 3 third color transformation