Abstract

We present a method to analytically compute the light distribution of triangles directly in frequency space. This allows for fast evaluation, shading, and propagation of light from 3D mesh objects using angular spectrum methods. The algorithm complexity is only dependent on the hologram resolution and the polygon count of the 3D model. In contrast to other polygon based computer generated holography methods we do not need to perform a Fourier transform per surface. The theory behind the approach is derived, and a suitable algorithm to compute a digital hologram from a general triangle mesh is presented. We review some first results rendered on a spatial-light-modulator-based display by our proof-of-concept software.

© 2008 Optical Society of America

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  1. V. M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging 2, 28-34 (1993).
    [CrossRef]
  2. M. Bove Jr., W. J. Plesniak, T. Quentmeyer, and J. Barabas, Real-time holographic video images with commodity PC hardware in Stereoscopic Displays and Virtual Reality Systems XII, A. J. Woods, M. T. Bolas, J. O. Merritt, and I. E. McDowell, eds., Proc. SPIE 5664, 255-262 (2005).
  3. T. Ito, N. Masuda, K. Yoshimura, A. Shiraki, T. Shimobaba, and T. Sugie, “Special-purpose computer horn-5 for a real-time electroholography,” Opt. Express 13, 1923-1932 (2005).
    [CrossRef] [PubMed]
  4. T. Haist, M. Reicherter, M. Wu, and L. Seifert, “Using graphics boards to compute holograms,” Comput. in Sci. and Eng. 8, 8-13 (2006); http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1563956.
    [CrossRef]
  5. 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]
  6. L. Ahrenberg, P. Benzie, M. Magnor, and J. Watson, “Computer generated holography using parallel commodity graphics hardware,” Opt. Express 14, 7636-7641 (2006).
    [CrossRef] [PubMed]
  7. D. Leseberg and C. Frère, “Computer-generated holograms of 3-D objects composed of tilted planar segments,” Appl. Opt. 27, 3020-3024 (1988); http://www.opticsinfobase.org/viewmedia.cfm?id=61773&seq=0.
    [CrossRef] [PubMed]
  8. K. Matsushima and A. Kondoh, “Wave optical algorithm for creating digitally synthetic holograms of three-dimensional surface objects,” in Volume 5005 Practical Holography XVII and Holographic Materials IX, T. H. Jeong and S. H. Stevenson, eds. Proc. SPIE 5005, 190-197 (2003).
  9. K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. 44, 4607-4614 (2005).
    [CrossRef] [PubMed]
  10. K. Matsushima, “Performance of the polygon-source method for creating computer-generated holograms of surface objects,” in ICO Topical Meeting on Optoinfomatics/Information Photonics 2006 ICO, (2006), pp. 99-100.
  11. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  12. R. N. Bracewell, K.-Y. Chang, A. K. Jha, and Y.-H. Wang, “Affine theorem for two-dimensional Fourier transform,” Electron. Lett. 29, 304-306 (1993).
    [CrossRef]
  13. T. Kreis, Handbook of Holographic Interfereometry (Wiley-VCH, 2005), pp. 116, 135.
  14. T. Tommasi and B. Bianco, “Frequency analysis of light diffraction between rotated planes,” Opt. Lett. 17, 556-558 (1992); http://www.opticsinfobase.org/viewmedia.cfm?id=11103&seq=0.
    [CrossRef] [PubMed]
  15. T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. Am. A 10, 299-305 (1993); http://www.opticsinfobase.org/viewmedia.cfm?id=4534&seq=0.
    [CrossRef]
  16. R. Ziegler, P. Kaufmann, and M. Gross, “A framework for holographic scene representation and image synthesis,” Tech. Rep. (Swiss Federal Institute of Technology Zurich, 2006).
  17. 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]
  18. J. Stam, “Diffraction shaders,” in SIGGRAPH '99: Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques (ACM Addison-Wesley, 1999), pp. 101-110.
    [CrossRef]
  19. K. Matsushima, “Exact hidden-surface removal in digitally synthetic full-parallax holograms,” in Practical Holography XIX: Materials and Applications, T. H. Jeong and H. I. Bjelkhagen, Proc. SPIE, 572425-32 (2005).

2006 (3)

2005 (2)

2003 (1)

1993 (3)

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

R. N. Bracewell, K.-Y. Chang, A. K. Jha, and Y.-H. Wang, “Affine theorem for two-dimensional Fourier transform,” Electron. Lett. 29, 304-306 (1993).
[CrossRef]

T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. Am. A 10, 299-305 (1993); http://www.opticsinfobase.org/viewmedia.cfm?id=4534&seq=0.
[CrossRef]

1992 (1)

1988 (1)

Ahrenberg, L.

Barabas, J.

M. Bove Jr., W. J. Plesniak, T. Quentmeyer, and J. Barabas, Real-time holographic video images with commodity PC hardware in Stereoscopic Displays and Virtual Reality Systems XII, A. J. Woods, M. T. Bolas, J. O. Merritt, and I. E. McDowell, eds., Proc. SPIE 5664, 255-262 (2005).

Benzie, P.

Bianco, B.

Bove, M.

M. Bove Jr., W. J. Plesniak, T. Quentmeyer, and J. Barabas, Real-time holographic video images with commodity PC hardware in Stereoscopic Displays and Virtual Reality Systems XII, A. J. Woods, M. T. Bolas, J. O. Merritt, and I. E. McDowell, eds., Proc. SPIE 5664, 255-262 (2005).

Bracewell, R. N.

R. N. Bracewell, K.-Y. Chang, A. K. Jha, and Y.-H. Wang, “Affine theorem for two-dimensional Fourier transform,” Electron. Lett. 29, 304-306 (1993).
[CrossRef]

Chang, K.-Y.

R. N. Bracewell, K.-Y. Chang, A. K. Jha, and Y.-H. Wang, “Affine theorem for two-dimensional Fourier transform,” Electron. Lett. 29, 304-306 (1993).
[CrossRef]

Frère, C.

Goodman, J. W.

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

Gross, M.

R. Ziegler, P. Kaufmann, and M. Gross, “A framework for holographic scene representation and image synthesis,” Tech. Rep. (Swiss Federal Institute of Technology Zurich, 2006).

Haist, T.

T. Haist, M. Reicherter, M. Wu, and L. Seifert, “Using graphics boards to compute holograms,” Comput. in Sci. and Eng. 8, 8-13 (2006); http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1563956.
[CrossRef]

Ito, T.

Jha, A. K.

R. N. Bracewell, K.-Y. Chang, A. K. Jha, and Y.-H. Wang, “Affine theorem for two-dimensional Fourier transform,” Electron. Lett. 29, 304-306 (1993).
[CrossRef]

Kaufmann, P.

R. Ziegler, P. Kaufmann, and M. Gross, “A framework for holographic scene representation and image synthesis,” Tech. Rep. (Swiss Federal Institute of Technology Zurich, 2006).

Kondoh, A.

K. Matsushima and A. Kondoh, “Wave optical algorithm for creating digitally synthetic holograms of three-dimensional surface objects,” in Volume 5005 Practical Holography XVII and Holographic Materials IX, T. H. Jeong and S. H. Stevenson, eds. Proc. SPIE 5005, 190-197 (2003).

Kreis, T.

T. Kreis, Handbook of Holographic Interfereometry (Wiley-VCH, 2005), pp. 116, 135.

Leseberg, D.

Lucente, V. M.

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

Magnor, M.

Masuda, N.

Matsushima, K.

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

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]

K. Matsushima, “Exact hidden-surface removal in digitally synthetic full-parallax holograms,” in Practical Holography XIX: Materials and Applications, T. H. Jeong and H. I. Bjelkhagen, Proc. SPIE, 572425-32 (2005).

K. Matsushima and A. Kondoh, “Wave optical algorithm for creating digitally synthetic holograms of three-dimensional surface objects,” in Volume 5005 Practical Holography XVII and Holographic Materials IX, T. H. Jeong and S. H. Stevenson, eds. Proc. SPIE 5005, 190-197 (2003).

K. Matsushima, “Performance of the polygon-source method for creating computer-generated holograms of surface objects,” in ICO Topical Meeting on Optoinfomatics/Information Photonics 2006 ICO, (2006), pp. 99-100.

Plesniak, W. J.

M. Bove Jr., W. J. Plesniak, T. Quentmeyer, and J. Barabas, Real-time holographic video images with commodity PC hardware in Stereoscopic Displays and Virtual Reality Systems XII, A. J. Woods, M. T. Bolas, J. O. Merritt, and I. E. McDowell, eds., Proc. SPIE 5664, 255-262 (2005).

Quentmeyer, T.

M. Bove Jr., W. J. Plesniak, T. Quentmeyer, and J. Barabas, Real-time holographic video images with commodity PC hardware in Stereoscopic Displays and Virtual Reality Systems XII, A. J. Woods, M. T. Bolas, J. O. Merritt, and I. E. McDowell, eds., Proc. SPIE 5664, 255-262 (2005).

Reicherter, M.

T. Haist, M. Reicherter, M. Wu, and L. Seifert, “Using graphics boards to compute holograms,” Comput. in Sci. and Eng. 8, 8-13 (2006); http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1563956.
[CrossRef]

Schimmel, H.

Seifert, L.

T. Haist, M. Reicherter, M. Wu, and L. Seifert, “Using graphics boards to compute holograms,” Comput. in Sci. and Eng. 8, 8-13 (2006); http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1563956.
[CrossRef]

Shimobaba, T.

Shiraki, A.

Stam, J.

J. Stam, “Diffraction shaders,” in SIGGRAPH '99: Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques (ACM Addison-Wesley, 1999), pp. 101-110.
[CrossRef]

Sugie, T.

Tanaka, T.

Tommasi, T.

Wang, Y.-H.

R. N. Bracewell, K.-Y. Chang, A. K. Jha, and Y.-H. Wang, “Affine theorem for two-dimensional Fourier transform,” Electron. Lett. 29, 304-306 (1993).
[CrossRef]

Watson, J.

Wu, M.

T. Haist, M. Reicherter, M. Wu, and L. Seifert, “Using graphics boards to compute holograms,” Comput. in Sci. and Eng. 8, 8-13 (2006); http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1563956.
[CrossRef]

Wyrowski, F.

Yoshimura, K.

Ziegler, R.

R. Ziegler, P. Kaufmann, and M. Gross, “A framework for holographic scene representation and image synthesis,” Tech. Rep. (Swiss Federal Institute of Technology Zurich, 2006).

Appl. Opt. (2)

Comput. in Sci. and Eng. (1)

T. Haist, M. Reicherter, M. Wu, and L. Seifert, “Using graphics boards to compute holograms,” Comput. in Sci. and Eng. 8, 8-13 (2006); http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?arnumber=1563956.
[CrossRef]

Electron. Lett. (1)

R. N. Bracewell, K.-Y. Chang, A. K. Jha, and Y.-H. Wang, “Affine theorem for two-dimensional Fourier transform,” Electron. Lett. 29, 304-306 (1993).
[CrossRef]

J. Electron. Imaging (1)

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

J. Opt. Soc. Am. A (2)

Opt. Express (3)

Opt. Lett. (1)

Other (8)

M. Bove Jr., W. J. Plesniak, T. Quentmeyer, and J. Barabas, Real-time holographic video images with commodity PC hardware in Stereoscopic Displays and Virtual Reality Systems XII, A. J. Woods, M. T. Bolas, J. O. Merritt, and I. E. McDowell, eds., Proc. SPIE 5664, 255-262 (2005).

K. Matsushima, “Performance of the polygon-source method for creating computer-generated holograms of surface objects,” in ICO Topical Meeting on Optoinfomatics/Information Photonics 2006 ICO, (2006), pp. 99-100.

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

T. Kreis, Handbook of Holographic Interfereometry (Wiley-VCH, 2005), pp. 116, 135.

K. Matsushima and A. Kondoh, “Wave optical algorithm for creating digitally synthetic holograms of three-dimensional surface objects,” in Volume 5005 Practical Holography XVII and Holographic Materials IX, T. H. Jeong and S. H. Stevenson, eds. Proc. SPIE 5005, 190-197 (2003).

R. Ziegler, P. Kaufmann, and M. Gross, “A framework for holographic scene representation and image synthesis,” Tech. Rep. (Swiss Federal Institute of Technology Zurich, 2006).

J. Stam, “Diffraction shaders,” in SIGGRAPH '99: Proceedings of the 26th Annual Conference on Computer Graphics and Interactive Techniques (ACM Addison-Wesley, 1999), pp. 101-110.
[CrossRef]

K. Matsushima, “Exact hidden-surface removal in digitally synthetic full-parallax holograms,” in Practical Holography XIX: Materials and Applications, T. H. Jeong and H. I. Bjelkhagen, Proc. SPIE, 572425-32 (2005).

Supplementary Material (4)

» Media 1: MOV (4051 KB)     
» Media 2: MOV (2904 KB)     
» Media 3: MOV (3807 KB)     
» Media 4: MOV (1380 KB)     

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Figures (5)

Fig. 1
Fig. 1

Triangle model. Each triangle is defined by three vertices: v 1 , v 2 , v 3 , which also span the triangle plane, Π j . n j is normal to the triangle plane, and the vector l is the direction toward a distant light source.

Fig. 2
Fig. 2

Two triangles. (a) Right triangle with vertices in ( 0 , 0 ) , ( 1 , 0 ) , ( 1 , 1 ) . (b) General triangle with vertices in ( s 1 , t 1 ) , ( s 2 , t 2 ) , ( s 3 , t 3 ) .

Fig. 3
Fig. 3

Stepwise transformation and rendering of the angular spectrum from a triangle. The angular spectrum of the triangle is computed in the triangle plane Π j , having normal n j . Thereafter the angular spectrum is transformed using the rotation transform R to a plane, Π j normal to the optical axis, n. In the last step a propagation transform P is used to propagate the light to the hologram plane Π H .

Fig. 4
Fig. 4

Images of reconstructions on an SLM, using a 633 nm laser and reconstruction distance of 0.5 m . The hologram resolution is 1220 × 1220 samples with a pixel pitch of 8.1 μm . (a) Simple text: 216 triangles (associated movie file, 4.0   MBytes ). (b) Bird mesh: 448 triangles (associated movie file, 2.9   MBytes ). (c) Knot: 480 triangles (associated movie file, 3.8     MBytes ). (d) Eight shape: 1536 triangles (associated movie file, 1.4   MBytes ).

Fig. 5
Fig. 5

Using a backfacing threshold to prune aliasing. (a) Numerical reconstruction with standard backface culling. (b) Numerical reconstruction using a backface-threshold of 0.2. (c) Closeup of marked area in (a). (d) Closeup of marked area in (b).

Tables (1)

Tables Icon

Table 1 Hologram Construction Times for a Few Different Triangular Models

Equations (26)

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W H = j = 1 T U j ,
A j = F { W j } .
U j = F 1 { T j { A j } } .
W H = i = 1 T F 1 { T j { F { W j } } } = F 1 { i = 1 T T j { F { W j } } } .
a j = n j · l .
f Δ ( x , y ) = { 1 , if     ( x , y )   lies inside   Δ 0 , else .
F Δ ( u , v ) f Δ ( x , y ) exp ( 2 π i ( x v + y v ) ) d y d x .
F Δ ( u , v ) = Δ exp ( 2 π i ( x u + y v ) ) d y d x = 0 1 0 x exp ( 2 π i ( x u + y v ) ) d y d x .
F Δ ( u , v ) = exp ( 2 π i u ) 1 ( 2 π ) 2 u v + 1 exp ( 2 π i ( u + v ) ) ( 2 π ) 2 v ( u + v ) .
F Δ = 0 1 0 x exp ( 2 π i y v ) d y d x = 1 exp ( 2 π i v ) ( 2 π v ) 2 i 2 π v .
F Δ = 0 1 0 x exp ( 2 π i x u ) d y d x = exp ( 2 π i u ) 1 ( 2 π u ) 2 + i exp ( 2 π i u ) 2 π u .
F Δ = 0 1 0 x exp ( 2 π i v ( y x ) ) d y d x = 1 exp ( 2 π i v ) ( 2 π v ) 2 + i 2 π v .
F Δ = 0 1 0 x exp ( 0 ) d y d x = 1 2 .
F Δ ( u , v ) = { 1 2 , u = v = 0 1 exp ( 2 π i v ) ( 2 π v ) 2 i 2 π v , u = 0 , v 0 exp ( 2 π i u ) 1 ( 2 π u ) 2 + i exp ( 2 π i u ) 2 π u , u 0 , v = 0 1 exp ( 2 π i v ) ( 2 π v ) 2 + i 2 π v , u = v , v 0 exp ( 2 π i u ) 1 ( 2 π ) 2 u v + 1 exp ( 2 π i ( u + v ) ) ( 2 π ) 2 v ( u + v ) , else
[ s t ] = [ a 11 a 12 a 21 a 22 ] [ x y ] + [ a 13 a 23 ] .
[ s t ] = 1 J [ ( t 3 t 2 ) ( s 2 s 3 ) ( t 1 t 2 ) ( s 2 s 1 ) ] [ x y ] + 1 J [ ( ( t 3 t 2 ) s 1 + ( s 2 s 3 ) t 1 ) ( ( t 1 t 2 ) s 1 + ( s 2 s 1 ) t 1 ) ] .
J = | a 11 a 12 a 21 a 22 | = a 11 a 22 a 12 a 21 .
f Γ ( x , y ) = f Δ ( s , t ) = f Δ ( a 11 x + a 12 y + a 13 , a 21 x + a 22 y + a 23 ) .
F Γ ( u , v ) = 1 | J | exp { 2 π i J [ ( a 22 a 13 a 12 a 23 ) u + ( a 11 a 23 a 13 a 21 ) v ] } × F Δ ( a 22 u a 12 v J , a 12 u + a 11 v J ) .
A j = A j exp ( 2 π i λ r ( 1 ( u λ ) 2 ( v λ ) 2 ) 1 / 2 ) ,
A j ( u , v ) = A j ( r 11 u + r 12 v + r 13 d ( u , v ) , r 21 u + r 22 v + r 23 d ( u , v ) ) J ( u , v ) .
J ( u , v ) = ( r 12 r 23 r 13 r 22 ) u d ( u , v ) + ( r 13 r 22 r 11 r 23 ) v d ( u , v ) + ( r 11 r 22 r 12 r 21 ) .
T = P { R } .
s = n j · n ,
W = W H + a R exp ( i k ) ,
I = ( W 2 min { W 2 } max { W 2 } min { W 2 } ) ( o max o min ) + o min ,

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