Abstract

Three-dimensional (3D) object tomography from a two-dimensional recorded hologram is a process of high-dimensional data inference from undersampled data. As such, recently, techniques developed in the field of compressive sensing and sparse representation have been applied for this task. While many applications of compressive sensing for tomography from digital holograms have been demonstrated in the past few years, the fundamental limits involved have not yet been addressed. We formulate the guarantees for compressive sensing-based recovery of 3D objects and show their relation to the physical attributes of the recording setup.

© 2013 Optical Society of America

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References

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2013

2011

2010

2009

2008

E. Candes and M. Wakin, IEEE Signal Process. Mag. 25, 21 (2008).
[CrossRef]

2007

M. Elad, IEEE Trans. Signal Process. 55, 5695 (2007).
[CrossRef]

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
[CrossRef]

Banerjee, P.

Barbastathis, G.

L. Tian, Y. Liu, and G. Barbastathis, in Biomedical Optics and 3-D Imaging, OSA Technical Digest (Optical Society of America, 2012), paper DW4C.3.

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
[CrossRef]

Brady, D.

Brady, D. J.

Bruckstein, A. M.

A. M. Bruckstein, D. L. Donoho, and M. Elad, SIAM Rev. 51, 34 (2009).
[CrossRef]

Candes, E.

E. Candes and M. Wakin, IEEE Signal Process. Mag. 25, 21 (2008).
[CrossRef]

Choi, K.

Coskun, A. F.

Denis, L.

Donoho, D. L.

A. M. Bruckstein, D. L. Donoho, and M. Elad, SIAM Rev. 51, 34 (2009).
[CrossRef]

Elad, M.

A. M. Bruckstein, D. L. Donoho, and M. Elad, SIAM Rev. 51, 34 (2009).
[CrossRef]

M. Elad, IEEE Trans. Signal Process. 55, 5695 (2007).
[CrossRef]

Figueiredo, M. A. T.

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
[CrossRef]

Fournier, C.

Horisaki, R.

Javidi, B.

Kreis, T.

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

Lam, E. Y.

Lim, S.

Liu, Y.

L. Tian, Y. Liu, and G. Barbastathis, in Biomedical Optics and 3-D Imaging, OSA Technical Digest (Optical Society of America, 2012), paper DW4C.3.

Lorenz, D.

Marks, D.

Marks, D. L.

Nehmetallah, G.

Ozcan, A.

Rivenson, Y.

Rosen, J.

Sencan, I.

Stern, A.

Su, T.-W.

Thiébaut, E.

Tian, L.

L. Tian, Y. Liu, and G. Barbastathis, in Biomedical Optics and 3-D Imaging, OSA Technical Digest (Optical Society of America, 2012), paper DW4C.3.

Trede, D.

Wakin, M.

E. Candes and M. Wakin, IEEE Signal Process. Mag. 25, 21 (2008).
[CrossRef]

Williams, L.

Zhang, X.

Appl. Opt.

IEEE Signal Process. Mag.

E. Candes and M. Wakin, IEEE Signal Process. Mag. 25, 21 (2008).
[CrossRef]

IEEE Trans. Image Process.

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
[CrossRef]

IEEE Trans. Signal Process.

M. Elad, IEEE Trans. Signal Process. 55, 5695 (2007).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Express

Opt. Lett.

SIAM Rev.

A. M. Bruckstein, D. L. Donoho, and M. Elad, SIAM Rev. 51, 34 (2009).
[CrossRef]

Other

L. Tian, Y. Liu, and G. Barbastathis, in Biomedical Optics and 3-D Imaging, OSA Technical Digest (Optical Society of America, 2012), paper DW4C.3.

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

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Figures (3)

Fig. 1.
Fig. 1.

Schematic description of the framework. A plane wave illuminating a volume (shaded gray). The wavefront scattered from the different particles is holographically recorded on a CCD.

Fig. 2.
Fig. 2.

Gram matrix for the 3D–2D forward Fresnel sensing operator. The partition to nine submatrices, marked by dashed lines, shows the coherence between the point spread function of two different planes.

Fig. 3.
Fig. 3.

Simulation results showing the normalized number of reconstructed 3D object’s particles ( MSE < 10 8 ) as a function of number of objects planes, for a constant volume length, L z , given a sensor with M pixels. The theoretical exact reconstruction guarantee according to Eq. (8) is placed in the inset.

Equations (9)

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g = Φ f ,
μ = max k l | ϕ k , ϕ l | / { ϕ k 2 ϕ l 2 } ,
S 0.5 ( 1 + 1 / μ ) .
g ( u Δ x , v Δ y ) = r = 1 N z F 2 D 1 { e j π λ r Δ z [ ( Δ υ x m ) 2 + ( Δ υ y n ) 2 ] e j 2 π λ r Δ z × F 2 D [ f ( p Δ x , q Δ y ; r Δ z ) ] } ,
g = [ H Δ z ; ; H N z Δ z ] [ f Δ z ; ; f N Z Δ z ] = Φ f , H r Δ z = F 2 D 1 Q r Δ z F 2 D .
H ˜ k Δ z * H ˜ l Δ z = F 2 D 1 Q k Δ z * F 2 D F 2 D 1 Q l Δ z F 2 D = F 2 D 1 Q ( l k ) Δ z F 2 D .
μ = max k l | H ˜ k Δ z * H ˜ l Δ z | = max k l | F 2 D 1 Q ( k l ) Δ z F 2 D | max k l { Δ 2 λ ( l k ) Δ z } = Δ 2 λ Δ z .
S 0.5 ( 1 + λ Δ z / Δ 2 ) .
f ^ = arg min f g Φ f 2 + τ f 1 .

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