Abstract

We develop a mathematical model of triangle-mesh-modeled three-dimensional (3D) surface objects for digital holography. The proposed mathematical model includes the analytic angular spectrum representation of image light fields emitted from 3D surface objects with occlusion and the computation method for the developed light field representation. Reconstruction of computer-generated holograms synthesized by using the developed model is demonstrated experimentally.

© 2008 Optical Society of America

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References

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  1. L. Yaroslavsky, “Digital holography: 30 years later,” Proc. SPIE 4659, 1-11 (2002).
    [CrossRef]
  2. L. Yaroslavsky, Digital Holography and Digital Image Processing (Kluwer Scientific, 2004).
  3. T. C. Poon, ed., Digital Holography and Three-Dimensional Display (Springer, 2006).
    [CrossRef]
  4. C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46-53 (2005).
    [CrossRef]
  5. C. Slinger, C. Cameron, S. Coomber, R. Miller, D. Pain, A. Smith, M. Smith, M. Stanley, and P. Watson, “Recent developments in computer-generated holography: toward a practical electroholography system for interactive 3D visualization,” Proc. SPIE , 5290, 27-41 (2004).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  8. M. Lucente, “Diffraction-specific fringe computation for electro-holography,” Ph.D. dissertation (MIT, 1994).
  9. K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. 44, 4607-4614 (2005).
    [CrossRef] [PubMed]
  10. 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).
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  11. H. Yoshikawa, “Fast computation of Fresnel holograms employing difference,” Opt. Rev. 8, 331-335 (2001).
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  14. H. Kim and B. Lee, “Optimal non-monotonic convergence of iterative Fourier transform algorithm,” Opt. Lett. 30, 296-298(2005).
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  15. H. Kim, B. Yang, and B. Lee, “Iterative Fourier transform algorithm with regularization for the optimal design of diffractive optical elements,” J. Opt. Soc. Am A 21, 2353-2365(2004).
    [CrossRef]
  16. K. Matsushima, “Exact hidden-surface removal in digitally synthetic full-parallax holograms,” SPIE 5742, 25-32 (2005).
    [CrossRef]
  17. Y. Sando, M. Itoh, and T. Yatagai, “Color computer-generated holograms from projection images,” Opt. Express 12, 2487-2493 (2004).
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  18. T. Yatagai, “Stereoscopic approach to 3-D display using computer-generated holograms,” Appl. Opt. 15, 2722-2729(1976).
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2005

2004

Y. Sando, M. Itoh, and T. Yatagai, “Color computer-generated holograms from projection images,” Opt. Express 12, 2487-2493 (2004).
[CrossRef] [PubMed]

H. Kim, B. Yang, and B. Lee, “Iterative Fourier transform algorithm with regularization for the optimal design of diffractive optical elements,” J. Opt. Soc. Am A 21, 2353-2365(2004).
[CrossRef]

C. Slinger, C. Cameron, S. Coomber, R. Miller, D. Pain, A. Smith, M. Smith, M. Stanley, and P. Watson, “Recent developments in computer-generated holography: toward a practical electroholography system for interactive 3D visualization,” Proc. SPIE , 5290, 27-41 (2004).
[CrossRef]

2003

2002

2001

H. Yoshikawa, “Fast computation of Fresnel holograms employing difference,” Opt. Rev. 8, 331-335 (2001).
[CrossRef]

2000

C. Cameron, D. Pain, M. Stanley, and C. Slinger, “Computational challenges of emerging novel true 3D holographic displays,” Proc. SPIE 4109, 129-140 (2000).
[CrossRef]

1976

Abookasis, D.

Cameron, C.

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46-53 (2005).
[CrossRef]

C. Slinger, C. Cameron, S. Coomber, R. Miller, D. Pain, A. Smith, M. Smith, M. Stanley, and P. Watson, “Recent developments in computer-generated holography: toward a practical electroholography system for interactive 3D visualization,” Proc. SPIE , 5290, 27-41 (2004).
[CrossRef]

C. Cameron, D. Pain, M. Stanley, and C. Slinger, “Computational challenges of emerging novel true 3D holographic displays,” Proc. SPIE 4109, 129-140 (2000).
[CrossRef]

Coomber, S.

C. Slinger, C. Cameron, S. Coomber, R. Miller, D. Pain, A. Smith, M. Smith, M. Stanley, and P. Watson, “Recent developments in computer-generated holography: toward a practical electroholography system for interactive 3D visualization,” Proc. SPIE , 5290, 27-41 (2004).
[CrossRef]

Gillet, J.-N.

Gundu, P. N.

Hark, E.

Itoh, M.

Kim, H.

H. Kim and B. Lee, “Optimal non-monotonic convergence of iterative Fourier transform algorithm,” Opt. Lett. 30, 296-298(2005).
[CrossRef] [PubMed]

H. Kim, B. Yang, and B. Lee, “Iterative Fourier transform algorithm with regularization for the optimal design of diffractive optical elements,” J. Opt. Soc. Am A 21, 2353-2365(2004).
[CrossRef]

Lee, B.

H. Kim and B. Lee, “Optimal non-monotonic convergence of iterative Fourier transform algorithm,” Opt. Lett. 30, 296-298(2005).
[CrossRef] [PubMed]

H. Kim, B. Yang, and B. Lee, “Iterative Fourier transform algorithm with regularization for the optimal design of diffractive optical elements,” J. Opt. Soc. Am A 21, 2353-2365(2004).
[CrossRef]

Lucente, M.

M. Lucente, “Diffraction-specific fringe computation for electro-holography,” Ph.D. dissertation (MIT, 1994).

Matsushima, K.

K. Matsushima, “Exact hidden-surface removal in digitally synthetic full-parallax holograms,” SPIE 5742, 25-32 (2005).
[CrossRef]

K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. 44, 4607-4614 (2005).
[CrossRef] [PubMed]

Miller, R.

C. Slinger, C. Cameron, S. Coomber, R. Miller, D. Pain, A. Smith, M. Smith, M. Stanley, and P. Watson, “Recent developments in computer-generated holography: toward a practical electroholography system for interactive 3D visualization,” Proc. SPIE , 5290, 27-41 (2004).
[CrossRef]

Pain, D.

C. Slinger, C. Cameron, S. Coomber, R. Miller, D. Pain, A. Smith, M. Smith, M. Stanley, and P. Watson, “Recent developments in computer-generated holography: toward a practical electroholography system for interactive 3D visualization,” Proc. SPIE , 5290, 27-41 (2004).
[CrossRef]

C. Cameron, D. Pain, M. Stanley, and C. Slinger, “Computational challenges of emerging novel true 3D holographic displays,” Proc. SPIE 4109, 129-140 (2000).
[CrossRef]

Rosen, J.

Sando, Y.

Sheng, Y.

Slinger, C.

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46-53 (2005).
[CrossRef]

C. Slinger, C. Cameron, S. Coomber, R. Miller, D. Pain, A. Smith, M. Smith, M. Stanley, and P. Watson, “Recent developments in computer-generated holography: toward a practical electroholography system for interactive 3D visualization,” Proc. SPIE , 5290, 27-41 (2004).
[CrossRef]

C. Cameron, D. Pain, M. Stanley, and C. Slinger, “Computational challenges of emerging novel true 3D holographic displays,” Proc. SPIE 4109, 129-140 (2000).
[CrossRef]

Smith, A.

C. Slinger, C. Cameron, S. Coomber, R. Miller, D. Pain, A. Smith, M. Smith, M. Stanley, and P. Watson, “Recent developments in computer-generated holography: toward a practical electroholography system for interactive 3D visualization,” Proc. SPIE , 5290, 27-41 (2004).
[CrossRef]

Smith, M.

C. Slinger, C. Cameron, S. Coomber, R. Miller, D. Pain, A. Smith, M. Smith, M. Stanley, and P. Watson, “Recent developments in computer-generated holography: toward a practical electroholography system for interactive 3D visualization,” Proc. SPIE , 5290, 27-41 (2004).
[CrossRef]

Stanley, M.

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46-53 (2005).
[CrossRef]

C. Slinger, C. Cameron, S. Coomber, R. Miller, D. Pain, A. Smith, M. Smith, M. Stanley, and P. Watson, “Recent developments in computer-generated holography: toward a practical electroholography system for interactive 3D visualization,” Proc. SPIE , 5290, 27-41 (2004).
[CrossRef]

C. Cameron, D. Pain, M. Stanley, and C. Slinger, “Computational challenges of emerging novel true 3D holographic displays,” Proc. SPIE 4109, 129-140 (2000).
[CrossRef]

Watson, P.

C. Slinger, C. Cameron, S. Coomber, R. Miller, D. Pain, A. Smith, M. Smith, M. Stanley, and P. Watson, “Recent developments in computer-generated holography: toward a practical electroholography system for interactive 3D visualization,” Proc. SPIE , 5290, 27-41 (2004).
[CrossRef]

Yang, B.

H. Kim, B. Yang, and B. Lee, “Iterative Fourier transform algorithm with regularization for the optimal design of diffractive optical elements,” J. Opt. Soc. Am A 21, 2353-2365(2004).
[CrossRef]

Yaroslavsky, L.

L. Yaroslavsky, “Digital holography: 30 years later,” Proc. SPIE 4659, 1-11 (2002).
[CrossRef]

L. Yaroslavsky, Digital Holography and Digital Image Processing (Kluwer Scientific, 2004).

Yatagai, T.

Yoshikawa, H.

H. Yoshikawa, “Fast computation of Fresnel holograms employing difference,” Opt. Rev. 8, 331-335 (2001).
[CrossRef]

Appl. Opt.

Computer

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38, 46-53 (2005).
[CrossRef]

J. Opt. Soc. Am A

H. Kim, B. Yang, and B. Lee, “Iterative Fourier transform algorithm with regularization for the optimal design of diffractive optical elements,” J. Opt. Soc. Am A 21, 2353-2365(2004).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

Opt. Rev.

H. Yoshikawa, “Fast computation of Fresnel holograms employing difference,” Opt. Rev. 8, 331-335 (2001).
[CrossRef]

Proc. SPIE

C. Cameron, D. Pain, M. Stanley, and C. Slinger, “Computational challenges of emerging novel true 3D holographic displays,” Proc. SPIE 4109, 129-140 (2000).
[CrossRef]

C. Slinger, C. Cameron, S. Coomber, R. Miller, D. Pain, A. Smith, M. Smith, M. Stanley, and P. Watson, “Recent developments in computer-generated holography: toward a practical electroholography system for interactive 3D visualization,” Proc. SPIE , 5290, 27-41 (2004).
[CrossRef]

L. Yaroslavsky, “Digital holography: 30 years later,” Proc. SPIE 4659, 1-11 (2002).
[CrossRef]

SPIE

K. Matsushima, “Exact hidden-surface removal in digitally synthetic full-parallax holograms,” SPIE 5742, 25-32 (2005).
[CrossRef]

Other

M. Lucente, “Diffraction-specific fringe computation for electro-holography,” Ph.D. dissertation (MIT, 1994).

L. Yaroslavsky, Digital Holography and Digital Image Processing (Kluwer Scientific, 2004).

T. C. Poon, ed., Digital Holography and Three-Dimensional Display (Springer, 2006).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

(a) Triangle mesh modeling of 3D object (two hands). (b) Triangle mesh surface-polyhedron objects in the global coordinate system. (c) Local coordinate system of the kth unit triangle. (d) The relationship between the local coordinate system and the global coordinate system.

Fig. 2
Fig. 2

Diffusive facet surface parameterized by the factor M: (a) upward-oriented triangles, (b) downward-oriented triangles.

Fig. 3
Fig. 3

Images of three triangles with the diffusiveness factor of M = 4 observed at observation angles (a)  φ v = 0 ° , (b)  φ v = 1 ° , (c)  φ v = 2 ° . Images of three triangles with the dif fusiveness factor of M = 64 observed at observation angles (d)  φ v = 0 ° , (e)  φ v = 5 ° , (f)  φ v = 10 ° .

Fig. 4
Fig. 4

(a) Projection image of triangle-mesh-modeled two hands toward an observation direction ( φ v , ψ v ) = ( 0 ° , 0 ° ) ;(b) projection image of triangle-mesh-modeled dinosaur toward an observation direction ( φ v , ψ v ) = ( 20 ° , 0 ° ) .

Fig. 5
Fig. 5

Images of three triangles, focused on the left-hand triangle, observed at the observation angles (a)  φ v = 5 ° and (b)  φ v = 10 ° . Images of three triangles focused on the right-hand triangle, observed at the observation angles (c)  φ v = 5 ° , (d)  φ v = 10 ° .

Fig. 6
Fig. 6

Schematic of CGH reconstruction and observation.

Fig. 7
Fig. 7

CGH of two hands: (a) amplitude pro file | Π ( u , v ) | , (b) phase profile Π ( u , v ) . Image of two hands focused on the left hand: (c) simulation and (d) experiment. Image of two hands focused on the right hand: (e) simulation and (f) experiment.

Fig. 8
Fig. 8

CGH of a dinosaur: (a) amplitude profile | Π ( u , v ) | and (b) phase profile Π ( u , v ) . Image of the dinosaur focused on the head: (c) simulation and (d) experiment. Image of the dinosaur focused on the tail: (e) simulation and (f) experiment.

Fig. 9
Fig. 9

Fourier transform of a triangle by simple geometric transform.

Equations (38)

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a k x + b k y + c k z + d k = 0 ,
( a k , b k , c k ) = ( cos ϕ k sin θ k , sin ϕ k sin θ k , cos θ k ) ,
d k = a k x k b k y k c k z k .
( x y z ) = ( cos θ k cos ϕ k cos θ k sin ϕ k sin θ k sin ϕ k cos ϕ k 0 sin θ k cos ϕ k sin θ k sin ϕ k cos θ k ) ( x x k y y k z z k ) .
W k ( x , y , 0 ) = η 0 exp ( j 2 π ( α 0 ( x + x k ) + β 0 ( y + y k ) + γ 0 z k ) ) U k ( x , y ) = η 0 exp ( j 2 π ( α 0 ( x + x k ) + β 0 ( y + y k ) + γ 0 z k ) ) A k ( α , β ) exp ( j 2 π ( α x + β y ) ) d α d β ,
W k ( x , y , 0 ) = η 0 exp ( j 2 π ( α 0 x k + β 0 y k + γ 0 z k ) ) A k ( α α 0 , β β 0 ) exp ( j 2 π ( α x + β y ) ) d α d β .
W k ( x , y , z ) = η 0 exp ( j 2 π ( α 0 x k + β 0 y k + γ 0 z k ) ) A k ( α α 0 , β β 0 ) exp ( j 2 π ( α x + β y + γ z ) ) d α d β ,
U k ( x , y ) = U k , ( x , y ) + U k , ( x , y ) ,
U k , ( x , y ) = m = 1 M n = 1 m Λ k , ( x x ˜ k , , m n , y y ˜ k , , m n ) ϒ k , , m n exp ( j Γ k , , m n ) ,
U k , ( x , y ) = m = 1 M 1 n = 1 m Λ k , ( x x ˜ k , , m n , y y ˜ k , , m n ) ϒ k , , m n exp ( j Γ k , , m n ) ,
x ˜ k , , m n = ( n 1 ) ( x 2 x 3 ) / M + ( m n ) ( x 1 x 3 ) / M ,
y ˜ k , , m n = ( n 1 ) ( y 2 y 3 ) / M + ( m n ) ( y 1 y 3 ) / M ,
x ˜ k , , m n = n ( x 2 x 3 ) / M + ( m n + 1 ) ( x 1 x 3 ) / M ,
y ˜ k , , m n = n ( y 2 y 3 ) / M + ( m n + 1 ) ( y 1 y 3 ) / M ,
A k ( α , β ) = U k ( x , y ) exp ( j 2 π ( α x + β y ) ) d x d y = ( U k , ( x , y ) + U k , ( x , y ) ) exp ( j 2 π ( α x + β y ) ) d x d y = A k , ( α , β ) + A k , ( α , β ) ,
m = 1 M n = 1 m ϒ k , , m n exp ( j Γ k , , m n ) exp ( j 2 π ( α x ˜ k , , m n + β y ˜ k , , m n ) ) A e , k , ( α , β ) ,
m = 1 M 1 n = 1 m ϒ k , , m n exp ( j Γ k , , m n ) exp ( j 2 π ( α x ˜ k , , m n + β y ˜ k , , m n ) ) A e , k , ( α , β ) ,
α ( k ) ( α , β ) = cos θ k cos ϕ k α + cos θ k sin ϕ k β sin θ k γ ,
β ( k ) ( α , β ) = sin ϕ k α + cos ϕ k β ,
γ ( k ) ( α , β ) = sin θ k cos ϕ k α + sin θ k sin ϕ k β + cos θ k γ .
[ d α ( k ) ( α , β ) d β ( k ) ( α , β ) ] = ( cos θ k cos ϕ k + α sin θ k γ cos θ k sin ϕ k + β sin θ k γ sin ϕ k cos ϕ k ) [ d α d β ] .
d α ( k ) ( α , β ) d β ( k ) ( α , β ) = | cos θ k + sin θ k ( cos ϕ k α + sin ϕ k β ) γ | d α d β .
W k ( x , y , z ) = η 0 exp ( j 2 π ( α 0 x k + β 0 y k + γ 0 z k ) ) × { A k ( α ( k ) ( α , β ) α 0 ( k ) ( α 0 , β 0 ) , β ( k ) ( α , β ) β 0 ( k ) ( α 0 , β 0 ) ) H ( γ ( k ) ( α , β ) ) exp ( j 2 π ( α ( x x k ) + β ( y y k ) + γ ( z z k ) ) ) } | cos θ k + sin θ k ( cos ϕ k α + sin ϕ k β ) γ | d α d β ,
A G , k ( α , β ) = η 0 exp ( j 2 π ( α 0 x k + β 0 y k + γ 0 z k ) ) A k ( α ( k ) ( α , β ) α 0 ( k ) ( α 0 , β 0 ) , β ( k ) ( α , β ) β 0 ( k ) ( α 0 , β 0 ) ) H ( γ ( k ) ( α , β ) ) | cos θ k + sin θ k ( cos ϕ k α + sin ϕ k β ) γ | exp ( j 2 π [ α ( x k ) + β ( y k ) + γ ( z k ) ] ) .
A total ( α , β ) = k Γ A G , k ( α , β ) ,
T ( ) ( φ v , ψ v , k , m , n ) = { 1 if the   ( m , n ) up (down) triangle of the   k th facet is   obser vable 0 if not .
A k ( α , β ) = m = 1 M n = 1 m T ( φ v , ψ v , k , m , n ) ϒ k , , m n exp ( j Γ k , , m n ) exp ( j 2 π ( α x ˜ k , , m n + β y ˜ k , , m n ) ) A e , k , ( α , β ) + m = 1 M 1 n = 1 m T ( φ v , ψ v , k , m , n ) ϒ k , , m n exp ( j Γ k , , m n ) exp ( j 2 π ( α x ˜ k , , m n + β y ˜ k , , m n ) ) A e , k , ( α , β ) = A k , ( α , β ) + A k , ( α , β ) .
W total ( x , y , z ) = A G total ( α , β ) exp ( j 2 π ( α x + β y + γ z ) ) d α d β .
Π ( u , v ) = A G total ( u λ f , v λ f ) ,
F ( u , v ) = 1 j λ f Π ( u , v ) = 1 j λ f k A G , k ( u λ f , v λ f ) .
W ˜ rec ( x , y ; Δ z ) = 1 j λ f Π ( u , v ) exp ( j π λ f [ ( Δ z f ) ( u 2 + v 2 ) 2 ( u x + v y ) ] ) d u d v .
W ˜ rec ( x , y ; 0 ) = A G total ( u λ f , v λ f ) exp ( j 2 π ( x d u λ f + y d v λ f ) ) ( d u λ f ) ( d v λ f ) = W total ( x , y ) .
W ¯ rec ( x , y ; Δ z ) = 1 j λ f exp ( j Π ( u , v ) ) exp ( j π λ f [ ( Δ z f ) ( u 2 + v 2 ) 2 ( u x + v y ) ] ) d u d v .
( x c y c ) = 1 ( y 2 y 1 ) 2 + ( x 2 x 1 ) 2 ( ( x 3 x 1 ) ( x 2 x 1 ) ( y 2 y 1 ) + y 3 ( y 2 y 1 ) 2 + y 1 ( x 2 x 1 ) 2 ( y 3 y 1 ) ( x 2 x 1 ) ( y 2 y 1 ) + x 3 ( x 2 x 1 ) 2 + x 1 ( y 2 y 1 ) 2 ) ,
( cos ( ψ ) , sin ( ψ ) ) = ( y 3 y c , x 3 x c ) / ( x 3 x c ) 2 + ( y 3 y c ) 2 .
( a , b , c ) = ( ( x 1 x c ) cos ψ + ( y 1 y c ) sin ψ , ( x 2 x c ) cos ψ ( y 2 y c ) sin ψ , ( x 3 x c ) sin ψ + ( y 3 y c ) cos ψ ) .
F ( f x , f y ) = { ( j c 2 π f x ) { e j 2 π f x b e j π ( f x b f y c ) sinc ( f x b f y c ) + e j 2 π f x a e j π ( f x a + f y c ) sinc ( f x a + f y c ) } for     f x 0 ( a + b c ) e i 2 π f y c { 1 ( j 2 π c f y 1 ) e j 2 π f y c ( 2 π f y ) 2 } for     f x = 0 , f y 0 ( a + b ) c 2 for     f x = 0 , f y = 0 .
F ¯ ( f x , f y ) = F ( f x cos ψ f y sin ψ , f x sin ψ + f y cos ψ ) exp ( j 2 π ( f x x c + f y y c ) ) .

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