Abstract

Images reconstructed from computer-generated holograms synthesized by the iterative direct binary search (DBS) technique suffer from noise due to leakage from uncontrolled parts of the diffraction pattern. The statistics of the nonhomogeneous leakage noise are analyzed under the assumption of independent, identically distributed (iid) addressable hologram cell transmittances. Three measures for the effect of the leakage, root-mean-squared (rms) error, efficiency, and contrast are defined and compared for ensemble statistics based on the model and for sample statistics of the leakage noise in images reconstructed from iid holograms and from actual DBS holograms. To assess the relative significance of the leakage noise, these measures are compared with the rms representation-related error and the total efficiency in images reconstructed from DBS holograms. Except for a cross artifact due to a nonzero mean hologram transmittance, the leakage noise is found to exhibit specklelike unity contrast.

© 1989 Optical Society of America

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References

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  1. W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer-Verlag, New York, 1980), pp. 291–366.
  2. W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics XVI, E. Wolf, ed. (North-Holland, New York, 1978), pp. 118–232.
  3. Special issue on computer-generated holograms, Appl. Opt. 26, 4350–4399 (1987).
  4. M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26, 2788–2798 (1987).
    [CrossRef] [PubMed]
  5. J. P. Allebach, “Representation-related errors in binary digital holograms: a unified analysis,” Appl. Opt. 20, 290–299 (1981).
    [CrossRef] [PubMed]
  6. N. C. Gallagher, B. Liu, “Statistical properties of the Fourier transform of random phase diffusers,” Optik 42, 65–86 (1975).
  7. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
    [CrossRef]
  8. D. J. Sakrison, “On the role of the observer and a distortion measure in image transmission,”IEEE Trans. Commun. COM-25, 1251–1267 (1977).
    [CrossRef]
  9. E. Parzen, Stochastic Processes (Holden-Day, Oakland, Calif., 1962), p. 93.

1987 (2)

1981 (1)

1977 (1)

D. J. Sakrison, “On the role of the observer and a distortion measure in image transmission,”IEEE Trans. Commun. COM-25, 1251–1267 (1977).
[CrossRef]

1975 (1)

N. C. Gallagher, B. Liu, “Statistical properties of the Fourier transform of random phase diffusers,” Optik 42, 65–86 (1975).

Allebach, J. P.

Dallas, W. J.

W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer-Verlag, New York, 1980), pp. 291–366.

Gallagher, N. C.

N. C. Gallagher, B. Liu, “Statistical properties of the Fourier transform of random phase diffusers,” Optik 42, 65–86 (1975).

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
[CrossRef]

Lee, W. H.

W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics XVI, E. Wolf, ed. (North-Holland, New York, 1978), pp. 118–232.

Liu, B.

N. C. Gallagher, B. Liu, “Statistical properties of the Fourier transform of random phase diffusers,” Optik 42, 65–86 (1975).

Parzen, E.

E. Parzen, Stochastic Processes (Holden-Day, Oakland, Calif., 1962), p. 93.

Sakrison, D. J.

D. J. Sakrison, “On the role of the observer and a distortion measure in image transmission,”IEEE Trans. Commun. COM-25, 1251–1267 (1977).
[CrossRef]

Seldowitz, M. A.

Sweeney, D. W.

Appl. Opt. (3)

IEEE Trans. Commun. (1)

D. J. Sakrison, “On the role of the observer and a distortion measure in image transmission,”IEEE Trans. Commun. COM-25, 1251–1267 (1977).
[CrossRef]

Optik (1)

N. C. Gallagher, B. Liu, “Statistical properties of the Fourier transform of random phase diffusers,” Optik 42, 65–86 (1975).

Other (4)

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
[CrossRef]

W. J. Dallas, “Computer-generated holograms,” in The Computer in Optical Research, B. R. Frieden, ed. (Springer-Verlag, New York, 1980), pp. 291–366.

W. H. Lee, “Computer-generated holograms: techniques and applications,” in Progress in Optics XVI, E. Wolf, ed. (North-Holland, New York, 1978), pp. 118–232.

E. Parzen, Stochastic Processes (Holden-Day, Oakland, Calif., 1962), p. 93.

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Figures (9)

Fig. 1
Fig. 1

Optical reconstruction of a 64 × 64 DBS hologram. The reconstruction region R is located off axis.

Fig. 2
Fig. 2

Optical reconstruction of the 64 × 64 DBS hologram in Fig. 1 replicated 2 × 2 times.

Fig. 3
Fig. 3

Plot of sinc8,5(x, y).

Fig. 4
Fig. 4

(a) Reconstructed image of three samples with no leakage and (b) reconstructed image with leakage from four uncontrolled samples for the one-dimensional case. Symbols: —, reconstructed image; ×, controlled samples; •, uncontrolled samples.

Fig. 5
Fig. 5

Binary transmittance function of the 64 × 64 DBS hologram reconstructed in Fig. 1.

Fig. 6
Fig. 6

Mean leakage intensity for a 64 × 64 binary amplitude hologram with μ = 1/2. The 16 × 16 reconstruction region R is centered at (16, 0). Note that the vertical scale is logarithmic.

Fig. 7
Fig. 7

Mean leakage intensity for a 64 × 64 binary phase hologram with μ = 0. The 16 × 16 reconstruction region R is centered at (16, 0). The vertical axis is inverted to illustrate the smaller leakage in R.

Fig. 8
Fig. 8

Mean leakage intensity for a 64 × 64 binary phase hologram in the 16 × 16 reconstruction region R centered at (16, 0).

Fig. 9
Fig. 9

Mean leakage intensity for a 64 × 64 binary phase hologram in a 4 × 4 subregion centered in the 16 × 16 reconstruction region R.

Tables (2)

Tables Icon

Table 1 Statistics for 64 × 64 Hologram, 16 × 16 Object

Tables Icon

Table 2 Statistics for 32 × 16 to 64 × 64 Holograms, 8 × 8 Object

Equations (32)

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H ( u , v ) = k = - M / 2 M / 2 - 1 l = - N / 2 N / 2 - 1 H k l rect ( u - k R R , v - l S S ) ,
rect ( x , y ) = { 1 if x , y < 1 / 2 0 otherwise
h ( x , y ) = R S sinc ( R x , S y ) k = - M / 2 M / 2 - 1 l = - N / 2 N / 2 - 1 H k l × exp [ i 2 π ( k R x + l S y ) ] ,
sinc ( x , y ) = sin ( π x ) sin ( π y ) π 2 x y .
h ( m X , n Y ) = ( R S X Y ) 1 / 2 sinc ( m M , n N ) h m n ,
h m n = 1 M N k = - M / 2 M / 2 - 1 l = - N / 2 N / 2 - 1 H k l exp [ i 2 π ( m k M + n l N ) ] .
h ( x , y ) = ( R S X Y ) 1 / 2 sinc ( R x , S y ) × m = - M / 2 M / 2 - 1 n = - N / 2 N / 2 - 1 h m n sinc M N ( x - m X X , y - n Y Y ) × exp [ - i π ( x - m X M X + y - n Y N Y ) ] ,
sinc M N ( x , y ) = sinc ( x , y ) sinc ( x M , y N ) .
h L ( x , y ) = ( R S X Y ) 1 / 2 sinc ( R x , S y ) × ( m , n ) ± R s h m n sinc M N ( x - m X X , y - n Y Y ) × exp [ - i π ( x - m X M X + y - n Y N Y ) ] ,
E ( H k l ) = μ , Var ( H k l ) = σ 2 .
E [ h m n ( r ) ] = ( M N ) 1 / 2 μ δ m n , E [ h m n ( i ) ] = 0 , Var [ h m n ( r ) ] = σ 2 2 ( 1 + δ m n + δ m + ( M / 2 ) , n + δ m , n + ( N / 2 ) + δ m + ( M / 2 ) , n + ( N / 2 ) ) , Var [ h m n ( i ) ] = σ 2 2 ( 1 - δ m n - δ m + ( M / 2 ) , n - δ m , n + ( N / 2 ) - δ m + ( M / 2 ) , n + ( N / 2 ) ) , Cov [ h m n ( r ) , h p q ( r ) ] = { σ 2 / 2 if ( m , n ) = ( - p , - q ) 0 otherwise , Cov [ h m n ( i ) , h p q ( i ) ] = { - σ 2 / 2 if ( m , n ) = ( - p , - q ) 0 otherwise , Cov [ h m n ( r ) , h p q ( i ) ] = 0 ,
E [ I L ( x , y ) ] = R S X Y sinc 2 ( R x , S y ) [ σ 2 α ( x , y ) + M N μ 2 sinc M N 2 ( x X , y Y ) ] , Var [ I L ( x , y ) ] = σ 2 ( R S X Y ) 2 sinc 4 ( R x , S y ) { σ 2 α 2 ( x , y ) + σ 2 β 2 ( x , y ) + 2 M N μ 2 × sinc M N 2 ( x X , y Y ) × [ α ( x , y ) + β ( x , y ) ] } ,
α ( x , y ) = ( m , n ) ± R s sinc M N 2 ( x - m X X , y - n Y Y ) , β ( x , y ) = ( m , n ) ± R s sinc M N ( x - m X X , y - n Y Y ) × sinc M N ( x + m X X , y + n Y Y ) + sinc M N 2 ( x X + M 2 , y Y + N 2 ) - sinc M N 2 ( x X , y Y + N 2 ) - sinc M N 2 ( x X + M 2 , y Y ) .
e L = { 1 A X B Y R E [ I L ( x , y ) ] d x d y } 1 / 2 .
e ¯ rms = ( 1 A B ( m , n ) R s γ f m n - h m n 2 ) 1 / 2 ,
η tot = 1 M N ( m , n ) R s h m n 2 ,
η L = 1 M R N S R E [ I L ( x , y ) ] d x d y .
C = 1 A X B Y R ( { Var [ I L ( x , y ) ] } 1 / 2 E [ I L ( x , y ) ] ) d x d y .
Φ M N ( ω ¯ ) = E { exp [ i ( ω 1 h m 1 n 1 ( r ) + ω 2 h m 1 n 1 ( i ) + + ω 2 P Q - 1 h m P n Q ( r ) + ω 2 P Q h m P n Q ( i ) ) ] } ,
Φ M N ( ω ¯ ) = k = - M / 2 M / 2 - 1 l = - N / 2 N / 2 - 1 Φ M N k l ( ω ¯ ) ,
Φ M N k l ( ω ¯ ) = E { exp [ i ( M N ) 1 / 2 ω ¯ a ¯ M N k l T ] } ,
a ¯ M N k l = H k l { cos [ 2 π ( m 1 k M + n 1 l N ) ] , , sin [ 2 π ( m P k M + n Q l N ) ] } ,
lim M , N Φ M N k l ( ω ¯ ) - 1 = 0 ,
lim M , N ln [ Φ M N ( ω ¯ ) ] = lim M , N { - 1 2 M N × k = - M / 2 M / 2 - 1 l = - N / 2 N / 2 - 1 E [ ( ω ¯ a ¯ M N k l T ) 2 ] } .
k = - M / 2 M / 2 - 1 l = - N / 2 N / 2 - 1 cos [ 2 π ( m i k M + n j l N ) ] cos [ 2 π ( m p k M + n q l N ) ] , k = - M / 2 M / 2 - 1 l = - N / 2 N / 2 - 1 cos [ 2 π ( m i k M + n j l N ) ] sin [ 2 π ( m p k M + n q l N ) ] , k = - M / 2 M / 2 - 1 l = - N / 2 N / 2 - 1 sin [ 2 π ( m i k M + n j l N ) ] sin [ 2 π ( m p k M + n q l N ) ] .
I L ( x , y ) = ( m , n ) , ( p , q ) ± R s h ˜ m n h ˜ p q * × exp [ i π ( m - p M + n - q N ) ] × sinc M N ( x - m X X , y - n Y Y ) × sinc M N ( x - p X X , y - q Y Y ) + 2 ( M N ) 1 / 2 μ sinc M N ( x X , y Y ) × ( m , n ) ± R s { h ˜ m n ( r ) cos [ π ( m M + n N ) ] - h ˜ m n ( i ) sin [ π ( m M + n N ) ] } × sinc M N ( x - m X X , y - n Y Y ) + M N μ 2 sinc M N 2 ( x X , y Y ) ,
E ( { ( m , n ) , ( p , q ) ± R s h ˜ m n h ˜ p q * exp [ i π ( m - p M + n - q N ) ] sinc M N ( x - m X X , y - n Y Y ) sinc M N ( x - p X X , y - q Y Y ) } 2 ) = ( m , n ) , ( p , q ) , ( r , s ) , ( u , v ) ± R s E ( h ˜ m n h ˜ p q * h ˜ r s h ˜ u v * ) exp [ i π ( m - p + r - u M + n - q + s - v N ) ] × sinc M N ( x - m X X , y - n Y Y ) sinc M N ( x - p X X , y - q Y Y ) sinc M N ( x - r X X , y - s Y Y ) sinc M N ( x - u X X , y - v Y Y ) .
E ( h ˜ m n h ˜ p q * h ˜ r s h ˜ u v * ) = E [ h ˜ m n ( r ) h ˜ p q ( r ) ] E [ h ˜ r s ( r ) h ˜ u v ( r ) ] + E [ h ˜ m n ( r ) h ˜ r s ( r ) ] E [ h ˜ p q ( r ) h ˜ u v ( r ) ] + E [ h ˜ m n ( r ) h ˜ u v ( r ) ] E [ h ˜ p q ( r ) h ˜ r s ( r ) ] + E [ h ˜ m n ( i ) h ˜ p q ( i ) ] E [ h ˜ r s ( i ) h ˜ u v ( i ) ] + E [ h ˜ m n ( i ) h ˜ r s ( i ) ] E [ h ˜ p q ( i ) h ˜ u v ( i ) ] + E [ h ˜ m n ( i ) h ˜ u v ( i ) ] E [ h ˜ p q ( i ) h ˜ r s ( i ) ] + E [ h ˜ m n ( r ) h ˜ p q ( r ) ] E [ h ˜ r s ( i ) h ˜ u v ( i ) ] + E [ h ˜ m n ( i ) h ˜ p q ( i ) ] E [ h ˜ r s ( r ) h ˜ u v ( r ) ] + E [ h ˜ m n ( r ) h ˜ u v ( r ) ] E [ h ˜ p q ( i ) h ˜ r s ( i ) ] + E [ h ˜ m n ( i ) h ˜ u v ( i ) ] E [ h ˜ p q ( r ) h ˜ r s ( r ) ] - E [ h ˜ m n ( i ) h ˜ r s ( i ) ] E [ h ˜ p q ( r ) h ˜ u v ( r ) ] - E [ h ˜ m n ( r ) h ˜ r s ( r ) ] E [ h ˜ p q ( i ) h ˜ u v ( i ) ] .
{ ( m , n ) , ( p , q ) ± R s E [ h ˜ m n ( r ) h ˜ p q ( r ) + h ˜ m n ( i ) h ˜ p q ( i ) ] exp [ i π ( m - p M + n - q N ) ] sinc M N ( x - m X X , y - n Y Y ) × sinc M N ( x - p X X , y - q Y Y ) } 2 + { ( m , n ) , ( u , v ) ± R s E [ h ˜ m n ( r ) h ˜ u v ( r ) + h ˜ m n ( i ) h ˜ u v ( i ) ] exp [ i π ( m - u M + n - v N ) ] × sinc M N ( x - m X X , y - n Y Y ) sinc M N ( x - u X X , y - v Y Y ) } 2 + | ( m , n ) , ( r , s ) ± R s E [ h ˜ m n ( r ) h ˜ r s ( r ) - h ˜ m n ( i ) h ˜ r s ( i ) ] exp [ i π ( m + r M + n + s N ) ] sinc M N ( x - m X X , y - n Y Y ) sinc M N ( x - r X X , y - s Y Y ) | 2
2 σ 4 [ ( m , n ) ± R s sinc M N 2 ( x - m X X , y - n Y Y ) ] 2 + σ 4 [ ( m , n ) ± R s sinc M N ( x - m X X , y - n Y Y ) × sinc M N ( x + m X X , y + n Y Y ) + sinc M N 2 ( x X + M 2 , y Y + N 2 ) - sinc M N 2 ( x X , y Y + N 2 ) - sinc M N 2 ( x X + M 2 , y Y ) ] 2 ,
2 M N μ 2 σ 2 sinc M N 2 ( x X , y Y ) [ ( m , n ) ± R s sinc M N 2 ( x - m X X , y - n Y Y ) + ( m , n ) ± R s sinc M N ( x - m X X , y - n Y Y ) sinc M N ( x + m X X , y + n Y Y ) + sinc M N 2 ( x X + M 2 , y Y + N 2 ) - sinc M N 2 ( x X , y Y + N 2 ) - sinc M N 2 ( x X + M 2 , y Y ) ] .
4 ( M N ) 1 / 2 μ sinc M N ( x X , y Y ) ( m , n ) , ( p , q ) , ( r , s ) ± R s E ( h ˜ m n h ˜ p q * { h ˜ r s ( r ) cos [ π ( r M + s N ) ] - h ˜ r s ( i ) sin [ π ( r M + s N ) ] } ) exp [ i π ( m - p M + n - q N ) ] sinc M N ( x - m X X , y - n Y Y ) sinc M N ( x - p X X , y - q Y Y ) × sinc M N ( x - r X X , y - s Y Y ) .

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