Abstract

Calculating computer-generated holograms takes a tremendous amount of computation time. We propose a fast method for calculating object lights for Fresnel holograms without the use of a Fourier transform. This method generates object lights of variously shaped patches from a basic object light for a fixed-shape patch by using three-dimensional affine transforms. It can thus calculate holograms that display complex objects including patches of various shapes. Computer simulations and optical experiments demonstrate the effectiveness of this method. The results show that it performs twice as fast as a method that uses a Fourier transform.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A 20, 1755-1762 (2003).
    [CrossRef]
  2. K. Matsushima and S. Nakahara, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. 44, 4607-4614 (2005).
    [CrossRef] [PubMed]
  3. H. Yoshikawa, T. Yamaguchi, and R. Kitayama, “Real-time generation of full color image hologram with compact distance look-up table,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2009), paper DWC4.
  4. A. Ritter, J. Böttger, O. Deussen, and Th. Strothotte, “Hardware-based rendering of full-parallax synthetic holograms,” Appl. Opt. 38 , 1364-1369 (1999).
    [CrossRef]
  5. L. Ahrenberg, P. Benzie, M. Magnor, and J. Waston, “Computer generated holograms from three dimensional meshes using an analytic light transport model,” Appl. Opt. 47, 1567-1574 (2008).
    [CrossRef] [PubMed]
  6. S. Kim and E. Kim, “Fast computation of hologram patterns of a 3D object using run-length encoding and novel look-up table methods,” Appl. Opt. 48, 1030-1041 (2009).
    [CrossRef]
  7. Y. Sakamoto and T. Nagao, “A fast computational method for computer-generated Fourier hologram using patch model,” Electron. Commun. Jpn. Part 2 Electron. 85, 16-24 (2002).
    [CrossRef]
  8. T. Ito, N. Masuda, K. Yoshimura, A. Shirake, T. Shimobaba, and T. Sugie, “Special-purpose computer HORN-5 for a real-time electroholography,” Opt. Express 13, 1923-1932(2005).
    [CrossRef] [PubMed]
  9. N. Masuda, T. Ito, T. Tanaka, A. Shiraki, and T. Sugie, “Computer generated holography using a graphics processing unit,” Opt. Express 14, 603-608 (2006).
    [CrossRef] [PubMed]
  10. M. Lucente, “Interactive computation of hologram using a look-up table,” J. Electron. Imaging 2, 28-34 (1993).
    [CrossRef]
  11. K. Matsushima, “Formulation of the rotational transformation of wave fields and their application to digital holography,” Appl. Opt. 47, D110-D116 (2008).
    [CrossRef] [PubMed]
  12. FFTW homepage: http://www.fftw.org/

2009 (1)

2008 (2)

2006 (1)

2005 (2)

2003 (1)

2002 (1)

Y. Sakamoto and T. Nagao, “A fast computational method for computer-generated Fourier hologram using patch model,” Electron. Commun. Jpn. Part 2 Electron. 85, 16-24 (2002).
[CrossRef]

1999 (1)

1993 (1)

M. Lucente, “Interactive computation of hologram using a look-up table,” J. Electron. Imaging 2, 28-34 (1993).
[CrossRef]

Ahrenberg, L.

Benzie, P.

Böttger, J.

Deussen, O.

Ito, T.

Kim, E.

Kim, S.

Kitayama, R.

H. Yoshikawa, T. Yamaguchi, and R. Kitayama, “Real-time generation of full color image hologram with compact distance look-up table,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2009), paper DWC4.

Lucente, M.

M. Lucente, “Interactive computation of hologram using a look-up table,” J. Electron. Imaging 2, 28-34 (1993).
[CrossRef]

Magnor, M.

Masuda, N.

Matsushima, K.

Nagao, T.

Y. Sakamoto and T. Nagao, “A fast computational method for computer-generated Fourier hologram using patch model,” Electron. Commun. Jpn. Part 2 Electron. 85, 16-24 (2002).
[CrossRef]

Nakahara, S.

Ritter, A.

Sakamoto, Y.

Y. Sakamoto and T. Nagao, “A fast computational method for computer-generated Fourier hologram using patch model,” Electron. Commun. Jpn. Part 2 Electron. 85, 16-24 (2002).
[CrossRef]

Schimmel, H.

Shimobaba, T.

Shirake, A.

Shiraki, A.

Strothotte, Th.

Sugie, T.

Tanaka, T.

Waston, J.

Wyrowski, F.

Yamaguchi, T.

H. Yoshikawa, T. Yamaguchi, and R. Kitayama, “Real-time generation of full color image hologram with compact distance look-up table,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2009), paper DWC4.

Yoshikawa, H.

H. Yoshikawa, T. Yamaguchi, and R. Kitayama, “Real-time generation of full color image hologram with compact distance look-up table,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2009), paper DWC4.

Yoshimura, K.

Appl. Opt. (5)

Electron. Commun. Jpn. Part 2 Electron. (1)

Y. Sakamoto and T. Nagao, “A fast computational method for computer-generated Fourier hologram using patch model,” Electron. Commun. Jpn. Part 2 Electron. 85, 16-24 (2002).
[CrossRef]

J. Electron. Imaging (1)

M. Lucente, “Interactive computation of hologram using a look-up table,” J. Electron. Imaging 2, 28-34 (1993).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Express (2)

Other (2)

FFTW homepage: http://www.fftw.org/

H. Yoshikawa, T. Yamaguchi, and R. Kitayama, “Real-time generation of full color image hologram with compact distance look-up table,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2009), paper DWC4.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (18)

Fig. 1
Fig. 1

Coordinate system for light propagation.

Fig. 2
Fig. 2

Algorithm for our method.

Fig. 3
Fig. 3

Algorithm for the conventional method.

Fig. 4
Fig. 4

Slide and rotation transform.

Fig. 5
Fig. 5

Distance transform.

Fig. 6
Fig. 6

Tilt transform.

Fig. 7
Fig. 7

Scaling transform.

Fig. 8
Fig. 8

Skew transform.

Fig. 9
Fig. 9

Object locations from slide and rotation transform and distance transform.

Fig. 10
Fig. 10

Results of distance transform.

Fig. 11
Fig. 11

Computer-simulated distance transform.

Fig. 12
Fig. 12

Propagation simulation.

Fig. 13
Fig. 13

Results of tilt transform by propagation simulation as in Fig. 12.

Fig. 14
Fig. 14

Results of scaling transform.

Fig. 15
Fig. 15

Results of skew transform.

Fig. 16
Fig. 16

Object locations from combined transforms.

Fig. 17
Fig. 17

Results of combined transforms.

Fig. 18
Fig. 18

Calculation times versus number of pixels on a side.

Tables (2)

Tables Icon

Table 1 Parameters for the Experiments

Tables Icon

Table 2 Times for Calculating Object Lights

Equations (30)

Equations on this page are rendered with MathJax. Learn more.

u ( x h , y h ) = j λ g ( x , y , z ) exp ( j k r ) r d x d y d z ,
r = { ( x h x ) 2 + ( y h y ) 2 + ( z z 0 ) 2 } 1 / 2 .
u = i = 1 p u i ,
r z 0 + ( x h x ) 2 2 z 0 + ( y h y ) 2 2 z 0 = z 0 + x h 2 + y h 2 2 z 0 x h x + y h y z 0 + x 2 + y 2 2 z 0 ,
u ( x h , y h ) = j λ z 0 exp ( j k z 0 ) g ( x , y ) exp [ j π λ z 0 { ( x h x ) 2 + ( y h y ) 2 } ] d x d y .
r z 0 + x h 2 + y h 2 2 z 0 x h x + y h y z 0 .
u ( x h , y h ) = j λ z 0 exp ( j k z 0 ) exp ( j k x h 2 + y h 2 2 z 0 ) g ( x , y ) exp { j k z 0 ( x h x + y h y ) } d x d y ,
u h i = T i [ u b ] ,
u h   all = i = 1 P u h i .
I = | u h   all + R | 2 .
u b ( x b , y b ) = j λ z 0 exp ( j k z 0 ) g ( x , y ) exp [ j π λ z 0 { ( x b x ) 2 + ( y b y ) 2 } ] d x d y = j λ z 0 exp ( j k z 0 ) exp { j k x b 2 + y b 2 2 z 0 } g ( x , y ) exp ( j k x 2 + y 2 2 z 0 ) exp { j k ( x b x + y b y ) z 0 } d x d y .
u h ( x h , y h ) = u b ( x h cos θ + y h sin θ Δ x , y h cos θ x h sin θ Δ y ) ,
u ( x h , y h ) = j λ ( z 0 + Δ z ) exp { j k ( z 0 + Δ z ) } exp { j k ( x h 2 + y h 2 ) 2 ( z 0 + Δ z ) } g ( x , y ) exp { j k z 0 + Δ z ( x h x + y h y ) } d x d y = j λ z 0 z 0 z 0 + Δ z exp ( j k z 0 ) exp ( j k Δ z ) exp { j k ( x h 2 + y h 2 ) 2 ( z 0 + Δ z ) } g ( x , y ) exp [ j k z 0 { z 0 z 0 + Δ z ( x h x + y h y ) } ] d x d y = z 0 z 0 + Δ z exp [ j k { Δ z + x h 2 + y h 2 2 ( z 0 + Δ z ) z 0 ( x h 2 + y h 2 ) 2 ( z 0 + Δ z ) 2 } ] j λ z 0 exp ( j k z 0 ) exp ( j k x b 2 + y b 2 2 z 0 ) g ( x , y ) exp { j k z 0 ( x b x + y b y ) } d x d y = C 1 exp ( j k L 1 ) u b ( x b , y b ) ,
x b = z 0 z + Δ z x h , y b = z 0 z + Δ z y h , L 1 = Δ z + x h 2 + y h 2 2 ( z 0 + Δ z ) z 0 ( x h 2 + y h 2 ) 2 ( z 0 + Δ z ) 2 ,
u ( x h , y h ) = j λ ( z 0 + Δ z ) exp { j k ( z 0 + Δ z ) } exp { j k ( x h 2 + y h 2 ) 2 ( z 0 + Δ z ) } g ( x , y ) exp { j k x 2 + y 2 2 ( z 0 + Δ z ) } exp { j k z 0 + Δ z ( x h x + y h y ) } d x d y = z 0 z 0 + Δ z exp ( j k L 1 ) j λ z 0 exp ( j k z 0 ) exp { j k x b 2 + y b 2 2 z 0 } g ( x , y ) exp { j k x 2 + y 2 2 Δ z z 0 ( z 0 + Δ z ) } exp ( j k x 2 + y 2 2 ) exp { j k z 0 ( x b x + y b y ) } d x d y = g ( x , y ) exp ( j k x 2 + y 2 2 ) exp { j k z 0 ( x b x + y b y ) } d x d y ,
g ( x , y ) = g ( x , y ) e rror 1 ( x , y ) ,
e rror 1 ( x , y ) = exp { j k x 2 + y 2 2 Δ z z 0 ( z 0 + Δ z ) } .
u h ( x h , y h ) z 0 z 0 + Δ z exp ( j k L 1 ) j λ z 0 exp ( j k z 0 ) exp { j k x b 2 + y b 2 2 z 0 } g ( x , y ) exp ( j k x 2 + y 2 2 ) exp { j k z 0 ( x b x + y b y ) } d x d y = C 1 exp ( j k L 1 ) u b ( x b , y b ) .
u h ( x h , y h ) = j λ z 0 exp ( j k z 0 ) exp ( j k x h 2 + y h 2 2 z 0 ) g ( x cos ϕ , y ) exp { j k x 2 + y 2 2 ( z 0 + x sin ϕ ) } exp { j k ( x h x + y h y ) z 0 + x sin ϕ } d x d y C 2 exp ( j k L 2 ) u b ( x b , y b ) ,
x b x h z 0 sin ϕ 2 cos ϕ , y b x 0 z 0 ( 2 cos ϕ ) y h , L 2 x sin ϕ + x h 2 { 1 1 ( 2 cos ϕ ) 2 } + 2 x h + z 0 sin ϕ + z 0 sin 2 ϕ ( 2 cos ϕ ) 2 + y h 2 { 1 z 0 2 z 0 2 ( 2 cos ϕ ) 2 } , e rror 2 exp [ j k 2 z 0 ( z 0 + x tan ϕ ) { x 3 + z 0 x 2 ( 1 cos 2 ϕ 1 ) x y 2 tan ϕ } ] .
u h ( x h , y h ) = j λ z 0 exp ( j k z 0 ) exp { j k x h 2 + y h 2 2 z 0 } g ( x R x , y ) exp ( j k x 2 + y 2 2 z 0 ) exp { j k ( x h x + y h y ) z 0 } d x d y C 3 exp ( j k L 3 ) u b ( x b , y b ) ,
x b = R x x h , y b = y h , L 3 = x h 2 ( 1 R x 2 ) 2 z 0 , e rror 2 = exp { j k x 2 2 z 0 ( 1 R x 2 ) } ,
u h ( x h , y h ) = j λ z 0 exp ( j k z 0 ) exp { j k x h 2 + y h 2 2 z 0 } g ( x S x y , y ) exp ( j k x 2 + y 2 2 z 0 ) exp { j k ( x h x + y h y ) z 0 } d x d y C 4 exp ( j k L 4 ) u b ( x b , y b ) ,
x b = x h , y b = y h + S x x h , L 4 = S x x h ( S x x h + 2 y h ) 2 z 0 , e rror 4 = exp { j k S x y ( S x y + 2 x ) 2 z 0 } .
0.0065 [ m ] Δ x 0.0065 [ m ] ,
0.0065 [ m ] Δ y 0.0065 [ m ] ,
0.075 [ m ] Δ z ,
2.4 ° ϕ 2.4 ° ,
0.0 < R x 2.0 ,
1.0 S x 1.0 .

Metrics