Abstract

Multiple view projection holography is a method to obtain a digital hologram by recording different views of a 3D scene with a conventional digital camera. Those views are digitally manipulated in order to create the digital hologram. The method requires a simple setup and operates under white light illuminating conditions. The multiple views are often generated by a camera translation, which usually involves a scanning effort. In this work we apply a compressive sensing approach to the multiple view projection holography acquisition process and demonstrate that the 3D scene can be accurately reconstructed from the highly subsampled generated Fourier hologram. It is also shown that the compressive sensing approach, combined with an appropriate system model, yields improved sectioning of the planes of different depths.

© 2011 OSA

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References

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2010 (7)

2009 (4)

2008 (1)

2007 (5)

2006 (2)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[CrossRef]

2003 (2)

2001 (1)

2000 (1)

Abookasis, D.

Aguet, F.

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new twist: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16(12), 2992–3004 (2007).
[CrossRef] [PubMed]

Bourquard, A.

Brady, D. J.

Brooker, G.

Candès, E.

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[CrossRef]

Chen, N.

Choi, K.

Cull, C. F.

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[CrossRef]

Eldar, Y. C.

Figueiredo, M. A. T.

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new twist: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16(12), 2992–3004 (2007).
[CrossRef] [PubMed]

Gazit, S.

Hahn, J.

Horisaki, R.

Indebetouw, G.

Itoh, M.

Javidi, B.

Y. Rivenson, A. Stern, and B. Javidi, “Single exposure super-resolution compressive imaging by double phase encoding,” Opt. Express 18(14), 15094–15103 (2010).
[CrossRef] [PubMed]

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel Holography,” Disp. Tech, Journalism 506–509(10), 6 (2010).

Katz, B.

Kim, N.

Kim, T.

Klysubun, P.

Lam, E. Y.

Li, Y.

Lim, S.

Mait, J. N.

Marks, D. L.

Mattheiss, M.

Park, J.-H.

Poon, T.-C.

Rivenson, Y.

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel Holography,” Disp. Tech, Journalism 506–509(10), 6 (2010).

Y. Rivenson, A. Stern, and B. Javidi, “Single exposure super-resolution compressive imaging by double phase encoding,” Opt. Express 18(14), 15094–15103 (2010).
[CrossRef] [PubMed]

Y. Rivenson and A. Stern, “Compressed imaging with separable sensing operator,” IEEE Signal Process. Lett. 16(6), 449–452 (2009).
[CrossRef]

Romberg, J.

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[CrossRef]

Rosen, J.

Sando, Y.

Schulz, T. J.

Segev, M.

Shaked, N. T.

Stern, A.

Szameit, A.

Tao, T.

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[CrossRef]

Unser, M.

Wikner, D. A.

Yatagai, T.

Zhang, X.

Appl. Opt. (5)

Disp. Tech, Journalism (1)

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel Holography,” Disp. Tech, Journalism 506–509(10), 6 (2010).

IEEE Signal Process. Lett. (1)

Y. Rivenson and A. Stern, “Compressed imaging with separable sensing operator,” IEEE Signal Process. Lett. 16(6), 449–452 (2009).
[CrossRef]

IEEE Trans. Image Process. (1)

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new twist: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16(12), 2992–3004 (2007).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory (2)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[CrossRef]

E. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Express (7)

Opt. Lett. (3)

Other (4)

M. Lustig, “Sparse MRI,” Ph.D. dissertation, Dept. Elect. Eng., Stanford Univ., Palo Alto, CA, 2008.

http://sites.google.com/site/igorcarron2/compressedsensinghardware .

J. W. Goodman, Introduction to Fourier optics, 3rd Ed., (Roberts and Company Publishers, 2005).

Y. Rivenson, A. Stern, and J. Rosen, “Compressive Sensing Approach for Reducing the Number of Exposures in Multiple View Projection Holography,” in Frontiers in Optics, OSA Technical Digest (CD) (Optical Society of America, 2010), paper FThM2.

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Figures (4)

Fig. 1
Fig. 1

Illustration of CMVP hologram acquisition. Acquisition of only ≈KlogNx projections results in a heavily undersampled Fourier hologram. Each sample in the hologram plane corresponds to a nonuniformly randomly picked projection.

Fig. 2
Fig. 2

Reconstruction examples of the B and U planes of simulated data. (a) Reconstruction of the B plane from 100% of the projections. (b) CS reconstruction of the B plane from 6% of the projections. (c) Reconstruction of the U plane from 100% of the projections. (d) CS reconstruction of the U plane from 6% of the projections.

Fig. 3
Fig. 3

CMVP reconstruction results of experimental data. (a) Reconstruction from 100% of the projections. (b) Reconstruction from 25% of the projections using the CS framework.

Fig. 4
Fig. 4

Applying CMVP tomographic sectioning to experimental data. (a) Reconstruction from 100% of the projections. (b) Compressive holography approach applied to CMVP, with only 25% of the nominal number of projections.

Equations (12)

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u i ( x , y ) = 1 { h ( υ x , υ y ) exp [ j π λ z i ( υ x 2 + υ y 2 ) ] } ,
u i ( p , q ) = m n h ( m , n ) exp { j π λ z i [ ( Δ υ x m ) 2 + ( Δ υ y n ) 2 ] } exp { j 2 π ( m p N x + n q N y ) } ,
u i = F 1 Q - λ 2 z i h ,
min { u i F Q 2 z i h M 2 2 + γ Ψ i u i 1 } ,
h ( υ x , υ y ) = i = 1 N z exp [ j π λ z i ( υ x 2 + υ y 2 ) ] { u i } = i = 1 N z { u i exp [ j π λ z i ( x 2 + y 2 ) ] } ,
h ( m , n ) = i = 1 N z Q λ 2 z i F u i .
h = [ Q λ 2 z 1 F ; ... ; Q λ 2 z N z F ] [ u 1 ; ... ; u N Z ] T = Φ u T .
min { h M Φ u T 2 2 + τ u T V } ,
u T V = l i , j ( u i + 1 , j , l u i , j , l ) 2 + ( u i , j + 1 , l u i , j , l ) 2
Δ x = max { Δ x o p t i c a l , Δ x g e o m e t r i c a l , Δ x h o log r a m } = max { λ z o A , Δ s M T , N p Δ s L 2 + z o 2 b L M T }
Δ z M A = z 0 L Δ x .
Δ z g a i n N u m b e r o f p r o j e c t i o n s = L / A K log N = N K log N .

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