Abstract

Digital halftoning is the process of converting a continuous-tone image into an arrangement of black and white dots for binary display devices such as digital ink-jet and electrophotographic printers. As printers are achieving print resolutions exceeding 1,200 dots per inch, it is becoming increasingly important for halftoning algorithms to consider the variations and interactions in the size and shape of printed dots between neighboring pixels. In the case of lenticular screening where statistically independent images are spatially multiplexed together, ignoring these variations and interactions, such as dot overlap, will result in poor lenticular image quality. To this end, we describe our use of model-based error-diffusion for the lenticular screening problem where statistical independence between component images is achieved by restricting the diffusion of error to only those pixels of the same component image where, in order to avoid instabilities, the proposed approach involves a novel error-clipping procedure.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. X. Didik, “A brief history of stereo images, printing and photography from 1692–2001,” Tech. Rep., Didik.com/Vari-Vue.com, 2001.
  2. M. Lake, “An art form that’s precise but friendly enough to wink,” New York Times, G11, May 20, 1999.
  3. R. A. Ulichney, “Dithering with blue noise,” Proc. of the IEEE 76, 56–79 (1988).
    [Crossref]
  4. J. E. Adamcewicz, “A study on the effects of dot gain, print contrast and tone reproduction as it relates to increased solid ink density on stochastically screened images with conventionally screened images,” M.S. thesis, Rochester Institute of Technology, 1994.
  5. M. Rodriguez, “Graphic arts perspective on digital halftoning,” in Human Vision, Visual Processing, and Digital Display V, B. E. Rogowitz and J. P. Allebach, eds., Proc. SPIE 2179, pp. 144–149 (1994).
    [Crossref]
  6. M. Rodriguez, “Promises and pitfalls of stochastic screening in the graphic arts industry,” IS&T’s Eighth International Congress on Advances in Non-Impact Printing Technologies, 1992.
  7. D. L. Lau and G. R. Arce, Modern Digital Halftoning, (Marcel Dekker, Inc., New York, New York, 2001).
  8. P. Fink, PostScript Screening: Adobe Accurate Screens, (Adobe Press, Mountain View, Calif., 1992).
  9. R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial gray-scale,” Proceedings Society Information Display17, 75–78 (1976).
  10. P. Stucki, “Mecca-a multiple-error correcting computation algorithm for bilevel image hardcopy reproduction,” Tech. Rep. RZ1060, IBM Research Laboratory, Zurich, Switzerland, 1981.
  11. R. A. Ulichney, Digital Halftoning, (MIT Press, Cambridge, MA, 1987).
  12. T. P. Goggins, “Method of producing multidimensional lithographic separations free of moire interference,” U.S. Patent No. 5,488,451 assigned to National Graphics, Inc., 1996.
  13. T. P. Goggins, “Method of producing a multidimensional composite image,” U.S. Patent No. 5,847,808 assigned to National Graphics, Inc., 1998.
  14. T. P. Goggins, “Method of producing multidimensional lithographic separations free of moire interference,” U.S. Patent No. 5,617,178 assigned to National Graphics, Inc., 2002.
  15. M. Pilu, “Halftoning of lenticular images,” U.S. Patent Application No. 20,030,011,824 filed by Hewlett Packard Co., January 2003.
  16. K. Yano, “Image forming system, apparatus, and method,” U.S. Patent Application No. 20,030,067,638 filed by Canon Kabushik Kaisha, April 2003.
  17. K. Iwano, H. Akahori, K. Nobori, K. Kayashima, M. Fujimoto, and Y. Kawamata, “Image printer for controlling the shape of pixels based upon correlation values,” U.S. Patent No. 6,459,470 filed by Matsushita Electric Industrial Co., Ltd., October 2002.
  18. P. G. Roetling and T. M. Holladay, “Tone reproduction and screen design for pictorial electrophotographic printing,” J. Appl. Photogr. Eng. 15, 179–182 (1979).
  19. T. N. Pappas and D. L. Neuhoff, “Printer models and error diffusion,” IEEE Trans. Image Process. 4, 66–79 (1995).
    [Crossref] [PubMed]
  20. J. H. Kim, T. I. Chung, H. S. Kim, K. S. Son, and Y. S. Kim, “A new edge-enhanced error diffusion algorithm based on the error sum criterion,” J. Electron. Imaging 4, 172–178 (1995).
    [Crossref]
  21. F. A. Baqai and J. P. Allebach, “Halftoning via direct binary search using analytical and stochastic printer model,” IEEE Trans. Image Process 12, 1–12 (2003).
    [Crossref]
  22. S. Wang, K. T. Knox, and N. George, “Novel centering method for overlapping correction in halftoning,” in Recent Progress in Digital Halftoning, R. Eschbach , ed., (Society for Imaging Science and Technology, Springfield, VA , 1994), pp. 56–60.
  23. T. N. Pappas, C. Dong, and D. L. Neuhoff, “Measurement of printer parameters for model-based halftoning,” J. Electron. Imaging 2, 193–204 (1993).
    [Crossref]
  24. M. Broja, R. Eschbach, and O. Bryngdahl, “Stability of active binarization processes,” Opt. Commun. 60, 353–358 (1985).
    [Crossref]
  25. S. Weissbach and F. Wyrowski, “Numerical stability of the error diffusion concept,” Opt. Commun. 93, 151–155 (1992).
    [Crossref]
  26. Z. Fan, “Stability analysis of error diffusion,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 321–324.
  27. Z. Fan, “Stability analysis of color error diffusion,” in Proceedings of the Symposium on Electronic Imaging Science and Technology, (Society for Imaging Science and Technology, Springfield, VA, 2000), pp. 483–488

2003 (1)

F. A. Baqai and J. P. Allebach, “Halftoning via direct binary search using analytical and stochastic printer model,” IEEE Trans. Image Process 12, 1–12 (2003).
[Crossref]

1995 (2)

T. N. Pappas and D. L. Neuhoff, “Printer models and error diffusion,” IEEE Trans. Image Process. 4, 66–79 (1995).
[Crossref] [PubMed]

J. H. Kim, T. I. Chung, H. S. Kim, K. S. Son, and Y. S. Kim, “A new edge-enhanced error diffusion algorithm based on the error sum criterion,” J. Electron. Imaging 4, 172–178 (1995).
[Crossref]

1994 (1)

M. Rodriguez, “Graphic arts perspective on digital halftoning,” in Human Vision, Visual Processing, and Digital Display V, B. E. Rogowitz and J. P. Allebach, eds., Proc. SPIE 2179, pp. 144–149 (1994).
[Crossref]

1993 (1)

T. N. Pappas, C. Dong, and D. L. Neuhoff, “Measurement of printer parameters for model-based halftoning,” J. Electron. Imaging 2, 193–204 (1993).
[Crossref]

1992 (1)

S. Weissbach and F. Wyrowski, “Numerical stability of the error diffusion concept,” Opt. Commun. 93, 151–155 (1992).
[Crossref]

1988 (1)

R. A. Ulichney, “Dithering with blue noise,” Proc. of the IEEE 76, 56–79 (1988).
[Crossref]

1985 (1)

M. Broja, R. Eschbach, and O. Bryngdahl, “Stability of active binarization processes,” Opt. Commun. 60, 353–358 (1985).
[Crossref]

1979 (1)

P. G. Roetling and T. M. Holladay, “Tone reproduction and screen design for pictorial electrophotographic printing,” J. Appl. Photogr. Eng. 15, 179–182 (1979).

Adamcewicz, J. E.

J. E. Adamcewicz, “A study on the effects of dot gain, print contrast and tone reproduction as it relates to increased solid ink density on stochastically screened images with conventionally screened images,” M.S. thesis, Rochester Institute of Technology, 1994.

Akahori, H.

K. Iwano, H. Akahori, K. Nobori, K. Kayashima, M. Fujimoto, and Y. Kawamata, “Image printer for controlling the shape of pixels based upon correlation values,” U.S. Patent No. 6,459,470 filed by Matsushita Electric Industrial Co., Ltd., October 2002.

Allebach, J. P.

F. A. Baqai and J. P. Allebach, “Halftoning via direct binary search using analytical and stochastic printer model,” IEEE Trans. Image Process 12, 1–12 (2003).
[Crossref]

Arce, G. R.

D. L. Lau and G. R. Arce, Modern Digital Halftoning, (Marcel Dekker, Inc., New York, New York, 2001).

Baqai, F. A.

F. A. Baqai and J. P. Allebach, “Halftoning via direct binary search using analytical and stochastic printer model,” IEEE Trans. Image Process 12, 1–12 (2003).
[Crossref]

Broja, M.

M. Broja, R. Eschbach, and O. Bryngdahl, “Stability of active binarization processes,” Opt. Commun. 60, 353–358 (1985).
[Crossref]

Bryngdahl, O.

M. Broja, R. Eschbach, and O. Bryngdahl, “Stability of active binarization processes,” Opt. Commun. 60, 353–358 (1985).
[Crossref]

Chung, T. I.

J. H. Kim, T. I. Chung, H. S. Kim, K. S. Son, and Y. S. Kim, “A new edge-enhanced error diffusion algorithm based on the error sum criterion,” J. Electron. Imaging 4, 172–178 (1995).
[Crossref]

Didik, F. X.

F. X. Didik, “A brief history of stereo images, printing and photography from 1692–2001,” Tech. Rep., Didik.com/Vari-Vue.com, 2001.

Dong, C.

T. N. Pappas, C. Dong, and D. L. Neuhoff, “Measurement of printer parameters for model-based halftoning,” J. Electron. Imaging 2, 193–204 (1993).
[Crossref]

Eschbach, R.

M. Broja, R. Eschbach, and O. Bryngdahl, “Stability of active binarization processes,” Opt. Commun. 60, 353–358 (1985).
[Crossref]

Fan, Z.

Z. Fan, “Stability analysis of error diffusion,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 321–324.

Z. Fan, “Stability analysis of color error diffusion,” in Proceedings of the Symposium on Electronic Imaging Science and Technology, (Society for Imaging Science and Technology, Springfield, VA, 2000), pp. 483–488

Fink, P.

P. Fink, PostScript Screening: Adobe Accurate Screens, (Adobe Press, Mountain View, Calif., 1992).

Floyd, R. W.

R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial gray-scale,” Proceedings Society Information Display17, 75–78 (1976).

Fujimoto, M.

K. Iwano, H. Akahori, K. Nobori, K. Kayashima, M. Fujimoto, and Y. Kawamata, “Image printer for controlling the shape of pixels based upon correlation values,” U.S. Patent No. 6,459,470 filed by Matsushita Electric Industrial Co., Ltd., October 2002.

George, N.

S. Wang, K. T. Knox, and N. George, “Novel centering method for overlapping correction in halftoning,” in Recent Progress in Digital Halftoning, R. Eschbach , ed., (Society for Imaging Science and Technology, Springfield, VA , 1994), pp. 56–60.

Goggins, T. P.

T. P. Goggins, “Method of producing multidimensional lithographic separations free of moire interference,” U.S. Patent No. 5,488,451 assigned to National Graphics, Inc., 1996.

T. P. Goggins, “Method of producing a multidimensional composite image,” U.S. Patent No. 5,847,808 assigned to National Graphics, Inc., 1998.

T. P. Goggins, “Method of producing multidimensional lithographic separations free of moire interference,” U.S. Patent No. 5,617,178 assigned to National Graphics, Inc., 2002.

Holladay, T. M.

P. G. Roetling and T. M. Holladay, “Tone reproduction and screen design for pictorial electrophotographic printing,” J. Appl. Photogr. Eng. 15, 179–182 (1979).

Iwano, K.

K. Iwano, H. Akahori, K. Nobori, K. Kayashima, M. Fujimoto, and Y. Kawamata, “Image printer for controlling the shape of pixels based upon correlation values,” U.S. Patent No. 6,459,470 filed by Matsushita Electric Industrial Co., Ltd., October 2002.

Kawamata, Y.

K. Iwano, H. Akahori, K. Nobori, K. Kayashima, M. Fujimoto, and Y. Kawamata, “Image printer for controlling the shape of pixels based upon correlation values,” U.S. Patent No. 6,459,470 filed by Matsushita Electric Industrial Co., Ltd., October 2002.

Kayashima, K.

K. Iwano, H. Akahori, K. Nobori, K. Kayashima, M. Fujimoto, and Y. Kawamata, “Image printer for controlling the shape of pixels based upon correlation values,” U.S. Patent No. 6,459,470 filed by Matsushita Electric Industrial Co., Ltd., October 2002.

Kim, H. S.

J. H. Kim, T. I. Chung, H. S. Kim, K. S. Son, and Y. S. Kim, “A new edge-enhanced error diffusion algorithm based on the error sum criterion,” J. Electron. Imaging 4, 172–178 (1995).
[Crossref]

Kim, J. H.

J. H. Kim, T. I. Chung, H. S. Kim, K. S. Son, and Y. S. Kim, “A new edge-enhanced error diffusion algorithm based on the error sum criterion,” J. Electron. Imaging 4, 172–178 (1995).
[Crossref]

Kim, Y. S.

J. H. Kim, T. I. Chung, H. S. Kim, K. S. Son, and Y. S. Kim, “A new edge-enhanced error diffusion algorithm based on the error sum criterion,” J. Electron. Imaging 4, 172–178 (1995).
[Crossref]

Knox, K. T.

S. Wang, K. T. Knox, and N. George, “Novel centering method for overlapping correction in halftoning,” in Recent Progress in Digital Halftoning, R. Eschbach , ed., (Society for Imaging Science and Technology, Springfield, VA , 1994), pp. 56–60.

Lake, M.

M. Lake, “An art form that’s precise but friendly enough to wink,” New York Times, G11, May 20, 1999.

Lau, D. L.

D. L. Lau and G. R. Arce, Modern Digital Halftoning, (Marcel Dekker, Inc., New York, New York, 2001).

Neuhoff, D. L.

T. N. Pappas and D. L. Neuhoff, “Printer models and error diffusion,” IEEE Trans. Image Process. 4, 66–79 (1995).
[Crossref] [PubMed]

T. N. Pappas, C. Dong, and D. L. Neuhoff, “Measurement of printer parameters for model-based halftoning,” J. Electron. Imaging 2, 193–204 (1993).
[Crossref]

Nobori, K.

K. Iwano, H. Akahori, K. Nobori, K. Kayashima, M. Fujimoto, and Y. Kawamata, “Image printer for controlling the shape of pixels based upon correlation values,” U.S. Patent No. 6,459,470 filed by Matsushita Electric Industrial Co., Ltd., October 2002.

Pappas, T. N.

T. N. Pappas and D. L. Neuhoff, “Printer models and error diffusion,” IEEE Trans. Image Process. 4, 66–79 (1995).
[Crossref] [PubMed]

T. N. Pappas, C. Dong, and D. L. Neuhoff, “Measurement of printer parameters for model-based halftoning,” J. Electron. Imaging 2, 193–204 (1993).
[Crossref]

Pilu, M.

M. Pilu, “Halftoning of lenticular images,” U.S. Patent Application No. 20,030,011,824 filed by Hewlett Packard Co., January 2003.

Rodriguez, M.

M. Rodriguez, “Graphic arts perspective on digital halftoning,” in Human Vision, Visual Processing, and Digital Display V, B. E. Rogowitz and J. P. Allebach, eds., Proc. SPIE 2179, pp. 144–149 (1994).
[Crossref]

M. Rodriguez, “Promises and pitfalls of stochastic screening in the graphic arts industry,” IS&T’s Eighth International Congress on Advances in Non-Impact Printing Technologies, 1992.

Roetling, P. G.

P. G. Roetling and T. M. Holladay, “Tone reproduction and screen design for pictorial electrophotographic printing,” J. Appl. Photogr. Eng. 15, 179–182 (1979).

Son, K. S.

J. H. Kim, T. I. Chung, H. S. Kim, K. S. Son, and Y. S. Kim, “A new edge-enhanced error diffusion algorithm based on the error sum criterion,” J. Electron. Imaging 4, 172–178 (1995).
[Crossref]

Steinberg, L.

R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial gray-scale,” Proceedings Society Information Display17, 75–78 (1976).

Stucki, P.

P. Stucki, “Mecca-a multiple-error correcting computation algorithm for bilevel image hardcopy reproduction,” Tech. Rep. RZ1060, IBM Research Laboratory, Zurich, Switzerland, 1981.

Ulichney, R. A.

R. A. Ulichney, “Dithering with blue noise,” Proc. of the IEEE 76, 56–79 (1988).
[Crossref]

R. A. Ulichney, Digital Halftoning, (MIT Press, Cambridge, MA, 1987).

Wang, S.

S. Wang, K. T. Knox, and N. George, “Novel centering method for overlapping correction in halftoning,” in Recent Progress in Digital Halftoning, R. Eschbach , ed., (Society for Imaging Science and Technology, Springfield, VA , 1994), pp. 56–60.

Weissbach, S.

S. Weissbach and F. Wyrowski, “Numerical stability of the error diffusion concept,” Opt. Commun. 93, 151–155 (1992).
[Crossref]

Wyrowski, F.

S. Weissbach and F. Wyrowski, “Numerical stability of the error diffusion concept,” Opt. Commun. 93, 151–155 (1992).
[Crossref]

Yano, K.

K. Yano, “Image forming system, apparatus, and method,” U.S. Patent Application No. 20,030,067,638 filed by Canon Kabushik Kaisha, April 2003.

IEEE Trans. Image Process (1)

F. A. Baqai and J. P. Allebach, “Halftoning via direct binary search using analytical and stochastic printer model,” IEEE Trans. Image Process 12, 1–12 (2003).
[Crossref]

IEEE Trans. Image Process. (1)

T. N. Pappas and D. L. Neuhoff, “Printer models and error diffusion,” IEEE Trans. Image Process. 4, 66–79 (1995).
[Crossref] [PubMed]

J. Appl. Photogr. Eng. (1)

P. G. Roetling and T. M. Holladay, “Tone reproduction and screen design for pictorial electrophotographic printing,” J. Appl. Photogr. Eng. 15, 179–182 (1979).

J. Electron. Imaging (2)

T. N. Pappas, C. Dong, and D. L. Neuhoff, “Measurement of printer parameters for model-based halftoning,” J. Electron. Imaging 2, 193–204 (1993).
[Crossref]

J. H. Kim, T. I. Chung, H. S. Kim, K. S. Son, and Y. S. Kim, “A new edge-enhanced error diffusion algorithm based on the error sum criterion,” J. Electron. Imaging 4, 172–178 (1995).
[Crossref]

Opt. Commun. (2)

M. Broja, R. Eschbach, and O. Bryngdahl, “Stability of active binarization processes,” Opt. Commun. 60, 353–358 (1985).
[Crossref]

S. Weissbach and F. Wyrowski, “Numerical stability of the error diffusion concept,” Opt. Commun. 93, 151–155 (1992).
[Crossref]

Proc. of the IEEE (1)

R. A. Ulichney, “Dithering with blue noise,” Proc. of the IEEE 76, 56–79 (1988).
[Crossref]

Proc. SPIE (1)

M. Rodriguez, “Graphic arts perspective on digital halftoning,” in Human Vision, Visual Processing, and Digital Display V, B. E. Rogowitz and J. P. Allebach, eds., Proc. SPIE 2179, pp. 144–149 (1994).
[Crossref]

Other (18)

M. Rodriguez, “Promises and pitfalls of stochastic screening in the graphic arts industry,” IS&T’s Eighth International Congress on Advances in Non-Impact Printing Technologies, 1992.

D. L. Lau and G. R. Arce, Modern Digital Halftoning, (Marcel Dekker, Inc., New York, New York, 2001).

P. Fink, PostScript Screening: Adobe Accurate Screens, (Adobe Press, Mountain View, Calif., 1992).

R. W. Floyd and L. Steinberg, “An adaptive algorithm for spatial gray-scale,” Proceedings Society Information Display17, 75–78 (1976).

P. Stucki, “Mecca-a multiple-error correcting computation algorithm for bilevel image hardcopy reproduction,” Tech. Rep. RZ1060, IBM Research Laboratory, Zurich, Switzerland, 1981.

R. A. Ulichney, Digital Halftoning, (MIT Press, Cambridge, MA, 1987).

T. P. Goggins, “Method of producing multidimensional lithographic separations free of moire interference,” U.S. Patent No. 5,488,451 assigned to National Graphics, Inc., 1996.

T. P. Goggins, “Method of producing a multidimensional composite image,” U.S. Patent No. 5,847,808 assigned to National Graphics, Inc., 1998.

T. P. Goggins, “Method of producing multidimensional lithographic separations free of moire interference,” U.S. Patent No. 5,617,178 assigned to National Graphics, Inc., 2002.

M. Pilu, “Halftoning of lenticular images,” U.S. Patent Application No. 20,030,011,824 filed by Hewlett Packard Co., January 2003.

K. Yano, “Image forming system, apparatus, and method,” U.S. Patent Application No. 20,030,067,638 filed by Canon Kabushik Kaisha, April 2003.

K. Iwano, H. Akahori, K. Nobori, K. Kayashima, M. Fujimoto, and Y. Kawamata, “Image printer for controlling the shape of pixels based upon correlation values,” U.S. Patent No. 6,459,470 filed by Matsushita Electric Industrial Co., Ltd., October 2002.

J. E. Adamcewicz, “A study on the effects of dot gain, print contrast and tone reproduction as it relates to increased solid ink density on stochastically screened images with conventionally screened images,” M.S. thesis, Rochester Institute of Technology, 1994.

F. X. Didik, “A brief history of stereo images, printing and photography from 1692–2001,” Tech. Rep., Didik.com/Vari-Vue.com, 2001.

M. Lake, “An art form that’s precise but friendly enough to wink,” New York Times, G11, May 20, 1999.

S. Wang, K. T. Knox, and N. George, “Novel centering method for overlapping correction in halftoning,” in Recent Progress in Digital Halftoning, R. Eschbach , ed., (Society for Imaging Science and Technology, Springfield, VA , 1994), pp. 56–60.

Z. Fan, “Stability analysis of error diffusion,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing, (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 321–324.

Z. Fan, “Stability analysis of color error diffusion,” in Proceedings of the Symposium on Electronic Imaging Science and Technology, (Society for Imaging Science and Technology, Springfield, VA, 2000), pp. 483–488

Supplementary Material (5)

» Media 1: MOV (267 KB)     
» Media 2: MOV (326 KB)     
» Media 3: MOV (309 KB)     
» Media 4: MOV (320 KB)     
» Media 5: MOV (320 KB)     

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1.
Fig. 1.

Illustration of the lenticular imaging process (provided by http://www.lenstar.org).

Fig. 2.
Fig. 2.

The component images of a lenticular image as seen through a lenticular lens array. [Media 1]

Fig. 3.
Fig. 3.

The component images of a lenticular image as seen through a lenticular lens array where error diffusion has been applied after spatial multiplexing. [Media 2]

Fig. 4.
Fig. 4.

The component images of a lenticular image as seen through a lenticular lens array where error diffusion has been applied prior to spatial multiplexing. [Media 3]

Fig. 5.
Fig. 5.

The component images of a lenticular image as seen through a lens array, using a printed dot model, where error diffusion has been applied prior to spatial multiplexing with tone correction. [Media 4]

Fig. 6.
Fig. 6.

Model-based error-diffusion.

Fig. 7.
Fig. 7.

Calculation of the error term in model-based error-diffusion.

Fig. 8.
Fig. 8.

The (top) traditional and (bottom) lenticular Stucki error-diffusion filters for a four component, lenticular image where halftoning can now be done after the spatially multiplexing of images but with the same results as if done prior to. Color coding has been used to indicate which component image each pixel belongs while the arrows indicate the raster scanning direction.

Fig. 9.
Fig. 9.

The component images of a lenticular sequence as seen through a lens array, using a printed-dot model, where the proposed model-based error diffusion technique has been applied after spatial multiplexing. [Media 5]

Equations (7)

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

y [ n ] = { 1 , if ( x [ n ] + x e [ n ] ) 0 0 , else
x e [ n ] = i = 1 M b i y e [ n i ]
y [ n ] = { 1 , if ( x [ n ] + e [ n ] ) 0 0 , else , where
e update [ n ] = y [ n ] ( x [ n ] + e [ n ] ) and
e update [ n + i ] = e [ n + i ] + b i e update [ n ] .
e [ n ] = i = 1 M b i ( y ˜ [ n i ] x [ n i ] )
y [ n ] = { 1 , if ( x [ n ] + i = 1 M b i ( y ˜ [ n i ] x [ n i ] ) ) 0 0 , else .

Metrics