Abstract

Hexagonal sampling offers substantial computational efficiency in processing circularly band-limited and/or circularly symmetric functions and also offers a significant reduction of required data storage compared with rectangular sampling. Both Fraunhofer and Fresnel computer-generated holograms utilizing hexagonal sampling are capable of exploiting these properties. Hexagonal fast Fourier transform (FFT) and inverse fast Fourier transform (IFFT) algorithms have been developed along with efficient algorithms for generating the lens transmittance and free-space-transfer matrices that comprise software packages designed to accept hexagonally sampled input arrays and generate hexagonally sampled output arrays suitable for direct encoding into digital holograms.

© 1982 Optical Society of America

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References

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  1. A. W. Lohmann and D. P. Paris, “Binary Fraunhofer holograms, generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
    [CrossRef] [PubMed]
  2. B. R. Brown and A. W. Lohmann, “Complex spatial filtering with binary masks,” Appl. Opt. 5, 967–969 (1966).
    [CrossRef] [PubMed]
  3. D. P. Petersen and D. Middleton, “Sampling and reconstruction of wave-number-limited functions in N-dimensional Euclidean spaces,” Inf. Control 5, 279–323 (1962).
    [CrossRef]
  4. A. Rosenfeld and A. Kak, Digital Picture Processing (Academic, New York, 1976).
    [CrossRef]
  5. R. M. Mersereau, “The processing of hexagonally sampled two-dimensional signals,” Proc. IEEE 67, 930–949 (1979).
    [CrossRef]
  6. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  7. W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. 71, 7–14 (1981).
    [CrossRef]
  8. A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  9. D. B. Harris, J. H. McClellan, D. S. K. Chan, and H. W. Schuessler, “Vector radix fast Fourier transform,” presented at IEEE Conference on Acoustics, Speech and Signal Processing, May 9–11, 1977, Hartford, Connecticut.
  10. B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev.160–168 (1969).
    [CrossRef]
  11. L. R. Robiner and B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  12. A. E. Siegman, “Quasi fast Hankel transform,” Opt. Lett. 1, 13–15 (1977).
    [CrossRef] [PubMed]
  13. G. P. Agrawal and M. Lax, “End correction in the quasi-fast Hankel transform for optical-propagation problems,” Opt. Lett. 6, 171–173 (1981).
    [CrossRef] [PubMed]

1981 (2)

1979 (1)

R. M. Mersereau, “The processing of hexagonally sampled two-dimensional signals,” Proc. IEEE 67, 930–949 (1979).
[CrossRef]

1977 (1)

1969 (1)

B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev.160–168 (1969).
[CrossRef]

1967 (1)

1966 (1)

1962 (1)

D. P. Petersen and D. Middleton, “Sampling and reconstruction of wave-number-limited functions in N-dimensional Euclidean spaces,” Inf. Control 5, 279–323 (1962).
[CrossRef]

Agrawal, G. P.

Brown, B. R.

B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev.160–168 (1969).
[CrossRef]

B. R. Brown and A. W. Lohmann, “Complex spatial filtering with binary masks,” Appl. Opt. 5, 967–969 (1966).
[CrossRef] [PubMed]

Chan, D. S. K.

D. B. Harris, J. H. McClellan, D. S. K. Chan, and H. W. Schuessler, “Vector radix fast Fourier transform,” presented at IEEE Conference on Acoustics, Speech and Signal Processing, May 9–11, 1977, Hartford, Connecticut.

Gold, B.

L. R. Robiner and B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Harris, D. B.

D. B. Harris, J. H. McClellan, D. S. K. Chan, and H. W. Schuessler, “Vector radix fast Fourier transform,” presented at IEEE Conference on Acoustics, Speech and Signal Processing, May 9–11, 1977, Hartford, Connecticut.

Kak, A.

A. Rosenfeld and A. Kak, Digital Picture Processing (Academic, New York, 1976).
[CrossRef]

Lax, M.

Lohmann, A. W.

McClellan, J. H.

D. B. Harris, J. H. McClellan, D. S. K. Chan, and H. W. Schuessler, “Vector radix fast Fourier transform,” presented at IEEE Conference on Acoustics, Speech and Signal Processing, May 9–11, 1977, Hartford, Connecticut.

Mersereau, R. M.

R. M. Mersereau, “The processing of hexagonally sampled two-dimensional signals,” Proc. IEEE 67, 930–949 (1979).
[CrossRef]

Middleton, D.

D. P. Petersen and D. Middleton, “Sampling and reconstruction of wave-number-limited functions in N-dimensional Euclidean spaces,” Inf. Control 5, 279–323 (1962).
[CrossRef]

Oppenheim, A. V.

A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Paris, D. P.

Petersen, D. P.

D. P. Petersen and D. Middleton, “Sampling and reconstruction of wave-number-limited functions in N-dimensional Euclidean spaces,” Inf. Control 5, 279–323 (1962).
[CrossRef]

Robiner, L. R.

L. R. Robiner and B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Rosenfeld, A.

A. Rosenfeld and A. Kak, Digital Picture Processing (Academic, New York, 1976).
[CrossRef]

Schafer, R. W.

A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Schuessler, H. W.

D. B. Harris, J. H. McClellan, D. S. K. Chan, and H. W. Schuessler, “Vector radix fast Fourier transform,” presented at IEEE Conference on Acoustics, Speech and Signal Processing, May 9–11, 1977, Hartford, Connecticut.

Siegman, A. E.

Southwell, W. H.

Appl. Opt. (2)

IBM J. Res. Dev. (1)

B. R. Brown and A. W. Lohmann, “Computer-generated binary holograms,” IBM J. Res. Dev.160–168 (1969).
[CrossRef]

Inf. Control (1)

D. P. Petersen and D. Middleton, “Sampling and reconstruction of wave-number-limited functions in N-dimensional Euclidean spaces,” Inf. Control 5, 279–323 (1962).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (2)

Proc. IEEE (1)

R. M. Mersereau, “The processing of hexagonally sampled two-dimensional signals,” Proc. IEEE 67, 930–949 (1979).
[CrossRef]

Other (5)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

A. V. Oppenheim and R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

D. B. Harris, J. H. McClellan, D. S. K. Chan, and H. W. Schuessler, “Vector radix fast Fourier transform,” presented at IEEE Conference on Acoustics, Speech and Signal Processing, May 9–11, 1977, Hartford, Connecticut.

A. Rosenfeld and A. Kak, Digital Picture Processing (Academic, New York, 1976).
[CrossRef]

L. R. Robiner and B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

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Figures (15)

Fig. 1
Fig. 1

Rectangular sampling grid.

Fig. 2
Fig. 2

Regular hexagonal sampling grid.

Fig. 3
Fig. 3

Hexagonal band region associated with hexagonal sampling.

Fig. 4
Fig. 4

Densest packing of circular spectra generates regular hexagonal sampling.

Fig. 5
Fig. 5

Hexagonally periodic sequence with (a) a hexagonal fundamental period and (b) a parallelogram-shaped fundamental period.

Fig. 6
Fig. 6

Decimation-in-time fast Fourier transform butterfly operation in (a) one dimension and (b) two dimensions.

Fig. 7
Fig. 7

Optical setup for image reconstruction using a Fourier transform hologram.

Fig. 8
Fig. 8

Lohmann cells located at hexagonal sampling locations.

Fig. 9
Fig. 9

Image reconstruction of a hexagonal Fourier transform hologram (N = 128).

Fig. 10
Fig. 10

Image reconstruction of a self-focusing (lens-incorporation) hologram (N = 128).

Fig. 11
Fig. 11

Optical setup required for a hologram generated by propagating the image backward through free space a distance z.

Fig. 12
Fig. 12

Image reconstruction of a self-focusing (backpropagation) hologram (N = 128).

Fig. 13
Fig. 13

Hexagonal fast Fourier transform response to (a) a single vertical impulse sheet 2δ(x), (b) a vertical cosine sheet 2δ(x)cos(2πf0y), (c) a single horizontal impulse sheet 2δ(y), and (d) a horizontal cosine sheet 2δ(y)cos(2πf0x).

Fig. 14
Fig. 14

Hexagonal periodicity with (a) period N and a parallelogram-shaped fundamental period and (b) period N/2.

Fig. 15
Fig. 15

The wraparound effect produced by nested hexagonal periodicity.

Tables (1)

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Equations (29)

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T 1 1 A ,             T 2 1 B .
r n 1 n 2 = 2 n 1 - n 2 2 T 1 i x + n 2 T 2 i y .
T 1 2 2 W 1 + W 3 ,             T 2 1 2 W 2 ,
ŵ k 1 , k 2 = k 1 d ŵ 1 + k 2 d ŵ 2             ( k 1 , k 2 integers ) ,
( a , b ) , ( b , a ) , ( - a , - b ) , ( - b , - a ) , ( a - b , a ) , ( a , a - b ) , ( b - a , - a ) , ( - a , b - a ) , ( b , b - a ) , ( b - a , b ) , ( - b , a - b ) , ( a - b , - b ) .
F ( u ) = F ( f x , f y ) = n 1 n 2 f ( 2 n 1 - n 2 2 T 1 , n 2 3 2 T 1 ) × exp { - i 2 π [ f x ( 2 n 1 - n 2 2 ) T 1 + f y n 2 3 2 T 1 ] }
f ( r n 1 n 2 ) = T 1 2 3 / 2 H F ( f x , f y ) × exp { i 2 π [ 2 n 1 - n 2 2 T 1 f x + n 2 3 2 T 1 f y ] } d f x d f y ,
F ( u k 1 k 2 ) = n 1 , n 2 H f ( r n 1 n 2 ) × exp { - i π [ ( 2 k 1 - k 2 ) ( 2 n 1 - n 2 ) ( 2 N 1 + N 2 ) + k 2 n 2 N 2 ] } .
f ( r n 1 n 2 ) = 1 ( 2 N 1 + N 2 ) N 2 k 1 , k 2 H F ( u k 1 k 2 ) × exp { i π [ ( 2 n 1 - n 2 ) ( 2 k 1 - k 2 ) ( 2 N 1 + N 2 ) + n 2 k 2 N 2 ] } .
F ( u k 1 , k 2 ) = n 1 , n 2 H f ( r n 1 n 2 ) × exp { - i π [ ( 2 k 1 - k 2 ) ( 2 n 1 - n 2 ) 3 N + n 2 k 2 N ] } , f ( r n 1 n 2 ) = 1 3 N 2 k 1 , k 2 H F ( u k 1 k 2 ) × exp { i π [ ( 2 k 1 - k 2 ) ( 2 n 1 - n 2 ) 3 N + n 2 k 2 N ] } .
I ( f x , f y ) L ( f x , f y ) = FFT [ i ( x , y ) ] exp [ - i π λ f ( f x 2 + f y 2 ) 1 / 2 ] ,
H ( f x , f y ) = exp { i 2 π z λ [ 1 - λ 2 ( f x 2 + f y 2 ) ] 1 / 2 } .
IFFT [ I ( f x , f y ) 1 H ( f x , f y ) ] ,
IFFT [ I ( f x , f y ) H * ( f x , f y ) ] ,
H ^ ( f x , f y ) = exp ( i 2 π z λ ) exp [ - i π λ z ( f x 2 + f y 2 ) ] .
F - 1 [ H ^ ( f x , f y ) ] = exp ( i 2 π z λ ) i λ z exp [ i π λ z ( x 2 + y 2 ) ] .
w R ( n 1 , n 2 ) = w [ 2 3 ( n 1 2 + n 2 2 - n 1 n 2 ) 1 / 2 ] .
w k ( n ) = I 0 ( β { 1 - [ 2 n / ( N - 1 ) ] 2 } 1 / 2 ) I 0 ( β ) , - ( N - 1 2 ) n N - 1 2 ,
f 1 ( x , y ) = 2 δ ( x ) cos 2 π f 0 y , f 2 ( x , y ) = 2 δ ( y ) cos 2 π f 0 x ,
F 1 ( f x , f y ) = δ ( f y - f 0 ) + δ ( f y + f 0 ) , F 2 ( f x , f y ) = δ ( f x - f 0 ) + δ ( f x + f 0 ) .
f 2 ( x , y ) = 2 δ ( y ) cos 2 π f 0 x ,
2 τ sin π τ f x π τ f x [ δ ( f x + f 0 ) + δ ( f x + f 0 ) ] ,
X ( u k 1 , k 2 ) = n 1 , n 2 H x ( r n 1 n 2 ) × exp { - i π [ ( 2 k 1 - k 2 ) ( 2 n 1 - n 2 ) 3 N + n 2 k 2 N ] } .
X ( u k 1 , k 2 ) = n 1 = 0 3 N - 1 n 2 = 0 N - 1 x ( r n 1 n 2 ) × exp { - i π [ ( 2 k 1 - k 2 ) ( 2 n 1 - n 2 ) 3 N + n 2 k 2 N ] } .
X ( u k 1 , k 2 ) = X ( k 1 , k 2 ) = HDFT N [ x ( r n 1 n 2 ) ] = HDFT N [ x ( n 1 , n 2 ) ] ,
X ( k 1 , k 2 ) = HDFT N / 2 [ x ( 2 r 1 , 2 r 2 ) ] + exp [ - i 2 π 3 N ( 2 k 2 - k 1 ) ] HDFT N / 2 [ x ( 2 r 1 , 2 r 2 + 1 ) ] + exp [ - i 2 π 3 N ( 2 k 1 - k 2 ) ] HDFT N / 2 [ x ( 2 r 1 + 1 , 2 r 2 ) ] + exp [ - i 2 π 3 N ( k 1 + k 2 ) ] HDFT N / 2 [ x ( 2 r 1 + 1 , 2 r 2 + 1 ) ] ,
X ( k 1 , k 2 ) = E E ( k 1 , k 2 ) + W 3 N 2 k 2 - k 1 E O ( k 1 , k 2 ) + W 3 N 2 k 1 - k 2 O E ( k 1 , k 2 ) + W 3 N k 1 + k 2 O O ( k 1 , k 2 ) , X ( k 1 + 3 N / 2 , k 2 ) = E E ( k 1 , k 2 ) - W 3 N 2 k 2 - k 1 E O ( k 1 , k 2 ) + W 3 N 2 k 1 - k 2 O E ( k 1 , k 2 ) - W 3 N k 1 + k 2 O O ( k 1 , k 2 ) , x ( k 1 + N , k 2 + N / 2 ) = E E ( k 1 , k 2 ) + W 3 N 2 k 2 - k 1 E O ( k 1 , k 2 ) - W 3 N 2 k 1 - k 2 O E ( k 1 , k 2 ) - W 3 N k 1 + k 2 O O ( k 1 , k 2 ) , X ( k 1 + 5 N / 2 , k 2 + N / 2 ) = E E ( k 1 , k 2 ) - W 3 N 2 k 2 - k 1 E O ( k 1 , k 2 ) - W 3 N 2 k 1 - k 2 O E ( k 1 , k 2 ) + W 3 N k 1 + k 2 O O ( k 1 , k 2 ) ,
N a 0 + ( N / 2 ) a 1 + ( N / 4 ) a 2 + .
X ( 2 r 1 , 2 r 2 ) = HDFT N / 2 [ x ( n 1 , n 2 ) + x ( n 1 + 3 N 2 , n 2 ) + x ( n 1 + N , n 2 + N 2 ) + x ( n 1 + 5 N 2 , n 2 + N 2 ) ] , X ( 2 r 2 , 2 r 2 + 1 ) = HDFT N / 2 { [ x ( n 1 , n 2 ) - x ( n 1 + 3 N 2 , n 2 ) + x ( n 1 + N , n 2 + N 2 ) - x ( n 1 + 5 N 2 , n 2 + N 2 ) ] W 3 N 2 n 2 - n 1 } , X ( 2 r 1 + 1 , 2 r 2 ) = HDFT N / 2 { [ x ( n 1 , n 2 ) + x ( n 1 + 3 N 2 , n 2 ) - x ( n 1 + N , n 2 + N 2 ) - x ( n 1 + 5 N 2 , n 2 + N 2 ) ] W 3 N 2 n 1 - n 2 } , X ( 2 r 1 + 1 , 2 r 2 + 1 ) = HDFT N / 2 { [ x ( n 1 , n 2 ) - ( n 1 + 3 N 2 , n 2 ) - x ( n 1 + N , n 2 + N 2 ) + x ( n 1 + 5 N 2 , n 2 + N 2 ) ] W 3 N n 1 + n 2 } .