Abstract

Optical scanning holography (OSH) records a three-dimensional object into a two-dimensional hologram through two-dimensional optical scanning. The recovery of sectional images from the hologram, termed as an inverse problem, has been previously implemented by conventional methods as well as the use of l2 norm. However, conventional methods require time consuming processing of section by section without eliminating the defocus noise and the l2norm method often suffers from the drawback of over-smoothing. Moreover, these methods require the whole hologram data (real and imaginary parts) to eliminate the twin image noise, whose computation complexity and the sophisticated post-processing are far from desirable. To handle these difficulties, an adaptively iterative shrinkage-thresholding (AIST) algorithm, characterized by fast computation and adaptive iteration, is proposed in this paper. Using only a half hologram data, the proposed method obtained satisfied on-axis reconstruction free of twin image noise. The experiments of multi-planar reconstruction and improvement of depth of focus further validate the feasibility and flexibility of our proposed AIST algorithm.

© 2012 OSA

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References

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    [CrossRef]
  23. A. Yang, A. Ganesh, S. Sastry, and Y. Ma, “Fast L1-minimization algorithms and an application in robust face recognition: a review,” in proceedings of IEEE International Conference on Image Processing, (Institute of Electrical and Electronics Engineers, California, 2010), pp. 1849–1852.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2011 (1)

Q. Zhang, H. Zhao, D. Chen, X. Qu, X. Chen, X. He, W. Li, Z. Hu, J. Liu, J. Liang, and J. Tian, “Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography,” Opt. Commun. 284(24), 5871–5876 (2011).
[CrossRef]

2010 (3)

2009 (4)

2008 (2)

2007 (1)

S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularizedleast squares,” IEEE J. Sel. Top. Signal Process. 1(4), 606–617 (2007).
[CrossRef]

2006 (3)

2005 (2)

2004 (1)

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57(11), 1413–1457 (2004).
[CrossRef]

2002 (2)

2001 (1)

S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43(1), 129–159 (2001).
[CrossRef]

1997 (1)

I. F. Gorodnitsky and B. D. Rao, “Sparse signal reconstruction from limited data using focuss: a re-weighted minimum norm algorithm,” IEEE Trans. Signal Process. 45(3), 600–616 (1997).
[CrossRef]

1992 (1)

1985 (1)

Bovik, A. C.

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9(3), 81–84 (2002).
[CrossRef]

Boyd, S.

S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularizedleast squares,” IEEE J. Sel. Top. Signal Process. 1(4), 606–617 (2007).
[CrossRef]

Candès, E.

E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59(8), 1207–1223 (2006).
[CrossRef]

Chan, T. F.

Chatziioannou, A. F.

Chen, D.

Q. Zhang, H. Zhao, D. Chen, X. Qu, X. Chen, X. He, W. Li, Z. Hu, J. Liu, J. Liang, and J. Tian, “Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography,” Opt. Commun. 284(24), 5871–5876 (2011).
[CrossRef]

X. He, J. Liang, X. Wang, J. Yu, X. Qu, X. Wang, Y. Hou, D. Chen, F. Liu, and J. Tian, “Sparse reconstruction for quantitative bioluminescence tomography based on the incomplete variables truncated conjugate gradient method,” Opt. Express 18(24), 24825–24841 (2010).
[CrossRef] [PubMed]

Chen, S.

S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43(1), 129–159 (2001).
[CrossRef]

Chen, X.

Q. Zhang, H. Zhao, D. Chen, X. Qu, X. Chen, X. He, W. Li, Z. Hu, J. Liu, J. Liang, and J. Tian, “Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography,” Opt. Commun. 284(24), 5871–5876 (2011).
[CrossRef]

Combettes, P.

P. Combettes and V. Wajs, “Signal recovery by proximal forward backward splitting,” Multiscale Model. Simul. 4(4), 1168–1200 (2005).
[CrossRef]

Daubechies, I.

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57(11), 1413–1457 (2004).
[CrossRef]

De Mol, C.

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57(11), 1413–1457 (2004).
[CrossRef]

Defrise, M.

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57(11), 1413–1457 (2004).
[CrossRef]

Donoho, D.

S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43(1), 129–159 (2001).
[CrossRef]

Douraghy, A.

Duncan, B. D.

El Maghnouji, A.

Feng, J.

Figueiredo, M. A.

S. J. Wright, R. D. Nowak, and M. A. Figueiredo, “Sparse Reconstruction by Separable Approximation,” IEEET, Signal Process. 57, 2479–2493 (2009).

Foster, R.

Gorinevsky, D.

S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularizedleast squares,” IEEE J. Sel. Top. Signal Process. 1(4), 606–617 (2007).
[CrossRef]

Gorodnitsky, I. F.

I. F. Gorodnitsky and B. D. Rao, “Sparse signal reconstruction from limited data using focuss: a re-weighted minimum norm algorithm,” IEEE Trans. Signal Process. 45(3), 600–616 (1997).
[CrossRef]

Han, D.

He, X.

Q. Zhang, H. Zhao, D. Chen, X. Qu, X. Chen, X. He, W. Li, Z. Hu, J. Liu, J. Liang, and J. Tian, “Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography,” Opt. Commun. 284(24), 5871–5876 (2011).
[CrossRef]

X. He, J. Liang, X. Wang, J. Yu, X. Qu, X. Wang, Y. Hou, D. Chen, F. Liu, and J. Tian, “Sparse reconstruction for quantitative bioluminescence tomography based on the incomplete variables truncated conjugate gradient method,” Opt. Express 18(24), 24825–24841 (2010).
[CrossRef] [PubMed]

Hou, Y.

Hu, Z.

Q. Zhang, H. Zhao, D. Chen, X. Qu, X. Chen, X. He, W. Li, Z. Hu, J. Liu, J. Liang, and J. Tian, “Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography,” Opt. Commun. 284(24), 5871–5876 (2011).
[CrossRef]

Huisken, J.

Indebetouw, G.

Kim, H.

Kim, S.-J.

S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularizedleast squares,” IEEE J. Sel. Top. Signal Process. 1(4), 606–617 (2007).
[CrossRef]

Kim, T.

Kim, Y. S.

Koh, K.

S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularizedleast squares,” IEEE J. Sel. Top. Signal Process. 1(4), 606–617 (2007).
[CrossRef]

Lam, E. Y.

Lee, B.

Li, W.

Q. Zhang, H. Zhao, D. Chen, X. Qu, X. Chen, X. He, W. Li, Z. Hu, J. Liu, J. Liang, and J. Tian, “Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography,” Opt. Commun. 284(24), 5871–5876 (2011).
[CrossRef]

Liang, J.

Q. Zhang, H. Zhao, D. Chen, X. Qu, X. Chen, X. He, W. Li, Z. Hu, J. Liu, J. Liang, and J. Tian, “Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography,” Opt. Commun. 284(24), 5871–5876 (2011).
[CrossRef]

X. He, J. Liang, X. Wang, J. Yu, X. Qu, X. Wang, Y. Hou, D. Chen, F. Liu, and J. Tian, “Sparse reconstruction for quantitative bioluminescence tomography based on the incomplete variables truncated conjugate gradient method,” Opt. Express 18(24), 24825–24841 (2010).
[CrossRef] [PubMed]

Liu, F.

Liu, J.

Q. Zhang, H. Zhao, D. Chen, X. Qu, X. Chen, X. He, W. Li, Z. Hu, J. Liu, J. Liang, and J. Tian, “Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography,” Opt. Commun. 284(24), 5871–5876 (2011).
[CrossRef]

Lu, Y.

Lustig, M.

S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularizedleast squares,” IEEE J. Sel. Top. Signal Process. 1(4), 606–617 (2007).
[CrossRef]

Martínez-Corral, M.

Min, S.-W.

Nowak, R. D.

S. J. Wright, R. D. Nowak, and M. A. Figueiredo, “Sparse Reconstruction by Separable Approximation,” IEEET, Signal Process. 57, 2479–2493 (2009).

Poon, T.-C.

Qin, C.

Qu, X.

Q. Zhang, H. Zhao, D. Chen, X. Qu, X. Chen, X. He, W. Li, Z. Hu, J. Liu, J. Liang, and J. Tian, “Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography,” Opt. Commun. 284(24), 5871–5876 (2011).
[CrossRef]

X. He, J. Liang, X. Wang, J. Yu, X. Qu, X. Wang, Y. Hou, D. Chen, F. Liu, and J. Tian, “Sparse reconstruction for quantitative bioluminescence tomography based on the incomplete variables truncated conjugate gradient method,” Opt. Express 18(24), 24825–24841 (2010).
[CrossRef] [PubMed]

Rao, B. D.

I. F. Gorodnitsky and B. D. Rao, “Sparse signal reconstruction from limited data using focuss: a re-weighted minimum norm algorithm,” IEEE Trans. Signal Process. 45(3), 600–616 (1997).
[CrossRef]

Romberg, J.

E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59(8), 1207–1223 (2006).
[CrossRef]

Saunders, M.

S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43(1), 129–159 (2001).
[CrossRef]

Stelzer, E. H.

Stout, D.

Swoger, J.

Tao, T.

E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59(8), 1207–1223 (2006).
[CrossRef]

Tian, J.

Vo, H.

Wajs, V.

P. Combettes and V. Wajs, “Signal recovery by proximal forward backward splitting,” Multiscale Model. Simul. 4(4), 1168–1200 (2005).
[CrossRef]

Wang, X.

Wang, Z.

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9(3), 81–84 (2002).
[CrossRef]

Wright, S. J.

S. J. Wright, R. D. Nowak, and M. A. Figueiredo, “Sparse Reconstruction by Separable Approximation,” IEEET, Signal Process. 57, 2479–2493 (2009).

Yang, X.

Yu, J.

Zhang, B.

Zhang, Q.

Q. Zhang, H. Zhao, D. Chen, X. Qu, X. Chen, X. He, W. Li, Z. Hu, J. Liu, J. Liang, and J. Tian, “Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography,” Opt. Commun. 284(24), 5871–5876 (2011).
[CrossRef]

Zhang, X.

Zhao, H.

Q. Zhang, H. Zhao, D. Chen, X. Qu, X. Chen, X. He, W. Li, Z. Hu, J. Liu, J. Liang, and J. Tian, “Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography,” Opt. Commun. 284(24), 5871–5876 (2011).
[CrossRef]

Zhong, W.

Zhu, S.

Appl. Opt. (3)

Commun. Pure Appl. Math. (2)

E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59(8), 1207–1223 (2006).
[CrossRef]

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57(11), 1413–1457 (2004).
[CrossRef]

IEEE J. Sel. Top. Signal Process. (1)

S.-J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularizedleast squares,” IEEE J. Sel. Top. Signal Process. 1(4), 606–617 (2007).
[CrossRef]

IEEE Signal Process. Lett. (1)

Z. Wang and A. C. Bovik, “A universal image quality index,” IEEE Signal Process. Lett. 9(3), 81–84 (2002).
[CrossRef]

IEEE Trans. Signal Process. (1)

I. F. Gorodnitsky and B. D. Rao, “Sparse signal reconstruction from limited data using focuss: a re-weighted minimum norm algorithm,” IEEE Trans. Signal Process. 45(3), 600–616 (1997).
[CrossRef]

IEEET, Signal Process. (1)

S. J. Wright, R. D. Nowak, and M. A. Figueiredo, “Sparse Reconstruction by Separable Approximation,” IEEET, Signal Process. 57, 2479–2493 (2009).

J. Opt. Soc. Am. A (6)

Multiscale Model. Simul. (1)

P. Combettes and V. Wajs, “Signal recovery by proximal forward backward splitting,” Multiscale Model. Simul. 4(4), 1168–1200 (2005).
[CrossRef]

Opt. Commun. (1)

Q. Zhang, H. Zhao, D. Chen, X. Qu, X. Chen, X. He, W. Li, Z. Hu, J. Liu, J. Liang, and J. Tian, “Source sparsity based primal-dual interior-point method for three-dimensional bioluminescence tomography,” Opt. Commun. 284(24), 5871–5876 (2011).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

SIAM Rev. (1)

S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43(1), 129–159 (2001).
[CrossRef]

Other (5)

E. Hale, W. Yin, and Y. Zhang, “A fixed-point continuation method for L1-regularized minimization with applications to compressed sensing,” Tech. Rep. TR07–07 (Rice Univ., Houston, TX, 2007).

T.-C. Poon, Optical Scanning Holography with MATLAB (Springer, New York, 2007).

X. Zhang, E. Y. Lam, and T.-C. Poon, “Fast iterative sectional image reconstruction in optical scanning holography,” in Digital Holography and Three-Dimensional Imaging, Technical Digest (CD) (Optical Society of America, 2009), paper DMA3.

S. Wei and H. Xu, “Staircasing reduction model applied to total variation based image reconstruction,”in 17th European Signal Processing Conference (EUSIPCO, Glasgow, Scotland, 2009), pp. 2579–2583.

A. Yang, A. Ganesh, S. Sastry, and Y. Ma, “Fast L1-minimization algorithms and an application in robust face recognition: a review,” in proceedings of IEEE International Conference on Image Processing, (Institute of Electrical and Electronics Engineers, California, 2010), pp. 1849–1852.

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Figures (7)

Fig. 1
Fig. 1

Complex hologram obtained by the physical OSH system. (a) Real part of the hologram. (b) Imaginary part of the hologram.

Fig. 2
Fig. 2

Reconstruction results of OSH sectional images. (a) and (b) are results obtained by conventional method using the whole data and a half data. (c) and (d) are results obtained by l 2 norm method using the whole data and a half data. (e) and (f) are results obtained by the AIST method using the whole data and a half data.

Fig. 3
Fig. 3

Hologram of object with three planar sections. (a) Real part of the hologram. (b) Imaginary part of the hologram.

Fig. 4
Fig. 4

Reconstruction results of the l 2 norm method and the AIST method. (a) Results of l 2 norm method with the whole hologram. (b) Results of the AIST method with the whole hologram. (c) Results of the AIST method with the real part of the hologram.

Fig. 5
Fig. 5

Sectional images illustration. The first sectional image is fixed at the depth of 1.0mm, while the second sectional image varies from0.2mm to 49.5mm.

Fig. 6
Fig. 6

Reconstruction results by the the l 2 norm method and the AIST method. The left two columns are the results of the l 2 norm method, and the right two columns are the results of the AIST method. The distance between two sectional images is (a) (b) 0.2 mm , (c) (d) 1.0 mm , (e) (f) 9.0 mm , and (g) (h) 49.5 mm respectively.

Fig. 7
Fig. 7

ImQ of the AIST method and the l 2 norm method.

Tables (1)

Tables Icon

Table 1 ImQ’s of the Reconstruction Results

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

g(x,y)= + φ(x,y,z)h(x,y;z) dz i=1 M φ(x,y, z i ) h(x,y; z i ),
h(x,y; z i )=j 1 λ 0 z i exp(j π λ 0 z i ( x 2 + y 2 )),
G=[ H 1 H 2 H M ][ Φ 1 Φ 2 Φ M ]+ε=HΦ+ε,
H T H=[ H 1 T H 2 T H M T ][ H 1 H 2 H M ] = [ H 1 T H 1 H 1 T H 2 H 1 T H M H 2 T H 1 H 2 T H 2 H 2 T H M H M T H 1 H M T H 2 H M T H M ],
h ~ (x,y; z i )=Rv(T( h (x,y; z i ))),
H i T H j =lex( h ~ (x,y; z i ) h (x,y; z j )), 1i,jM,
HG=[ H 1 H 2 H M ]G=[ H 1 G H 2 G H M G ],
H i G=lex(h(x,y; z i )g(x,y)).
min Φ 1 , subject to GHΦ 2 δ.
Φ * =arg min Φ F(x)=arg min Φ 1 2 GHΦ 2 +λ Φ 1 ,
f(Φ)f( Φ k )+ (Φ Φ k ) T f( Φ k )+ 1 2 Φ Φ k 2 2 2 f( Φ k ).
α k+1 = ( Φ k+1 Φ k ) T (f( Φ k+1 )f( Φ k )) ( Φ k+1 Φ k ) T ( Φ k+1 Φ k ) .
Φ k+1 =arg min Φ f(x)+g(x) arg min Φ { (Φ Φ k ) T f( Φ K )+ α K 2 Φ Φ k 2 2 + λ k Φ 1 },
f( Φ K )=[ lex( h ~ 1 γ k ) lex( h ~ 2 γ k ) lex( h ~ M γ k ) ][ lex( h ~ 1 g) lex( h ~ 2 g) lex( h ~ M g) ],
Φ k+1 =arg min Φ 1 2 Φθ 2 + λ k α k Φ 1 ,
Φ k+1 = arg min Φ Φ θ k 2 2 2 + λ Φ 1 α k = soft( θ k , λ k α k ) = { sgn( θ k )max{ | θ | sep λ k α k ,0 }, if | θ | sep > λ k α k 0, otherwise, ,
λ k+1,p+1 = λ 0,p exp( ζ 1 p ζ 2 k), k0,p0,
ImQ= 4 σ ϕ ϕ ^ E(ϕ)E( ϕ ^ ) ( σ ϕ 2 + σ ϕ ^ 2 )( E 2 (ϕ)+ E 2 ( ϕ ^ )) ,

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