Abstract

A theory of binary images (two density levels) is presented. By considering the spatial frequency response function of the viewer, it is shown that binary images can provide a good representation of band-limited, continuous-tone images. The analytical formulation given permits a quantitative evaluation of effective image errors for a large class of binary images. As the spacing between binary elements decreases, the image improves until the spacing is about one-third the period of the highest spatial frequency in the continuous image. The analysis is applied to the design of transparency masks for spatial modulation of light beams. These shaped beams are used for input and output image scanning as part of digital image-processing operations. A binary gaussian function has been generated which effectively approximates a continuous gaussian to within ±0.5%.

© 1971 Optical Society of America

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References

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  1. F. Scott, Photogr. Sci. Eng. 11, 169 (1967).
  2. O. Bryndahl, E. Riseberg, Optica Acta 11, 117 (1964).
    [CrossRef]
  3. J. J. DePalma, E. M. Lowry, J. Opt. Soc. Am. 52, 328 (1962).
    [CrossRef]
  4. R. E. Kinzly, R. C. Hoos, P. G. Roetling, in SPIE Image Information Recovery Seminar Proceedings (1968), p. 97.
  5. R. V. Shack, in SPIE 14th Annual Technology Symposium, Proceedings, Redondo Beach, Calif. (1969), p. 392.
  6. W. Swindell, Appl. Opt. 9, 2459 (1970).
    [CrossRef] [PubMed]

1970 (1)

1967 (1)

F. Scott, Photogr. Sci. Eng. 11, 169 (1967).

1964 (1)

O. Bryndahl, E. Riseberg, Optica Acta 11, 117 (1964).
[CrossRef]

1962 (1)

Bryndahl, O.

O. Bryndahl, E. Riseberg, Optica Acta 11, 117 (1964).
[CrossRef]

DePalma, J. J.

Hoos, R. C.

R. E. Kinzly, R. C. Hoos, P. G. Roetling, in SPIE Image Information Recovery Seminar Proceedings (1968), p. 97.

Kinzly, R. E.

R. E. Kinzly, R. C. Hoos, P. G. Roetling, in SPIE Image Information Recovery Seminar Proceedings (1968), p. 97.

Lowry, E. M.

Riseberg, E.

O. Bryndahl, E. Riseberg, Optica Acta 11, 117 (1964).
[CrossRef]

Roetling, P. G.

R. E. Kinzly, R. C. Hoos, P. G. Roetling, in SPIE Image Information Recovery Seminar Proceedings (1968), p. 97.

Scott, F.

F. Scott, Photogr. Sci. Eng. 11, 169 (1967).

Shack, R. V.

R. V. Shack, in SPIE 14th Annual Technology Symposium, Proceedings, Redondo Beach, Calif. (1969), p. 392.

Swindell, W.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

Optica Acta (1)

O. Bryndahl, E. Riseberg, Optica Acta 11, 117 (1964).
[CrossRef]

Photogr. Sci. Eng. (1)

F. Scott, Photogr. Sci. Eng. 11, 169 (1967).

Other (2)

R. E. Kinzly, R. C. Hoos, P. G. Roetling, in SPIE Image Information Recovery Seminar Proceedings (1968), p. 97.

R. V. Shack, in SPIE 14th Annual Technology Symposium, Proceedings, Redondo Beach, Calif. (1969), p. 392.

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Figures (6)

Fig. 1
Fig. 1

Typical amplitude spectra and MTF.

Fig. 2
Fig. 2

Binary Mask.

Fig. 3
Fig. 3

Gaussian image spectra.

Fig. 4
Fig. 4

Direct space error functions.

Fig. 5
Fig. 5

Regular hexagonal lattice.

Fig. 6
Fig. 6

Reciprocal lattice.

Equations (24)

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f n ¯ [ ( x ¯ - g ¯ n ¯ ) / α n ¯ ] d 2 x ¯ = α n ¯ 2 .
h ( x ¯ ) = n ¯ f n ¯ [ ( x ¯ - g ¯ n ¯ ) / α n ¯ ] .
H ( k ¯ ) = n ¯ α n ¯ 2 F n ¯ ( α n ¯ k ¯ ) exp [ - 2 π i ( g ¯ n ¯ · k ¯ ) ] .
H ( k ¯ ) = n ¯ r n ¯ F n ¯ ( r n ¯ k ¯ ) exp [ - 2 π i ( g ¯ n ¯ · k ¯ ) ]
H eff ( k ¯ ) V ( k ¯ ) H ( k ¯ ) .
E ( k ¯ ) V ( k ¯ ) [ R ( k ¯ ) - H ( k ¯ ) ] .
A ( k ¯ ) n ¯ r n ¯ exp [ - 2 π i ( g ¯ n ¯ · k ¯ ) ]
B ( k ¯ ) sinc ( a ¯ 1 · k ¯ ) sinc ( a ¯ 2 · k ¯ ) A ( k ¯ ) ,
Rect ( X ) 1 0 { X ½ , otherwise .
r n ¯ = R ( k ¯ ) exp [ 2 π i ( g ¯ n ¯ · k ¯ ) d 2 k ¯ ] .
A ( k ¯ ) = n ¯ R ( k ¯ ) exp { - 2 π i [ g ¯ n ¯ · ( k ¯ - k ¯ ) ] d 2 k ¯ } .
n ¯ exp { - 2 π i [ g ¯ n ¯ · ( k ¯ - k ¯ ) ] } = δ 2 [ ( k ¯ - k ¯ ) + Q ¯ m ¯ ] ,
A ( k ¯ ) = m ¯ R ( k ¯ - Q ¯ m ¯ ) ,
Λ H ( k ¯ ) = V ( k ¯ ) n ¯ Δ r n ¯ F n ¯ ( r n ¯ k ¯ ) exp [ - 2 π i ( g ¯ n ¯ · k ¯ ) ] .
Λ H ( k ¯ ) = V ( k ¯ ) n ¯ Δ r n ¯ F n ¯ ( r n ¯ k ¯ ) exp [ - 2 π i ( g ¯ n ¯ · k ¯ ) ] ,
VAR [ Λ H ( k ¯ ) ] = Λ H ( k ¯ ) - Λ H ( k ¯ ) 2 .
VAR [ Λ H ( k ¯ ) ] = ( Δ r n ¯ ) 2 V ( k ¯ ) n ¯ F n ¯ ( r n ¯ k ¯ ) .
b ¯ 1 = ( a ¯ 2 × a ¯ 3 ) / [ a ¯ 1 · ( a ¯ 2 × a ¯ 3 ) ] , b ¯ 2 = ( a ¯ 3 × a ¯ 1 ) / [ a ¯ 1 · ( a ¯ 2 × a ¯ 3 ) ] .
R ( k ) = - r ( x ) exp [ - 2 π i ( x · k ) ] d 2 x ,
r ( x ¯ ) = - R ( k ¯ ) exp [ 2 π i ( x ¯ · k ¯ ) ] d 2 k ¯ .
H ( k ¯ ) = n ¯ r n ¯ F n ¯ ( r n ¯ k ¯ ) exp [ - 2 π i ( g ¯ n ¯ · k ¯ ) ] ,
A ( k ¯ ) = n ¯ r n ¯ exp [ - 2 π i ( g ¯ n ¯ · k ¯ ) ] ,
B ( k ¯ ) = sinc ( a ¯ 1 · k ¯ ) sinc ( a ¯ 2 · k ¯ ) n ¯ r n ¯ exp [ - 2 π i ( g ¯ n ¯ - k ¯ ) ] .
sinc ( a 1 · k ) sinc ( a 2 · k ) F n ¯ ( r n ¯ k ) 1.

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