Abstract

A novel fast frequency-based method to estimate the focus distance of digital hologram for a single object is proposed. The focus distance is computed by analyzing the distribution of intersections of smoothed-rays. The smoothed-rays are determined by the directions of energy flow which are computed from local spatial frequency spectrum based on the windowed Fourier transform. So our method uses only the intrinsic frequency information of the optical field on the hologram and therefore does not require any sequential numerical reconstructions and focus detection techniques of conventional photography, both of which are the essential parts in previous methods. To show the effectiveness of our method, numerical results and analysis are presented as well.

© 2014 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  6. P. Memmolo, C. Distante, M. Paturzo, A. Finizio, P. Ferraro, and B. Javidi, “Automatic focusing in digital holog-raphy and its application to stretched holograms,” Opt. Lett. 36, 1945–1947 (2011).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
  10. F. Hlawatsch and G. Boudreauz-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9, 21–67 (1992).
    [Crossref]
  11. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley & Sons, 2000).
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    [Crossref]

2011 (1)

2010 (1)

2009 (1)

2008 (1)

2006 (1)

2004 (3)

1992 (1)

F. Hlawatsch and G. Boudreauz-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9, 21–67 (1992).
[Crossref]

1989 (2)

K. Toraichi, M. Kamada, S. Itahashi, and R. Mori, “Window functions represented by b-spline functions,” IEEE T. Acoust. Speech 37, 145–147 (1989).
[Crossref]

J. Gillespie and R. A. King, “The use of self-entropy as a focus measure in digital holography,” Pattern Recogn. Lett. 9, 19–25 (1989).
[Crossref]

Boudreauz-Bartels, G.

F. Hlawatsch and G. Boudreauz-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9, 21–67 (1992).
[Crossref]

Callens, N.

Distante, C.

Dubois, F.

Ferraro, P.

Finizio, A.

Gillespie, J.

J. Gillespie and R. A. King, “The use of self-entropy as a focus measure in digital holography,” Pattern Recogn. Lett. 9, 19–25 (1989).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2005), 3rd ed.

Hlawatsch, F.

F. Hlawatsch and G. Boudreauz-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9, 21–67 (1992).
[Crossref]

Itahashi, S.

K. Toraichi, M. Kamada, S. Itahashi, and R. Mori, “Window functions represented by b-spline functions,” IEEE T. Acoust. Speech 37, 145–147 (1989).
[Crossref]

Itoh, M.

Javidi, B.

Jin, H.

L. Ma, H. Wang, Y. Li, and H. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A: Pure Appl. Opt. 6, 396–400 (2004).
[Crossref]

Kamada, M.

K. Toraichi, M. Kamada, S. Itahashi, and R. Mori, “Window functions represented by b-spline functions,” IEEE T. Acoust. Speech 37, 145–147 (1989).
[Crossref]

Kemao, Q.

Kim, T.

King, R. A.

J. Gillespie and R. A. King, “The use of self-entropy as a focus measure in digital holography,” Pattern Recogn. Lett. 9, 19–25 (1989).
[Crossref]

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley & Sons, 2000).

Li, Y.

L. Ma, H. Wang, Y. Li, and H. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A: Pure Appl. Opt. 6, 396–400 (2004).
[Crossref]

Liebling, M.

Ma, L.

L. Ma, H. Wang, Y. Li, and H. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A: Pure Appl. Opt. 6, 396–400 (2004).
[Crossref]

Memmolo, P.

Mori, R.

K. Toraichi, M. Kamada, S. Itahashi, and R. Mori, “Window functions represented by b-spline functions,” IEEE T. Acoust. Speech 37, 145–147 (1989).
[Crossref]

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley & Sons, 2000).

Paturzo, M.

Poon, T.-C.

Rajshekhar, G.

Rastogi, P.

Schockaert, C.

Tachiki, M.

Toraichi, K.

K. Toraichi, M. Kamada, S. Itahashi, and R. Mori, “Window functions represented by b-spline functions,” IEEE T. Acoust. Speech 37, 145–147 (1989).
[Crossref]

Unser, M.

Wang, H.

L. Ma, H. Wang, Y. Li, and H. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A: Pure Appl. Opt. 6, 396–400 (2004).
[Crossref]

Yatagai, T.

Yourassowsky, C.

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley & Sons, 2000).

Appl. Opt. (3)

IEEE Signal Process. Mag. (1)

F. Hlawatsch and G. Boudreauz-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9, 21–67 (1992).
[Crossref]

IEEE T. Acoust. Speech (1)

K. Toraichi, M. Kamada, S. Itahashi, and R. Mori, “Window functions represented by b-spline functions,” IEEE T. Acoust. Speech 37, 145–147 (1989).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

L. Ma, H. Wang, Y. Li, and H. Jin, “Numerical reconstruction of digital holograms for three-dimensional shape measurement,” J. Opt. A: Pure Appl. Opt. 6, 396–400 (2004).
[Crossref]

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

Opt. Express (1)

Opt. Lett. (1)

Pattern Recogn. Lett. (1)

J. Gillespie and R. A. King, “The use of self-entropy as a focus measure in digital holography,” Pattern Recogn. Lett. 9, 19–25 (1989).
[Crossref]

Other (2)

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley & Sons, 2000).

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2005), 3rd ed.

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Figures (8)

Fig. 1
Fig. 1 Focus estimation error for spherical wave with the source at (a) 0.3 ≤ z0 ≤ 20 and (b) short distances z0 ≤ 0.4.
Fig. 2
Fig. 2 The wave profiles, (a) the amplitude and (b) the phase, for a hologram of for a planar cross-shaped object with the focus distance z0 = 1 and (c) the numerical reconstruction at the estimated focus distance = 1.00736.
Fig. 3
Fig. 3 The intersection density for a planar cross-shaped object.
Fig. 4
Fig. 4 Estimation error comparison between the smoothed-rays and the non-smoothed rays for planar cross-shaped objects of size (a) 10mm and (b) 20mm.
Fig. 5
Fig. 5 Projected rays to the hologram plane for the planar cross-shaped object of size 20mm with the Z-depth at z = 1.3: (a) smoothed ray and (b) non-smoothed ray.
Fig. 6
Fig. 6 Projected rays to the hologram plane for the planar cross-shaped object of size 10mm with the Z-depth at z = 1.3: (a) smoothed ray and (b) non-smoothed ray.
Fig. 7
Fig. 7 The wave profiles, (a) the amplitude and (b) the phase, for a phase-shifting hologram of a real coin and (c) the numerical reconstruction result at the estimated focus distance = 0.356507(brightness enhanced).
Fig. 8
Fig. 8 Intersection density for a phase-shifting hologram of a real coin.

Equations (5)

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W F f ( x , y , α , β ) = f ( x , y ) W ( x x , y y ) exp [ j 2 π ( α x + β y ) ] d x d y ,
d ( x , y ) = w 1 d ˜ 1 + + w n d ˜ n w 1 d ˜ 1 + + w n d ˜ n ,
A = [ 3 a k i 2 a k i b k i a k i c k i a k i b k i 3 b k i 2 b k i c k i a k i c k i b k i c k i 3 c k i 2 ] and b = [ ( x k i x k i a k i 2 y k i a k i b k i ) ( y k i x k i a k i b k i y k i b k i 2 ) ( x k i a k i c k i y k i b k i c k i ) ] ,
z ˜ = i = 1 M d i c i i = 1 M d i .
W ˜ ( r ) = 10 7 π { 1 3 2 r 2 + 3 4 r 3 : r 1 1 4 ( 2 r ) 4 : 1 < r 2 . 0 : r > 2

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