Abstract

Images that have been halftoned with a periodic screen may suffer from several kinds of defects, including Moiré patterns which result when the original image contains periodic structures with periods near that of the screen. A random quasiperiodic halftone process is proposed to eliminate these defects. The process can be implemented either as a contact screen or electronically. The statistical mean and covariance of the spectrum of the halftone image are derived. Experimental work confirms the results of the analysis and shows the effectiveness of the random quasiperiodic process in eliminating the defects found in periodically halftoned images.

© 1976 Optical Society of America

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References

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  1. J. W. Wesner, “Screen Patterns Used in Reproduction of Continuous-Tone Graphics,” Appl. Opt. 13, 1703–1710 (1974).
    [Crossref] [PubMed]
  2. H. Kato and J. W. Goodman, “Nonlinear Filtering in Coherent Optical Systems Through Halftone Screen Processes,” Appl. Opt. 14, 1813–1824 (1975).
    [Crossref] [PubMed]
  3. S. R. Dashiell and A. A. Sawchuk, “Optical Synthesis of Nonlinear Nonmonotonic Functions,” J. Opt. Soc. Am. 65, 1177A (1975).
  4. J. A. C. Yule, Principles of Color Reproduction (Wiley, New York, 1967), p. 328.
  5. O. Bryngdahl, “Random Carrier Photography,” J. Opt. Soc. Am. 63, 1064–1070 (1973).
    [Crossref]
  6. A. C. Hardy and F. L. Wurzburg, “A Photoelectric Method for Preparing Printing Plates,” J. Opt. Soc. Am. 38, 295–300 (1948).
    [Crossref] [PubMed]
  7. P. G. Roetling, “Halftone Method with Edge Enhancement and Moiré Suppression,” J. Opt. Soc. Am. 65, 1177A (1975).
  8. D. Kermisch and P. G. Roetling, “Fourier Spectrum of Halftone Images,” J. Opt. Soc. Am. 65, 716–723 (1975).
    [Crossref]
  9. A. Papoulis, System and Transforms with Application in Optics (McGraw-Hill, New York, 1968), pp. 287–293.

1975 (4)

H. Kato and J. W. Goodman, “Nonlinear Filtering in Coherent Optical Systems Through Halftone Screen Processes,” Appl. Opt. 14, 1813–1824 (1975).
[Crossref] [PubMed]

S. R. Dashiell and A. A. Sawchuk, “Optical Synthesis of Nonlinear Nonmonotonic Functions,” J. Opt. Soc. Am. 65, 1177A (1975).

P. G. Roetling, “Halftone Method with Edge Enhancement and Moiré Suppression,” J. Opt. Soc. Am. 65, 1177A (1975).

D. Kermisch and P. G. Roetling, “Fourier Spectrum of Halftone Images,” J. Opt. Soc. Am. 65, 716–723 (1975).
[Crossref]

1974 (1)

1973 (1)

1948 (1)

Bryngdahl, O.

Dashiell, S. R.

S. R. Dashiell and A. A. Sawchuk, “Optical Synthesis of Nonlinear Nonmonotonic Functions,” J. Opt. Soc. Am. 65, 1177A (1975).

Goodman, J. W.

Hardy, A. C.

Kato, H.

Kermisch, D.

Papoulis, A.

A. Papoulis, System and Transforms with Application in Optics (McGraw-Hill, New York, 1968), pp. 287–293.

Roetling, P. G.

D. Kermisch and P. G. Roetling, “Fourier Spectrum of Halftone Images,” J. Opt. Soc. Am. 65, 716–723 (1975).
[Crossref]

P. G. Roetling, “Halftone Method with Edge Enhancement and Moiré Suppression,” J. Opt. Soc. Am. 65, 1177A (1975).

Sawchuk, A. A.

S. R. Dashiell and A. A. Sawchuk, “Optical Synthesis of Nonlinear Nonmonotonic Functions,” J. Opt. Soc. Am. 65, 1177A (1975).

Wesner, J. W.

Wurzburg, F. L.

Yule, J. A. C.

J. A. C. Yule, Principles of Color Reproduction (Wiley, New York, 1967), p. 328.

Appl. Opt. (2)

J. Opt. Soc. Am. (5)

S. R. Dashiell and A. A. Sawchuk, “Optical Synthesis of Nonlinear Nonmonotonic Functions,” J. Opt. Soc. Am. 65, 1177A (1975).

O. Bryngdahl, “Random Carrier Photography,” J. Opt. Soc. Am. 63, 1064–1070 (1973).
[Crossref]

A. C. Hardy and F. L. Wurzburg, “A Photoelectric Method for Preparing Printing Plates,” J. Opt. Soc. Am. 38, 295–300 (1948).
[Crossref] [PubMed]

P. G. Roetling, “Halftone Method with Edge Enhancement and Moiré Suppression,” J. Opt. Soc. Am. 65, 1177A (1975).

D. Kermisch and P. G. Roetling, “Fourier Spectrum of Halftone Images,” J. Opt. Soc. Am. 65, 716–723 (1975).
[Crossref]

Other (2)

A. Papoulis, System and Transforms with Application in Optics (McGraw-Hill, New York, 1968), pp. 287–293.

J. A. C. Yule, Principles of Color Reproduction (Wiley, New York, 1967), p. 328.

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Figures (9)

FIG. 1
FIG. 1

Periodic halftone of an image with constant absorptance.

FIG. 2
FIG. 2

Random quasiperiodic halftone of an image with constant absorptance.

FIG. 3
FIG. 3

Random quasiperiodic halftones of low-frequency bar pattern with λ = 0.0, 0.5, and 1.0, left to right. The pattern is oriented at 10° above horizontal and has period 5X.

FIG. 4
FIG. 4

Random quasiperiodic halftones of high-frequency bar pattern with λ = 0.0, 0.5, and 1.0, left to right. The pattern is oriented at 5° above horizontal and has period 1.25X. Note the spurious low-frequency pattern at 18° below horizontal that appears when λ = 0.

FIG. 5
FIG. 5

Superposition of constant absorptance quasiperiodic halftones oriented 15° apart for λ = 0.0, 0.5, and 1.0, left to right.

FIG. 6
FIG. 6

Fourier spectrum of periodic halftone of high-frequency bar pattern (Fig. 4, λ = 0).

FIG. 7
FIG. 7

Key to Fig. 6 identifying the components belonging to each diffraction order on the horizonal axis. Arrows mark the spectrum of the original image in the zeroeth diffraction order. The spectrum of the spurious low-frequency pattern in Fig. 4, λ = 0 is indicated by the dashed line.

FIG. 8
FIG. 8

Fourier spectrum of random quasiperiodic halftone of high-frequency bar pattern (Fig. 4, λ = 1). All the components of the zeroeth diffraction order shown in Fig. 7 are visible.

FIG. 9
FIG. 9

Supplementary figure for the benefit of the readers to give a better impression of how Figs. 35 would appear under normal viewing conditions.

Equations (66)

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h ( x ) = g [ x ; f ( x ) ] .
g [ x ; a ] = n = p [ x - n X ; a ] .
p [ x ; a ] = { 0 or 1 , x < 1 2 X , 0 , elsewhere
1 X - X / 2 X / 2 p [ x ; a ] d x = a .
g [ x ; a ] = n = - p n [ x - n X ; a ] .
p n [ x ; a ] = rect ( x X ) j = - p [ x - U n - j X ; a ] ,
p [ x ; a ] = rect ( x / X a ) .
{ H ( u ) } = F ( u ) + 1 X n 0 - P ¯ [ n X ; f ( ξ ) ] × exp [ - i 2 π ( u - n X ) ξ ] d ξ .
c H H ( u 1 , u 2 ) = { ( H ( u 1 ) - { H ( u 1 ) } ) ( H * ( u 2 ) - { H * ( u 2 ) } ) } .
c H H ( u 1 , u 2 ) = 1 X n c P P [ μ , μ - n X ; f ( ξ ) , f ( η ) ] × exp { - i 2 π [ ( u 1 - μ ) ξ - ( u 2 - μ + n X ) η ] } × d μ d ξ d η ,
c P P [ u 1 , u 2 ; a 1 , a 2 ] = { ( P n [ u 1 ; a 1 ] - P ¯ [ u 1 ; a 1 ] ) × ( P n * [ u 2 ; a 2 ] - P ¯ * [ u 2 ; a 2 ] ) }
{ H ( u ) } = F ( u ) + 1 X n 0 Φ ( n X ) - P [ n X ; f ( ξ ) ] × exp [ - i 2 π ( u - n X ) ξ ] d ξ
c H H ( u 1 , u 2 ) = m n [ Φ ( m - n X ) - Φ ( m X ) Φ ( n X ) ] × k F m , k ( u 1 - m X ) F n , k * ( u 2 - n X ) .
F m , k ( u ) = 1 X ( k - 1 / 2 ) X ( k + 1 / 2 ) X P [ m X ; f ( ξ ) ] exp ( - i 2 π u ξ ) d ξ .
h ( x ) = f ( x ) + e ( x ) .
H ( u ) = F ( u ) + E ( u ) .
E 1 ( u ) = 1 X n 0 - P [ n X ; f ( ξ ) ] exp [ - i 2 π ( u - n X ) ξ ] d ξ .
{ E 2 ( u ) } = 0 , c E 2 E 2 ( u 1 , u 2 ) = m 0 k F m , k ( u 1 - m X ) F m , k * ( u 2 - m X ) .
{ E 2 ( u ) 2 } = m 0 k | F m , k ( u - m X ) | 2 .
ϕ ( x ) = ( 1 - λ ) ϕ 1 ( x ) + λ ϕ 2 ( x ) ,             0 λ 1 ,
{ E ( u ) 2 } = ( 1 - λ ) E 1 ( u ) 2 - λ ( 1 - λ ) E 12 ( u ) + λ { E 2 ( u ) 2 } , E 12 ( u ) = m m , n 0 n k k l l F m , k ( u - m X ) F n , l * ( u - n X ) .
{ E 2 ( u ) 2 } = m 0 1 Y 2 n | F m ( n Y ) | 2 × sinc 2 [ X ( u - n Y - m X ) ] .
h ( x , y ) = g [ x , y ; f ( x , y ) ] ,
g [ x , y ; a ] = m n p m , n [ x - m X , y - n X ; a ] .
p m , n [ x , y ; a ] = { 0 or 1 , x , y < 1 2 X , 0 , elsewhere , 1 X 2 - X / 2 X / 2 p m , n [ x , y ; a ] d x d y = a .
p m , n [ x , y ; a ] = rect ( x X ) rect ( y X ) × i j p [ x - U m , n - i X , y - V m , n - j X ; a ] .
{ H ( u , v ) } = F ( u , v ) + 1 X 2 m not both = 0 n Φ ( m X , n X ) × P [ m X , n X ; f ( ξ , η ) ] exp { - i 2 π [ ( u - m X ) ξ + ( v - n X ) η ] } d ξ d η ,
c H H ( u 1 , u 2 ; v 1 , v 2 ) = k l m n i j F k , l , i , j ( u 1 - k X , v 1 - l X ) × F m , n , i , j * ( u 2 - m X , v 2 - n X ) × { Φ ( k - m X , l - n X ) - Φ ( k X , l X ) Φ ( m X , n X ) } .
F k , l . m , n ( u , v ) = 1 X 2 ( m - 1 / 2 ) X ( m + 1 / 2 ) X ( n - 1 / 2 ) X ( n + 1 / 2 ) X P [ k X , l X ; f ( ξ , η ) ] × exp [ - i 2 π ( u ξ + v η ) ] d ξ d η .
E 1 ( u , v ) = 1 X 2 m m , n not n both = 0 P [ m X , n X ; f ( ξ , η ) ] × exp { - i 2 π [ ( u - m X ) ξ + ( v - n X ) η ] } d ξ d η .
ϕ 2 ( x , y ) = 1 / X 2 rect ( x / X ) rect ( y / X )
{ E 2 ( u , v ) } = 0 , E { E 2 ( u , v ) 2 } = m m , n n not i both j = 0 | F m , n , i , j ( u - m X , v - n X ) | 2
h ( x , y ) = { 1 , f ( x , y ) + s ( x , y ) T , 0 , elsewhere .
s ( x , y ) = m n σ m , n ( x - m X , y - n X ) ;
σ m , n ( x , y ) = rect ( x X ) rect ( y X ) × i j σ [ x - U m , n - i X , y - V m , n - j X ] .
σ ( x , y ) = T - a 1 ,
σ ( x , y ) = T / a 1 .
p [ x , y ; a ] = { 1 , x , y 1 1 X a , 0 , elsewhere ,
a 1 = ( 2 / X max [ x , y ] ) 2 .
ϕ ( x , y ) = ( 1 - λ ) δ ( x ) δ ( y ) + ( λ / X 2 ) rect ( x / X ) rect ( y / X ) ,             0 λ 1.
h ( x ) = - g [ x ; f ( ξ ) ] δ ( x - ξ ) d ξ .
{ h ( x ) } = - { g [ x ; f ( ξ ) ] } δ ( x - ξ ) d ξ .
{ g [ x ; a ] } - n { p n [ x - n X ; a ] } .
{ g [ x ; a ] } = n p ¯ [ x - n X ; a ] ,
{ H ( u ) } = F ( u ) + 1 X n 0 - P ¯ [ n X ; f ( ξ ) ] × exp [ - i 2 π ( u - n X ) ξ ] d ξ ,
c h h ( x 1 , x 2 ) = { [ h ( x 1 ) - { h ( x 1 ) } ] [ h ( x 2 ) - { h ( x 2 ) } ] }
c h h ( x 1 , x 2 ) = c g g [ x 1 , x 2 ; f ( ξ ) , f ( η ) ] × δ ( x 1 - ξ ) δ ( x 2 - η ) d ξ d η ,
c g g [ x 1 , x 2 ; a 1 , a 2 ] = ( ( g [ x 1 ; a 1 ] - { g [ x 1 ; a 1 ] } ) × ( g [ x 2 ; a 2 ] - { g [ x 2 ; a 2 ] } ) ) = m n [ ( p m [ x 1 - m X ; a 1 ] - p ¯ [ x 1 - m X ; a 1 ] ) × ( p n [ x 2 - n X ; a 2 ] - p ¯ [ x 2 - n X ; a 2 ] ) } .
c g g [ x 1 , x 2 ; a 1 , a 2 ] = n c p p [ x 1 - n X , x 2 - n X ; a 1 , a 2 ] ,
c g g [ u 1 , u 2 ; a 1 , a 2 ] = C p p [ u 1 , u 2 ; a 1 , a 2 ] 1 X n δ ( u 1 + u 2 - n X ) ,
c H H ( u 1 , u 2 ) = 1 X n c P P [ μ , μ - n X ; f ( ξ ) , f ( η ) ] × exp { - i 2 π [ ( u 1 - μ ) ξ - ( u 2 - μ + n X ) η ] } d μ d ξ d η .
p * [ x ; a ] = m p [ x - m X ; a ] ,
p n [ x ; a ] = rect ( x / X ) p * [ x - U n ; a ] .
p ¯ [ x ; a ] = rect ( x / X ) p ¯ * [ x : a ] ,
p ¯ * [ x ; a ] = - X / 2 X / 2 p * [ x - ξ ; a ] ϕ ( ξ ) d ξ ,
P ¯ [ u ; a ] = m P [ m X ; a ] Φ ( m X ) sinc [ X ( u - m X ) ]
P ¯ [ n / X ; a ] = P [ n / X ; a ] Φ ( n / X ) .
r p p [ x 1 , x 2 ; a 1 , a 2 ] = { p n [ x 1 ; a 1 ] p n [ x 2 ; a 2 ] } = rect ( x 1 / X ) rect ( x 2 / X ) r p * p * [ x 1 , x 2 ; a 1 , a 2 ] ,
r p * p * [ x 1 , x 2 ; a 1 , a 2 ] = - X / 2 X / 2 p * [ x 1 - ξ ; a 1 ] × p * [ x 2 - ξ ; a 2 ] ϕ ( ξ ) d ξ .
R p p [ u 1 , u 2 ; a 1 , a 2 ] = m n P [ m X ; a 1 ] P [ n X ; a 2 ] Φ ( m + n X ) × sinc [ X ( u 1 - m X ) ] sinc [ X ( u 2 - n X ) ] .
c P P [ u 1 , u 2 ; a 1 , a 2 ] = R p p [ u 1 , - u 2 ; a 1 , a 2 ] - P ¯ [ u 1 ; a 1 ] P ¯ * [ u 2 ; a 2 ] = m n P [ m X ; a 1 ] P [ - n X ; a 2 ] [ Φ ( m - n X ) - Φ ( m X ) Φ ( - n X ) ] × sinc [ X ( u 1 - m X ) ] sinc [ X ( - u 2 + n X ) ] .
F m ( N ) ( u ) 2 = 1 N X 2 × n = - ( N - 1 ) / 2 ( N - 1 ) / 2 | ( n - 1 / 2 ) X ( n + 1 / 2 ) X f m ( x ) exp ( - i 2 π u x ) d x | 2 ,
{ N 2 ( u ) 2 } m m 0 | F m ( N ) ( u - m X ) | 2 .
( n - 1 / 2 ) X ( n + 1 / 2 X ) f m ( x ) exp ( - i 2 π u x ) d x = n X - p Y - X / 2 n X - p Y + X / 2 f m ( x ) exp ( - i 2 π u x ) d x exp ( - i 2 π p Y u ) ,
F m ( N ) ( u ) 2 = 1 N X 2 × n = - ( N - 1 ) / 2 ( N - 1 ) / 2 | n Y / N - X / 2 n Y / N + X / 2 f m ( x ) exp ( - i 2 π u x ) d x | 2 .
F m ( u ) 2 = 1 Y - Y / 2 Y / 2 | 1 X y - X / 2 y + X / 2 f m ( x ) exp ( - i 2 π u x ) d x | 2 d y .