Abstract

Computer-generated holograms (CGH’s) synthesized by the iterative direct-binary-search (DBS) algorithm yield lower reconstruction error and higher diffraction efficiency than do CGH’s designed by conventional methods, but the DBS algorithm is computationally intensive. A fast algorithm for DBS is developed that recursively computes the error measure to be minimized. For complex amplitude-based error, the required computation for an L-point CGH is reduced by a factor of (L/log2L)1/2. The fast intensity-based algorithm is substantially more complicated, and modifications are considered in order to make the algorithm more efficient. An acceleration technique that attempts to increase the rate of convergence of the DBS algorithm is also investigated.

© 1991 Optical Society of America

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References

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  1. 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]
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    [CrossRef]
  3. R. H. Squires, J. P. Allebach, “Digital holograms: a guide to reducing quantization and phase encoding errors,” in International Conference on Computer-Generated Holography, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.437, 12–18 (1983).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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  8. E. Cinlar, Introduction To Stochastic Processes (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  9. J. P. Allebach, B. Liu, “Minimax spectrum shaping with a bandwidth constraint,” Appl. Opt. 14, 3062–3072 (1975).
    [CrossRef] [PubMed]
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    [CrossRef]

1989

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

1988

1987

1984

1980

1975

1973

Allebach, J. P.

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

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]

J. P. Allebach, B. Liu, “Minimax spectrum shaping with a bandwidth constraint,” Appl. Opt. 14, 3062–3072 (1975).
[CrossRef] [PubMed]

R. H. Squires, J. P. Allebach, “Digital holograms: a guide to reducing quantization and phase encoding errors,” in International Conference on Computer-Generated Holography, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.437, 12–18 (1983).
[CrossRef]

Barnard, E.

Bryngdahl, O.

Bucklew, J. A.

Cinlar, E.

E. Cinlar, Introduction To Stochastic Processes (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Gallagher, N. C.

Hauck, R.

Jennison, B. K.

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

Liu, B.

Parzen, E.

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

Seldowitz, M. A.

Squires, R. H.

R. H. Squires, J. P. Allebach, “Digital holograms: a guide to reducing quantization and phase encoding errors,” in International Conference on Computer-Generated Holography, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.437, 12–18 (1983).
[CrossRef]

Sweeney, D. W.

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

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]

Wyrowski, F.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Eng.

B. K. Jennison, J. P. Allebach, D. W. Sweeney, “Iterative approaches to computer-generated holography,” Opt. Eng. 28, 629–637 (1989).
[CrossRef]

Other

R. H. Squires, J. P. Allebach, “Digital holograms: a guide to reducing quantization and phase encoding errors,” in International Conference on Computer-Generated Holography, S. H. Lee, ed., Proc. Soc. Photo-Opt. Instrum. Eng.437, 12–18 (1983).
[CrossRef]

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

E. Cinlar, Introduction To Stochastic Processes (Prentice-Hall, Englewood Cliffs, N.J., 1975).

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Figures (6)

Fig. 1
Fig. 1

Wave-front shaping with an M × N binary CGH.

Fig. 2
Fig. 2

Transition diagram for the homogeneous, first-order, discrete-parameter Markov chain describing the acceptance of trial inversions since the last direct update of H ˜ k l = DFT[rmnhmn]. Here, p is the probability of accepting an inversion, Ci is the constant increment in the cost of evaluating the sum in Eq. (10) when an additional inversion is retained, Cu is the cost of directly recomputing H ˜ k l, and Smax is the number of retained inversions at which H ˜ k l is recomputed.

Fig. 3
Fig. 3

Binary transmittance function of a 1024 × 1024 DBS hologram for a matched-filtering application synthesized with the fast complex-amplitude-based algorithm.

Fig. 4
Fig. 4

Digitally reconstructed 256 × 256 region from the 1024 × 1024 DBS hologram shown in Fig. 3.

Fig. 5
Fig. 5

Image reconstructed digitally from a 512 × 512 DBS hologram synthesized by using the fast complex-amplitude-based algorithm. The 128 × 128 object has zero phase.

Fig. 6
Fig. 6

Image reconstructed digitally from a 512 × 512 DBS hologram synthesized by using the fast complex-amplitude-based algorithm. The 128 × 128 object has random phase.

Tables (5)

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Table 1 Asymptotic Number of Operations Required for a Single Iteration of the Complex-Amplitude Direct-Binary-Search Algorithm

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Table 2 Average CPU Times (min) Required for the Complex-Amplitude Direct-Binary-Search Algorithm

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Table 3 Test for Second-Order Independence of Trial Inversions for a 64 × 64 Hologram

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Table 4 Average CPU Times (min) Required for the Intensity-Based Direct-Binary-Search Algorithm

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Table 5 Effect of Initial Hologram Configuration on Reconstruction Quality

Equations (24)

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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 ) ] .
e = 1 A B f - λ h 2
λ = f , h h 2
h m n = h m n + a M N exp [ i 2 π ( m k M + n l N ) ] ,
e = 1 A B ( f 2 - f , h 2 h 2 ) .
f , h = f , h + a M N ( m , n ) R f m n exp [ - i 2 π ( m k M + n l N ) ] .
h 2 = h 2 + A B M N + 2 a Re { 1 M N ( m , n ) R h m n × exp [ - i 2 π ( m k M + n l N ) ] } = h 2 + A B M N + 2 a Re ( H ˜ k l ) ,
h m n = h m n i - S + 1 M N s = 1 S a s exp [ i 2 π ( m k s M + n l s N ) ] ,
H ˜ k l = H ˜ k l i - S + 1 M N s = 1 S a s R k - k s , l - l s ,
h 2 = h 2 + A B M N + 2 a Re ( H ˜ k l i - S + 1 M N s = 1 S a s R k - k s , l - l s ) .
E [ C ] = q = 0 S max n = 0 S max c q n ρ q n ,
c q n = { q C i 0 n = q S max or 0 < n = q + 1 S max q C i + C u q = S max , n = 0 0 else
ρ q n = Pr [ S j + 1 = n , S j = q ] = Pr [ S j + 1 = n S j = q ] Pr [ S j = q ] p q n π q
P = [ p q n 0 ] = [ 1 - p p 0 0 0 0 0 1 - p p 0 0 0 0 0 1 - p p 0 0 0 0 0 0 1 - p p p 0 0 0 0 1 - p ] ,
E [ C ] = S max C i 2 + p C u S max + 1 ,
S max * = ( 2 p C u / C i ) 1 / 2 - 1.
E [ C ] min = ( 2 p C u C i ) 1 / 2 - C i / 2.
e = 1 A B [ f 2 2 - ( f 2 , h 2 ) 2 h 2 2 ] .
f 2 , h 2 = f 2 , h 2 + 1 M N f 2 + 2 a Re { DFT [ f 2 h i - S ] ( k , l ) + 1 M N s = 1 S a s DFT [ f 2 ] ( k - k s , l - l s ) } .
h 2 2 = h 2 2 + A B ( M N ) 2 + 4 M N h 2 + 4 a M N Re ( H ˜ k l ) + 4 a Re { DFT [ r h 2 h ] ( k , l ) } + 2 M N Re { DFT [ r h 2 ] ( 2 k , 2 l ) } .
DFT [ r h 2 h ] ( k , l ) = DFT [ r h i - S 2 h i - S ] ( k , l ) + 2 M N s = 1 S a s DFT [ r h i - S 2 ] ( k - k s , l - l s ) + 1 M N s = 1 S a s DFT [ r ( h i - S ) 2 ] ( k + k s , l + l s ) + 2 M N s = 1 S t = 1 S a s a t H ˜ ( k - k s + k t , l - l s + l t ) i - S + 1 M N s = 1 S t = 1 S a s a t H ˜ ( k s + k t - k , l s + l t - l ) i - S + 1 ( M N ) 3 / 2 s = 1 S t = 1 S u = 1 S a s a t a u R ( k - k s - k t + k u , l - l s - l t + l u )
DFT [ r h 2 ] ( 2 k , 2 l ) = DFT [ r ( h i - S ) 2 ] ( 2 k , 2 l ) + 2 M N s = 1 S a s H ˜ ( 2 k - k s , 2 l - l s ) i - S + 1 M N s = 1 S t = 1 S a s a t R ( 2 k - k s - k t , 2 l - l s - l t ) .
c q n = { q C i ( 1 ) + q 2 C i ( 2 ) + q 3 C i ( 3 ) 0 n = q S max or 0 < n = q + 1 S max q C i ( 1 ) + q 2 C i ( 2 ) + q 3 C i ( 3 ) + C u q = S max , n = 0 0 else ,
E [ C ] = C i ( 3 ) 4 S max 3 + [ C i ( 2 ) 3 + C i ( 3 ) 4 ] S max 2 + [ C i ( 1 ) 2 + C i ( 2 ) 6 ] S max + p C u S max + 1 .

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