Abstract

Inline digital holograms are classically reconstructed using linear operators to model diffraction. It has long been recognized that such reconstruction operators do not invert the hologram formation operator. Classical linear reconstructions yield images with artifacts such as distortions near the field-of-view boundaries or twin images. When objects located at different depths are reconstructed from a hologram, in-focus and out-of-focus images of all objects superimpose upon each other. Additional processing, such as maximum-of-focus detection, is thus unavoidable for any successful use of the reconstructed volume. In this Letter, we consider inverting the hologram formation model in a Bayesian framework. We suggest the use of a sparsity-promoting prior, verified in many inline holography applications, and present a simple iterative algorithm for 3D object reconstruction under sparsity and positivity constraints. Preliminary results with both simulated and experimental holograms are highly promising.

© 2009 Optical Society of America

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References

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2007 (2)

2006 (1)

D. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

2004 (3)

2003 (1)

M. Liebling, T. Blu, and M. Unser, IEEE Trans. Image Process. 12, 29 (2003).
[CrossRef]

2000 (1)

C. Buraga, S. Coëtmellec, D. Lebrun, and C. Özkul, Opt. Lasers Eng. 33, 409 (2000).
[CrossRef]

1997 (1)

T. Kreis, M. Adams, and W. Jueptner, Proc. SPIE 3098, 224 (1997).
[CrossRef]

1993 (2)

H. Ozaktas and D. Mendlovic, J. Opt. Soc. Am. A 10, 2522 (1993).
[CrossRef]

S. Mallat and Z. Zhang, IEEE Trans. Inf. Theory 41, 3397 (1993).

1987 (1)

L. Onural and P. Scott, Opt. Eng. (Bellingham) 26, 1124 (1987).

Adams, M.

T. Kreis, M. Adams, and W. Jueptner, Proc. SPIE 3098, 224 (1997).
[CrossRef]

Blu, T.

M. Liebling, T. Blu, and M. Unser, IEEE Trans. Image Process. 12, 29 (2003).
[CrossRef]

Buraga, C.

C. Buraga, S. Coëtmellec, D. Lebrun, and C. Özkul, Opt. Lasers Eng. 33, 409 (2000).
[CrossRef]

Coëtmellec, S.

C. Buraga, S. Coëtmellec, D. Lebrun, and C. Özkul, Opt. Lasers Eng. 33, 409 (2000).
[CrossRef]

Daubechies, I.

I. Daubechies, M. Defrise, and C. De Mol, Commun. Pure Appl. Math. 57, 1413 (2004).
[CrossRef]

De Mol, C.

I. Daubechies, M. Defrise, and C. De Mol, Commun. Pure Appl. Math. 57, 1413 (2004).
[CrossRef]

Defrise, M.

I. Daubechies, M. Defrise, and C. De Mol, Commun. Pure Appl. Math. 57, 1413 (2004).
[CrossRef]

Denis, L.

Donoho, D.

D. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

Fessler, J.

Fournier, C.

Goepfert, C.

Jueptner, W.

T. Kreis, M. Adams, and W. Jueptner, Proc. SPIE 3098, 224 (1997).
[CrossRef]

Kreis, T.

T. Kreis, M. Adams, and W. Jueptner, Proc. SPIE 3098, 224 (1997).
[CrossRef]

Lebrun, D.

C. Buraga, S. Coëtmellec, D. Lebrun, and C. Özkul, Opt. Lasers Eng. 33, 409 (2000).
[CrossRef]

Liebling, M.

M. Liebling and M. Unser, J. Opt. Soc. Am. A 21, 2424 (2004).
[CrossRef]

M. Liebling, T. Blu, and M. Unser, IEEE Trans. Image Process. 12, 29 (2003).
[CrossRef]

Mallat, S.

S. Mallat and Z. Zhang, IEEE Trans. Inf. Theory 41, 3397 (1993).

Mendlovic, D.

Onural, L.

L. Onural and P. Scott, Opt. Eng. (Bellingham) 26, 1124 (1987).

Ozaktas, H.

Özkul, C.

C. Buraga, S. Coëtmellec, D. Lebrun, and C. Özkul, Opt. Lasers Eng. 33, 409 (2000).
[CrossRef]

Scott, P.

L. Onural and P. Scott, Opt. Eng. (Bellingham) 26, 1124 (1987).

Sotthivirat, S.

Soulez, F.

Thiébaut, É.

Unser, M.

M. Liebling and M. Unser, J. Opt. Soc. Am. A 21, 2424 (2004).
[CrossRef]

M. Liebling, T. Blu, and M. Unser, IEEE Trans. Image Process. 12, 29 (2003).
[CrossRef]

Zhang, Z.

S. Mallat and Z. Zhang, IEEE Trans. Inf. Theory 41, 3397 (1993).

Commun. Pure Appl. Math. (1)

I. Daubechies, M. Defrise, and C. De Mol, Commun. Pure Appl. Math. 57, 1413 (2004).
[CrossRef]

IEEE Trans. Image Process. (1)

M. Liebling, T. Blu, and M. Unser, IEEE Trans. Image Process. 12, 29 (2003).
[CrossRef]

IEEE Trans. Inf. Theory (2)

S. Mallat and Z. Zhang, IEEE Trans. Inf. Theory 41, 3397 (1993).

D. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Eng. (Bellingham) (1)

L. Onural and P. Scott, Opt. Eng. (Bellingham) 26, 1124 (1987).

Opt. Lasers Eng. (1)

C. Buraga, S. Coëtmellec, D. Lebrun, and C. Özkul, Opt. Lasers Eng. 33, 409 (2000).
[CrossRef]

Proc. SPIE (1)

T. Kreis, M. Adams, and W. Jueptner, Proc. SPIE 3098, 224 (1997).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Reconstruction of a simulated hologram (b) of two opaque disks located respectively at 100 mm and 110 mm from the hologram plane (a). (c), (d) conventional reconstruction ϑ ̂ adj . (e), (f) sparse reconstruction ϑ ̂ sparse (200 iterations). Superimposed is the transmittance profile over the line between the ▶ ◀ signs.

Fig. 2
Fig. 2

Reconstruction of an experimental hologram of a plane object. (a), (b) The hologram and a truncation with three-fourths of the pixels missing. (c), (d) Conventional reconstruction ϑ ̂ adj from the (c) full or (d) truncated hologram. (e), (f) Sparse reconstruction (50 iterations) ϑ ̂ sparse from the (e) full or (f) truncated hologram.

Equations (4)

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d = c 1 H ϑ + ϵ .
ϑ ̂ sparse = arg min ϑ , c 1 / 2 c 1 H ϑ d 2 2 + τ ϑ 1 ,
ϑ ̂ sparse = arg min ϑ 1 / 2 H ¯ ϑ d ¯ 2 2 + τ ϑ 1 ,
ϑ ̂ ( k + 1 ) = S τ + [ ϑ ̂ ( k ) + H ¯ ( d ¯ H ¯ ϑ ̂ ( k ) ) ] ,

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