Abstract

A compressive Fresnel holography approach is suggested for the recovery of partially occluded objects. Reconstruction guarantees are analyzed and the effectiveness of the method is demonstrated using simulations and an experimental result showing the reconstruction of a partially occluded resolution chart.

© 2012 Optical Society of America

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References

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  1. B. Javidi, R. Ponce-Díaz, and S.-H. Hong, Opt. Lett. 31, 1106 (2006).
    [CrossRef]
  2. Y. S. Hwang, S.-H. Hong, and B. Javidi, J. Display Tech. 3, 64 (2007).
    [CrossRef]
  3. N. Joshi, S. Avidan, W. Matusik, and D. J. Kriegman, in IEEE 11th International Conference on Computer Vision (2007).
  4. M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, ACM Trans. Graphics 23, 825 (2004).
  5. J. Maycock, C. P. McElhinney, B. M. Hennelly, T. J. Naughton, J. B. McDonald, and B. Javidi, Appl. Opt. 45, 2975 (2006).
    [CrossRef]
  6. Y. Rivenson, A. Stern, and B. Javidi, J. Display Tech. 6, 506 (2010).
    [CrossRef]
  7. M. M. Marim, M. Atlan, E. Angelini, and J.-C. Olivo-Marin, Opt. Lett. 35, 871 (2010).
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  8. Y. Rivenson, A. Stern, and J. Rosen, Opt. Express 19, 6109 (2011).
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  11. T. Kreis, Handbook of Holographic Interferometry, 1st ed. (Wiley-VCH, 2004), Chap. 3.
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    [CrossRef]
  14. L. Yaroslavsky, in Advances in Signal Transforms: Theory and Applications, J. Astola and L. Yaroslavsky, eds., EURASIP Book Series on Signal Processing and Communication (2007).
  15. Available at http://www.cs.ubc.ca/labs/scl/sparco .

2011 (3)

2010 (3)

2007 (2)

Y. S. Hwang, S.-H. Hong, and B. Javidi, J. Display Tech. 3, 64 (2007).
[CrossRef]

M. Lustig, D. L. Donoho, and J. M. Pauly, Magn. Reson. Med. 58, 1182 (2007).
[CrossRef]

2006 (2)

2004 (1)

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, ACM Trans. Graphics 23, 825 (2004).

Aguet, F.

Angelini, E.

Atlan, M.

Avidan, S.

N. Joshi, S. Avidan, W. Matusik, and D. J. Kriegman, in IEEE 11th International Conference on Computer Vision (2007).

Bolas, M.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, ACM Trans. Graphics 23, 825 (2004).

Bourquard, A.

Brady, D. J.

Chen, B.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, ACM Trans. Graphics 23, 825 (2004).

Donoho, D. L.

M. Lustig, D. L. Donoho, and J. M. Pauly, Magn. Reson. Med. 58, 1182 (2007).
[CrossRef]

Hennelly, B. M.

Hong, S.-H.

Y. S. Hwang, S.-H. Hong, and B. Javidi, J. Display Tech. 3, 64 (2007).
[CrossRef]

B. Javidi, R. Ponce-Díaz, and S.-H. Hong, Opt. Lett. 31, 1106 (2006).
[CrossRef]

Horowitz, M.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, ACM Trans. Graphics 23, 825 (2004).

Hwang, Y. S.

Y. S. Hwang, S.-H. Hong, and B. Javidi, J. Display Tech. 3, 64 (2007).
[CrossRef]

Javidi, B.

Joshi, N.

N. Joshi, S. Avidan, W. Matusik, and D. J. Kriegman, in IEEE 11th International Conference on Computer Vision (2007).

Kreis, T.

T. Kreis, Handbook of Holographic Interferometry, 1st ed. (Wiley-VCH, 2004), Chap. 3.

Kriegman, D. J.

N. Joshi, S. Avidan, W. Matusik, and D. J. Kriegman, in IEEE 11th International Conference on Computer Vision (2007).

Levoy, M.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, ACM Trans. Graphics 23, 825 (2004).

Lim, S.

Lustig, M.

M. Lustig, D. L. Donoho, and J. M. Pauly, Magn. Reson. Med. 58, 1182 (2007).
[CrossRef]

Marim, M. M.

Marks, D. L.

Matusik, W.

N. Joshi, S. Avidan, W. Matusik, and D. J. Kriegman, in IEEE 11th International Conference on Computer Vision (2007).

Maycock, J.

McDonald, J. B.

McDowall, I.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, ACM Trans. Graphics 23, 825 (2004).

McElhinney, C. P.

Naughton, T. J.

Olivo-Marin, J.-C.

Pauly, J. M.

M. Lustig, D. L. Donoho, and J. M. Pauly, Magn. Reson. Med. 58, 1182 (2007).
[CrossRef]

Ponce-Díaz, R.

Rivenson, Y.

Rosen, J.

Stern, A.

Unser, M.

Vaish, V.

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, ACM Trans. Graphics 23, 825 (2004).

Yaroslavsky, L.

L. Yaroslavsky, in Advances in Signal Transforms: Theory and Applications, J. Astola and L. Yaroslavsky, eds., EURASIP Book Series on Signal Processing and Communication (2007).

ACM Trans. Graphics (1)

M. Levoy, B. Chen, V. Vaish, M. Horowitz, I. McDowall, and M. Bolas, ACM Trans. Graphics 23, 825 (2004).

Appl. Opt. (2)

J. Display Tech. (2)

Y. Rivenson, A. Stern, and B. Javidi, J. Display Tech. 6, 506 (2010).
[CrossRef]

Y. S. Hwang, S.-H. Hong, and B. Javidi, J. Display Tech. 3, 64 (2007).
[CrossRef]

Magn. Reson. Med. (1)

M. Lustig, D. L. Donoho, and J. M. Pauly, Magn. Reson. Med. 58, 1182 (2007).
[CrossRef]

Opt. Express (2)

Opt. Lett. (3)

Other (4)

L. Yaroslavsky, in Advances in Signal Transforms: Theory and Applications, J. Astola and L. Yaroslavsky, eds., EURASIP Book Series on Signal Processing and Communication (2007).

Available at http://www.cs.ubc.ca/labs/scl/sparco .

N. Joshi, S. Avidan, W. Matusik, and D. J. Kriegman, in IEEE 11th International Conference on Computer Vision (2007).

T. Kreis, Handbook of Holographic Interferometry, 1st ed. (Wiley-VCH, 2004), Chap. 3.

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Figures (4)

Fig. 1.
Fig. 1.

A schematic setup for a partially occluded object’s wavefield acquisition.

Fig. 2.
Fig. 2.

(a), (c) Partially occluding planes. (b), (d) Corresponding cross sections of |ϕm,ϕl*|/{ϕm2ϕl2}, of (a) and (c), respectively. μFF is marked with a dotted line.

Fig. 3.
Fig. 3.

Reconstruction PSNR as a function of occluding percentage for occluding planes with different square sizes, which yield different μFF.

Fig. 4.
Fig. 4.

(a) Reconstruction of the USAF chart without the occluding plane, using standard Fresnel backpropagation, for reference. (b) Focusing on the occluding plane, from which the occluding function was extracted from. (c) Same as (a), but this time the occluding plane distorts the reconstructed object plane. (d) Same experiment as (c), but applying the CS framework to reconstruct the object plane. (e), (f), and (g) zoom in on the highlighted parts of (a), (c), and (d) respectively. (h), (i), and (j) correspond to the highlighted cross sections of (e)–(g).

Equations (10)

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uccd(x)={uin*exp[jπx2/(λz1)]λz1p(x)}*exp[jπx2/(λz2)]λz2.
uccd=Φuin,
μ=μ(Φ)=maxml|ϕm,ϕl*|/{ϕm2ϕl2},
μ=maxml|ϕ˜m,ϕ˜l*|/{ϕ˜m2ϕ˜l2},
ϕ˜l=δ(xl)*exp[jπx2/(λz1)]/λz1p(x).
ϕ˜l(qΔxz1)=1λz1p(qΔxz1)ejπλz1(lΔxoqΔxz1)2=1λz1p(qΔxz1)ejπλz1(lΔxo)2ejπλz1(qΔxz1)2ej2πλz1lqΔxoΔxz1,
μFF=maxml1λz1|q=1N|p(qΔxz1)|2ej2πλz1ΔxoΔxz1q(ml)|1λ2z12q=1N|p(qΔxz1)|2q=1N|p(qΔxz1)|2=maxml|P͡(mlλz1ΔxoΔxz1)P͡(mlλz1ΔxoΔxz1)|/q=1N|p(qΔxz1)|2.
ϕ˜l=p(qΔxo)F1{exp{jπλz1(nΔυ)2}F{δ[(sl)Δxo]}}ejπ4Δxoλz1p(qΔxo)ejπλz1Δxo2(ql)2rect[(ql)NΔxo2λz1(N1)],
μNFmaxml|Δxo2λz1q=1N|p(qΔxo)|2rect[(ql)NΔxo2(N1)λz1]×rect[(qm)NΔxo2(N1)λz1]ej2πΔxo2λz1q(ml)|/ϕ˜m2ϕl2.
minΨuin1+αTV(uin)s.t.uoutΦuin2<ε,

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