Abstract:
A color management system using distributed profiles includes a color printer adapted to print controlled color using a custom profile. The color printer system creates a custom profile by reading its initial characterization data, the profiles of the inks and the profile of the paper used in the printer. The color printing system includes a spectral measurement module adapted to generate spectral measurement of an output of the color printer. The printing system computes its initial characterization data by using the output of the color printer, measured by the spectral measurement module. The printing system updates the paper profile, the ink profile and/or the printer characterization data based on the measurement of the spectral measurement module.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS  
       [0001]     This application claims priority under 35 U.S.C. §119 to European Application 03 013 650.1 filed in Europe on 16 Jun. 2003, the entire contents of which are hereby incorporated by reference in their entirety. 
     
    
     BACKGROUND  
       [0002]     This present invention relates generally to a color printing system, and more specifically to a system that allows for automatically managing of colors in a printing system with the ability to choose the amount of user intervention in the process.  
         [0003]     When printing with a color printing system, variations in color images produced by the system are very common. Such variations can occur, for example, when printing the same image at different times, when printing with different inks or when printing on different papers. To overcome these difficulties, color management techniques offer the possibility to print a target containing several hundreds patches, to measure these patches with a dedicated instrument, in order to characterize the printer system. The printing system includes a printer, an ink set and a specific paper. If one of the printer, ink of paper changes, the characterization has to be repeated. This characterization process is time consuming and generates costs in material as well as in resource allocation. Time consumption, or more generally resource allocation, is currently one of the most critical aspects for color management in printing facilities.  
         [0004]     Color management is based on the use of International Color Consortium (ICC) profiles, which relate the color outputted by a device to the color expressed in a device independent space. This device independent space can be either CIELAB or XYZ color space. This workflow is summarized in  FIG. 0 .  
         [0005]     The data is acquired by an input device  1000 , whose ICC profile  1010  is known. The ICC profile is used by a color management module  1111  (CMM) to convert the input data  1100  into the CIELAB (or XYZ) color space, resulting in the device independent data  1101 . To print this data, the CMM  1111  uses the ICC profile  1011  of the output device  1001 , and transforms the device independent data  1101  into an output device specific data  1102 . The output device specific data  1102  is sent to the output device  1001 , resulting in a print  1002  with the desired colors.  
         [0006]     In order to successfully apply color management one needs an input ICC profile, an output ICC profile associated with output device as well as a CMM.  
       SUMMARY  
       [0007]     A system is disclosed that allows managing colors in an efficient way, minimizing the user intervention time and resource allocations.  
         [0008]     Exemplary embodiments include an ability to integrate characterization data from several sources and merge them together to build a color characterization ICC profile.  
         [0009]     A characterization of the printing process is performed by printing an adequate target on the printer only once, with a paper and ink set in accordance with the user needs. This characterization can be done by the printer manufacturer or by the user, but has to be done at least once for each individual printer. In this characterization process, the system is able to characterize each component of the printing process: the inks, the paper, and the printer. Then, if a change occurs, the system is able to perform color management with the characteristics of the added elements and the characteristics of the removed element, without the need of a new print. For example, if the cyan ink is replaced by a blue one, the system only needs the characteristics of the blue ink to be able to manage the colors of the whole system; the characteristics of the cyan ink being known through the initial characterization process.  
         [0010]     If a change of paper is performed in the system, the paper characteristics may be given by the paper manufacturer, or can be measured with a spectrophotometer outside the printing line. If a change of ink is performed in the system, the ink characteristics may be given by the ink manufacturer, or can be measured on a sample of the ink that has been printed on a well known, or measurable paper. This print is performed in general by the ink manufacturer. It may also be done by the user, but outside the printing line. In this way, the printing line has never to be stopped to perform a color profile. In addition to this, measuring a paper or an ink characteristics requires less than 10 measurement actions, and is much more time efficient than a standard color characterization process.  
         [0011]     Additionally, the system is able to enhance the precision of the color management if a print of the new system configuration is available. This improvement distinguishes itself from the currently available techniques in the number of color patches needed. The present invention is able to enhance the color management even if only one patch is printed on the printer. In general, the more patches are printed, the better the color management. But instead of using several hundreds patches, this technique requires from one to about hundred patches to work.  
         [0012]     The current technique is also able to integrate some information coming from a printer of the same family. This information comes in general from the printer manufacturer.  
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0013]     The invention is described below in more detail in connection with the drawings. In the drawings:  
         [0014]      FIG. 0  shows a flow diagram of the workflow for creating International Color Consortium (ICC) profiles,  
         [0015]      FIG. 1  shows a flow diagram of a first step of the color management workflow according to an exemplary embodiment of the invention,  
         [0016]      FIG. 2  shows a flow diagram of a second step of the color management workflow according to an exemplary embodiment of the invention,  
         [0017]      FIG. 3  shows how an exemplary inventive system computes an ICC profile when an ink is replaced in the printer,  
         [0018]      FIG. 4  shows an exemplary technique that allows improving the precision of the color management by printing one or several color patches with the printer,  
         [0019]      FIG. 5  depicts an exemplary method of enabling a paper change in the color management framework of this invention,  
         [0020]      FIG. 6  shows an example of how the paper and ink characteristics can be computed from a sample of paper and printed ink, and  
         [0021]      FIGS. 7-16  show details of workflow and data elements of an exemplary embodiment of the inventive system as outlined in  FIGS. 1-6 . 
     
    
     DETAILED DESCRIPTION  
       [0022]     A system and method are disclosed for improving the efficiency of a color management workflow. Exemplary embodiments allow performing color management with much less resource allocations than the current techniques. Currently available physical models of ink and paper can also be improved.  
         [0023]     A new color management workflow is disclosed that is performed in two steps. The first step is described in  FIG. 1  and includes characterizing a printer system. The second step includes integrating a new element, and computing an ICC profile without the need of a print.  
         [0024]     In the first step, the user chooses the ink set  102 , the printer  101  and the paper  100 . A test chart  103  composed of several color patches is printed, according to the printer input values  120 , and measured with a device  104  that extracts spectral information  113  from the print  103 . By spectral information, we refer to a vector of at least three components that are associated to a part of the spectrum in continuous space. The measured test chart  113  is given as input to the estimation engine  106  that computes the printer system characteristics  108 . The estimation engine  106  also uses some a-priori knowledge coming from the printer family characteristics  105 .  
         [0025]     The printer system characteristics  108  contains a set of one or more files for each of the paper characteristics  109 , the ink characteristics  111 , the printer characteristics  110 , the joint paper, ink and printer characteristics  112 , the measured initial test chart  113 , and the difference test chart  114  (representing a difference file, or fitting file, which can be used to adjust spectral files).  
         [0026]     The difference test chart  114  can be computed according to the method described in  FIG. 2 .  
         [0027]     The distributed profile assembler  200  combines the characteristics of the various elements of the printer system  108  (the paper characteristics  109 , the printer characteristics  110 , the ink characteristics  111  and the joint paper, ink and printer characteristics  112 ) to build the International Color Consortium (ICC) profile  201 . It also re-computes the color patches of the test chart  103  resulting in the synthetic test chart  202 . The synthetic test chart  202  is compared to the initial test chart measurement  113  with the test chart differentiator  203  resulting in the difference test chart  114 . The test chart differentiator computes the spectral difference between each of the corresponding patches in  113  and  202 .  
         [0028]      FIG. 3  describes how the system can compute an ICC profile when an ink (or several inks) is replaced in the printer. The ink characteristics replacement module  302  modifies the ink characteristics  111 , by replacing the characteristics of the removed ink with the characteristics of the new ink  300 . The user has to tell the system which ink gets replaced (information  301 ). The output of module  302  embodies the new ink characteristics, that combined with the paper characteristics  109 , the printer characteristics  110  the joint paper, ink and printer characteristics  112  in the distributed profile assembler  200  result in an ICC profile  303  and a synthetic test chart  304 . The system assumes that the joint paper ink and printer characteristics  112  have not changed. The synthetic test chart  304  can be corrected by the test chart correction module  305  with the difference test chart  114 : the patches of synthetic test chart  304  that are equal to the patches of the synthetic test chart  202  get corrected by addition of the corresponding patch in the difference test chart  114 . The test chart correction module  305  outputs a corrected synthetic test chart  306 . A standard profile creation module  307  uses the corrected synthetic test chart  306  to create a corrected ICC profile  308 .  
         [0029]     The user can choose either to use the profile  303  or the profile  308 .  
         [0030]      FIG. 4  describes a technique that allows improving the precision of the color management by printing one or several color patches with the printer. An ink set  402  and a paper  400  are fed into the printer  101  that prints the color patch(es)  403 . The ink set  402  may differ, but is not required to differ, from the ink set  102 . The paper  400  may differ, but is not required to differ, from the paper  100 . The color patch(es)  403  are measured with the measurement device  104  resulting in the patch(es) measurement(s)  413 . The estimation refinement engine  406  uses the printer system characteristics  108 , the printer family characteristics  105  and the patch(es) measurement(s)  413  to compute a corrected printer system characteristics  408 .  
         [0031]     If the ink set  402  differs from the ink set  102 , the estimation refinement engine  406  also uses the characteristics of the changed inks  411 . If the ink set  402  is the same than the ink set  102 , the system keeps the ink set characteristics  111  (i.e. the ink set characteristics  411  is equal to the ink set characteristics  111 ) unless the estimation refinement engine  406  determines, based on the information coming from the patch(es) measurement(s)  413 , that the ink set characteristics  111  have changed over time and performs an appropriate correction which will result in a new ink set characteristics  411 .  
         [0032]     If the paper  400  differs from the paper  100 , the estimation refinement engine  406  also uses the characteristics of the new paper  409 . If the paper  400  is the same than the paper  100 , the system keeps the paper characteristics  109  (i.e. the paper characteristics  409  are equal to the paper characteristics  109 ) unless the estimation refinement engine  406  determines, based on the information coming from the patch(es) measurement(s)  413 , that the paper characteristics  109  have changed over time and performs an appropriate correction which will result in a new paper characteristics  409 .  
         [0033]     The corrected printer system characteristics  408  contains the paper characteristics  409 , the printer characteristics  110 , the ink characteristics  411  the corrected joint paper, ink and printer characteristics  412 , the initial test chart measurement  113 , the difference test chart  114  and the patches measurement  413 .  
         [0034]     An exemplary purpose of the processing in  FIG. 4  is the update of the joint paper, ink and printer characteristics  112  with the corrected joint paper, ink and printer characteristics  412 .  
         [0035]      FIG. 5  depicts a method enabling a paper change in the color management framework of this invention: the distributed profile assembler  200  combines the new paper characteristics  509  with the printer characteristics  110 , the ink characteristics  111  and the joint paper, ink and printer characteristics  112  to build the ICC profile  503 . It also re-computes the color patches of the test chart  103  resulting in the synthetic test chart  504 . The system assumes that the joint paper ink and printer characteristics  112  have not changed. The synthetic test chart  504  can be corrected by the test chart correction module  505  with the difference test chart  114 : the patches of synthetic test chart  504  get corrected by addition of the corresponding patch in the difference test chart  114 . The test chart correction module  505  outputs a corrected synthetic test chart  506 . A standard profile creation module  307  uses the corrected synthetic test chart  506  to create a corrected ICC profile  508 .  
         [0036]     The user can choose either to use the profile  503  or the profile  508 .  
         [0037]      FIG. 6  describes how to compute the paper and ink characteristics from a sample of paper and printed ink. The paper  600  is measured by the measurements device  104 . The paper measurement  605  is input in the paper characteristics estimation module  606  that outputs the new paper characteristics  409 . The concatenation of the measurement device  104 , the paper measurement  605  and the paper characteristics estimation module  606  is denoted as the paper module  607 .  
         [0038]     To characterize the ink, the ink  610  is printed on paper  611  and measured with the measurement device  104  that outputs the ink on paper measurement  615 . The paper  611  alone—with no ink on it—is characterized by the paper module  607 . Using the paper module  607  output and the ink on paper measurement  615 , the ink characteristics estimation module  616  computes the ink characteristics  411 .  
         [0039]     The following details an internal functioning of the distributed profile assembler  200 . Let ink 001 be the first ink printed on the paper and ink 002 be the ink printed after ink 001. In other words, ink 002 can lie either on ink 001, or on the paper at the location where no ink 001 has been deposited, as depicted in  FIG. 13 . The order in which the inks are deposited on paper are detailed in the printing order information  811 .  
         [0040]     The distributed profile assembler  200  simulates the ink deposition on paper to compute, for each input to the printer, the reflection spectrum as well as a CIELAB value. The ink deposition refers to an ink coverage percentage, and to an ink thickness or concentration.  
         [0041]     Given an input value, the distributed profile assembler  200  simulates ink deposition on paper in the order that corresponds to the printing order. The deposition of ink 001 is computed using the tone reproduction curve of ink 001, contained in the Set of tone reproduction curves  820 . If the tone reproduction curve of ink 001 cannot be found in  820 , then the distributed profile assembler  200  uses the mean tone reproduction curve  812 . The ink deposition is further corrected using the trapping ratio to reference paper  805 .  
         [0042]     For each subsequent ink, the ink deposition is computed using a statistical model. The statistical model enumerates each possible overlapping case, and determines its importance—or weight—for the final calculation. For example, in a two inks printer as shown in  FIG. 16 , the distributed profile assembler  200  computes the ink 001 deposition on paper (case 1), the ink 002 deposition on paper (case 2), as well as the ink 002 deposition on top of ink 001 (case 3). Then, it computes the spectra associated with ink 001 on paper, ink 002 on paper, ink 002 on ink 001, and adds the results in an add-hoc manner, by using the probabilities of the events: ink 001 on paper (case 1), ink 002 on paper (case 2), ink 002 on ink 001 (case 3), and no ink at all (case 0), respectively.  
         [0043]     When printing ink X on top of another ink Y, the ink deposition X is computed using the tone reproduction curve of ink X stored in the set of tone reproduction curves  820 , and the trapping parameter of the ink couple X/Y, stored in the ink superposition matrix  821 . If the tone reproduction curve of ink X is not available in the set of tone reproduction curves  820 , the distributed profile assembler  200  uses the mean tone reproduction curve  812 . If the trapping parameter of the ink couple X/Y is not available, the distributed profile assembler uses the mean trapping parameter  834  of ink Y. If the mean trapping parameter  834  of ink Y is not available, the distributed profile assembler  200  uses the mean trapping parameter  813 .  
         [0044]     The distributed profile assembler  200  can use the trapping parameter in two ways, depending on the type of printer  810 . For gravure or offset printer types, the trapping parameter refers to the ratio of the ink thickness when printed on ink with respect to the ink thickness when printed on the reference paper  100 . For ink jet printer type the trapping parameter refers to a joint thickness ratio and area ratio of ink with respect to the ink printed on paper  100 .  
         [0045]     The estimation engine  106  is composed of the distributed profile assembler  200 , the comparator  900  and the parameter updater  901 . From a given state of the ink characteristics  111 , the paper characteristics  109 , the printer characteristics  110 , the joint paper, ink and printer characteristics  112  and the printer family characteristics  105 , the distributed profile assembler  200  computes a spectrum  902  given an printer input value  120 . The comparator  900  compares the spectrum  902  with the corresponding measured spectrum found in the initial test chart measurement  113 . The result of the comparison is put in the parameter updater  901  who corrects the appropriate characteristics in printer system characteristics  108 . The workflow is repeated until the comparator  900  decides that the spectrum  902  matches the initial test chart measurement  113  patch. This recursive workflow is referred to as the update procedure  999  in the text.  
         [0046]     The association of the characteristics with the patches in the initial test chart measurement  113  is overviewed in  FIG. 8 , and detailed in  FIGS. 9, 10 ,  11 ,  12 ,  13 ,  14  and  15 . In  FIG. 8 , the elements are estimated in the order they appear on the figure, from top to bottom.  
         [0047]      FIG. 9  depicts the estimation of the paper characteristics  409 . The paper characteristics  109  and  409  are the same, but estimated in a different context. The estimation engine  106  uses the paper, polarized and UV filtered  920  spectrum to compute the diffuse reflection characteristics  801  of the paper. The estimation engine computes the surface reflection spectrum  802  of the paper, by using the difference between the paper, polarized and UV filtered  920  spectrum and the paper, UV filtered  921  spectrum. The estimation engine  106  computes the whitener spectrum  806  from the difference between the paper  922  spectrum and the paper, UV filtered  921  spectrum. The whitener spectrum  806  gets normalized and results in the Whitener spectrum shape  803 . The overall amplitude of the whitener spectrum  806  is stored in the whitener strength  804 .  
         [0048]     If the paper, UV filtered  921  spectrum is unavailable, the estimation engine  106  sets the value of the whitener spectrum  806  to a value known a-priori; the whitener strength  804  is determined using a smoothness criterion applied on the spectrum of the paper  922 , in the region of the spectrum where the whitener has its maximum. Any overshoot in that region of the spectrum is supposed to be caused by the whitener spectrum.  
         [0049]     If the polarized and UV filtered  920  spectrum is unavailable, it is set to a constant value, also known a priori.  
         [0050]      FIG. 10  depicts the estimation of the absorption spectrum  830  of an ink. The absorption spectrum  830  is first set to an initial value that is of no importance for this invention. The full tone patch, polarized and UV filtered  910  is extracted from the initial test chart measurement  113 . By disregarding the scattering spectrum  831 , the absorption of UV light factor  833 , the surface reflection spectrum  832  and the mean of trapping parameter  834 , the distributed profile assembler  200  computes the spectrum  902 . The comparator  900  compares the spectrum  902  with the full tone patch, polarized and UV filtered  910  and launches the update procedure to find the best values of the absorption spectrum  830 .  
         [0051]      FIG. 11  depicts the estimation of the surface reflection spectrum  832 . The surface reflection spectrum  832  is first set to an initial value that is of no great importance for this invention. The full tone patch, polarized and UV filtered  910  and the full tone patch, UV filtered  911  are input to the comparator  900 . By disregarding the scattering spectrum  831 , the absorption of UV light factor  833  and the mean of trapping parameter  834 , the distributed profile assembler  200  computes the spectrum  902  and spectrum  903 . The spectrum  902  is computed from the absorption spectrum  830  and the spectrum  903  computed from the combination of the absorption spectrum  830  and the surface reflection spectrum  832 . The comparator  900  compares the difference between spectrum  902  and spectrum  903  and compares it to the difference between spectrum  910  and spectrum  911 . Then, the comparator  900  launches the update procedure to find the best values of the surface reflection spectrum  832 .  
         [0052]     The paper characteristics estimation module  606  works like the estimation module  106  in  FIG. 11 . The estimation module  106  as well as the paper characteristics estimation module  606  set the trapping ratio to reference paper  805  equal to 1.  
         [0053]      FIG. 12  depicts the estimation of the absorption of the UV light factor  833 . The absorption of the UV light factor  833  is first set equal to 0. The full tone patch, UV filtered  911  is subtracted from the full tone patch  912  to result in the whitener contribution spectrum  913 . The distributed profile assembler  200  uses the whitener spectrum shape  802  and strength  804  and the absorption of the UV light factor  833 , and assumes that it embodies the energy of light that illuminates the ink from the bottom. The distributed profile assembler  200  outputs the associated computed whitener spectrum  905 . The comparator  900  compares the difference between the computed whitener spectrum  905  and whitener contribution spectrum  913 . Then, the comparator  900  launches the update procedure  999  to find the best value of the absorption of the UV light factor  833 .  
         [0054]      FIG. 13  depicts the estimation of the scattering spectrum  831 , and the trapping parameter that will be stored in the ink superposition matrix  821 . The scattering spectrum is first set to 0. The scattering spectrum of the first ink, ink 001, is assumed to be 0. The scattering spectrum  831  is computed from the overprint patch  930  of the color ink  931  on a darker color  932 . By darker color we refer to an ink who absorbs light in the region where the given ink scatters, or is transparent to the light. If available, black ink is used. Among the ink characteristics  111 , the distributed profile assembler uses the absorption spectrum and the scattering spectrum of the color ink  931  and the darker ink  932  to compute the spectrum  906  of the overprint patch. The comparator  900  compares the spectrum  906  with the overprint patch, polarized and UV filtered spectrum  930  and launches the update procedure  999  to find the best values of the scattering spectrum  831  and the trapping parameter in the ink superposition matrix  821 . Once every combination of ink overprint has been computed, the mean of trapping parameters  834  as well as the mean trapping parameter  813  are computed from the ink superposition matrix  821 .  
         [0055]      FIG. 14  depicts the estimation of the set of tone reproduction curves  820 . The tone reproduction curve  820  relate the printer input values  120  to the area coverage an ink thickness. The initial test chart measurement  113  contains the gradient of patches  950 , i.e. a set of patches with increasing area coverage. The set of tone reproduction curves  820  are set to an initial value irrelevant for this invention. From the printer input values  120 , and printer system characteristics  108 , the profile assembler engine  200  outputs a set of computed spectra  906 . The comparator  900  compares the difference between the set of computed spectra  906  and the gradient of patches  950 . Then, the comparator  900  launches the update procedure to find the best values of the set of tone reproduction curves  820 . The parameter updater  901  makes sure that the computed area coverage and ink thickness vary smoothly according to the input values  120 . Finally, the set of reproduction curves  820  is averaged and stored into the mean tone reproduction curve  812 .  
         [0056]     The ink characteristics estimation module  616  estimates the ink characteristics  411  in the same way than the estimation engine  106  estimates the ink characteristics  111  in  FIGS. 10, 11 ,  12  and  13 .  
         [0057]     The estimation refinement engine  406  is composed of the same elements as the estimation engine  106 . Its purpose is, when a print is available, to replace the characteristics elements computed in the context of  FIG. 6  with the elements computed as described in  FIGS. 10, 11 ,  12 ,  13  and  14 .  
         [0058]      FIG. 15  illustrates how a print of a single patch of color on a new paper can improve the system color description. The estimation refinement engine  406  uses a full tone patch, polarized and UV filtered  950  of an old ink—whose characteristics are contained in the ink characteristics  111 —on the new paper. The estimation refinement engine  406  is used to estimate variations of ink deposition between the ink on the old paper  100  and on the new paper  400 . This variation is embodied by a trapping parameter, computed using the processing of  FIG. 13 , and assuming the ink is printed on a full transparent and non scattering ink; the resulting trapping parameter is stored in the trapping ratio to reference paper  805  parameter.  
         [0059]     When printed patch(es) of a new ink are available, the estimation refinement engine  406  can recompute the ink characteristics using the method described in  FIGS. 10, 11 ,  12 ,  13  and  14 .  
         [0060]     The following is a description of an exemplary preferred embodiment of a mathematical model underlying the present invention.  
         [0061]     The color management is enabled through the modeling and computation of the spectrum that a combination of inks reflect.  
         [0062]     The reflectance spectrum R of ink printed on the paper is modeled by the following equation:  
         R   ⁡     (       k   λ     ,   l     )       =       R   sf     +           (     1   -     r   0       )     ⁢     I   Iλ     ⁢     R   pλ     ⁢     I   Aλ             sin   2     ⁡     (     α   1     )       ·     (     1   -       R   Pλ     ⁢     I   Sλ         )         ⁢     (   0.1   )             
 
 where 
    R sf  is a surface reflection spectrum  832  which depends on the paper, the inks and the wavelength.     R pλ  is the internal diffuse spectral characteristics  801  of the paper.     r 0  is the proportion of the light that gets reflected, thus not entering the paper.     sin 2 (α 1 ) is a normalization parameter that depends on the geometry of the measurement device used to characterize printed data.    
 
         [0067]     For a single ink printed on paper with no whitener, the integrals I Aλ , I Sλ  and I 1λ  are defined as follow:  
           I   Aλ     =       ∫     l   =   0     ∞     ⁢       ∫     θ   =   0       α   2       ⁢         P   ⁡     (   l   )       ·     (     1   -     r   21       )       ⁢     ⅇ       -     k   λ       ⁢     l   /     cos   ⁡     (     θ   2     )             ⁢           ⁢     sin   ⁡     (     2   ⁢   θ     )       ⁢     ⅆ   θ     ⁢           ⁢     ⅆ   l             ,     
     ⁢       I   Sλ     =       ∫     l   =   0     ∞     ⁢       ∫     θ   =   0       π   /   2       ⁢         P   ⁡     (   l   )       ·     r   21       ⁢     ⅇ       -   2     ⁢     k   λ     ⁢     l   /     cos   ⁡     (     θ   2     )             ⁢     sin   ⁡     (     2   ⁢   θ     )       ⁢     ⅆ   θ     ⁢     ⅆ   l             ,     
     ⁢       I   Iλ     =       ∫     l   =   0     ∞     ⁢         P   ⁡     (   l   )       ·     ⅇ       -     k   λ       ⁢     l   /     cos   ⁡     (     θ   2     )               ⁢           ⁢     ⅆ   l           ,       
 
 where k λ  is the ink transmittance index ( 830 ), and depends only on the ink, l is the ink layer thickness expressed in arbitrary units, P(l) is the ink layer thickness probability density function, λ is the wavelength, and θ 2 , α 2  two angles given by the measurement device. r 21  is a fixed parameter given by the measurement instrument geometry. 
 
         [0068]     For several inks, the integrals I Aλ , I Sλ  and I 1λ  are defined as follow:  
           I   Aλ     =       ∫       l   1     =   0     ∞     ⁢           ⁢     …   ⁢           ⁢       ∫       l   n     =   0     ∞     ⁢       ∫     θ   =   0       α   2       ⁢         P   ⁡     (       l   1     ,   …   ⁢           ,     l   n       )       ·     
     ⁢           ⁢     (     1   -     r   21       )       ⁢     ⅇ     -           k     λ   ,   1       ⁢     l   1       +       k     λ   ,   2       ⁢     l   2       +   …   +       k     λ   ,   n       ⁢     l   n           cos   ⁡     (     θ   2     )             ⁢           ⁢     sin   ⁡     (     2   ⁢   θ     )       ⁢     ⅆ   θ     ⁢           ⁢     ⅆ     l   n       ⁢           ⁢   …   ⁢           ⁢     ⅆ     l   1                   ,     
     ⁢       I   IUV     =       ∫       l   1     =   0     ∞     ⁢           ⁢     …   ⁢           ⁢       ∫       l   n     =   0     ∞     ⁢         P   ⁡     (       l   1     ,   …   ⁢           ,     l   n       )       ·     
     ⁢           ⁢     ⅇ     -           k     uv   ,   1       ⁢     l   1       +       k     uv   ,   2       ⁢     l   2       +   …   +       k     uv   ,   n       ⁢     l   n           cos   ⁡     (     θ   2     )               ⁢           ⁢     ⅆ     l   1       ⁢           ⁢   …   ⁢           ⁢     ⅆ     l   n                 ,     
     ⁢       I   Sλ     =       (     1   +       I   IUV     ·     W   λ         )     ·       ∫       l   0     =   0     ∞     ⁢           ⁢     …   ⁢           ⁢       ∫       l   n     =   0     ∞     ⁢       ∫     θ   =   0       π   /   2       ⁢         P   ⁡     (       l   1     ,   …   ⁢           ,     l   n       )       ·     
     ⁢           ⁢     r   21       ⁢     ⅇ       -   2     ⁢           k     λ   ,   1       ⁢     l   1       +       k     λ   ,   2       ⁢     l   2       +   …   +       k     λ   ,   n       ⁢     l   n           cos   ⁡     (     θ   2     )             ⁢     sin   ⁡     (     2   ⁢   θ     )       ⁢     ⅆ   θ     ⁢     ⅆ     l   n       ⁢           ⁢   …   ⁢           ⁢     ⅆ     l   1                     ,     
     ⁢       I   Iλ     =         ∫       l   1     =   0     ∞     ⁢           ⁢     …   ⁢           ⁢       ∫       l   n     =   0     ∞     ⁢         (     1   +     W   λ       )     ·     P   ⁡     (       l   1     ,   …   ⁢           ,     l   n       )       ·     
     ⁢           ⁢     ⅇ     -           k     λ   ,   1       ⁢     l   1       +       k     λ   ,   2       ⁢     l   2       +   …   +       k     λ   ,   n       ⁢     l   n           cos   ⁡     (     θ   2     )               ⁢           ⁢     ⅆ     l   n       ⁢           ⁢   …   ⁢           ⁢     ⅆ     l   1               +     I   IUV         ,       
 
 where n is the number of ink involved, and l 1 , . . . ,l n  the thickness of the first ink, second ink, . . . , and nth ink. k λ,i  denotes the transmittance index of ink number i, at wavelength λ. k uv,i  denotes the transmittance index of ink number i, to ultra-violet light ( 833 ). Each parameter that depends on λ is a vector that has as many components as the measured spectrum. In the current implementation  36  components are used, representing wavelengths ranging from 380 nm (nanometers) to 730 nm in steps of 10 nm. 
 
         [0069]     The ink characteristics are composed of 
        The transmittance indexes vector k λ  ( 830 )     The transmittance value k uv  to ultra-violet light ( 833 )     R sf  the surface reflectance factor ( 832 )     μ the mean trapping parameter ( 834 ), see below        
 
         [0074]     The paper characteristics are composed of 
        The internal paper reflection parameter vector R pλ  ( 801 )     The paper whitener spectrum vector W λ  ( 803  and  804 )     A trapping correction factor μ ( 805 )        
 
         [0078]     The joint paper, printer and ink characteristics are composed of 
        The dependence of P(l 1 , . . . ,l n ) with the input command sent to the printer to generate the color patch, i.e. the tone reproduction curves ( 820 ).        
 
         [0080]     The printer characteristics are composed of 
        A parameterization of the probability density function P(l 1 , . . . ,l n ) reducing the span of the possible functions for a particular printer family, affecting the tone reproduction curves ( 820 ).     μ a trapping parameter for each ink superposition combination ( 821 )        
 
         [0083]     For example, for an offset printer, the probability density function can be expressed as 
 
 P ( l   1   , . . . ,l   n )= P   1 ( l   1 )* P   2 (μ(l 1 )·l 2 )* P   3 (μ( l   1   +l   2 )· l   3 )* . . . 
 
 where P i (·) is the probability density function of the thickness of ink number i. By convention, ink number i it the i th  ink printed on the paper. The symbol * denotes the convolution operation. 
 
         [0084]     P i (l) can be defined as  
           P   i     ⁡     (   l   )       =         a   i     ·     δ   ⁡     (   l   )         +       (     1   -     a   i       )     ·     ⅇ     -       l   -   1       2   ⁢     σ   i                     
 
 and the trapping parameter μ(l) is defined as  
         μ   ⁡     (   l   )       =     {         μ           if   ⁢           ⁢   l     &gt;   0             1       else               
 
         [0085]     In this configuration, μ and α i  belong to the joint paper, printer and ink characteristics.  
         [0086]     It will be appreciated by those skilled in the art that the present invention can be embodied in other specific forms without departing from the spirit or essential characteristics thereof. The presently disclosed embodiments are therefore considered in all respects to be illustrative and not restricted. The scope of the invention is indicated by the appended claims rather than the foregoing description and all changes that come within the meaning and range and equivalence thereof are intended to be embraced therein.