Abstract

In color halftoning, moiré is the low-frequency spatial artifact generated by the interference of superimposed primary color dot screens that adds an unwanted artificial texture to the printed image. When these overlapping dot screens are irregular, as in the case of stochastic dot screens, this interference pattern follows a random spatial distribution resulting in “stochastic” moiré. This stochastic moiré is at its most visible when the overlapping dither patterns have the same relative spacing between dots. We study the occurrence of stochastic moiré in green-noise halftones where dither patterns are composed of clusters of varying sizes and where the visibility of stochastic moiré can be reduced by varying the coarseness of dither patterns between the component cyan, magenta, yellow, and black colors.

© 2002 Optical Society of America

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References

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  1. R. A. Ulichney, Digital Halftoning (MIT Press, Cambridge, Mass., 1987).
  2. D. L. Lau, G. R. Arce, Modern Digital Halftoning (Marcel Dekker, New York, 2001).
  3. H. R. Kang, Digital Color Halftoning (SPIE Press, Bellingham, Wash., 1999).
  4. D. L. Lau, A. M. Khan, G. R. Arce, “Stochastic moiré,” in Proceedings of the IS&T’s PICS 2001: Image Processing, Image Quality, Image Capture Systems Conference (Society for Imaging Science and Technology, Springfield, Va., 2001), pp. 96–100.
  5. L. Glass, “Moiré effect from random dots,” Nature 223, 578–580 (1969).
    [Crossref] [PubMed]
  6. L. Glass, “Perception of random dot interference patterns,” Nature 246, 360–362 (1973).
    [Crossref] [PubMed]
  7. D. L. Lau, G. R. Arce, N. C. Gallagher, “Green-noise digital halftoning,” in Proc. IEEE 86, 2424–2444 (1998).
    [Crossref]
  8. T. Mitsa, K. J. Parker, “Digital halftoning technique using a blue noise mask,” J. Opt. Soc. Am. A 9, 1920–1929 (1992).
    [Crossref]
  9. D. L. Lau, G. R. Arce, N. C. Gallagher, “Digital halftoning by means of green-noise masks,” J. Opt. Soc. Am. A 16, 1575–1586 (1999).
    [Crossref]
  10. I. Amidror, R. D. Hersch, V. Ostromoukhov, “Spectral analysis and minimization of moiré patterns in color separation,” J. Electron. Imaging 3, 295–317 (1994).
    [Crossref]
  11. F. W. Campbell, R. H. Carpenter, J. Levinson, “Visibility of aperiodic patterns compared with that of sinusoidal gratings,” J. Physiol. 190, 283–298 (1969).
  12. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1994).
  13. J. Sullivan, L. Ray, R. Miller, “Design of minimum visual modulation halftone patterns,” IEEE Trans. Syst. Man Cybern. 21, 33–38 (1991).
    [Crossref]
  14. R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial gray-scale,” Proc. Soc. Inf. Display 17, 75–78 (1976).
  15. J. F. Jarvis, C. N. Judice, W. H. Ninke, “A survey of techniques for the display of continuous-tone pictures on bilevel displays,” Comput. Graph. Image Process. 5, 13–40 (1976).
    [Crossref]
  16. P. Stucki, “Mecca-a multiple-error correcting computation algorithm for bilevel image hard-copy reproduction,” (IBM Research Laboratory, Zurich, Switzerland, 1981).
  17. R. A. Ulichney, “Dithering with blue noise,” Proc. IEEE 76, 56–79 (1988).
    [Crossref]
  18. M. Yao, K. J. Parker, “Application of the blue-noise mask in color halftoning,” in Visual Communications and Image Processing, R. Ansari, M. J. Smith, eds., Proc. SPIE2727, 876–880 (1996).
    [Crossref]
  19. D. L. Lau, G. R. Arce, N. C. Gallagher, “Digital color halftoning via generalized error-diffusion and multi-channel green-noise masks,” IEEE Trans. Image Process. 9, 923–935 (2000).
    [Crossref]
  20. D. J. Lieberman, J. P. Allebach, “Model based direct binary search halftone optimization with a dual interpretation,” in Proceedings of the 1998 IEEE International Conference on Image Processing, R. Werner, ed. (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 44–48.
  21. R. Levien, “Output dependent feedback in error diffusion halftoning,” in Proceedings of the IS&T’s Eighth International Congress on Advances in Non-Impact Printing Technologies (Society for Imaging Science and Technology, Springfield, Va., 1992), pp. 280–282.
  22. F. Faheem, D. L. Lau, G. R. Arce, “Multilevel halftoning using bi-level quantization,” in Proceedings of the IS&T’s PICS 2001: Image Processing, Image Quality, Image Capture Systems Conference (Society for Imaging Science and Technology, Springfield, Va., 2001), pp. 64–67.

2000 (1)

D. L. Lau, G. R. Arce, N. C. Gallagher, “Digital color halftoning via generalized error-diffusion and multi-channel green-noise masks,” IEEE Trans. Image Process. 9, 923–935 (2000).
[Crossref]

1999 (1)

1998 (1)

D. L. Lau, G. R. Arce, N. C. Gallagher, “Green-noise digital halftoning,” in Proc. IEEE 86, 2424–2444 (1998).
[Crossref]

1994 (1)

I. Amidror, R. D. Hersch, V. Ostromoukhov, “Spectral analysis and minimization of moiré patterns in color separation,” J. Electron. Imaging 3, 295–317 (1994).
[Crossref]

1992 (1)

1991 (1)

J. Sullivan, L. Ray, R. Miller, “Design of minimum visual modulation halftone patterns,” IEEE Trans. Syst. Man Cybern. 21, 33–38 (1991).
[Crossref]

1988 (1)

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

1976 (2)

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial gray-scale,” Proc. Soc. Inf. Display 17, 75–78 (1976).

J. F. Jarvis, C. N. Judice, W. H. Ninke, “A survey of techniques for the display of continuous-tone pictures on bilevel displays,” Comput. Graph. Image Process. 5, 13–40 (1976).
[Crossref]

1973 (1)

L. Glass, “Perception of random dot interference patterns,” Nature 246, 360–362 (1973).
[Crossref] [PubMed]

1969 (2)

L. Glass, “Moiré effect from random dots,” Nature 223, 578–580 (1969).
[Crossref] [PubMed]

F. W. Campbell, R. H. Carpenter, J. Levinson, “Visibility of aperiodic patterns compared with that of sinusoidal gratings,” J. Physiol. 190, 283–298 (1969).

Allebach, J. P.

D. J. Lieberman, J. P. Allebach, “Model based direct binary search halftone optimization with a dual interpretation,” in Proceedings of the 1998 IEEE International Conference on Image Processing, R. Werner, ed. (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 44–48.

Amidror, I.

I. Amidror, R. D. Hersch, V. Ostromoukhov, “Spectral analysis and minimization of moiré patterns in color separation,” J. Electron. Imaging 3, 295–317 (1994).
[Crossref]

Arce, G. R.

D. L. Lau, G. R. Arce, N. C. Gallagher, “Digital color halftoning via generalized error-diffusion and multi-channel green-noise masks,” IEEE Trans. Image Process. 9, 923–935 (2000).
[Crossref]

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

D. L. Lau, G. R. Arce, N. C. Gallagher, “Green-noise digital halftoning,” in Proc. IEEE 86, 2424–2444 (1998).
[Crossref]

F. Faheem, D. L. Lau, G. R. Arce, “Multilevel halftoning using bi-level quantization,” in Proceedings of the IS&T’s PICS 2001: Image Processing, Image Quality, Image Capture Systems Conference (Society for Imaging Science and Technology, Springfield, Va., 2001), pp. 64–67.

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

D. L. Lau, A. M. Khan, G. R. Arce, “Stochastic moiré,” in Proceedings of the IS&T’s PICS 2001: Image Processing, Image Quality, Image Capture Systems Conference (Society for Imaging Science and Technology, Springfield, Va., 2001), pp. 96–100.

Campbell, F. W.

F. W. Campbell, R. H. Carpenter, J. Levinson, “Visibility of aperiodic patterns compared with that of sinusoidal gratings,” J. Physiol. 190, 283–298 (1969).

Carpenter, R. H.

F. W. Campbell, R. H. Carpenter, J. Levinson, “Visibility of aperiodic patterns compared with that of sinusoidal gratings,” J. Physiol. 190, 283–298 (1969).

Faheem, F.

F. Faheem, D. L. Lau, G. R. Arce, “Multilevel halftoning using bi-level quantization,” in Proceedings of the IS&T’s PICS 2001: Image Processing, Image Quality, Image Capture Systems Conference (Society for Imaging Science and Technology, Springfield, Va., 2001), pp. 64–67.

Floyd, R. W.

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial gray-scale,” Proc. Soc. Inf. Display 17, 75–78 (1976).

Gallagher, N. C.

D. L. Lau, G. R. Arce, N. C. Gallagher, “Digital color halftoning via generalized error-diffusion and multi-channel green-noise masks,” IEEE Trans. Image Process. 9, 923–935 (2000).
[Crossref]

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

D. L. Lau, G. R. Arce, N. C. Gallagher, “Green-noise digital halftoning,” in Proc. IEEE 86, 2424–2444 (1998).
[Crossref]

Glass, L.

L. Glass, “Perception of random dot interference patterns,” Nature 246, 360–362 (1973).
[Crossref] [PubMed]

L. Glass, “Moiré effect from random dots,” Nature 223, 578–580 (1969).
[Crossref] [PubMed]

Hersch, R. D.

I. Amidror, R. D. Hersch, V. Ostromoukhov, “Spectral analysis and minimization of moiré patterns in color separation,” J. Electron. Imaging 3, 295–317 (1994).
[Crossref]

Jarvis, J. F.

J. F. Jarvis, C. N. Judice, W. H. Ninke, “A survey of techniques for the display of continuous-tone pictures on bilevel displays,” Comput. Graph. Image Process. 5, 13–40 (1976).
[Crossref]

Judice, C. N.

J. F. Jarvis, C. N. Judice, W. H. Ninke, “A survey of techniques for the display of continuous-tone pictures on bilevel displays,” Comput. Graph. Image Process. 5, 13–40 (1976).
[Crossref]

Kang, H. R.

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

Khan, A. M.

D. L. Lau, A. M. Khan, G. R. Arce, “Stochastic moiré,” in Proceedings of the IS&T’s PICS 2001: Image Processing, Image Quality, Image Capture Systems Conference (Society for Imaging Science and Technology, Springfield, Va., 2001), pp. 96–100.

Lau, D. L.

D. L. Lau, G. R. Arce, N. C. Gallagher, “Digital color halftoning via generalized error-diffusion and multi-channel green-noise masks,” IEEE Trans. Image Process. 9, 923–935 (2000).
[Crossref]

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

D. L. Lau, G. R. Arce, N. C. Gallagher, “Green-noise digital halftoning,” in Proc. IEEE 86, 2424–2444 (1998).
[Crossref]

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

D. L. Lau, A. M. Khan, G. R. Arce, “Stochastic moiré,” in Proceedings of the IS&T’s PICS 2001: Image Processing, Image Quality, Image Capture Systems Conference (Society for Imaging Science and Technology, Springfield, Va., 2001), pp. 96–100.

F. Faheem, D. L. Lau, G. R. Arce, “Multilevel halftoning using bi-level quantization,” in Proceedings of the IS&T’s PICS 2001: Image Processing, Image Quality, Image Capture Systems Conference (Society for Imaging Science and Technology, Springfield, Va., 2001), pp. 64–67.

Levien, R.

R. Levien, “Output dependent feedback in error diffusion halftoning,” in Proceedings of the IS&T’s Eighth International Congress on Advances in Non-Impact Printing Technologies (Society for Imaging Science and Technology, Springfield, Va., 1992), pp. 280–282.

Levinson, J.

F. W. Campbell, R. H. Carpenter, J. Levinson, “Visibility of aperiodic patterns compared with that of sinusoidal gratings,” J. Physiol. 190, 283–298 (1969).

Lieberman, D. J.

D. J. Lieberman, J. P. Allebach, “Model based direct binary search halftone optimization with a dual interpretation,” in Proceedings of the 1998 IEEE International Conference on Image Processing, R. Werner, ed. (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 44–48.

Miller, R.

J. Sullivan, L. Ray, R. Miller, “Design of minimum visual modulation halftone patterns,” IEEE Trans. Syst. Man Cybern. 21, 33–38 (1991).
[Crossref]

Mitsa, T.

Ninke, W. H.

J. F. Jarvis, C. N. Judice, W. H. Ninke, “A survey of techniques for the display of continuous-tone pictures on bilevel displays,” Comput. Graph. Image Process. 5, 13–40 (1976).
[Crossref]

Ostromoukhov, V.

I. Amidror, R. D. Hersch, V. Ostromoukhov, “Spectral analysis and minimization of moiré patterns in color separation,” J. Electron. Imaging 3, 295–317 (1994).
[Crossref]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1994).

Parker, K. J.

T. Mitsa, K. J. Parker, “Digital halftoning technique using a blue noise mask,” J. Opt. Soc. Am. A 9, 1920–1929 (1992).
[Crossref]

M. Yao, K. J. Parker, “Application of the blue-noise mask in color halftoning,” in Visual Communications and Image Processing, R. Ansari, M. J. Smith, eds., Proc. SPIE2727, 876–880 (1996).
[Crossref]

Ray, L.

J. Sullivan, L. Ray, R. Miller, “Design of minimum visual modulation halftone patterns,” IEEE Trans. Syst. Man Cybern. 21, 33–38 (1991).
[Crossref]

Steinberg, L.

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial gray-scale,” Proc. Soc. Inf. Display 17, 75–78 (1976).

Stucki, P.

P. Stucki, “Mecca-a multiple-error correcting computation algorithm for bilevel image hard-copy reproduction,” (IBM Research Laboratory, Zurich, Switzerland, 1981).

Sullivan, J.

J. Sullivan, L. Ray, R. Miller, “Design of minimum visual modulation halftone patterns,” IEEE Trans. Syst. Man Cybern. 21, 33–38 (1991).
[Crossref]

Ulichney, R. A.

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

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

Yao, M.

M. Yao, K. J. Parker, “Application of the blue-noise mask in color halftoning,” in Visual Communications and Image Processing, R. Ansari, M. J. Smith, eds., Proc. SPIE2727, 876–880 (1996).
[Crossref]

Comput. Graph. Image Process. (1)

J. F. Jarvis, C. N. Judice, W. H. Ninke, “A survey of techniques for the display of continuous-tone pictures on bilevel displays,” Comput. Graph. Image Process. 5, 13–40 (1976).
[Crossref]

IEEE Trans. Image Process. (1)

D. L. Lau, G. R. Arce, N. C. Gallagher, “Digital color halftoning via generalized error-diffusion and multi-channel green-noise masks,” IEEE Trans. Image Process. 9, 923–935 (2000).
[Crossref]

IEEE Trans. Syst. Man Cybern. (1)

J. Sullivan, L. Ray, R. Miller, “Design of minimum visual modulation halftone patterns,” IEEE Trans. Syst. Man Cybern. 21, 33–38 (1991).
[Crossref]

J. Electron. Imaging (1)

I. Amidror, R. D. Hersch, V. Ostromoukhov, “Spectral analysis and minimization of moiré patterns in color separation,” J. Electron. Imaging 3, 295–317 (1994).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Physiol. (1)

F. W. Campbell, R. H. Carpenter, J. Levinson, “Visibility of aperiodic patterns compared with that of sinusoidal gratings,” J. Physiol. 190, 283–298 (1969).

Nature (2)

L. Glass, “Moiré effect from random dots,” Nature 223, 578–580 (1969).
[Crossref] [PubMed]

L. Glass, “Perception of random dot interference patterns,” Nature 246, 360–362 (1973).
[Crossref] [PubMed]

Proc. IEEE (2)

D. L. Lau, G. R. Arce, N. C. Gallagher, “Green-noise digital halftoning,” in Proc. IEEE 86, 2424–2444 (1998).
[Crossref]

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

Proc. Soc. Inf. Display (1)

R. W. Floyd, L. Steinberg, “An adaptive algorithm for spatial gray-scale,” Proc. Soc. Inf. Display 17, 75–78 (1976).

Other (10)

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1994).

M. Yao, K. J. Parker, “Application of the blue-noise mask in color halftoning,” in Visual Communications and Image Processing, R. Ansari, M. J. Smith, eds., Proc. SPIE2727, 876–880 (1996).
[Crossref]

P. Stucki, “Mecca-a multiple-error correcting computation algorithm for bilevel image hard-copy reproduction,” (IBM Research Laboratory, Zurich, Switzerland, 1981).

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

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

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

D. L. Lau, A. M. Khan, G. R. Arce, “Stochastic moiré,” in Proceedings of the IS&T’s PICS 2001: Image Processing, Image Quality, Image Capture Systems Conference (Society for Imaging Science and Technology, Springfield, Va., 2001), pp. 96–100.

D. J. Lieberman, J. P. Allebach, “Model based direct binary search halftone optimization with a dual interpretation,” in Proceedings of the 1998 IEEE International Conference on Image Processing, R. Werner, ed. (Institute of Electrical and Electronics Engineers, New York, 1998), pp. 44–48.

R. Levien, “Output dependent feedback in error diffusion halftoning,” in Proceedings of the IS&T’s Eighth International Congress on Advances in Non-Impact Printing Technologies (Society for Imaging Science and Technology, Springfield, Va., 1992), pp. 280–282.

F. Faheem, D. L. Lau, G. R. Arce, “Multilevel halftoning using bi-level quantization,” in Proceedings of the IS&T’s PICS 2001: Image Processing, Image Quality, Image Capture Systems Conference (Society for Imaging Science and Technology, Springfield, Va., 2001), pp. 64–67.

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Figures (20)

Fig. 1
Fig. 1

Illustration of (right) periodic moiré produced by superimposing two AM halftone patterns (left and center) with a screen angle offset of 15°.

Fig. 2
Fig. 2

Moiré-minimized rosette pattern created by superimposing CMYK screens with screen angles 15°, 75°, 0°, and 45°.

Fig. 3
Fig. 3

Two examples of aperiodic or stochastic moiré produced by superimposing two FM halftone patterns.

Fig. 4
Fig. 4

Illustration of (left) AM halftoning, (center) blue-noise or FM halftoning, and (right) green-noise halftoning.

Fig. 5
Fig. 5

Periodic moiré approximated by the superposition of two cosinusoidal gratings such that the resulting interference pattern is modeled as the convolution of impulses in the Fourier domain.

Fig. 6
Fig. 6

Periodic moiré created by superimposing two cosinusoidal gratings where the peaks and valleys of each grating are indicated as a solid or dashed line, respectively.

Fig. 7
Fig. 7

Diagram showing the location of in-phase (X’s) and out-of-phase (O’s) components where the minimum distance between X’s and O’s is r.

Fig. 8
Fig. 8

In-phase and out-of-phase points for periodic moiré created by superimposing (left) two 2D cosinusoidal gratings and (right) two binary AM halftone patterns.

Fig. 9
Fig. 9

Three two-color halftone patterns illustrating A, perfectly overlapping or in-phase pixels; B, uncorrelated pixels; and C, perfectly nonoverlapping or out-of-phase pixels.

Fig. 10
Fig. 10

Two examples of the stochastic moiré surface created by superimposing two FM halftone patterns.

Fig. 11
Fig. 11

Diagram of the spectral domain representations of (a) the Poisson point process defining the sampling grid, (b) the circular-bandlimited stochastic moiré surface with bandwidth less than half the principal frequency, and (c) the power spectrum of the sampled surface with no aliasing.

Fig. 12
Fig. 12

Plots of the visual cost of stochastic moiré versus minority-pixel intensity level for several error-diffusion schemes: (a) Floyd and Steinberg,14 (b) Jarvis et al.,15 (c) Stucki,16 (d) Ulichney.17

Fig. 13
Fig. 13

Selected stochastic moiré surfaces for several error-diffusion techniques. (a) Floyd and Steinberg,14 (b) Jarvis et al.,15 (c) Stucki,16 (d) Ulichney.17

Fig. 14
Fig. 14

Binary halftones created by combining the two channels by using a pixelwise logical or function (1=black) for halftones created by error-diffusion: (a) Floyd and Steinberg,14 (b) Jarvis et al.,15 (c) Stucki,16 (d) Ulichney.17

Fig. 15
Fig. 15

The binary image created (left) before and (right) after applying a one-pixel vertical shift to one of the two channels forming a perfectly out-of-phase stochastic moiré surface.

Fig. 16
Fig. 16

(Left) green-noise halftone representing gray level 0.25 and (right) its corresponding parent process created by replacing each cluster with a single minority pixel at the cluster’s centroid.

Fig. 17
Fig. 17

Composite halftones created by superimposing a dither pattern generated by using Ulichney’s error diffusion onto a pattern generated by using Levien’s error diffusion with output-dependent feedback with hysteresis constants (left) h=1.0 and (right) h=2.0.

Fig. 18
Fig. 18

Composite halftones created by (left) two uncorrelated blue-noise masks and (right) an uncorrelated combination of a blue-noise and a green-noise mask.

Fig. 19
Fig. 19

Image “Violin” halftoned with four uncorrelated blue-noise masks.

Fig. 20
Fig. 20

Image “Violin” halftoned with four uncorrelated green-noise masks with varying levels of coarseness.

Equations (9)

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

r=cos(θ),
D[n]=1λbi=1Ndiδ[n-ai],
di=minjai-bj2.
X˜(f )=1gnx[n]exp(-j2πfn)
VC(ϕA, ϕB)=averageacpower(D(f )×HVS(f )),
Pr(ai+1=bj+1)
=x=ai+Pr(ai+1=bj+1|bj+1=x)Pr(bj+1=x)dx.
 
λg=M¯/g,

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