Abstract

An analysis of kinoform image reconstruction error is presented. This analysis considers the effects of the error introduced by the kinoform approximation and the quantization effects of plotting. The error measure developed is applied to a proposed method for computing kinoforms. Numerical results indicate that images produced by this method offer considerable reduction in the error when compared with images produced from kinoforms made with the random phase method.

© 1973 Optical Society of America

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References

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  1. L. B. Lesen, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
    [CrossRef]
  2. J. C. Patau, L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 14, 485 (1970).
    [CrossRef]
  3. J. C. Patau, submitted to Geophys. 00 000 (197x).
  4. J. W. Cooley, P. A. W. Lewis, P. D. Welch, IEEE Trans. Audio Electroacoust. 15, 79 (1967).
    [CrossRef]
  5. R. A. Gabel, B. Liu, Appl. Opt. 9, 1180 (1970).
    [CrossRef] [PubMed]
  6. J. Katzenelson, IRE Trans. Automatic Control 7, 58 (1962). See also D. G. Watts, J. Katzenelson, IEEE Trans. Autom. Control 8, 187 (1963).
    [CrossRef]

1970 (2)

J. C. Patau, L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 14, 485 (1970).
[CrossRef]

R. A. Gabel, B. Liu, Appl. Opt. 9, 1180 (1970).
[CrossRef] [PubMed]

1969 (1)

L. B. Lesen, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

1967 (1)

J. W. Cooley, P. A. W. Lewis, P. D. Welch, IEEE Trans. Audio Electroacoust. 15, 79 (1967).
[CrossRef]

1962 (1)

J. Katzenelson, IRE Trans. Automatic Control 7, 58 (1962). See also D. G. Watts, J. Katzenelson, IEEE Trans. Autom. Control 8, 187 (1963).
[CrossRef]

Cooley, J. W.

J. W. Cooley, P. A. W. Lewis, P. D. Welch, IEEE Trans. Audio Electroacoust. 15, 79 (1967).
[CrossRef]

Gabel, R. A.

Hirsch, P. M.

J. C. Patau, L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 14, 485 (1970).
[CrossRef]

L. B. Lesen, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Jordan, J. A.

J. C. Patau, L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 14, 485 (1970).
[CrossRef]

L. B. Lesen, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Katzenelson, J.

J. Katzenelson, IRE Trans. Automatic Control 7, 58 (1962). See also D. G. Watts, J. Katzenelson, IEEE Trans. Autom. Control 8, 187 (1963).
[CrossRef]

Lesem, L. B.

J. C. Patau, L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 14, 485 (1970).
[CrossRef]

Lesen, L. B.

L. B. Lesen, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

Lewis, P. A. W.

J. W. Cooley, P. A. W. Lewis, P. D. Welch, IEEE Trans. Audio Electroacoust. 15, 79 (1967).
[CrossRef]

Liu, B.

Patau, J. C.

J. C. Patau, L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 14, 485 (1970).
[CrossRef]

J. C. Patau, submitted to Geophys. 00 000 (197x).

Welch, P. D.

J. W. Cooley, P. A. W. Lewis, P. D. Welch, IEEE Trans. Audio Electroacoust. 15, 79 (1967).
[CrossRef]

Appl. Opt. (1)

IBM J. Res. Dev. (2)

L. B. Lesen, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 13, 150 (1969).
[CrossRef]

J. C. Patau, L. B. Lesem, P. M. Hirsch, J. A. Jordan, IBM J. Res. Dev. 14, 485 (1970).
[CrossRef]

IEEE Trans. Audio Electroacoust. (1)

J. W. Cooley, P. A. W. Lewis, P. D. Welch, IEEE Trans. Audio Electroacoust. 15, 79 (1967).
[CrossRef]

IRE Trans. Automatic Control (1)

J. Katzenelson, IRE Trans. Automatic Control 7, 58 (1962). See also D. G. Watts, J. Katzenelson, IEEE Trans. Autom. Control 8, 187 (1963).
[CrossRef]

Other (1)

J. C. Patau, submitted to Geophys. 00 000 (197x).

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Figures (8)

Fig. 1
Fig. 1

Phase quantizing function.

Fig. 2
Fig. 2

Image error at the point (x,y).

Fig. 3
Fig. 3

D-chart of the method for producing optimum kinoforms.

Fig. 4
Fig. 4

One-dimensional image function.

Fig. 5
Fig. 5

Image reconstruction error for τ(x).

Fig. 6
Fig. 6

Sample values of reconstructed image compared with original image τ(x): (a) thirty-two quantizing levels; (b) eight quantizing levels; (c) four quantizing levels; (d) two quantizing levels.

Fig. 7
Fig. 7

(a) Original sampled image. (b) Reconstructed image from kinoform made by the random phase method. (c) Reconstructed image from kinoform made by the method of Sect. V. (d) Key to plots in Figs. 7(a), 7(b), 7(c).

Fig. 8
Fig. 8

Image reconstruction error for P.

Equations (56)

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A ( χ , ξ ) exp [ i ψ ( χ , ξ ) ] = - τ ( x , y ) × exp [ i φ ( x , y ) ] exp [ - i 2 π λ ζ ( x χ + y ξ ) ] d x d y ,
A ( χ , ξ ) = { A ξ , χ ( W / 2 ) 0 otherwise ;
A ( χ , ξ ) exp [ i ψ ( χ , ξ ) ] = - τ ( x , y ) × exp [ i φ ( x , y ) ] exp [ - i 2 π ( x χ + y ξ ) ] d x d y ,
S = { ( x , y ) x , y x / 2 } .
τ ( x , y ) exp [ i φ ( x , y ) ] = ∑∑ m , n = - 1 X 2 A m n × exp ( i ψ ) m n exp [ i ( 2 π / X ) ( m x + n y ) ] .
A m n = A ( m / X , n / X ) ψ m n = ψ ( m / X , n / X ) .
| m X | , | n X | W 2
h ( x , y ) exp [ i α ( x , y ) ] = 2 X 2 ∑∑ m , n I A × exp [ i ( ψ m n + θ m n ) ] exp [ i 2 π X ( m x + n y ) ] .
Φ = ɛ { 1 X 2 S [ τ ( x , y ) ] - h ( x , y ) ] 2 d x d y } ,
e ( x , y ) = τ ( x , y ) exp [ i φ ( x , y ) ] - h ( x , y ) exp [ i α ( x , y ) ] .
Φ = 1 / 2 Φ 0 ,
Φ 0 = 1 X 4 [ ∑∑ m , n I ( A m n - A ) 2 + 2 ( 1 - sin Θ / 2 Θ / 2 ) × A ∑∑ m , n I A m n + ∑∑ m , n I A m n 2 ] .
Φ = [ 2 ( area of S 1 ) + ( area of S 2 ) 2 ( area of S ) ] Φ 0 .
Φ 0 = 1 X 4 [ ∑∑ m , n I ( A m n - A ) 2 + ∑∑ m , n ∈⃥ I A m n 2 ] .
A = 1 N 0 2 sin Θ / 2 Θ / 2 ∑∑ m , n I A m n ,
N 0 2 = ∑∑ m , n I 1 .
Φ 0 = 1 X 4 ( ∑∑ m , n = - A m n 2 - N 0 2 A 2 ) .
- X / 2 X / 2 τ 2 ( x , y ) d x d y = 1 X 2 ∑∑ m , n = - A m n 2 ,
Φ 0 = 1 X 2 - X / 2 X / 2 τ 2 ( x , y ) d x d y - N 0 2 X 4 A 2 .
Σ Σ A p q p , q I .
A p q exp ( i ψ p q ) = 1 N 2 ∑∑ m , n = 0 N - 1 τ m n exp ( i φ m n ) × exp [ - i 2 π N ( m p + n q ) ]
τ m n exp ( i φ m n ) = ∑∑ p = q = 0 N - 1 A p q exp ( i ψ p q ) × exp [ i 2 π N ( m p + n q ) ] .
X 2 A p q exp ( i ψ p q ) A p q exp ( i ψ p q ) .
∑∑ p , q l A p q = ( X / N ) 2 ∑∑ p , q I | ∑∑ m , n = 0 N - 1 τ m n exp ( i φ m n ) × exp [ - i 2 π N ( m p + n q ) ] | .
Φ 0 = 1 X 2 - X / 2 X / 2 τ 2 ( x , y ) d x d y - N 0 2 X 4 ɛ { A 2 } .
ɛ { A 2 } = ( 1 N 0 2 sin Θ / 2 Θ / 2 ) 2 ɛ { ( ∑∑ m , n I A m n ) 2 } .
ɛ { A 2 } ( ɛ { A } ) 2 .
ɛ { A } = 1 N 0 2 sin Θ / 2 Θ / 2 ∑∑ m , n I ɛ { A m n } .
ɛ { A m n 2 } ( ɛ { A m n } ) 2 .
ɛ { A m n 2 } > ( ɛ { A m n } ) 2 .
ɛ { A p q 2 } = 1 N 4 ∑∑ m , n = 0 N - 1 τ m n 2 .
ɛ { A p q 2 } = X 4 N 4 ∑∑ m , n = 0 N - 1 τ m n 2 .
ɛ { A 2 } < X 4 N 4 ( sin Θ / 2 Θ / 2 ) 2 ∑∑ m , n = 0 N - 1 τ m n 2 .
Φ 0 > Φ L = 1 X 2 S τ 2 ( x , y ) d x d y - N 0 2 N 4 ( sin Θ / 2 Θ / 2 ) 2 ∑∑ m , n = 0 N - 1 τ m n 2 .
A ˜ p q = { A p , q I 0 p , q ∈⃥ I .
τ ( x ) = { 1 x 3 / 16 0 x > 3 / 16.
Φ 0 = 1 X - X / 2 X / 2 τ 2 ( x ) d x - N 0 X 2 A 2 .
A = 1 N 0 sin Θ / 2 Θ / 2 n I A n
Φ L = 1 X - X / 2 X / 2 τ 2 ( x ) d x - N 0 N 2 ( sin Θ / 2 Θ / 2 ) n = 0 N - 1 τ n 2 .
( Θ = 0 ) 0.1806 , 0.1926 , 0.2072 , 0.2306 , 0.2270 0.2008 , 0.2280 , 0.2534 , 0.2296 , 0.1978.
h = τ [ 1 + e 2 / τ 2 - 2 e / τ cos γ ] 1 / 2 .
[ h ( x , y ) - τ ( x , y ) ] 2 e ( x , y ) 2 cos 2 γ ( x , y )
Φ = ɛ [ 1 X 2 S cos 2 γ ( x , y ) e ( x , y ) 2 d x d y ] .
Φ = ɛ { cos 2 γ } ɛ [ 1 X 2 S e ( x , y ) 2 d x d y ] .
e ( x , y ) = 1 X 2 ∑∑ m , n = - A m n exp [ i ψ m n + i ( 2 π X ) ( m x + n y ) ] - 1 X 2 ∑∑ m , n I A exp [ i ( ψ m n + θ m n ) ] exp [ i 2 π X ( m x + n y ) ] = 1 X 2 ∑∑ m , n = - E m n exp [ i 2 π X ( m x + n y ) ] .
E m n = { A m n exp ( i ψ m n ) m , n ∈⃥ I [ A m n - A exp ( i θ m n ) ] exp ( i ψ m n ) m , n I .
1 X 2 S e ( x , y ) 2 d x d y = 1 X 4 ∑∑ m , n = - E m n 2 .
ɛ { E m n 2 } = { A m n 2 m , n ∈⃥ I A m n 2 + A 2 - 2 A A m n ɛ { cos θ m n } m , n I .
ɛ { cos θ m n } = ( sin Θ / 2 ) / Θ / 2
ɛ [ 1 X 2 S e ( x , y ) 2 d x d y ] = 1 X 4 × [ ∑∑ m , n I ( A m n 2 + A 2 - 2 A A m n sin Θ / 2 Θ / 2 ) + ∑∑ m , n ∈⃥ I A m n 2 ] .
ɛ { cos 2 γ } = 1 / 2.
Φ = 1 / 2 Φ 0 .
e ( x , y ) = h ( x , y ) exp [ i α ( x , y ) ] ;
[ h ( x , y ) - τ ( x , y ) ] 2 = h 2 ( x , y ) = e ( x , y ) 2 .
Φ = ɛ [ 1 X 2 S 1 e ( x , y ) 2 d x d y ] + 1 / 2 ɛ [ 1 X 2 S 2 e ( x , y ) 2 d x d y ] .
Φ = { [ 2 ( area of S 1 ) + ( area of S 2 ) ] / 2 ( area of S ) } Φ 0 .

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