Abstract

Screening techniques are widely used in the binary display of continuous-tone images by digital output devices. The quality of the halftone image resulting from such a nonlinear transformation is dependent on the dot profile employed. In the Fourier domain, aliasing degrades the halftone image. The relationship between image quality and dot profile is studied from this point of view. For a well-known dot profile, it is shown analytically how aliasing is suppressed, and that quantization contouring may be eliminated without an appreciable increase in aliasing.

© 1977 Optical Society of America

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References

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  1. K. Knowlton and L. Harmon, “Computer-Produced Grey Scales,” Comput. Graph. Image Process. 1, 1–20 (1972).
    [Crossref]
  2. J. F. Jarvis, C. N. Judice, and 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]
  3. J. O. Limb, “Design of Dither Waveforms for Quantized Visual Signals,” Bell System Tech. J. 48, 2555–2582 (1969).
    [Crossref]
  4. B. Lippel and M. Kurland, “The Effect of Dither on Luminance Quantization of Pictures,” IEEE Trans. Commun. Tech. COM-19, 879–888 (1971).
    [Crossref]
  5. B. E. Bayer, “An Optimum Method for Two-Level Rendition of Continuous-Tone Pictures,” in IEEE International Conference on Communications, Conference Record, (26–11)−(26–15) (1973).
  6. C. N. Judice, J. F. Jarvis, and W. H. Ninke, “Using Ordered Dither to Display Continuous-Tone Pictures on an AC Plasma Panel,” Proc. SID 15, 161–169 (1974).
  7. J. W. Wesner, “Screen Patterns Used in Reproduction of Continuous-Tone Graphics,” Appl. Opt. 13, 1703–1710 (1974).
    [Crossref] [PubMed]
  8. D. Kermisch and P. G. Roetling, “Fourier Spectrum of Halftone Images,” J. Opt. Soc. Am. 65, 716–723 (1975).
    [Crossref]
  9. D. E. Pearson, Transmission and Display of Pictorial Information, (Wiley, New York, 1975), pp. 31–47.
  10. J. P. Allebach and B. Liu, “Random Quasi-Periodic Halftone Process,” J. Opt. Soc. Am. 66, 909–917 (1976).
    [Crossref]

1976 (2)

J. F. Jarvis, C. N. Judice, and 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]

J. P. Allebach and B. Liu, “Random Quasi-Periodic Halftone Process,” J. Opt. Soc. Am. 66, 909–917 (1976).
[Crossref]

1975 (1)

1974 (2)

C. N. Judice, J. F. Jarvis, and W. H. Ninke, “Using Ordered Dither to Display Continuous-Tone Pictures on an AC Plasma Panel,” Proc. SID 15, 161–169 (1974).

J. W. Wesner, “Screen Patterns Used in Reproduction of Continuous-Tone Graphics,” Appl. Opt. 13, 1703–1710 (1974).
[Crossref] [PubMed]

1973 (1)

B. E. Bayer, “An Optimum Method for Two-Level Rendition of Continuous-Tone Pictures,” in IEEE International Conference on Communications, Conference Record, (26–11)−(26–15) (1973).

1972 (1)

K. Knowlton and L. Harmon, “Computer-Produced Grey Scales,” Comput. Graph. Image Process. 1, 1–20 (1972).
[Crossref]

1971 (1)

B. Lippel and M. Kurland, “The Effect of Dither on Luminance Quantization of Pictures,” IEEE Trans. Commun. Tech. COM-19, 879–888 (1971).
[Crossref]

1969 (1)

J. O. Limb, “Design of Dither Waveforms for Quantized Visual Signals,” Bell System Tech. J. 48, 2555–2582 (1969).
[Crossref]

Allebach, J. P.

Bayer, B. E.

B. E. Bayer, “An Optimum Method for Two-Level Rendition of Continuous-Tone Pictures,” in IEEE International Conference on Communications, Conference Record, (26–11)−(26–15) (1973).

Harmon, L.

K. Knowlton and L. Harmon, “Computer-Produced Grey Scales,” Comput. Graph. Image Process. 1, 1–20 (1972).
[Crossref]

Jarvis, J. F.

J. F. Jarvis, C. N. Judice, and 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]

C. N. Judice, J. F. Jarvis, and W. H. Ninke, “Using Ordered Dither to Display Continuous-Tone Pictures on an AC Plasma Panel,” Proc. SID 15, 161–169 (1974).

Judice, C. N.

J. F. Jarvis, C. N. Judice, and 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]

C. N. Judice, J. F. Jarvis, and W. H. Ninke, “Using Ordered Dither to Display Continuous-Tone Pictures on an AC Plasma Panel,” Proc. SID 15, 161–169 (1974).

Kermisch, D.

Knowlton, K.

K. Knowlton and L. Harmon, “Computer-Produced Grey Scales,” Comput. Graph. Image Process. 1, 1–20 (1972).
[Crossref]

Kurland, M.

B. Lippel and M. Kurland, “The Effect of Dither on Luminance Quantization of Pictures,” IEEE Trans. Commun. Tech. COM-19, 879–888 (1971).
[Crossref]

Limb, J. O.

J. O. Limb, “Design of Dither Waveforms for Quantized Visual Signals,” Bell System Tech. J. 48, 2555–2582 (1969).
[Crossref]

Lippel, B.

B. Lippel and M. Kurland, “The Effect of Dither on Luminance Quantization of Pictures,” IEEE Trans. Commun. Tech. COM-19, 879–888 (1971).
[Crossref]

Liu, B.

Ninke, W. H.

J. F. Jarvis, C. N. Judice, and 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]

C. N. Judice, J. F. Jarvis, and W. H. Ninke, “Using Ordered Dither to Display Continuous-Tone Pictures on an AC Plasma Panel,” Proc. SID 15, 161–169 (1974).

Pearson, D. E.

D. E. Pearson, Transmission and Display of Pictorial Information, (Wiley, New York, 1975), pp. 31–47.

Roetling, P. G.

Wesner, J. W.

Appl. Opt. (1)

Bell System Tech. J. (1)

J. O. Limb, “Design of Dither Waveforms for Quantized Visual Signals,” Bell System Tech. J. 48, 2555–2582 (1969).
[Crossref]

Comput. Graph. Image Process. (2)

K. Knowlton and L. Harmon, “Computer-Produced Grey Scales,” Comput. Graph. Image Process. 1, 1–20 (1972).
[Crossref]

J. F. Jarvis, C. N. Judice, and 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 International Conference on Communications, Conference Record (1)

B. E. Bayer, “An Optimum Method for Two-Level Rendition of Continuous-Tone Pictures,” in IEEE International Conference on Communications, Conference Record, (26–11)−(26–15) (1973).

IEEE Trans. Commun. Tech. (1)

B. Lippel and M. Kurland, “The Effect of Dither on Luminance Quantization of Pictures,” IEEE Trans. Commun. Tech. COM-19, 879–888 (1971).
[Crossref]

J. Opt. Soc. Am. (2)

Proc. SID (1)

C. N. Judice, J. F. Jarvis, and W. H. Ninke, “Using Ordered Dither to Display Continuous-Tone Pictures on an AC Plasma Panel,” Proc. SID 15, 161–169 (1974).

Other (1)

D. E. Pearson, Transmission and Display of Pictorial Information, (Wiley, New York, 1975), pp. 31–47.

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Figures (8)

FIG. 1
FIG. 1

The diamond dot profile.

FIG. 2
FIG. 2

Three nonlinearities for the diamond dot profile of Fig. 1.

FIG. 3
FIG. 3

Grating dot profile for 3 powers of 4 when the halftone cell size M is 8. Left to right, the gratings are p[k,l;4j−3], j = 0, 1, 2.

FIG. 4
FIG. 4

Grating dot profile for absorptance â = 46/64. The dot profile is a superposition of the separately shown grating orders.

FIG. 5
FIG. 5

Structure of A(K0,…,Kλ;mλ,nλ) within a single period of length 4λ+1−N, illustrated for λ = 1. The 4 subintervals of the period correspond to Kλ = 0, 1, 2, and 3.

FIG. 6
FIG. 6

Halftone of high-frequency bar pattern with discrete diamond dot profile. The halftone cell size is 8.

FIG. 7
FIG. 7

Halftone of high-frequency bar pattern with grating dot profile. The halftone cell size is 8.

FIG. 8
FIG. 8

Change in spectrum of halftone image when M = X/R is increased from 2 to 4. The points marked with squares are for M = 2, and those marked with dots are added when M = 4. The υ axis indicates the location of the spectral orders corresponding to the |P[m, n; â]|.

Tables (1)

Tables Icon

TABLE I Maxima of the nonlinearities |P[m, n; a]| for the grating and diamond dot profiles.a

Equations (36)

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h ( x , y ) = m , n = p [ m , n ; f m n ] rect [ ( x m R ) / R ] rect [ ( y n R ) / R ] ,
p [ m , n ; a ] = 0 or 1 ,
1 M 2 m , n = M / 2 M / 2 1 p [ m , n ; a ] = â , â = [ a M 2 ] R / M 2 .
p [ m , n ; f m n ] = { 1 , f m n σ m n 0 , otherwise .
p [ m , n ; b ] = 1 p [ m , n ; a ] = 1 a b
σ m n = min [ a : p [ m , n ; a ] = 1 ] .
σ m n = â + 1 / ( 2 M 2 ) .
H ( u , υ ) = sinc ( R u ) sinc ( R υ ) m n F m n ( u m / X , υ n / X ) , F m n ( u , υ ) = P [ m , n ; f ( x , y ) ] exp [ i 2 π ( u x + υ y ) ] dx dy , P [ m , n ; a ] = 1 M 2 k , 1 = M / 2 M / 2 1 p [ k , l ; a ] exp [ i 2 π ( m k + n l ) / M ] .
E ( a ) = k , l = M / 2 | k | + | l | 0 M / 2 1 | P [ k , l ; a ] | 2 = â ( 1 â ) .
p [ k , l ; a ] = 1 p [ k α , l β ; 1 a ] .
| P [ m , n ; a ] | = | P [ m , n ; 1 a ] | , | m | + | n | 0 .
| P [ m , n ; a ] P [ m , n ; b ] | | â b ̂ | a , b .
â = j = 0 N 1 K j 4 j N , K j = 0 , 1 , 2 , 3 .
p [ k , l ; â ] = j = 0 N 1 q = 1 K j p [ k s 1 ( j , q , â ) , l s 2 ( j , q , â ) ; 4 j N ] , s i ( j , q , â ) = r = j + 1 N 1 c i ( K r + 1 ) 2 N r 1 + c i ( q ) 2 N j 1 , i = 1 , 2 ; c 1 ( 1 ) = 0 , c 1 ( 2 ) = 1 , c 1 ( 3 ) = 0 , c 1 ( 4 ) = 1 , c 2 ( 1 ) = 0 , c 2 ( 2 ) = 1 , c 2 ( 3 ) = 1 , c 2 ( 4 ) = 0 .
m = r = 0 N 1 m r 2 r , n = r = 0 N 1 n r 2 r , m r , n r = 0 , 1
λ = min [ r : m r or n r = 1 ] .
P [ m , n ; â ] = A ( K 0 , . K λ ; m λ ; n λ ) × exp [ i θ ( K λ + 1 , , K N 1 ; m λ , , m N 1 ; n λ , , n N 1 ) ] ,
A ( K 0 , , K λ ; m λ ; n λ ) = 4 λ N q = 1 K λ ( 1 ) t ( q ; m λ ; n λ ) + ( 1 ) t ( K λ + 1 ; m λ ; n λ ) j = 0 λ 1 K j 4 N j , θ ( K λ + 1 , , K N 1 ; m λ , , m N 1 ; n λ , , n N 1 ) = 2 π [ r = λ + 1 N 1 q = λ r t ( K r + 1 ; m q ; n q ) 2 q r 1 ] ,
t ( q ; r ; s ) = c 1 ( q ) r + c 2 ( q ) s .
| P [ m , n ; a ] | { 2 | m n | 4 N , m 0 , n 0 | n 2 | 4 N , m = 0 , n 0 | m 2 | 4 N , m 0 , n = 0 .
h ( x ) = k p [ k ; f k ] rect [ ( x k R ) / R ] .
h ( x ) = l k p [ k ; f l ] rect [ ( x k R ) / R ] δ l k , 0 .
H ( u ) = R sinc ( R u ) l k p [ k ; f l ] δ l k , 0 exp ( i 2 π kRu ) .
k p [ k ; f l ] exp ( i 2 π kRu ) = 1 R m P [ m ; f l ] δ ( u m / X ) ,
P [ m ; a ] = 1 M r = M / 2 M / 2 1 p [ r ; a ] exp ( i 2 π m r / M ) .
k δ l k l , 0 exp ( i 2 π kRu ) = exp ( i 2 π lRu ) .
H ( u ) = R sinc ( R u ) l 1 / 2 R 1 / 2 R m P [ m ; f l ] × δ ( u μ m / X ) exp ( i 2 π l R μ ) d μ .
l P [ m ; f l ] exp ( i 2 π l R μ ) = 1 R l F m ( μ l / R ) ,
F m ( u ) = P [ m ; f ( x ) ] exp ( i 2 π u x ) d x .
H ( u ) = sinc ( R u ) m F m ( u m / X ) .
p ( m + 2 N j / 2 , n ; â ) = p [ m , n + 2 N j / 2 ; â ] = p [ m , n ; a ] .
p [ m + 2 N ( j + 1 ) / 2 , n + 2 N ( j + 1 ) / 2 ; a ] = p [ m , n ; a ] .
P [ m , n , â ] = j = 0 N 1 P [ m , n ; 4 j N ] × exp [ i 2 π ( r = j + 1 n 1 [ m c 1 ( K r + 1 ) + n c 2 ( K r + 1 ) ] 2 r 1 ) ] × q = 1 K j exp { i 2 π [ m c 1 ( q ) + n c 2 ( q ) ] 2 j 1 } .
P [ m , n ; 4 j N ] = { 4 j N , 0 j λ 0 , else .
m r = j + 1 N 1 c 1 ( K r + 1 ) 2 r 1 = r = j + 1 N 1 l = λ N 1 m l c 1 ( K r + 1 ) 2 l r 1 , j λ
r = ( j ) N 1 l = λ r m l c 1 ( K r + 1 ) 2 l r 1 ,