Abstract

In Digital Holography there are applications where computing a few samples of a wavefield is sufficient to retrieve an image of the region of interest. In such cases, the sampling rate achieved by the direct and the spectral methods of the discrete Fresnel transform could be excessive. A few algorithmic methods have been proposed to numerically compute samples of propagated wavefields while allowing down-sampling control. Nevertheless, all of them require the computation of at least two 2D discrete Fourier transforms which increases the computational load. Here, we propose the use of an aliasing operator and a single discrete Fourier transform to achieve an efficient method to down-sample the wavefields obtained by the Fresnel transform.

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References

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  1. T. M. Kreis, M. Adams, and W. P. P. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE3098, 224–233 (1997).
    [CrossRef]
  2. U. Schnars and W. Jueptner, Digital Holography (Springer, 2005).
  3. B. M. Hennelly, D. P. Kelly, D. S. Monaghan, and N. Pandey, “Zoom Algorithms for Digital Holography,” in Information Optics and Photonics: Algorithms, Systems, and Applications, T. Fournel and B. Javidi, eds. (Springer, 2010), pp. 187–204.
  4. X. Deng, B. Bihari, J. Gan, F. Zhao, and R. T. Chen, “Fast algorithm for chirp transforms with zooming-in ability and its applications,” J. Opt. Soc. Am. A17(4), 762–771 (2000).
    [CrossRef] [PubMed]
  5. F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett.29(14), 1668–1670 (2004).
    [CrossRef] [PubMed]
  6. W. T. Rhodes, “Light tubes, Wigner diagrams and optical signal propagation simulation,” in Optical Information Processing: A Tribute to Adolf Lohmann, H. J. Caulfield, ed. (SPIE Press, 2002), pp. 343–356.
  7. J. C. Li, P. Tankam, Z. J. Peng, and P. Picart, “Digital holographic reconstruction of large objects using a convolution approach and adjustable magnification,” Opt. Lett.34(5), 572–574 (2009).
    [CrossRef] [PubMed]
  8. L. Yu and M. K. Kim, “Pixel resolution control in numerical reconstruction of digital holography,” Opt. Lett.31(7), 897–899 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  11. E. N. Leith and J. Upatnieks, “Reconstructed wavefronts and communication theory,” J. Opt. Soc. Am.52(10), 1123–1130 (1962).
    [CrossRef]
  12. S. Satake, T. Kunugi, K. Sato, and T. Ito, “Digital Holographic Particle Tracking Velocimetry for 3-D Transient Flow around an Obstacle in a Narrow Channel,” Opt. Rev.11, 162–164 (2004).
  13. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing (Prentice Hall, 1999).
  14. M. Frigo and S. G. Johnson, “The FFTW web page,” (2007), http://www.fftw.org/ .

2009

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng.48(9), 095801 (2009).
[CrossRef]

J. C. Li, P. Tankam, Z. J. Peng, and P. Picart, “Digital holographic reconstruction of large objects using a convolution approach and adjustable magnification,” Opt. Lett.34(5), 572–574 (2009).
[CrossRef] [PubMed]

2006

2004

2000

1997

T. M. Kreis, M. Adams, and W. P. P. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE3098, 224–233 (1997).
[CrossRef]

1962

Adams, M.

T. M. Kreis, M. Adams, and W. P. P. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE3098, 224–233 (1997).
[CrossRef]

Alfieri, D.

Bihari, B.

Chen, R. T.

Coppola, G.

De Nicola, S.

Deng, X.

Ferraro, P.

Finizio, A.

Gan, J.

Hennelly, B. M.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng.48(9), 095801 (2009).
[CrossRef]

Ito, T.

S. Satake, T. Kunugi, K. Sato, and T. Ito, “Digital Holographic Particle Tracking Velocimetry for 3-D Transient Flow around an Obstacle in a Narrow Channel,” Opt. Rev.11, 162–164 (2004).

Jüptner, W. P. P.

T. M. Kreis, M. Adams, and W. P. P. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE3098, 224–233 (1997).
[CrossRef]

Kelly, D. P.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng.48(9), 095801 (2009).
[CrossRef]

Kim, M. K.

Kreis, T. M.

T. M. Kreis, M. Adams, and W. P. P. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE3098, 224–233 (1997).
[CrossRef]

Kunugi, T.

S. Satake, T. Kunugi, K. Sato, and T. Ito, “Digital Holographic Particle Tracking Velocimetry for 3-D Transient Flow around an Obstacle in a Narrow Channel,” Opt. Rev.11, 162–164 (2004).

Leith, E. N.

Li, J. C.

Naughton, T. J.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng.48(9), 095801 (2009).
[CrossRef]

Pandey, N.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng.48(9), 095801 (2009).
[CrossRef]

Peng, Z. J.

Picart, P.

Pierattini, G.

Rhodes, W. T.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng.48(9), 095801 (2009).
[CrossRef]

Satake, S.

S. Satake, T. Kunugi, K. Sato, and T. Ito, “Digital Holographic Particle Tracking Velocimetry for 3-D Transient Flow around an Obstacle in a Narrow Channel,” Opt. Rev.11, 162–164 (2004).

Sato, K.

S. Satake, T. Kunugi, K. Sato, and T. Ito, “Digital Holographic Particle Tracking Velocimetry for 3-D Transient Flow around an Obstacle in a Narrow Channel,” Opt. Rev.11, 162–164 (2004).

Tankam, P.

Upatnieks, J.

Yamaguchi, I.

Yaroslavsky, L. P.

Yu, L.

Zhang, F.

Zhao, F.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

D. P. Kelly, B. M. Hennelly, N. Pandey, T. J. Naughton, and W. T. Rhodes, “Resolution limits in practical digital holographic systems,” Opt. Eng.48(9), 095801 (2009).
[CrossRef]

Opt. Lett.

Opt. Rev.

S. Satake, T. Kunugi, K. Sato, and T. Ito, “Digital Holographic Particle Tracking Velocimetry for 3-D Transient Flow around an Obstacle in a Narrow Channel,” Opt. Rev.11, 162–164 (2004).

Proc. SPIE

T. M. Kreis, M. Adams, and W. P. P. Jüptner, “Methods of digital holography: a comparison,” Proc. SPIE3098, 224–233 (1997).
[CrossRef]

Other

U. Schnars and W. Jueptner, Digital Holography (Springer, 2005).

B. M. Hennelly, D. P. Kelly, D. S. Monaghan, and N. Pandey, “Zoom Algorithms for Digital Holography,” in Information Optics and Photonics: Algorithms, Systems, and Applications, T. Fournel and B. Javidi, eds. (Springer, 2010), pp. 187–204.

W. T. Rhodes, “Light tubes, Wigner diagrams and optical signal propagation simulation,” in Optical Information Processing: A Tribute to Adolf Lohmann, H. J. Caulfield, ed. (SPIE Press, 2002), pp. 343–356.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time signal processing (Prentice Hall, 1999).

M. Frigo and S. G. Johnson, “The FFTW web page,” (2007), http://www.fftw.org/ .

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Figures (6)

Fig. 1
Fig. 1

Generation of g(m) from f(m): (a) example function f(m); (b) g(m) generated from the addition of shifted replicas of f(m).

Fig. 2
Fig. 2

Samples of the FT of f(m) = [1,2,3,4,4,4,3,2,1] (0≤m≤8) obtained by using the proposed down-sampling method with Ns = 7 (Ns<Ni), the 1D DFT of a truncated input sequence with Ns = 7 (Ns<Ni Trunc), the 1D DFT of the original input sequence (Ns = Ni), and the 1D DFT of a zero padded input sequence (Ns>>Ni): (a) magnitude of the frequency spectrum; (b) phase of the frequency spectrum.

Fig. 3
Fig. 3

Interference patterns: (a) H1 from a synthetic spoke target as object; (b) H2 from a real USAF test target as object.

Fig. 4
Fig. 4

Sampled in-focus real-images for Nξ = Nx and Nη = Ny: (a) from H1; (b) from H2.

Fig. 5
Fig. 5

Sampled in-focus real-images using truncated interference patterns for Δξ = Δη = 51.13 μm: (a) from truncated H1 with Nξ = Nη = 276; (b) from truncated H2 with Nξ = Nη = 597.

Fig. 6
Fig. 6

Sampled in-focus real-images using the proposed down-sampling method for Δξ = Δη = 51.13 μm: (a) from H1 with Nξ = Nη = 276; (b) from H2 with Nξ = Nη = 597.

Equations (13)

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Γ(k,l)=p(k,l)×DF T 2D { f(m,n) },
f(m,n)= h r (m,n)×exp[ iπ( m N x 2 +n N y 2 ) ]×exp{ i π λd [ ( m N x 2 ) 2 Δ x 2 + ( n N y 2 ) 2 Δ y 2 ] };
p(k,l)= i λd exp( i 2π λ d )×exp{ i π λd [ ( k N ξ 2 ) 2 Δ ξ 2 + ( l N η 2 ) 2 Δ η 2 ] }.
Δ ξ = λd N ξ Δ x = λd N x Δ x ; Δ η = λd N η Δ y = λd N y Δ y .
N ξ = λd Δ ξ Δ x ; N η = λd Δ η Δ y .
FT{f(m)} | ω=(2π/ N s )k = m=0 N i 1 f(m)exp( iωm ) | ω=(2π/ N s )k = m=0 N s 1 g(m)exp( i2π km N s ) for 0k N s 1 =DFT{g(m)},
g(m)= r= f(m+r N s ) for 0m N s 1
g(m)= r=0 ( N i 1) N s m N s f(m+r N s ) .
m( N i 1) N s N i 1 N s =[( N i -1) mod N s ].
g(m)={ r=0 ( N i 1) N s f(m+r N s ) for 0m[( N i -1) mod N s ] r=0 N i N s 1 f(m+r N s ) for [( N i -1) mod N s ]<m N s -1 .
FT{f(m)}= m=0 N i 1 f(m)exp [ iω(m N i 2 ) ]=exp( iω N i 2 ) m=0 N i 1 f(m)exp( iωm ) .
FT{f(m)} | ω=(2π/ N s )k = exp( iω N i 2 ) m=0 N i 1 f(m)exp( iωm ) | ω=(2π/ N s )k =exp( iπ N i N s k ) m=0 N s 1 g(m)exp( i2π km N s ) for 0kN1 =exp( iπ N i N s k )DFT{g(m)}.
Γ(k,l)=p(k,l)×DF T 2D { g(m,n) }×exp[ iπ( k N x N ξ +l N y N η ) ],

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