Abstract

An efficient method for obtaining modified Fresnel holograms of real existing three-dimensional (3-D) scenes illuminated by incoherent white light is presented. To calculate the hologram, the method uses multiple-viewpoint projections of the 3-D scene. However, contrary to other similar methods, this one is able to calculate the Fresnel hologram of the 3-D scene directly rather than calculating a Fourier hologram first. This significantly decreases the amount of calculations needed to obtain the hologram and also reduces the reconstruction errors. The proposed method is first mathematically introduced and then demonstrated by both simulations and experiments.

© 2008 Optical Society of America

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References

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  1. Y. Li, D. Abookasis, and J. Rosen, “Computer-generated holograms of three-dimensional realistic objects recorded without wave interference,” Appl. Opt. 40, 2864-2870 (2001).
    [CrossRef]
  2. D. Abookasis and J. Rosen, “Computer-generated holograms of three-dimensional objects synthesized from their multiple angular viewpoints,” J. Opt. Soc. Am. A 20, 1537-1545 (2003).
    [CrossRef]
  3. Y. Sando, M. Itoh, and T. Yatagai, “Holographic three-dimensional display synthesized from three-dimensional Fourier spectra of real existing objects,” Opt. Lett. 28, 2518-2520 (2003).
    [CrossRef] [PubMed]
  4. T. Yatagai, “Stereoscopic approach to 3-D display using computer-generated holograms,” Appl. Opt. 15, 2722-2729 (1976).
    [CrossRef] [PubMed]
  5. D. Abookasis and J. Rosen, “Three types of computer-generated hologram synthesized from multiple angular viewpoints of a three-dimensional scene,” Appl. Opt. 45, 6533-6538 (2006).
    [CrossRef] [PubMed]
  6. N. T. Shaked, J. Rosen, and A. Stern, “Integral holography: white-light single-shot hologram acquisition,” Opt. Express 15, 5754-5760 (2007).
    [CrossRef] [PubMed]
  7. B. Katz, N. T. Shaked, and J. Rosen, “Synthesizing computer generated holograms with reduced number of perspective projections,” Opt. Express 15, 13250-13255 (2007).
    [CrossRef] [PubMed]
  8. Y. Sando, M. Itoh, and T. Yatagai, “Full-color computer-generated holograms using 3-D Fourier spectra,” Opt. Express 12, 6246-6251 (2004).
    [CrossRef] [PubMed]
  9. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 67 and 353.

2007

N. T. Shaked, J. Rosen, and A. Stern, “Integral holography: white-light single-shot hologram acquisition,” Opt. Express 15, 5754-5760 (2007).
[CrossRef] [PubMed]

B. Katz, N. T. Shaked, and J. Rosen, “Synthesizing computer generated holograms with reduced number of perspective projections,” Opt. Express 15, 13250-13255 (2007).
[CrossRef] [PubMed]

2006

D. Abookasis and J. Rosen, “Three types of computer-generated hologram synthesized from multiple angular viewpoints of a three-dimensional scene,” Appl. Opt. 45, 6533-6538 (2006).
[CrossRef] [PubMed]

2004

Y. Sando, M. Itoh, and T. Yatagai, “Full-color computer-generated holograms using 3-D Fourier spectra,” Opt. Express 12, 6246-6251 (2004).
[CrossRef] [PubMed]

2003

D. Abookasis and J. Rosen, “Computer-generated holograms of three-dimensional objects synthesized from their multiple angular viewpoints,” J. Opt. Soc. Am. A 20, 1537-1545 (2003).
[CrossRef]

Y. Sando, M. Itoh, and T. Yatagai, “Holographic three-dimensional display synthesized from three-dimensional Fourier spectra of real existing objects,” Opt. Lett. 28, 2518-2520 (2003).
[CrossRef] [PubMed]

2001

1976

Abookasis, D.

D. Abookasis and J. Rosen, “Three types of computer-generated hologram synthesized from multiple angular viewpoints of a three-dimensional scene,” Appl. Opt. 45, 6533-6538 (2006).
[CrossRef] [PubMed]

D. Abookasis and J. Rosen, “Computer-generated holograms of three-dimensional objects synthesized from their multiple angular viewpoints,” J. Opt. Soc. Am. A 20, 1537-1545 (2003).
[CrossRef]

Y. Li, D. Abookasis, and J. Rosen, “Computer-generated holograms of three-dimensional realistic objects recorded without wave interference,” Appl. Opt. 40, 2864-2870 (2001).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 67 and 353.

Itoh, M.

Katz, B.

Li, Y.

Rosen, J.

N. T. Shaked, J. Rosen, and A. Stern, “Integral holography: white-light single-shot hologram acquisition,” Opt. Express 15, 5754-5760 (2007).
[CrossRef] [PubMed]

B. Katz, N. T. Shaked, and J. Rosen, “Synthesizing computer generated holograms with reduced number of perspective projections,” Opt. Express 15, 13250-13255 (2007).
[CrossRef] [PubMed]

D. Abookasis and J. Rosen, “Three types of computer-generated hologram synthesized from multiple angular viewpoints of a three-dimensional scene,” Appl. Opt. 45, 6533-6538 (2006).
[CrossRef] [PubMed]

D. Abookasis and J. Rosen, “Computer-generated holograms of three-dimensional objects synthesized from their multiple angular viewpoints,” J. Opt. Soc. Am. A 20, 1537-1545 (2003).
[CrossRef]

Y. Li, D. Abookasis, and J. Rosen, “Computer-generated holograms of three-dimensional realistic objects recorded without wave interference,” Appl. Opt. 40, 2864-2870 (2001).
[CrossRef]

Sando, Y.

Shaked, N. T.

B. Katz, N. T. Shaked, and J. Rosen, “Synthesizing computer generated holograms with reduced number of perspective projections,” Opt. Express 15, 13250-13255 (2007).
[CrossRef] [PubMed]

N. T. Shaked, J. Rosen, and A. Stern, “Integral holography: white-light single-shot hologram acquisition,” Opt. Express 15, 5754-5760 (2007).
[CrossRef] [PubMed]

Stern, A.

N. T. Shaked, J. Rosen, and A. Stern, “Integral holography: white-light single-shot hologram acquisition,” Opt. Express 15, 5754-5760 (2007).
[CrossRef] [PubMed]

Yatagai, T.

Appl. Opt.

D. Abookasis and J. Rosen, “Three types of computer-generated hologram synthesized from multiple angular viewpoints of a three-dimensional scene,” Appl. Opt. 45, 6533-6538 (2006).
[CrossRef] [PubMed]

Appl. Opt.

J. Opt. Soc. Am. A

D. Abookasis and J. Rosen, “Computer-generated holograms of three-dimensional objects synthesized from their multiple angular viewpoints,” J. Opt. Soc. Am. A 20, 1537-1545 (2003).
[CrossRef]

Opt. Express

N. T. Shaked, J. Rosen, and A. Stern, “Integral holography: white-light single-shot hologram acquisition,” Opt. Express 15, 5754-5760 (2007).
[CrossRef] [PubMed]

Y. Sando, M. Itoh, and T. Yatagai, “Full-color computer-generated holograms using 3-D Fourier spectra,” Opt. Express 12, 6246-6251 (2004).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Other

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 67 and 353.

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Figures (5)

Fig. 1
Fig. 1

Optical system for capturing multiple-viewpoint projections of the 3-D scene. The camera and its imaging lens move together into a different capturing viewpoint for each projection.

Fig. 2
Fig. 2

Top view of the optical system shown in Fig. 1.

Fig. 3
Fig. 3

Simulated results for generating a 1-D modified Fresnel hologram: (a) Several projections taken from the entire digitally obtained projection set, (b) magnitude and phase of the 1-D modified Fresnel hologram, (c) the six best in-focus reconstructed planes along the optical axis, (d) the six best in-focus reconstructed planes along the optical axis after the resampling process of the horizontal axis.

Fig. 4
Fig. 4

Experimental results for generating a 1-D modified Fresnel hologram: (a) Several projections taken from the entire experimentally obtained projection set, (b) magnitude and phase of the 1-D modified Fresnel hologram, (c) the six best in-focus reconstructed planes along the optical axis,(d) the six best in-focus reconstructed planes along the optical axis after the resampling process of the horizontal axis.

Fig. 5
Fig. 5

(Color online) Graph of the axial size of the reconstructed object z ¯ r versus the axial size of the observed object z ¯ s . Solid curve—predicted (calculated) curve; dashed curve—based on the experimental results. The diamond points represent each one of the six object planes.

Equations (15)

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H ( m , n ) = P m ( x p , y p ) E n ( x p , y p ) d x p d y p ,
E n ( x p , y p ) = exp ( j 2 π b x p 2 ) δ ( y p n Δ p ) ,
x p = f ( x s m α ) z s , y p = f y s z s ,
H S S P ( m , n ; x s , y s , z s ) = [ h ( x s , y s , z s ) Δ x s Δ y s Δ z s × δ ( x p x p , y p y p ) ] × E n ( x p , y p ) d x p d y p = h ( x s , y s , z s ) E n ( x p , y p ) × Δ x s Δ y s Δ z s .
H S S P ( m , n ; x s , y s , z s ) = h ( x s , y s , z s ) × exp [ j 2 π b f 2 ( x s m α ) 2 z s 2 ] × δ ( f y s z s n Δ p ) Δ x s Δ y s Δ z s .
H ( m , n ) = H S S P ( m , n ; x s , y s , z s ) d x s d y s d z s = h ( x s , y s , z s ) exp [ j 2 π b f 2 ( x s m α ) 2 z s 2 ] × δ ( f y s z s n Δ p ) d x s d y s d z s .
H ( m , n ) = 1 2 z s h ( x s , y s , z s ) × exp [ j 2 π b f 2 ( x s m α ) 2 z s ] × δ ( f y s z s n Δ p ) d x s d y s d z s .
s ( m , n ; z r ) = | H ( m , n ) exp ( j 2 π γ z r m 2 ) | ,
H ( m , n ) = h ( x s , y s , z s ) × exp [ j 2 π b ( M α Δ p ) 2 ( Δ p x s α m Δ p ) 2 ] × δ ( M y s n Δ p ) d x s d y s d z s ,
M x = x ¯ r x ¯ s = Δ p / α , M y = y ¯ r / y ¯ s = M = f / z s ,
z ¯ r = 1 γ b f 2 α 2 z ¯ s ( z ¯ s + 2 z s ) .
M z = z ¯ r z ¯ s = z ¯ s + 2 z s γ b f 2 α 2 .
b = [ ( K f α / z s , min ) 2 ( K f α / z s , min f α / z s , min ) 2 ] 1 z s , min 2 / ( 2 K f 2 α 2 ) .
Δ x s = max { 1.22 λ z s D , Δ p M , f α M z s , min } ,
Δ z s = Δ x s z s / ( K α ) .

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