Abstract

The Yule-Nielsen curve-fitting parameter n is derived in terms of the point spread function describing the scattering of light in paper substrates upon which are printed halftone dot or line patterns. The value of n is shown not to depend strongly on either screen frequency or area coverage at low area coverages or highlight regions. However, when the shadow regions are considered, n becomes increasingly dependent on both of these parameters. This dependence is not explicitly allowed for by the Yule-Nielsen equation. It is analytically shown that the value of n approaches 1.0 as the substrate approaches a specular surface and approaches 2.0 as the substrate becomes a perfect diffuser. The value of this analysis is the following. First, it indicates that the Yule-Nielsen equation is physically adequate in highlight and medium shadow regions. Second, an accurate determination of n at any screen frequency can be made from a single measurement of n at any other frequency using an inferred scattering length. Third, this analysis shows that the empirical (Yule-Nielsen) determination of n from dot density and fractional area coverages is valid only in the regions below about 50% area coverage.

© 1978 Optical Society of America

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References

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  1. F. W. Clulow, Color, Its Principles and Applications (Morgan and Morgan, New York, 1972).
  2. Pattern from Kodak Bulletin Q-3, “Halftone Methods for the Graphic Arts” (1972).
  3. A. Murray, Franklin Inst. 221, 721 (1936).
    [CrossRef]
  4. J. A. C. Yule, W. J. Nielsen, TAGA Proc. 3, 65 (1957).
  5. D. Lehmbeck, Xerox Corporation; private communication.
  6. J. A. C. Yule, D. J. Howe, J. H. Altman, TAPPI Proc., 50, 337–344 (1967).
  7. O. Lewis, Xerox Corporation; private communication.

1967 (1)

J. A. C. Yule, D. J. Howe, J. H. Altman, TAPPI Proc., 50, 337–344 (1967).

1957 (1)

J. A. C. Yule, W. J. Nielsen, TAGA Proc. 3, 65 (1957).

1936 (1)

A. Murray, Franklin Inst. 221, 721 (1936).
[CrossRef]

Altman, J. H.

J. A. C. Yule, D. J. Howe, J. H. Altman, TAPPI Proc., 50, 337–344 (1967).

Clulow, F. W.

F. W. Clulow, Color, Its Principles and Applications (Morgan and Morgan, New York, 1972).

Howe, D. J.

J. A. C. Yule, D. J. Howe, J. H. Altman, TAPPI Proc., 50, 337–344 (1967).

Lehmbeck, D.

D. Lehmbeck, Xerox Corporation; private communication.

Lewis, O.

O. Lewis, Xerox Corporation; private communication.

Murray, A.

A. Murray, Franklin Inst. 221, 721 (1936).
[CrossRef]

Nielsen, W. J.

J. A. C. Yule, W. J. Nielsen, TAGA Proc. 3, 65 (1957).

Yule, J. A. C.

J. A. C. Yule, D. J. Howe, J. H. Altman, TAPPI Proc., 50, 337–344 (1967).

J. A. C. Yule, W. J. Nielsen, TAGA Proc. 3, 65 (1957).

Franklin Inst. (1)

A. Murray, Franklin Inst. 221, 721 (1936).
[CrossRef]

TAGA Proc. (1)

J. A. C. Yule, W. J. Nielsen, TAGA Proc. 3, 65 (1957).

TAPPI Proc. (1)

J. A. C. Yule, D. J. Howe, J. H. Altman, TAPPI Proc., 50, 337–344 (1967).

Other (4)

O. Lewis, Xerox Corporation; private communication.

D. Lehmbeck, Xerox Corporation; private communication.

F. W. Clulow, Color, Its Principles and Applications (Morgan and Morgan, New York, 1972).

Pattern from Kodak Bulletin Q-3, “Halftone Methods for the Graphic Arts” (1972).

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Figures (8)

Fig. 1
Fig. 1

Gray levels created by halftone dots.2 These examples are reproduced in a scale much larger than normally used.

Fig. 2
Fig. 2

Comparison of halftone dot data with Eqs. (1) and (3) for a particular xerographic image/paper combination.

Fig. 3
Fig. 3

Comparison of Gg and Ge for (xx′) > 0. Normalization has been applied such that the areas under the curves are equal.

Fig. 4
Fig. 4

General 1-D line screen pattern.

Fig. 5
Fig. 5

Comparison of line screen data with Eq. (4) for photographically produced line screens on a particular paper.

Fig. 6
Fig. 6

Comparison of line screen data with Eq. (4) for a particular xerographic image/paper combination. Screen frequency is 40 lines/cm.

Fig. 7
Fig. 7

Yule-Nielsen constant as a predicted function of screen frequency. The two data points shown were taken from Fig. 5.

Fig. 8
Fig. 8

Comparison of halftone dot data with Eq. (4) using values of n obtained from Eq (29). Photographic samples were used.

Equations (49)

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R = c a R d R p + ( 1 c a ) R p = R p [ 1 c a ( 1 R d ) ] .
D = log 10 { 10 D p [ 1 c a ( 1 10 D d ) ] } .
R = c a R d R p [ ( 1 c a ) + c a R d ] + ( 1 c a ) R p [ ( 1 c a ) + c a R d ]
R = R p [ 1 c a ( 1 R d ) ] [ 1 c a ( 1 R d ) ] = R p [ 1 c a ( 1 R d ) ] 2 ,
D = 2 log 10 [ 1 c a ( 1 10 D d / 2 ) ] log R p
D = D p 2 log 10 [ 1 c a ( 1 10 D d / 2 ) ] .
D = D p n log 10 [ 1 c a ( 1 10 D d / n ) ] .
Δ L = R 0 [ I 0 ( x , y ) Δ x Δ y ] [ G ( x x , y y ) Δ x Δ y ] ,
I ( x , y ) = lim Δ x , Δ y 0 Δ L d x d y
I ( x , y ) = I ( x , y ) G ( x x , y y ) d x d y .
G g ( x x ) = 1 π σ exp [ ( x x ) 2 / σ 2 ] .
G e ( x x ) = ( σ / π ) / [ ( x x ) 2 + σ 2 ] .
I ( x ) = I ( x ) G ( x x ) d x .
J ( x ) = T ( x ) I 0 T ( x ) G ( x x ) d x .
R = 1 2 X X X J ( x ) d x .
R = R 0 π lim Y 1 2 Y Y Y T ( σ y ) T ( σ y ) × exp [ ( y y ) 2 ] d y d y ,
T ( σ x / L ) = 1 2 [ 1 + cos ( 2 π σ x L ) ] ,
R = ( R 0 4 ) [ 1 + 1 2 exp ( π 2 σ 2 L 2 ) ] .
R ( σ ) = R ( 0 ) · ( 2 3 ) [ 1 + 1 2 exp ( π 2 σ 2 L 2 ) ] .
R = ( R 0 4 ) [ 1 + 1 2 exp ( π σ L ) ] ,
R ( σ ) = R ( 0 ) · ( 2 3 ) [ 1 + 1 2 exp ( π σ L ) ] .
T ( x ) = T min + ( T max T min ) × [ 1 L + 2 π m = 1 ( 1 ) m m sin m π l 2 L cos 2 π m x L ] .
n , m A n A m d x = 0 for n m ; n , m A n A m d x = n a n 2 for n = m ,
a n 2 = A n 2 d x ; A m = cos 2 π m x / L .
R = [ ( 1 c a ) T max + c a T min ] 2 { 1 + c a ( 1 c a ) [ T max T min ( 1 c a ) T max + c a T min ] 2 × sin 2 ( m π l / L ) m 2 exp [ ( m π σ / L ) ] sin 2 ( m π l / L ) ] m 2 } .
R = [ 1 c a ( 1 R d ) ] 2 { 1 + c a ( 1 c a ) ( 1 R d ) 2 [ 1 c a ( 1 R d ) ] 2 · S } ,
R ( σ = 0 ) = [ 1 c a ( 1 R d ) ] ;
R ( σ = ) = [ 1 c a ( 1 R d ) ] 2 .
n log e [ 1 c a ( 1 R d 1 / n ) ] = 2 log e [ 1 c a ( 1 R d ) ] + log e { 1 + c a ( 1 c a ) ( 1 R d ) 2 [ 1 c a ( 1 R d ) ] 2 · S } .
n = 2 { log e [ 1 c a ( 1 R d ) ] log e [ 1 c a ( 1 R d 1 / n ) ] } + log e { 1 + c a ( 1 c a ) ( 1 R d ) 2 [ 1 c a ( 1 R d ) ] 2 · S } log e [ 1 c a ( 1 R d 1 / n ) ] ,
n = 2 + log e [ 1 + ( c a 1 c a ) S ] / log e ( 1 c a ) .
n 2 S .
n 2 exp ( π σ / L ) .
n = 2 + log e [ 1 + c a ( 1 c a ) ( 1 c a ) 2 · S ] / log e ( 1 c a ) .
lim c a 1 [ log e ( 1 + c a 1 c a · S ) c a / log e [ 1 c a ) c a ] = lim c a 1 [ S 1 + c a ( S 1 ) ] .
n 2 { S / [ 1 + c a ( S 1 ) ] }
n = 2 log e ( 1 + S ) / log e 2 ( c a = 0.5 ) .
S = exp ( π σ / L ) + exp ( 3 π σ / L ) / 9 + exp ( 5 π σ / L ) / 25 + 1 + 1 9 + 1 25 +
S ( 8 / π 2 ) exp ( π σ / L )
n 2 log e [ 1 + exp ( π σ / L ) / 1.23 ] / log e 2 ( c a = 0.5 ) .
T ( x , y ) = T ( x ) T ( y ) ,
R = lim X Y 1 4 X Y X X Y Y T ( x , y ) × T ( x , y ) G ( x x , y y ) d x d y d x d y .
R = lim X , Y 1 4 X Y T ( x ) T ( y ) × T ( x ) T ( y ) G ( x x ) G ( y y ) d x d y d x d y = [ lim x 1 2 x T ( x ) T ( x ) G ( x x ) d x d y ] 2 .
R = ( 1 c a ) [ S + 1 c a ) 1 / 2 ( 1 S ) ] 2 .
R ( σ = 0 ) = ( 1 c a ) ( T min = 0 , T max = 1 ) ;
R ( σ = ) = ( 1 c a ) 2 ( T min = 0 , T max = 1 ) .
n = 2 + 2 log e [ 1 + 1 ( 1 c a ) 1 / 2 ( 1 c a ) 1 / 2 S ] / log e ( 1 c a ) .
n 2 S
n 2 exp ( π σ / L ) .

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