Abstract

We compare three optical architectures for compressive imaging: sequential, parallel, and photon sharing. Each of these architectures is analyzed using two different types of projection: (a) principal component projections and (b) pseudo-random projections. Both linear and nonlinear reconstruction methods are studied. The performance of each architecture-projection combination is quantified in terms of reconstructed image quality as a function of measurement noise strength. Using a linear reconstruction operator we find that in all cases of (a) there is a measurement noise level above which compressive imaging is superior to conventional imaging. Normalized by the average object pixel brightness, these threshold noise standard deviations are 6.4, 4.9, and 2.1 for the sequential, parallel, and photon sharing architectures, respectively. We also find that conventional imaging outperforms compressive imaging using pseudo-random projections when linear reconstruction is employed. In all cases the photon sharing architecture is found to be more photon-efficient than the other two optical implementations and thus offers the highest performance among all compressive methods studied here. For example, with principal component projections and a linear reconstruction operator, the photon sharing architecture provides at least 17.6% less reconstruction error than either of the other two architectures for a noise strength of 1.6 times the average object pixel brightness. We also demonstrate that nonlinear reconstruction methods can offer additional performance improvements to all architectures for small values of noise.

© 2007 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  18. H. H. Barrett, D. W. Wilson, and B. M. W. Tsui, "Noise properties of the EM algorithm: I. Theory," Phys. Med. Biol. 39, 833-845 (1994).
    [CrossRef] [PubMed]

2006 (5)

J. Haupt and R. Nowak, "Signal reconstruction from noisy random projections," IEEE Trans. Info. Theory 52, 4036-4048 (2006).
[CrossRef]

M. F. Duarte, M. A. Davenport, M. B. Wakin, and R. G. Baraniuk, "Sparse signal detection from incoherent projections," in IEEE International Conference on Acoustics, Speech and Signal Processing, 2006. ICASSP 2006 Proceedings, Vol. 3 (IEEE, 2006).

E. Candes, J. Romberg, and T. Tao, "Stable signal reconvery from incomplete and inaccurate measurements," Commun. Pure Appl. Mathematics 59, 1207-1223 (2006).
[CrossRef]

D. Donoho, "Compressed sensing," IEEE Trans. Info. Theory 52, 1289-1306 (2006).
[CrossRef]

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, "A new compressive imaging camera using optical domain compression," Proc. SPIE 6065, 606509 (2006).
[CrossRef]

2005 (2)

H. S. Pal, D. Ganotra, and M. A. Neifeld, "Face recognition by using feature-specific imaging," Appl. Opt. 44, 3784-3794 (2005).
[CrossRef] [PubMed]

J. Ke, M. D. Stenner, and M. A. Neifeld, "Minimum reconstruction error in feature-specific imaging," Proc. SPIE 5817, 7-12 (2005).
[CrossRef]

2004 (1)

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley-Interscience, 2004).

2003 (2)

2002 (2)

2001 (2)

S. S. Chen, D. L. Donoho, and M. A. Saunders, "Atomic decomposition by basis pursuit," SIAM Soc. Ind. Appl. Math. J. Numer. Anal. 43, 129-159 (2001).

D. L. Marks, R. Stack, A. J. Johnson, D. J. Brady, and D. C. Munson, "Cone-beam tomography with a digital camera," Appl. Opt. 40, 1795-1805 (2001).
[CrossRef]

1999 (1)

W. Wenzel and K. Hamacher, "A stochastic tunneling approach for global minimization of complex potential energy landscapes," Phys. Rev. Lett. 82, 3003-3007 (1999).
[CrossRef]

1994 (1)

H. H. Barrett, D. W. Wilson, and B. M. W. Tsui, "Noise properties of the EM algorithm: I. Theory," Phys. Med. Biol. 39, 833-845 (1994).
[CrossRef] [PubMed]

Baraniuk, R. G.

M. F. Duarte, M. A. Davenport, M. B. Wakin, and R. G. Baraniuk, "Sparse signal detection from incoherent projections," in IEEE International Conference on Acoustics, Speech and Signal Processing, 2006. ICASSP 2006 Proceedings, Vol. 3 (IEEE, 2006).

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, "A new compressive imaging camera using optical domain compression," Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Baron, D.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, "A new compressive imaging camera using optical domain compression," Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Barrett, H. H.

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley-Interscience, 2004).

H. H. Barrett, D. W. Wilson, and B. M. W. Tsui, "Noise properties of the EM algorithm: I. Theory," Phys. Med. Biol. 39, 833-845 (1994).
[CrossRef] [PubMed]

Brady, D. J.

Candes, E.

E. Candes, J. Romberg, and T. Tao, "Stable signal reconvery from incomplete and inaccurate measurements," Commun. Pure Appl. Mathematics 59, 1207-1223 (2006).
[CrossRef]

Candes, E. J.

E. J. Candes and J. K. Romberg, "Practical signal recovery from random projections," IEEE Trans. Signal Process. (to be published).

Cathey, W. T.

Chen, S. S.

S. S. Chen, D. L. Donoho, and M. A. Saunders, "Atomic decomposition by basis pursuit," SIAM Soc. Ind. Appl. Math. J. Numer. Anal. 43, 129-159 (2001).

Davenport, M. A.

M. F. Duarte, M. A. Davenport, M. B. Wakin, and R. G. Baraniuk, "Sparse signal detection from incoherent projections," in IEEE International Conference on Acoustics, Speech and Signal Processing, 2006. ICASSP 2006 Proceedings, Vol. 3 (IEEE, 2006).

Donoho, D.

D. Donoho, "Compressed sensing," IEEE Trans. Info. Theory 52, 1289-1306 (2006).
[CrossRef]

Donoho, D. L.

S. S. Chen, D. L. Donoho, and M. A. Saunders, "Atomic decomposition by basis pursuit," SIAM Soc. Ind. Appl. Math. J. Numer. Anal. 43, 129-159 (2001).

Dowski, E. R.

Duarte, M. F.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, "A new compressive imaging camera using optical domain compression," Proc. SPIE 6065, 606509 (2006).
[CrossRef]

M. F. Duarte, M. A. Davenport, M. B. Wakin, and R. G. Baraniuk, "Sparse signal detection from incoherent projections," in IEEE International Conference on Acoustics, Speech and Signal Processing, 2006. ICASSP 2006 Proceedings, Vol. 3 (IEEE, 2006).

Ganotra, D.

Hamacher, K.

W. Wenzel and K. Hamacher, "A stochastic tunneling approach for global minimization of complex potential energy landscapes," Phys. Rev. Lett. 82, 3003-3007 (1999).
[CrossRef]

Haupt, J.

J. Haupt and R. Nowak, "Signal reconstruction from noisy random projections," IEEE Trans. Info. Theory 52, 4036-4048 (2006).
[CrossRef]

Johnson, A. J.

Jolliffe, I. T.

I. T. Jolliffe, Principle Component Analysis (Springer, 2002).

Kelly, K. F.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, "A new compressive imaging camera using optical domain compression," Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Laska, J. N.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, "A new compressive imaging camera using optical domain compression," Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Marks, D. L.

Munson, D. C.

Myers, K. J.

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley-Interscience, 2004).

Neifeld, M. A.

Nowak, R.

J. Haupt and R. Nowak, "Signal reconstruction from noisy random projections," IEEE Trans. Info. Theory 52, 4036-4048 (2006).
[CrossRef]

Pal, H.

Pal, H. S.

Romberg, J.

E. Candes, J. Romberg, and T. Tao, "Stable signal reconvery from incomplete and inaccurate measurements," Commun. Pure Appl. Mathematics 59, 1207-1223 (2006).
[CrossRef]

Romberg, J. K.

E. J. Candes and J. K. Romberg, "Practical signal recovery from random projections," IEEE Trans. Signal Process. (to be published).

Sarvotham, S.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, "A new compressive imaging camera using optical domain compression," Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Saunders, M. A.

S. S. Chen, D. L. Donoho, and M. A. Saunders, "Atomic decomposition by basis pursuit," SIAM Soc. Ind. Appl. Math. J. Numer. Anal. 43, 129-159 (2001).

Shankar, P.

Stack, R.

Stenner, J. Ke

J. Ke, M. D. Stenner, and M. A. Neifeld, "Minimum reconstruction error in feature-specific imaging," Proc. SPIE 5817, 7-12 (2005).
[CrossRef]

Takhar, D.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, "A new compressive imaging camera using optical domain compression," Proc. SPIE 6065, 606509 (2006).
[CrossRef]

Tao, T.

E. Candes, J. Romberg, and T. Tao, "Stable signal reconvery from incomplete and inaccurate measurements," Commun. Pure Appl. Mathematics 59, 1207-1223 (2006).
[CrossRef]

Tsui, B. M. W.

H. H. Barrett, D. W. Wilson, and B. M. W. Tsui, "Noise properties of the EM algorithm: I. Theory," Phys. Med. Biol. 39, 833-845 (1994).
[CrossRef] [PubMed]

Wakin, M. B.

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, "A new compressive imaging camera using optical domain compression," Proc. SPIE 6065, 606509 (2006).
[CrossRef]

M. F. Duarte, M. A. Davenport, M. B. Wakin, and R. G. Baraniuk, "Sparse signal detection from incoherent projections," in IEEE International Conference on Acoustics, Speech and Signal Processing, 2006. ICASSP 2006 Proceedings, Vol. 3 (IEEE, 2006).

Wenzel, W.

W. Wenzel and K. Hamacher, "A stochastic tunneling approach for global minimization of complex potential energy landscapes," Phys. Rev. Lett. 82, 3003-3007 (1999).
[CrossRef]

Wilson, D. W.

H. H. Barrett, D. W. Wilson, and B. M. W. Tsui, "Noise properties of the EM algorithm: I. Theory," Phys. Med. Biol. 39, 833-845 (1994).
[CrossRef] [PubMed]

Appl. Opt. (4)

Commun. Pure Appl. Mathematics (1)

E. Candes, J. Romberg, and T. Tao, "Stable signal reconvery from incomplete and inaccurate measurements," Commun. Pure Appl. Mathematics 59, 1207-1223 (2006).
[CrossRef]

IEEE Trans. Info. Theory (2)

J. Haupt and R. Nowak, "Signal reconstruction from noisy random projections," IEEE Trans. Info. Theory 52, 4036-4048 (2006).
[CrossRef]

D. Donoho, "Compressed sensing," IEEE Trans. Info. Theory 52, 1289-1306 (2006).
[CrossRef]

IEEE Trans. Signal Process. (1)

E. J. Candes and J. K. Romberg, "Practical signal recovery from random projections," IEEE Trans. Signal Process. (to be published).

Opt. Express (1)

Phys. Med. Biol. (1)

H. H. Barrett, D. W. Wilson, and B. M. W. Tsui, "Noise properties of the EM algorithm: I. Theory," Phys. Med. Biol. 39, 833-845 (1994).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

W. Wenzel and K. Hamacher, "A stochastic tunneling approach for global minimization of complex potential energy landscapes," Phys. Rev. Lett. 82, 3003-3007 (1999).
[CrossRef]

Proc. SPIE (2)

J. Ke, M. D. Stenner, and M. A. Neifeld, "Minimum reconstruction error in feature-specific imaging," Proc. SPIE 5817, 7-12 (2005).
[CrossRef]

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, "A new compressive imaging camera using optical domain compression," Proc. SPIE 6065, 606509 (2006).
[CrossRef]

SIAM Soc. Ind. Appl. Math. J. Numer. Anal. (1)

S. S. Chen, D. L. Donoho, and M. A. Saunders, "Atomic decomposition by basis pursuit," SIAM Soc. Ind. Appl. Math. J. Numer. Anal. 43, 129-159 (2001).

Other (4)

M. F. Duarte, M. A. Davenport, M. B. Wakin, and R. G. Baraniuk, "Sparse signal detection from incoherent projections," in IEEE International Conference on Acoustics, Speech and Signal Processing, 2006. ICASSP 2006 Proceedings, Vol. 3 (IEEE, 2006).

I. T. Jolliffe, Principle Component Analysis (Springer, 2002).

The University of Oulu Physics-Based Face Database, http://www.ee.oulu.fi/research/imag/color/pbfd.html.

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley-Interscience, 2004).

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Figures (13)

Fig. 1
Fig. 1

Examples used for this study: (a) examples from class X1, (b) first five PC basis vectors for class X1, (c) examples from class X2 with K = 1600, and (d) examples from class X2 with K = 400.

Fig. 2
Fig. 2

(Color online) Schematic optical system diagram for various compressive imaging architectures: (a) sequential, (b) parallel, (c) space-domain photon sharing, and (d) time-domain photon sharing.

Fig. 3
Fig. 3

(Color online) Reconstruction RMSE versus number of features for compressive imaging based on PC projections and several values of noise: (a) σ 0 = 1335 , (b) σ 0 = 751 , and (c) σ 0 = 422 .

Fig. 4
Fig. 4

(Color online) Optimal energy allocation η versus feature index for NP projections and three values of noise.

Fig. 5
Fig. 5

(Color online) Minimum reconstruction RMSE versus σ 0 for all compressive imaging architectures using PC projections.

Fig. 6
Fig. 6

Optimal reconstructions for σ 0 = 0.21 obtained from the (a) sequential architecture ( M opt = 207 ) , (b) parallel architecture ( M opt = 941 ) , and (c) pipeline architecture ( M opt = 2601 ) .

Fig. 7
Fig. 7

(Color online) Reconstruction RMSE versus number of features for PR projections with K = 1600 and various noise strengths: (a) σ 0 = 237 , (b) σ 0 = 133.5 , and (c) σ 0 = 1.34 .

Fig. 8
Fig. 8

(Color online) Reconstruction RMSE versus number of features for PR projections with noise strength σ 0 = 237 and two values of sparsity K 1 = 1600 and K 2 = 400 .

Fig. 9
Fig. 9

(Color online) Minimum reconstruction RMSE versus σ 0 for compressive imaging using PR projections.

Fig. 10
Fig. 10

Reconstruction RMSE versus σ 0 obtained using PR projections and linear (solid) and nonlinear (dotted) reconstruction methods: (a) various nonlinear algorithms compared using the PS architecture, and (b) comparison of US, UP, and PS architectures.

Fig. 11
Fig. 11

(Color online) RMSE versus number of features obtained from the PS architecture using PR projections with linear (diamonds) and nonlinear (squares) reconstructions for (a) K = 1600 and (b) K = 400 .

Fig. 12
Fig. 12

Example reconstructions for the PS architecture using PR projections and σ 0 = 0.21 from (a)–(c) K = 1600 and (d)–(f) K = 400 : (a) and (d) are the original objects, (b) and (e) are the optimal linear reconstructions, and (c) and (f) are the optimal nonlinear reconstructions.

Fig. 13
Fig. 13

RMSE versus number of features using PC projection and the PS architecture with linear (diamonds) and nonlinear (squares) reconstruction methods when (a) K = 1600 and (b) K = 400 .

Tables (1)

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Table 1 Threshold Values of Noise Strength above Which Compressive Imaging Is Superior to Conventional Imaging

Equations (2)

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m = F x + n ,
W = R x F T ( F R x F T + D n ) 1 ,

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