Abstract

The reproduction of color images by color halftoning can be characterized by the Neugebauer model/equation. However, the Neugebauer equation is not easy to solve because of the highly nonlinear relationship between the underlying Neugebauer primaries and the colorants. We attempt to solve the Neugebauer equation by vector space methods. The proposed method of solution is applicable to any number of colorants, although our experimental results are confined to the CMY and CMYK cases. Among the constraints we consider are those related to a bound on the permissible amount of total ink and a bound on the total cost of applying colorants to achieve a satisfactory level of color reproduction. Our results demonstrate that the vector space method is a feasible approach for solving for the required amounts of colorants in the constrained color halftoning problem.

© 2006 Optical Society of America

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References

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  1. J. A. C. Yule, Principles of Color Reproduction (Wiley, 1967).
  2. H. R. Kang, Digital Color Halftoning (SPIE, 1999).
  3. H. E. J. Neugebauer, "Die Theoretischen Grundlagen des Mehrfarbenbuchdrucks (The theoretical foundation for multicolor printing)," Z. Wiss. Photogr. 36, 73-89 (1937).
  4. Adobe ICC Profiles (Adobe, 2003), http://www.adobe.com/support/downloads.
  5. E. J. Giorgianni and T. E. Madden, Digital Color Management (Addison-Wesley, 1998).
  6. L. Liu, Y. Yang, and H. Stark, "Spatial processing in color reproduction," J. Opt. Soc. Am. A 8, 1482-1491 (2005).
    [CrossRef]
  7. H. J. Trussell, "Application of set theoretic methods to color system," Color Res. Appl. 16, 31-64 (1991).
    [CrossRef]
  8. Y. Yang and H. Stark, "Solutions of several color-matching problems using projection theory," J. Opt. Soc. Am. A 11, 89-96 (1994).
    [CrossRef]
  9. G. Sharma and H. J. Trussell, "Set theoretic estimation in color scanner characterization," J. Electron. Imaging 5, 479-489 (1996).
    [CrossRef]
  10. G. Sharma, "Set theoretic estimation for problems in subtractive color," Color Res. Appl. 25, 333-348 (2000).
    [CrossRef]
  11. H. Stark and Y. Yang, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics, Wiley Series in Telecommunications and Signal Processing (Wiley,1998).
    [PubMed]
  12. I. Pobboravsky and M. Pearson, "Computation of dot areas required to match a colorimetrically specified color using the modified Neugebauer equations," in Proceedings 1972 of Technical Association of Graphics Arts (Technical Association of Graphics Arts, 1972), pp. 65-77.
  13. M. Mahy, "Color separation method and apparatus for same," U.S. patent 5,878,195 (2 March 1999).
  14. R. K. Molla, Electronic Color Separation (R. K. Printing & Publishing, 1988).
  15. A. Levi and H. Stark, "Image restoration by the method of generalized projections with application to restoration from magnitude," J. Opt. Soc. Am. A 1, 932-943 (1984).
    [CrossRef]
  16. D. C. Youla, "Generalized image restoration by the method of alternating orthogonal projections," IEEE Trans. Circuits Syst. 25, 694-702 (1978).
    [CrossRef]
  17. D. C. Youla and H. Webb, "Image restoration by the method of convex projections: part 1--theory," IEEE Trans. Med. Imaging MI-1, 81-94 (1982).
    [CrossRef]
  18. M. I. Sezan and H. Stark, "Image restoration by the method of convex projections: part 2--applications," IEEE Trans. Med. Imaging MI-1, 95-101 (1982).
    [CrossRef]
  19. A. Levi and H. Stark, "Signal restoration from phase by projections onto convex sets," J. Opt. Soc. Am. 73, 810-822 (1983).
    [CrossRef]
  20. P. L. Combettes, "The foundations of set theoretic estimation," Proc. IEEE 81, 182-208 (1993).
    [CrossRef]
  21. H. J. Trussell and M. Civanlar, "The feasible solution in signal restoration," IEEE Trans. Acoust., Speech, Signal Process. 32, 201-212 (1984).
    [CrossRef]
  22. J. P. Allebach, "Reconstruction of continuous-tone from halftone by projections onto convex sets," in Proceedings of the 1988 International Conference on Advances in Communication and Control Systems (Optimization Software, 1988), pp. 469-478.
  23. Y. Yang and N. P. Galatsanos, "Removal of compression artifacts using projections onto convex sets and line process modeling," IEEE Trans. Image Process. 6, 1345-1357 (1997).
    [CrossRef] [PubMed]
  24. P. L. Combettes and J. C. Pesquet, "Image restoration subject to a total variation constraint," IEEE Trans. Image Process. 13, 1213-1222 (2004).
    [CrossRef] [PubMed]
  25. K. Sayanagi, "Black printer, UCR and UCA--gray component replacement," in Proceedings 1987 of Technical Association of Graphics Arts (Technical Association of Graphics Arts, 1987), pp. 711-724.
  26. ICC Specification ICC.1:2004-04 : Image Technology Colour Management--Architecture, Profile Format, and Data Structure (International Color Consortium, 2004).
    [PubMed]

2005 (1)

L. Liu, Y. Yang, and H. Stark, "Spatial processing in color reproduction," J. Opt. Soc. Am. A 8, 1482-1491 (2005).
[CrossRef]

2004 (1)

P. L. Combettes and J. C. Pesquet, "Image restoration subject to a total variation constraint," IEEE Trans. Image Process. 13, 1213-1222 (2004).
[CrossRef] [PubMed]

2000 (1)

G. Sharma, "Set theoretic estimation for problems in subtractive color," Color Res. Appl. 25, 333-348 (2000).
[CrossRef]

1997 (1)

Y. Yang and N. P. Galatsanos, "Removal of compression artifacts using projections onto convex sets and line process modeling," IEEE Trans. Image Process. 6, 1345-1357 (1997).
[CrossRef] [PubMed]

1996 (1)

G. Sharma and H. J. Trussell, "Set theoretic estimation in color scanner characterization," J. Electron. Imaging 5, 479-489 (1996).
[CrossRef]

1994 (1)

1993 (1)

P. L. Combettes, "The foundations of set theoretic estimation," Proc. IEEE 81, 182-208 (1993).
[CrossRef]

1991 (1)

H. J. Trussell, "Application of set theoretic methods to color system," Color Res. Appl. 16, 31-64 (1991).
[CrossRef]

1984 (2)

H. J. Trussell and M. Civanlar, "The feasible solution in signal restoration," IEEE Trans. Acoust., Speech, Signal Process. 32, 201-212 (1984).
[CrossRef]

A. Levi and H. Stark, "Image restoration by the method of generalized projections with application to restoration from magnitude," J. Opt. Soc. Am. A 1, 932-943 (1984).
[CrossRef]

1983 (1)

1982 (2)

D. C. Youla and H. Webb, "Image restoration by the method of convex projections: part 1--theory," IEEE Trans. Med. Imaging MI-1, 81-94 (1982).
[CrossRef]

M. I. Sezan and H. Stark, "Image restoration by the method of convex projections: part 2--applications," IEEE Trans. Med. Imaging MI-1, 95-101 (1982).
[CrossRef]

1978 (1)

D. C. Youla, "Generalized image restoration by the method of alternating orthogonal projections," IEEE Trans. Circuits Syst. 25, 694-702 (1978).
[CrossRef]

Allebach, J. P.

J. P. Allebach, "Reconstruction of continuous-tone from halftone by projections onto convex sets," in Proceedings of the 1988 International Conference on Advances in Communication and Control Systems (Optimization Software, 1988), pp. 469-478.

Civanlar, M.

H. J. Trussell and M. Civanlar, "The feasible solution in signal restoration," IEEE Trans. Acoust., Speech, Signal Process. 32, 201-212 (1984).
[CrossRef]

Combettes, P. L.

P. L. Combettes and J. C. Pesquet, "Image restoration subject to a total variation constraint," IEEE Trans. Image Process. 13, 1213-1222 (2004).
[CrossRef] [PubMed]

P. L. Combettes, "The foundations of set theoretic estimation," Proc. IEEE 81, 182-208 (1993).
[CrossRef]

Galatsanos, N. P.

Y. Yang and N. P. Galatsanos, "Removal of compression artifacts using projections onto convex sets and line process modeling," IEEE Trans. Image Process. 6, 1345-1357 (1997).
[CrossRef] [PubMed]

Giorgianni, E. J.

E. J. Giorgianni and T. E. Madden, Digital Color Management (Addison-Wesley, 1998).

Kang, H. R.

H. R. Kang, Digital Color Halftoning (SPIE, 1999).

Levi, A.

Liu, L.

L. Liu, Y. Yang, and H. Stark, "Spatial processing in color reproduction," J. Opt. Soc. Am. A 8, 1482-1491 (2005).
[CrossRef]

Madden, T. E.

E. J. Giorgianni and T. E. Madden, Digital Color Management (Addison-Wesley, 1998).

Mahy, M.

M. Mahy, "Color separation method and apparatus for same," U.S. patent 5,878,195 (2 March 1999).

Molla, R. K.

R. K. Molla, Electronic Color Separation (R. K. Printing & Publishing, 1988).

Neugebauer, H. E.

H. E. J. Neugebauer, "Die Theoretischen Grundlagen des Mehrfarbenbuchdrucks (The theoretical foundation for multicolor printing)," Z. Wiss. Photogr. 36, 73-89 (1937).

Pearson, M.

I. Pobboravsky and M. Pearson, "Computation of dot areas required to match a colorimetrically specified color using the modified Neugebauer equations," in Proceedings 1972 of Technical Association of Graphics Arts (Technical Association of Graphics Arts, 1972), pp. 65-77.

Pesquet, J. C.

P. L. Combettes and J. C. Pesquet, "Image restoration subject to a total variation constraint," IEEE Trans. Image Process. 13, 1213-1222 (2004).
[CrossRef] [PubMed]

Pobboravsky, I.

I. Pobboravsky and M. Pearson, "Computation of dot areas required to match a colorimetrically specified color using the modified Neugebauer equations," in Proceedings 1972 of Technical Association of Graphics Arts (Technical Association of Graphics Arts, 1972), pp. 65-77.

Sayanagi, K.

K. Sayanagi, "Black printer, UCR and UCA--gray component replacement," in Proceedings 1987 of Technical Association of Graphics Arts (Technical Association of Graphics Arts, 1987), pp. 711-724.

Sezan, M. I.

M. I. Sezan and H. Stark, "Image restoration by the method of convex projections: part 2--applications," IEEE Trans. Med. Imaging MI-1, 95-101 (1982).
[CrossRef]

Sharma, G.

G. Sharma, "Set theoretic estimation for problems in subtractive color," Color Res. Appl. 25, 333-348 (2000).
[CrossRef]

G. Sharma and H. J. Trussell, "Set theoretic estimation in color scanner characterization," J. Electron. Imaging 5, 479-489 (1996).
[CrossRef]

Stark, H.

L. Liu, Y. Yang, and H. Stark, "Spatial processing in color reproduction," J. Opt. Soc. Am. A 8, 1482-1491 (2005).
[CrossRef]

Y. Yang and H. Stark, "Solutions of several color-matching problems using projection theory," J. Opt. Soc. Am. A 11, 89-96 (1994).
[CrossRef]

A. Levi and H. Stark, "Image restoration by the method of generalized projections with application to restoration from magnitude," J. Opt. Soc. Am. A 1, 932-943 (1984).
[CrossRef]

A. Levi and H. Stark, "Signal restoration from phase by projections onto convex sets," J. Opt. Soc. Am. 73, 810-822 (1983).
[CrossRef]

M. I. Sezan and H. Stark, "Image restoration by the method of convex projections: part 2--applications," IEEE Trans. Med. Imaging MI-1, 95-101 (1982).
[CrossRef]

H. Stark and Y. Yang, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics, Wiley Series in Telecommunications and Signal Processing (Wiley,1998).
[PubMed]

Trussell, H. J.

G. Sharma and H. J. Trussell, "Set theoretic estimation in color scanner characterization," J. Electron. Imaging 5, 479-489 (1996).
[CrossRef]

H. J. Trussell, "Application of set theoretic methods to color system," Color Res. Appl. 16, 31-64 (1991).
[CrossRef]

H. J. Trussell and M. Civanlar, "The feasible solution in signal restoration," IEEE Trans. Acoust., Speech, Signal Process. 32, 201-212 (1984).
[CrossRef]

Webb, H.

D. C. Youla and H. Webb, "Image restoration by the method of convex projections: part 1--theory," IEEE Trans. Med. Imaging MI-1, 81-94 (1982).
[CrossRef]

Yang, Y.

L. Liu, Y. Yang, and H. Stark, "Spatial processing in color reproduction," J. Opt. Soc. Am. A 8, 1482-1491 (2005).
[CrossRef]

Y. Yang and N. P. Galatsanos, "Removal of compression artifacts using projections onto convex sets and line process modeling," IEEE Trans. Image Process. 6, 1345-1357 (1997).
[CrossRef] [PubMed]

Y. Yang and H. Stark, "Solutions of several color-matching problems using projection theory," J. Opt. Soc. Am. A 11, 89-96 (1994).
[CrossRef]

H. Stark and Y. Yang, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics, Wiley Series in Telecommunications and Signal Processing (Wiley,1998).
[PubMed]

Youla, D. C.

D. C. Youla and H. Webb, "Image restoration by the method of convex projections: part 1--theory," IEEE Trans. Med. Imaging MI-1, 81-94 (1982).
[CrossRef]

D. C. Youla, "Generalized image restoration by the method of alternating orthogonal projections," IEEE Trans. Circuits Syst. 25, 694-702 (1978).
[CrossRef]

Yule, J. A.

J. A. C. Yule, Principles of Color Reproduction (Wiley, 1967).

Color Res. Appl. (2)

G. Sharma, "Set theoretic estimation for problems in subtractive color," Color Res. Appl. 25, 333-348 (2000).
[CrossRef]

H. J. Trussell, "Application of set theoretic methods to color system," Color Res. Appl. 16, 31-64 (1991).
[CrossRef]

IEEE Trans. Acoust., Speech, Signal Process. (1)

H. J. Trussell and M. Civanlar, "The feasible solution in signal restoration," IEEE Trans. Acoust., Speech, Signal Process. 32, 201-212 (1984).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

D. C. Youla, "Generalized image restoration by the method of alternating orthogonal projections," IEEE Trans. Circuits Syst. 25, 694-702 (1978).
[CrossRef]

IEEE Trans. Image Process. (2)

Y. Yang and N. P. Galatsanos, "Removal of compression artifacts using projections onto convex sets and line process modeling," IEEE Trans. Image Process. 6, 1345-1357 (1997).
[CrossRef] [PubMed]

P. L. Combettes and J. C. Pesquet, "Image restoration subject to a total variation constraint," IEEE Trans. Image Process. 13, 1213-1222 (2004).
[CrossRef] [PubMed]

IEEE Trans. Med. Imaging (2)

D. C. Youla and H. Webb, "Image restoration by the method of convex projections: part 1--theory," IEEE Trans. Med. Imaging MI-1, 81-94 (1982).
[CrossRef]

M. I. Sezan and H. Stark, "Image restoration by the method of convex projections: part 2--applications," IEEE Trans. Med. Imaging MI-1, 95-101 (1982).
[CrossRef]

J. Electron. Imaging (1)

G. Sharma and H. J. Trussell, "Set theoretic estimation in color scanner characterization," J. Electron. Imaging 5, 479-489 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Proc. IEEE (1)

P. L. Combettes, "The foundations of set theoretic estimation," Proc. IEEE 81, 182-208 (1993).
[CrossRef]

Other (12)

J. P. Allebach, "Reconstruction of continuous-tone from halftone by projections onto convex sets," in Proceedings of the 1988 International Conference on Advances in Communication and Control Systems (Optimization Software, 1988), pp. 469-478.

H. Stark and Y. Yang, Vector Space Projections: A Numerical Approach to Signal and Image Processing, Neural Nets, and Optics, Wiley Series in Telecommunications and Signal Processing (Wiley,1998).
[PubMed]

I. Pobboravsky and M. Pearson, "Computation of dot areas required to match a colorimetrically specified color using the modified Neugebauer equations," in Proceedings 1972 of Technical Association of Graphics Arts (Technical Association of Graphics Arts, 1972), pp. 65-77.

M. Mahy, "Color separation method and apparatus for same," U.S. patent 5,878,195 (2 March 1999).

R. K. Molla, Electronic Color Separation (R. K. Printing & Publishing, 1988).

J. A. C. Yule, Principles of Color Reproduction (Wiley, 1967).

H. R. Kang, Digital Color Halftoning (SPIE, 1999).

H. E. J. Neugebauer, "Die Theoretischen Grundlagen des Mehrfarbenbuchdrucks (The theoretical foundation for multicolor printing)," Z. Wiss. Photogr. 36, 73-89 (1937).

Adobe ICC Profiles (Adobe, 2003), http://www.adobe.com/support/downloads.

E. J. Giorgianni and T. E. Madden, Digital Color Management (Addison-Wesley, 1998).

K. Sayanagi, "Black printer, UCR and UCA--gray component replacement," in Proceedings 1987 of Technical Association of Graphics Arts (Technical Association of Graphics Arts, 1987), pp. 711-724.

ICC Specification ICC.1:2004-04 : Image Technology Colour Management--Architecture, Profile Format, and Data Structure (International Color Consortium, 2004).
[PubMed]

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Figures (3)

Fig. 1
Fig. 1

Surface of γ 4 = γ 1 γ 2 .

Fig. 2
Fig. 2

Test images in L a b channels. (a) Lena, (b) peppers.

Fig. 3
Fig. 3

Error images in L , a , and b channels for the (a) VSP and (b) GCR methods.

Tables (7)

Tables Icon

Table 1 Eight NPs Resulting from the Mixing of Subtractive Colorants Cyan, Magenta, Yellow a

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Table 2 Relationship between the Amounts of Applied Colorants β i , 0 β i 1 , i = 1 ,2,3 and the Fractions α j of NP j , j = 1 , , 8

Tables Icon

Table 3 Error Histogram in CMY Reproduction Using VSP a

Tables Icon

Table 4 Error Histogram in CMYK Reproduction Using VSP and Gray-Component Replacement (GCR) Methods

Tables Icon

Table 5 Average Color Difference in CMYK Reproduction Using VSP and GCR Methods

Tables Icon

Table 6 Per-Pixel Cost in CMYK Reproduction Using VSP and GCR Methods a

Tables Icon

Table 7 Relationship between the Amounts of Applied Colorants β i , i = 1 , , 4 and the Fraction α j of the NP j , j = 1 , , 16 , in the Four-Colorant Case

Equations (58)

Equations on this page are rendered with MathJax. Learn more.

c t = P α .
γ 1 β 1 , γ 2 β 2 , γ 3 β 3 , γ 4 β 1 β 2 , γ 5 β 1 β 3 , γ 6 β 2 β 3 , γ 7 β 1 β 2 β 3 .
α = u + M γ ,
M = [ 1 1 1 1 1 1 1 0 0 1 0 1 1 1 0 1 0 1 0 1 1 0 0 0 0 0 1 1 1 0 0 1 1 0 1 0 0 0 0 1 0 1 0 0 0 1 0 0 1 0 0 0 0 0 0 1 ] .
c t = P ( u + M γ ) ,
x k + 1 = P n P n 1 P 1 x k
C 1 = { γ R 7 : P M γ = c t P u } .
C 1 , i { γ R 7 : a i T γ = b i } , i = 1 , 2 , 3 .
C 2 = { γ R 7 : 0 γ i 1 , i = 1 , , 7 } .
C 3 = i = 1 6 C 3 , i ,
C 3 , 1 = { γ : γ 4 = γ 1 γ 2 } , C 3 , 2 = { γ : γ 5 = γ 1 γ 3 } ,
C 3 , 3 = { γ : γ 6 = γ 2 γ 3 } , C 3 , 4 = { γ : γ 7 = γ 1 γ 6 } ,
C 3 , 5 = { γ : γ 7 = γ 2 γ 5 } , C 3 , 6 = { γ : γ 7 = γ 3 γ 4 } .
C T ( ϵ ) { γ R 15 : i = 1 4 γ i ϵ } .
C c ( ω , η ) { γ R 15 : j = 1 4 ω j γ j η } .
γ k + 1 = P ̃ 3 P 2 P ̃ 1 γ k ,
β 4 ( 4 ) = min { β i ( 3 ) , i = 1 , 2 , 3 } ,
β i ( 4 ) = ( β i ( 3 ) 0.95 β 4 ( 4 ) ) ( 1 0.95 β 4 ( 4 ) ) , i = 1 , 2 , 3 ,
C c ( ω , η ) = { γ ̂ R 15 N : n = 1 N p n ω T γ n η } ,
p 9 = ( 0.0184 0.0192 0.0149 ) T ,
p 10 = ( 0.0165 0.0178 0.0058 ) T ,
p 11 = ( 0.0120 0.0088 0.0077 ) T ,
p 12 = ( 0.0114 0.0087 0.0045 ) T ,
p 13 = ( 0.0081 0.0107 0.0135 ) T ,
p 14 = ( 0.0067 0.0097 0.0054 ) T ,
p 15 = ( 0.0057 0.0055 0.0071 ) T ,
p 16 = ( 0.0052 0.0054 0.0043 ) T .
γ 1 = β 1 , γ 2 = β 2 , γ 3 = β 3 , γ 4 = β 4 ,
γ 5 = β 1 β 2 , γ 6 = β 1 β 3 , γ 7 = β 1 β 4 , γ 8 = β 2 β 3 ,
γ 9 = β 2 β 4 , γ 10 = β 3 β 4 ,
γ 11 = β 1 β 2 β 3 , γ 12 = β 1 β 2 β 4 , γ 13 = β 1 β 3 β 4 ,
γ 14 = β 2 β 3 β 4 , γ 15 = β 1 β 2 β 3 β 4 .
M = [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 0 0 1 0 1 1 0 1 1 1 1 0 0 1 0 0 1 0 1 0 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 0 0 1 1 0 1 1 0 1 1 0 0 0 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 1 1 1 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 ] .
C 1 = { γ R 15 : A γ = b } ,
C 1 , i { γ R 15 : a i T γ = b i } , i = 1 , 2 , 3 ,
C 2 = { γ R 15 : 0 γ i 1 , i = 1 , , 15 } .
C 3 = i = 1 11 C 3 , i ,
C 3 , 1 = { γ R 15 : γ 5 = γ 1 γ 2 } , C 3 , 2 = { γ R 15 : γ 6 = γ 1 γ 3 } ,
C 3 , 3 = { γ R 15 : γ 7 = γ 1 γ 4 } , C 3 , 4 = { γ R 15 : γ 8 = γ 2 γ 3 } ,
C 3 , 5 = { γ R 15 : γ 9 = γ 2 γ 4 } , C 3 , 6 = { γ R 15 : γ 10 = γ 3 γ 4 } ,
C 3 , 7 = { γ R 15 : γ 11 = γ 1 γ 8 } , C 3 , 8 = { γ R 15 : γ 12 = γ 4 γ 5 } ,
C 3 , 9 = { γ R 15 : γ 13 = γ 4 γ 6 } , C 3 , 10 = { γ R 15 : γ 14 = γ 4 γ 8 } ,
C 3 , 11 = { γ R 15 : γ 15 = γ 5 γ 10 } .
y * = P 1 , i x = { x , if x , a i = b i x + b i x , a i a i 2 a i T , else } .
y * = P 2 x ( y 1 * , , y K * ) T ,
y i * = { 1 , if x i > 1 0 , if x i < 0 x i , else } for i = 1 , , K .
L ( y , x , λ ) = i = 1 7 ( y i x i ) 2 + λ ( y 4 y 1 y 2 ) .
y 1 * = 2 λ * x 2 4 x 1 ( 4 λ * 2 ) , y 2 * = 2 λ * x 1 4 x 2 ( 4 λ * 2 ) , y 4 * = x 4 λ * 2 ,
λ 5 2 x 3 λ 4 8 λ 3 + 8 ( 2 x 3 + x 1 x 2 ) λ 2 + 16 ( x 1 2 + x 2 2 + 1 ) λ 32 ( x 3 x 1 x 2 ) = 0 .
y * = P 3 , 1 x { x , when x 4 = x 1 x 2 ( y 1 * y 2 * x 3 y 4 * x 5 x 6 x 7 ) T , when x 4 x 1 x 2 } ,
y 1 * = 2 λ * x 2 4 x 1 ( 4 λ * 2 ) , y 2 * = 2 λ * x 1 4 x 2 ( 4 λ * 2 ) , y 4 * = x 4 λ * 2 .
y 1 * = ( 3 x 1 4 ) ( x 2 + x 3 + x 4 ϵ ) 4 ,
y 2 * = ( 3 x 2 4 ) ( x 1 + x 3 + x 4 ϵ ) 4 ,
y 3 * = ( 3 x 3 4 ) ( x 1 + x 2 + x 4 ϵ ) 4 ,
y 4 * = ( 3 x 4 4 ) ( x 1 + x 2 + x 3 ϵ ) 4 ,
y i * = x i , i = 5 , , 15 .
y * = P T x { x , if i = 1 4 x i ϵ ( y 1 * , , y 15 * ) T , else } ,
y * = P C c x = { x , if w , x η x + η x , w w , w w , else } .

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