Abstract

We have developed a new technique for generating homogeneously distributed irregular dot patterns useful for optical devices and digital halftoning technologies. To introduce irregularity, we use elaborately designed sequences called low-discrepancy sequences instead of pseudorandom numbers. We also use a molecular-dynamics redistribution method to improve the distribution of dots. Our method can produce arbitrary density distributions in accordance with a given design. The generated patterns are free from visible roughness as well as any moiré patterns when superimposed on other regular patterns. We demonstrate that our method effectively improves luminance uniformity and eliminates moiré patterns when used for a backlight unit of a liquid-crystal display.

© 2003 Optical Society of America

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  1. Y. Oki, “Novel backlight with high luminance and low power consumption by prism-on-light-pipe technology,” in Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 1998), pp. 157–160.
  2. K. Käläntär, S. Matsumoto, T. Onishi, K. Takizawa, “Optical micro deflector based functional light-guide plate for backlight unit,” in Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 2000), pp. 1029–1031 and references therein.
  3. I. Amidror, The theory of the Moiré Phenomenon (Kluwer Academic, Dordrecht, The Netherlands, 2000), Chap. 3.
  4. D. L. Lau, A. M. Khan, G. R. Arce, “Stochastic moiré,” in Proceedings of the 2001 Image Processing, Image Quality, Image Capture Systems Conference (Society for Imaging Science and Technology, Springfield, Va., 2001).
  5. R. A. Ulichney, “Dithering with blue noise,” Proc. IEEE 76, 56–79 (1988).
    [CrossRef]
  6. T. Mitsa, K. J. Parker, “Digital halftoning technique using blue-noise mask,” J. Opt. Soc. Am. A 9, 1920–1929 (1992).
    [CrossRef]
  7. D. L. Lau, G. R. Arce, N. C. Gallagher, “Digital halftoning via green noise masks,” J. Opt. Soc. Am. A 16, 1575–1586 (1999).
    [CrossRef]
  8. C. B. Atkins, J. P. Allebach, C. A. Bouman, “Halftone postprocessing for improved highlight rendition,” in Proceedings of the 1997 IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 1, pp. 791–794.
  9. S. Hiller, O. Deussen, A. Keller, “Tiled blue noise samples,” in Proceedings of Vision, Modeling, and Visualization 2001 (IOS, Amsterdam, The Netherlands, 2001), pp. 265–271.
  10. H. Taniguchi, Y. Hira, Y. Mori, “Liquid crystal display devices,” U.S. patent6,099,134 (1998).
  11. S. Tezuka, Uniform Random Numbers: Theory and Practice (Kluwer Academic, Boston, 1995).
  12. S. Ninomiya, S. Tezuka, “Toward real-time pricing of complex financial derivatives,” Appl. Math. Finance 3, 1–20 (1996).
    [CrossRef]
  13. S. Tezuka, “Polynomial arithmetic analogue of Halton sequences,” ACM (Assoc. Comput. Mach.) Trans. Model. Comput. Simul. 3, 99–107 (1993).
    [CrossRef]
  14. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 7.
  15. W. Purgathofer, R. F. Tobler, M. Geiler, “Forced random dithering: improved threshold matrices for ordered dithering,” in Proceedings of the First IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1994), pp. 1032–1035.
  16. T. Idé, H. Numata, H. Mizuta, Y. Taira, M. Suzuki, M. Noguchi, Y. Katsu, “Moiré-free collimating light-guide with low-discrepancy dot patterns,” in Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 2002), pp. 1232–1235.
  17. G. Lebanon, A. L. Bruckstein, “Variational approach to moiré pattern synthesis,” J. Opt. Soc. Am. A 18, 1371–1382 (2001).
    [CrossRef]

2001 (1)

1999 (1)

1996 (1)

S. Ninomiya, S. Tezuka, “Toward real-time pricing of complex financial derivatives,” Appl. Math. Finance 3, 1–20 (1996).
[CrossRef]

1993 (1)

S. Tezuka, “Polynomial arithmetic analogue of Halton sequences,” ACM (Assoc. Comput. Mach.) Trans. Model. Comput. Simul. 3, 99–107 (1993).
[CrossRef]

1992 (1)

1988 (1)

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

Allebach, J. P.

C. B. Atkins, J. P. Allebach, C. A. Bouman, “Halftone postprocessing for improved highlight rendition,” in Proceedings of the 1997 IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 1, pp. 791–794.

Amidror, I.

I. Amidror, The theory of the Moiré Phenomenon (Kluwer Academic, Dordrecht, The Netherlands, 2000), Chap. 3.

Arce, G. R.

D. L. Lau, G. R. Arce, N. C. Gallagher, “Digital halftoning via green noise masks,” J. Opt. Soc. Am. A 16, 1575–1586 (1999).
[CrossRef]

D. L. Lau, A. M. Khan, G. R. Arce, “Stochastic moiré,” in Proceedings of the 2001 Image Processing, Image Quality, Image Capture Systems Conference (Society for Imaging Science and Technology, Springfield, Va., 2001).

Atkins, C. B.

C. B. Atkins, J. P. Allebach, C. A. Bouman, “Halftone postprocessing for improved highlight rendition,” in Proceedings of the 1997 IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 1, pp. 791–794.

Bouman, C. A.

C. B. Atkins, J. P. Allebach, C. A. Bouman, “Halftone postprocessing for improved highlight rendition,” in Proceedings of the 1997 IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 1, pp. 791–794.

Bruckstein, A. L.

Deussen, O.

S. Hiller, O. Deussen, A. Keller, “Tiled blue noise samples,” in Proceedings of Vision, Modeling, and Visualization 2001 (IOS, Amsterdam, The Netherlands, 2001), pp. 265–271.

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 7.

Gallagher, N. C.

Geiler, M.

W. Purgathofer, R. F. Tobler, M. Geiler, “Forced random dithering: improved threshold matrices for ordered dithering,” in Proceedings of the First IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1994), pp. 1032–1035.

Hiller, S.

S. Hiller, O. Deussen, A. Keller, “Tiled blue noise samples,” in Proceedings of Vision, Modeling, and Visualization 2001 (IOS, Amsterdam, The Netherlands, 2001), pp. 265–271.

Hira, Y.

H. Taniguchi, Y. Hira, Y. Mori, “Liquid crystal display devices,” U.S. patent6,099,134 (1998).

Idé, T.

T. Idé, H. Numata, H. Mizuta, Y. Taira, M. Suzuki, M. Noguchi, Y. Katsu, “Moiré-free collimating light-guide with low-discrepancy dot patterns,” in Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 2002), pp. 1232–1235.

Käläntär, K.

K. Käläntär, S. Matsumoto, T. Onishi, K. Takizawa, “Optical micro deflector based functional light-guide plate for backlight unit,” in Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 2000), pp. 1029–1031 and references therein.

Katsu, Y.

T. Idé, H. Numata, H. Mizuta, Y. Taira, M. Suzuki, M. Noguchi, Y. Katsu, “Moiré-free collimating light-guide with low-discrepancy dot patterns,” in Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 2002), pp. 1232–1235.

Keller, A.

S. Hiller, O. Deussen, A. Keller, “Tiled blue noise samples,” in Proceedings of Vision, Modeling, and Visualization 2001 (IOS, Amsterdam, The Netherlands, 2001), pp. 265–271.

Khan, A. M.

D. L. Lau, A. M. Khan, G. R. Arce, “Stochastic moiré,” in Proceedings of the 2001 Image Processing, Image Quality, Image Capture Systems Conference (Society for Imaging Science and Technology, Springfield, Va., 2001).

Lau, D. L.

D. L. Lau, G. R. Arce, N. C. Gallagher, “Digital halftoning via green noise masks,” J. Opt. Soc. Am. A 16, 1575–1586 (1999).
[CrossRef]

D. L. Lau, A. M. Khan, G. R. Arce, “Stochastic moiré,” in Proceedings of the 2001 Image Processing, Image Quality, Image Capture Systems Conference (Society for Imaging Science and Technology, Springfield, Va., 2001).

Lebanon, G.

Matsumoto, S.

K. Käläntär, S. Matsumoto, T. Onishi, K. Takizawa, “Optical micro deflector based functional light-guide plate for backlight unit,” in Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 2000), pp. 1029–1031 and references therein.

Mitsa, T.

Mizuta, H.

T. Idé, H. Numata, H. Mizuta, Y. Taira, M. Suzuki, M. Noguchi, Y. Katsu, “Moiré-free collimating light-guide with low-discrepancy dot patterns,” in Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 2002), pp. 1232–1235.

Mori, Y.

H. Taniguchi, Y. Hira, Y. Mori, “Liquid crystal display devices,” U.S. patent6,099,134 (1998).

Ninomiya, S.

S. Ninomiya, S. Tezuka, “Toward real-time pricing of complex financial derivatives,” Appl. Math. Finance 3, 1–20 (1996).
[CrossRef]

Noguchi, M.

T. Idé, H. Numata, H. Mizuta, Y. Taira, M. Suzuki, M. Noguchi, Y. Katsu, “Moiré-free collimating light-guide with low-discrepancy dot patterns,” in Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 2002), pp. 1232–1235.

Numata, H.

T. Idé, H. Numata, H. Mizuta, Y. Taira, M. Suzuki, M. Noguchi, Y. Katsu, “Moiré-free collimating light-guide with low-discrepancy dot patterns,” in Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 2002), pp. 1232–1235.

Oki, Y.

Y. Oki, “Novel backlight with high luminance and low power consumption by prism-on-light-pipe technology,” in Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 1998), pp. 157–160.

Onishi, T.

K. Käläntär, S. Matsumoto, T. Onishi, K. Takizawa, “Optical micro deflector based functional light-guide plate for backlight unit,” in Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 2000), pp. 1029–1031 and references therein.

Parker, K. J.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 7.

Purgathofer, W.

W. Purgathofer, R. F. Tobler, M. Geiler, “Forced random dithering: improved threshold matrices for ordered dithering,” in Proceedings of the First IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1994), pp. 1032–1035.

Suzuki, M.

T. Idé, H. Numata, H. Mizuta, Y. Taira, M. Suzuki, M. Noguchi, Y. Katsu, “Moiré-free collimating light-guide with low-discrepancy dot patterns,” in Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 2002), pp. 1232–1235.

Taira, Y.

T. Idé, H. Numata, H. Mizuta, Y. Taira, M. Suzuki, M. Noguchi, Y. Katsu, “Moiré-free collimating light-guide with low-discrepancy dot patterns,” in Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 2002), pp. 1232–1235.

Takizawa, K.

K. Käläntär, S. Matsumoto, T. Onishi, K. Takizawa, “Optical micro deflector based functional light-guide plate for backlight unit,” in Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 2000), pp. 1029–1031 and references therein.

Taniguchi, H.

H. Taniguchi, Y. Hira, Y. Mori, “Liquid crystal display devices,” U.S. patent6,099,134 (1998).

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 7.

Tezuka, S.

S. Ninomiya, S. Tezuka, “Toward real-time pricing of complex financial derivatives,” Appl. Math. Finance 3, 1–20 (1996).
[CrossRef]

S. Tezuka, “Polynomial arithmetic analogue of Halton sequences,” ACM (Assoc. Comput. Mach.) Trans. Model. Comput. Simul. 3, 99–107 (1993).
[CrossRef]

S. Tezuka, Uniform Random Numbers: Theory and Practice (Kluwer Academic, Boston, 1995).

Tobler, R. F.

W. Purgathofer, R. F. Tobler, M. Geiler, “Forced random dithering: improved threshold matrices for ordered dithering,” in Proceedings of the First IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1994), pp. 1032–1035.

Ulichney, R. A.

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

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 7.

ACM (Assoc. Comput. Mach.) Trans. Model. Comput. Simul. (1)

S. Tezuka, “Polynomial arithmetic analogue of Halton sequences,” ACM (Assoc. Comput. Mach.) Trans. Model. Comput. Simul. 3, 99–107 (1993).
[CrossRef]

Appl. Math. Finance (1)

S. Ninomiya, S. Tezuka, “Toward real-time pricing of complex financial derivatives,” Appl. Math. Finance 3, 1–20 (1996).
[CrossRef]

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

Proc. IEEE (1)

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

Other (11)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992), Chap. 7.

W. Purgathofer, R. F. Tobler, M. Geiler, “Forced random dithering: improved threshold matrices for ordered dithering,” in Proceedings of the First IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1994), pp. 1032–1035.

T. Idé, H. Numata, H. Mizuta, Y. Taira, M. Suzuki, M. Noguchi, Y. Katsu, “Moiré-free collimating light-guide with low-discrepancy dot patterns,” in Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 2002), pp. 1232–1235.

C. B. Atkins, J. P. Allebach, C. A. Bouman, “Halftone postprocessing for improved highlight rendition,” in Proceedings of the 1997 IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1997), Vol. 1, pp. 791–794.

S. Hiller, O. Deussen, A. Keller, “Tiled blue noise samples,” in Proceedings of Vision, Modeling, and Visualization 2001 (IOS, Amsterdam, The Netherlands, 2001), pp. 265–271.

H. Taniguchi, Y. Hira, Y. Mori, “Liquid crystal display devices,” U.S. patent6,099,134 (1998).

S. Tezuka, Uniform Random Numbers: Theory and Practice (Kluwer Academic, Boston, 1995).

Y. Oki, “Novel backlight with high luminance and low power consumption by prism-on-light-pipe technology,” in Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 1998), pp. 157–160.

K. Käläntär, S. Matsumoto, T. Onishi, K. Takizawa, “Optical micro deflector based functional light-guide plate for backlight unit,” in Digest of Technical Papers (Society for Information Display, Santa Ana, Calif., 2000), pp. 1029–1031 and references therein.

I. Amidror, The theory of the Moiré Phenomenon (Kluwer Academic, Dordrecht, The Netherlands, 2000), Chap. 3.

D. L. Lau, A. M. Khan, G. R. Arce, “Stochastic moiré,” in Proceedings of the 2001 Image Processing, Image Quality, Image Capture Systems Conference (Society for Imaging Science and Technology, Springfield, Va., 2001).

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Figures (12)

Fig. 1
Fig. 1

Conventional structure of edge-lit backlight unit. There is a pattern of diffusing white spots on the bottom surface of the light guide. (a) Bird’s-eye view. The x and y axes, which are perpendicular to each other, show the directions of the prismatic grooves of the prism sheets. (b) Cross-section view. An example of the path of a light ray is shown. LG is short for light guide.

Fig. 2
Fig. 2

Broken permutation symmetry among pixels in error diffusion algorithms. The pixels are not treated symmetrically because the order of the error diffusion process is fixed for the pixels.

Fig. 3
Fig. 3

Comparison between (a) pseudorandom numbers and (b) LDS.

Fig. 4
Fig. 4

Schematic diagram of the relaxation method. The repulsive force from dots B and C is acting on A. The situation is not one-sided: B and C are also affected by the surrounding dots, showing the permutation symmetry in the relaxation algorithm.

Fig. 5
Fig. 5

Comparison between two initial patterns: (a) pseudorandom numbers and (b) LDS. The iteration number and the dynamical parameters are the same for both. We observe visible roughness in (a) and uniform irregularity in (b).

Fig. 6
Fig. 6

Example of DLDS patterns with steep density gradient: (a) DLDS pattern and (b) density distribution.

Fig. 7
Fig. 7

Square of the L2 discrepancy as a function of N.

Fig. 8
Fig. 8

Prototyped integrated-type light guides.

Fig. 9
Fig. 9

Shape of (a) the microscatterers and (b) the prismatic grooves in units of millimeters.

Fig. 10
Fig. 10

Snapshots of the microscatterer patterns with use of (a) the PRP method and (b) the DLDS method.

Fig. 11
Fig. 11

Snapshots through a liquid-crystal cell with use of (a) the PRP method and (b) the DLDS method.

Fig. 12
Fig. 12

Luminance fluctuation along the vertical axis in Fig. 11 for light guides with use of (a) the PRP method and (b) the DLDS method. For (a) and (b), the luminance is normalized with the luminance value at the base of Figs. 11(a) and 11(b), respectively.

Equations (18)

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

DN(2)=sup(x, y)[0, 1]2#E(x, y)N-xy,
TN(2)=[0, 1]2#E(x, y)N-xy2 dxdy1/2.
TN(2)=1N2i=1Nj=1N[1-max(xi, xj)][1-max(yi, yj)]-12Ni=1N(1-xi2)(1-yi2)+19,
DN(2)(LDS)C (log N)2N
TN(2)(LDS)=O(log N)2N
DN(2)(random)=Olog log NN1/2,
[TN(2)(random)]2=56N.
m d2ridt2+c dridt=j=1N fij(ri,rj)
ri(t)=ri(t0)+1c t0tdτFi(τ)1-exp-c(t-τ)m,
ri(t+Δt)=ri(t)+1c ΔtFi(t).
fij=rij|rij|×1forbij<Dexp[-(|rij|-bij)/L]forbijD,
bij2(kD)2=x2+y2(kx)2+y2,
Pk=ρkl=1Mρl,
l=1kPlU0<l=1k+1Pl.
x=xk+LxkU1,y=yk+LykU2,
λ(ρ)=aρ,
Dλ(ρ).
l=2π|k1-k2|=l1l2|l1-l2|,

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