System and method for regularizing an ink model for a color device

1. A method for printing an image with a printer that uses three inks defining a three-dimensional ink-space and for which is available a set of three-dimensional color measurements {p i }(i=1, . . . , M) of color patches with three-ink combinations in the three-dimensional ink-space of the printer, the method comprising the steps of:

decomposing at least a portion of the three dimensional ink-space into a union of tetrahedrons Δ j (j=1, . . . , N) of which vertices correspond with the three-ink combinations of the color patches;

determining a first piece-wise linear three-ink printer model by calculating for each tetrahedron a linear transformation A j +B j *C(j=1, . . . , N), wherein A j is a 3×1 column matrix, B j is a 3×3 matrix, and C is a 3×1 column matrix representing a three-ink combination, the linear transformation mapping the three-ink combination in such a tetrahedron onto a color in a three-dimensional color space;

determining for each tetrahedron Δ j (j=1, . . . , N) a value of a determinant of the matrix B j ;

counting a number n neg of negative determinants and a number n pos of positive determinants;

if n pos >n neg :

selecting a positive real threshold value ε that provides a minimum value for the determinants;

defining an error functional R having a set of M colors as arguments, the error functional being a sum of measured distances between the real threshold value ε and the determinants according to the following rule:

R=R ( p 1 , . . . , p m )=Σ j=1, . . . , N R j ( p 1 , . . . , p m ); wherein

R j =R j ( p 1 , . . . , p M )=0 if det B j ≧ε; and

R j =R j ( p 1 , . . . , p M )=(ε−det B j i ) 2 if det B j i<ε

if n pos <n neg :

selecting a negative real threshold value ε that provides a maximum value for the determinants;

defining an error functional R having a set of M colors as arguments, the error functional being a sum of measured distances between the real threshold ε and the determinants according to the following rule:

R=R ( p 1 , . . . , p m )=Σ j=1, . . . , N R j ( p 1 , . . . , p m ); wherein

R j =R j ( p 1 , . . . , p M )=0 if det B j ≧ε; and

R j =R j ( p 1 , . . . , p M )=(ε−det B j i ) 2 if det B j i<ε

searching for a set of colors {p i 0 }(i=1, . . . ; M) that minimizes the error functional;

using the set of colors {p i 0 } to replace the set of color measurements {p i };

obtaining a second piece-wise linear three-ink printer model from a second set of color measurements that is regular by calculating for each tetrahedron a linear transformation F|Δ j (c)=A j 0 +B j 0 xC, wherein A j 0 is a 3×1 column matrix, B j 0 is a 3×3 matrix, and C is a 3×1 column matrix representing a three-ink combination, the linear transformation mapping a three-ink combination in such a tetrahedron onto a color in the three-dimensional color space;

using the second piece-wise linear three-ink printer model to create separation tables to convert colors in an image into sets of three inks;

using the separation tables to separate an image that is to be printed into a set of separations; and

using the set of separations to print the image.

2. A method for printing an image with a printer that uses four inks defining a four-dimensional ink-space and for which is available a set of three-dimensional color measurements {p i }(i=1, . . . , M) of color patches with four-ink combinations (C k , k=1, 2 , 3, 4) in the four dimensional ink-space, wherein for a given color the ink C 4 can be exchanged for a combination of inks C 1 , C 2 , C 3 , the method comprising the steps of:

decomposing at least a portion of the four dimensional ink-space into a union of pentahedrons Δ j (j=1, . . . ,N) of which vertices correspond with the four-ink combinations of the color patches;

determining a first piece-wise linear four-ink printer model by calculating for each pentahedron a linear transformation A j +B j *C(j=1, . . . , N), wherein A j is a 3×1 column matrix, B j is a 3×4 matrix, and C is a 4×1 column matrix representing a four-ink combination, the linear transformation mapping a four-ink combination in such a pentahedron onto a color in a three-dimensional color space;

determining for each pentahedron Δ j (j=1, . . . , N) a characteristic vector χ j that represents a direction in the pentahedron along which all four-ink combinations map onto the same color in color space, the characteristic vector χ j having coordinates [χ j 1 , χ j 2 , χ j 3 , χ j 4 ]=[det B j 1 , −det B j 2 , det B j 3 , −det B j 4 ](j=1, . . . , N), wherein B jk (k=1, 2, 3, 4) are 3×3 matrices obtained by omitting the k th out of the four columns in the 3×4 matrix B j ;

counting a number n k neg of negative coordinates of the characteristic vector coordinates and a number n k pos of positive coordinates of the characteristic vectors;

if n k pos >n k neg for k=1, 2, 3 and if n 4 pos <n 4 neg :

selecting a positive real threshold value ε;

defining an error functional R having a set of N colors as arguments, the error functional being a sum of measured distances between the real threshold value ε and the coordinates according to the following rule:

R=R ( p 1 , . . . , p M )=Σ k=1, 2, 3, 4, j=1, . . . , N R j k ( p 1 , . . . , p M );

wherein for k=1, 2, 3:

R j k =R j k (p 1 , . . . , p M )=0 if χ j k ≧ε; and

R j k =R j k (p 1 , . . . , p M )=(ε−χ j k χ j k ) 2 if χ j k <ε;

wherein for k=4:

R j 4 =R j 4 (p 1 , . . . , p M )=0 if χ j 4 ≦−ε; and

R j 4 =R j 4 (p 1 , . . . , p M )=(ε−χ j ′) 2 if χ j 4 >−ε(j=1, . . . , N) ;

if n k pos <n k neg for i=1, 2, 3 and if n 4 pos >n 4 neg :

selecting a negative real threshold value ε;

defining an error functional R having a set of N colors as arguments, the error functional being a sum of measured distances between the real threshold ε and the coordinates according to the following rule:

R=R ( p 1 , . . . , p M )=Σ k=1, 2, 3, 4, j=1, . . . , N R j k ( p 1 , . . . , p M );

wherein for k=1, 2, 3:

R j k=R j k (p 1 , . . . , p M )=0 if χ j k ≦ε; and

R j k=R j k (p 1 , . . . , p M )=(ε+χ j k ) 2 if χ j k>ε;

wherein for k=4:

R j 4=R j 4 (p 1 , . . . , p M )=0 if χ j 4 ≧−ε; and

R j 4 =R j 4 (p 1 , . . . , p M )=(ε+χ j 4 )<−ε for j=1, . . . , N;

searching for a set of colors {p i 0 } that minimizes the error functional;

using the set of colors {p i 0 } to replace the set of color measurements {p i };

obtaining a second piece-wise linear four-ink printer model by calculating for each pentahedron a linear transformation F|Δ j (c)=A j 0 +B j 0 ×C, wherein A j 0 is a 3×1 column matrix, B j 0 is a 3×4 matrix, and C is a 4×1 column matrix representing a four-ink combination, the linear transformation mapping a four-ink combination in such a pentahedron onto a color in the three-dimensional color space;

using the second piece-wise linear four-ink printer model to create separation tables to convert colors in an image into sets of four inks;

using the separation tables to separate an image that is to be printed into a set of separations;

using the set of separations to print the image.

3. A method for printing an image with a printer that uses four inks defining a four-dimensional ink-space for which is available a set of three-dimensional color measurements {p i }(i=1, . . . , M) of color patches with four-ink combinations (C k, k= 1, 2, 3, 4) in the four dimensional ink-space, wherein for a given color the inks C 4 and C 4 can be exchanged for a combination of inks C 1 and C 4 , the method comprising the steps of:

decomposing at least a portion of the four dimensional ink-space into a union of pentahedrons Δ j (j=1, . . . , N) of which vertices correspond with the four-ink combinations of the color patches;

determining a first piece-wise linear four-ink printer model by calculating for each pentahedron a linear transformation A j +B j *C(j=1, . . . , N), wherein A j is a 3×1 column matrix, B j is a 3×4 matrix, and C is a 4×1 column matrix representing a four-ink combination, the linear transformation mapping a four-ink combination in such a pentahedron onto a color in a three-dimensional color space;

determining for each pentahedron Δ j (j=1, . . . , N) a characteristic vector χ j that represents a direction in the pentahedron along which all four-ink combinations map onto the same color in color space, the characteristic vector χ j having coordinates [χ j 1 , χ j 2 , χ j 3 , χ j 4 ]=[det B j 1 , −det B j 2 , det B j 3 , −det B j 4 ](j=1, . . . , N), wherein B jk (k=1, 2, 3, 4) are 3×3 matrices obtained by omitting the k th out of the four columns in the 3×4 matrix B j ;

counting a number n k neg of negative coordinates of the characteristic vector coordinates and a number n k pos of positive coordinates of the characteristic vectors;

if n k pos >n k neg for k=2, 3 and if n k pos<n k neg for k=1, 4:

selecting a positive real threshold value ε;

defining an error functional R having a set of N colors as arguments, the error functional being a sum of measured distances between the real threshold value ε and the coordinates according to the following rule:

R=R ( p 1 , . . . , p M )=Σ k=1, 2, 3, 4, j=1, . . . , N R j k ( p 1 , . . . , p M );

wherein for k=2, 3:

R j k =R j k (p 1 , . . . , p M )=0 if χ j k ≧ε; and

R j k =R j k (p 1 , . . . , p M )=(ε−χ j k χ j k ) 2 if χ j k <ε;

wherein for k=1, 4:

R j 4 =R j k (p 1 , . . . , p M )=0 if χ j k ≦−ε; and

R j 4 =R j k (p 1 , . . . , p M )=(ε−χ j k ) 2 if χ j k >ε(j=1, . . . , N);

if n k pos <n k neg for i=2, 3 and if n k pos >n k neg for k=1, 4:

selecting a negative real threshold value ε;

defining an error functional R having a set of N colors as arguments, the error functional being a sum of measured distances between the real threshold ε and the coordinates according to the following rule:

R=R (p 1 , . . . , p M )=Σ k=1, 2 , 3, 4, j=1, . . . , N R j k ( p 1 , . . . ,p M);

wherein for k=2, 3:

R j k =R j k (p 1 , . . . , p M )=0 if χ j k ≦ε; and

R j k =R j k (p 1 , . . . , p M )=(ε+χ j k ) 2 if χ j k >ε;

wherein for k=1, 4:

R j 4 =R j 4 (p 1 , . . . , p M )=0 if χ j 4 ≧ε; and

R j 4 =R j 4 (p 1 , . . . , p M )=(ε+χ j 4 ) 2 if χ j 4 <−ε for j=1, . . . , N;

searching for a set of colors {p i 0 } that minimizes the error functional;

using the set of colors {p i 0 } to replace the set of color measurements {p i };

obtaining a second piece-wise linear four-ink printer model by calculating for each pentahedron a linear transformation F|Δ j (c)=A j 0 +B j 0 ×C, wherein A j 0 is a 3×1 column matrix, B j 0 is a 3×4 matrix, and C is a 4×1 column matrix representing a four-ink combination, the linear transformation mapping a four-ink combination in such a pentahedron onto a color in the three-dimensional color space;

using the second piece-wise linear four-ink printer model to create separation tables to convert colors in an image into sets of four inks;

using the separation tables to separate an image that is to be printed into a set of separations; and

using the set of separations to print the image.

4. The method according to claim 1 , wherein the printer uses a CMY or an RGB three-ink process.

5. The method according to claim 2 , wherein the printer uses a CMYK four-ink process.

6. The method according to claim 3 , wherein the printer uses a OMYK process or a GCYK process.

7. A method for printing an image with a printer that uses n inks defining a n-dimensional ink-space and for which is available a first set of three-dimensional color measurements {p i}(i= 1, . . . , M) of color patches with n-ink combinations (C k , k=1, . . . n) in an the n-dimensional ink-space, the method comprising the steps of:

obtaining a plurality of piece-wise four-ink models, each piece-wise-linear four-ink model of the plurality of piece-wise-linear four-ink models is a subset of four of the n-inks, each of the four-ink printer model having a global ink exchange that defines its error functional;

minimizing an error functional that is a sum of the error functionals of the four-ink printer model using a method according to claim 2 depending on the global ink exchange to obtain a second set of three-dimensional color measurements {p 0 i }(i=1, . . . , M);

using the second set of three-dimensional color measurements {p 0 i }(i=1, . . . , M) to create separation tables to convert colors in an image into sets of n inks;

using the separation tables to separate an image that is to be printed into a set of separations; and

using the set of separations to print the image.

8. A method for printing an image with a printer that uses three inks defining a three-dimensional ink-space and for which is available a first set of three-dimensional color measurements {p i }(i=1, . . . , M) of color patches with n-ink combinations (C k , k=1, . . . n) in the n-dimensional ink-space, the method comprising the steps of:

obtaining a plurality of piece-wise four-ink models, each piece-wise-linear four-ink model of the plurality of piece-wise-linear four-ink models is a subset of four of the n-inks, each of the four-ink printer model having a global ink exchange that defines its error functional;

minimizing an error functional that is a sum of the error functionals of the four-ink printer model using a method according to claim 3 depending on the global ink exchange to obtain a second set of three-dimensional color measurements {p 0 i }(i=1, . . . , M);

using the second piece-wise linear three-ink printer model to create separation tables to convert colors in an image into sets of three inks;

using the separation tables to separate an image that is to be printed into a set of separations; and

using the set of separations to print the image with a printer that uses three inks and for which is available.