Abstract

A block-based compressive imaging (BCI) system using sequential architecture is presented in this paper. Feature measurements are collected using the principal component analysis (PCA) projection. The linear Wiener operator and a nonlinear method based on the Field-of-Expert (FoE) prior model are used for object reconstruction. Experimental results are given to demonstrate the superior reconstruction performance of the FoE-based method over the Wiener operator. In addition, the effects of system parameters, such as the object block size, the number of features per block, and the noise level to the BCI reconstruction performance are discussed with different kinds of objects. Then an optimal block size is defined and studied for BCI.

© 2012 OSA

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References

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  1. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
    [CrossRef]
  2. E. J. Candès, “Compressive sampling,” in Proceedings of the International Congress of Mathematicians, (European Mathematical Society, 2006), pp. 1433–1452.
  3. M. A. Neifeld and J. Ke, “Optical architectures for compressive imaging,” Appl. Opt. 46, 5293–5303 (2007).
    [CrossRef] [PubMed]
  4. M. Lustig, D. L. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
    [CrossRef] [PubMed]
  5. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
    [CrossRef] [PubMed]
  6. H. Di, K. Zheng, X. Zhang, E. Y. Lam, T. Kim, Y. S. Kim, T.-C. Poon, and C. Zhou, “Multiple-image encryption by compressive holography,” Appl. Opt. 51, 1000–1009 (2012).
    [CrossRef] [PubMed]
  7. X. Zhang and E. Y. Lam, “Sectional image reconstruction in optical scanning holography usingcompressed sensing,” in IEEE International Conference on Image Processing, (IEEE, 2010), pp. 3349–3352.
    [CrossRef]
  8. M. E. Gehm, R. John, D. J. Brady, R. M. Willett, and T. J. Schulz, “Single-shot compressive spectral imaging with a dual-disperser architecture,” Opt. Express 15, 14013–14027 (2007).
    [CrossRef] [PubMed]
  9. Z. Xu and E. Y. Lam, “Image reconstruction using spectroscopic and hyperspectral information for compressive terahertz imaging,” J. Opt. Soc. Am. A 27, 1638–1646 (2010).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. P. K. Baheti and M. A. Neifeld, “Recognition using information-optimal adaptive feature-specific imaging,” J. Opt. Soc. Am. A 26, 1055–1070 (2009).
    [CrossRef]
  13. X. Zhang and E. Y. Lam, “Edge-preserving sectional image reconstruction in optical scanning holography,” J. Opt. Soc. Am. A 27, 1630–1637 (2010).
    [CrossRef]
  14. D. Needell and J. A. Tropp, “CoSaMP: Iterative signal recovery from incomplete and inaccurate samples,” Appl. Comput. Harmon. Anal. 26, 301–321 (2009).
    [CrossRef]
  15. J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
    [CrossRef]
  16. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
    [CrossRef]
  17. J. Ke, A. Ashok, and M. A. Neifeld, “Object reconstruction from adaptive compressive measurements in feature-specific imaging,” Appl. Opt. 49, H27–H39 (2010).
    [CrossRef] [PubMed]
  18. J. Ke and E. Y. Lam, “Nonlinear image reconstruction in block-based compressive imaging,” in IEEE International Symposium on Circuits and Systems, (IEEE, 2012), pp. 2917–2920.
  19. S.-H. Cho, S.-H. Lee, N. Gung-Chan, S. Jun-Oh, J.-H. Son, H. Park, and C.-B. Ahn, “Fast terahertz reflection tomography using block-based compressed sensing,” Opt. Express 19, 16401–16409 (2011).
    [CrossRef] [PubMed]
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    [CrossRef]
  21. L. Sun, X. Wen, M. Lei, H. Xu, J. Zhu, and Y. Wei, “Signal reconstruction based on block compressed sensing,” Artificial Intelligence and Computational Intelligence, Lecture Notes in Computer Science pp. 312–319 (2011).
  22. S. Roth and M. J. Black, “Fields of experts,” Int. J. Comput. Vis. 82, 205–229 (2009).
    [CrossRef]
  23. I. T. Jolliffe, Principle Component Analysis (Springer, 2002).
  24. K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
    [CrossRef] [PubMed]
  25. M. Welling, G. Hinton, and S. Osindero, “Learning sparse topographic representations with products of Student-t distributions,” in Advances in Neural Information Processing Systems (MIT Press, 2003).
  26. G. E. Hinton, “Training products of experts by minimizing contrastive divergence,” Neural Comput. 14, 1771–1800 (2002).
    [CrossRef] [PubMed]
  27. J. S. Liu, Monte Carlo Strategies in Scientific Computing (Springer, 2003).
  28. G. H. Golub and C. F. V. Loan, Matrix Computations (The Johns Hopkins University Press, 1996).
  29. The Berkeley Segmentation Dataset and Benchmark, http://www.eecs.berkeley.edu/Research/Projects/CS/vision/bsds/ .
  30. J. Ke, M. D. Stenner, and M. A. Neifeld, “Minimum reconstruction error in feature-specific imaging,” in Visual Information Processing XIV, Proc. SPIE 5817,7–12(2005).
  31. National Optical Astronomy Observatory/Association of Universities for Research in Astronomy/National Science Foundation, http://www.noao.edu/image_gallery/ .

2012 (1)

2011 (2)

2010 (3)

2009 (5)

J. Ke, P. Shankar, and M. A. Neifeld, “Distributed imaging using an array of compressive cameras,” Opt. Commun. 282, 185–197 (2009).
[CrossRef]

D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
[CrossRef] [PubMed]

D. Needell and J. A. Tropp, “CoSaMP: Iterative signal recovery from incomplete and inaccurate samples,” Appl. Comput. Harmon. Anal. 26, 301–321 (2009).
[CrossRef]

S. Roth and M. J. Black, “Fields of experts,” Int. J. Comput. Vis. 82, 205–229 (2009).
[CrossRef]

P. K. Baheti and M. A. Neifeld, “Recognition using information-optimal adaptive feature-specific imaging,” J. Opt. Soc. Am. A 26, 1055–1070 (2009).
[CrossRef]

2007 (5)

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef] [PubMed]

J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
[CrossRef]

M. A. Neifeld and J. Ke, “Optical architectures for compressive imaging,” Appl. Opt. 46, 5293–5303 (2007).
[CrossRef] [PubMed]

M. Lustig, D. L. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef] [PubMed]

M. E. Gehm, R. John, D. J. Brady, R. M. Willett, and T. J. Schulz, “Single-shot compressive spectral imaging with a dual-disperser architecture,” Opt. Express 15, 14013–14027 (2007).
[CrossRef] [PubMed]

2006 (1)

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

2005 (1)

J. Ke, M. D. Stenner, and M. A. Neifeld, “Minimum reconstruction error in feature-specific imaging,” in Visual Information Processing XIV, Proc. SPIE 5817,7–12(2005).

2002 (1)

G. E. Hinton, “Training products of experts by minimizing contrastive divergence,” Neural Comput. 14, 1771–1800 (2002).
[CrossRef] [PubMed]

1992 (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

Ahn, C.-B.

Ashok, A.

J. Ke, A. Ashok, and M. A. Neifeld, “Block-wise motion detection using compressive imaging system,” Opt. Commun. 284, 1170–1180 (2011).
[CrossRef]

J. Ke, A. Ashok, and M. A. Neifeld, “Object reconstruction from adaptive compressive measurements in feature-specific imaging,” Appl. Opt. 49, H27–H39 (2010).
[CrossRef] [PubMed]

Baheti, P. K.

Black, M. J.

S. Roth and M. J. Black, “Fields of experts,” Int. J. Comput. Vis. 82, 205–229 (2009).
[CrossRef]

Brady, D. J.

Candès, E. J.

E. J. Candès, “Compressive sampling,” in Proceedings of the International Congress of Mathematicians, (European Mathematical Society, 2006), pp. 1433–1452.

Cho, S.-H.

Choi, K.

Dabov, K.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef] [PubMed]

Di, H.

Donoho, D. L.

M. Lustig, D. L. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef] [PubMed]

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

Egiazarian, K.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef] [PubMed]

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

Foi, A.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef] [PubMed]

Gan, L.

L. Gan, “Block compressed sensing of natural images,” in 2007 15th International Conference on Digital Signal Processing, (IEEE, 2007), pp. 403–406.
[CrossRef]

Gehm, M. E.

Gilbert, A. C.

J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
[CrossRef]

Golub, G. H.

G. H. Golub and C. F. V. Loan, Matrix Computations (The Johns Hopkins University Press, 1996).

Gung-Chan, N.

Hinton, G.

M. Welling, G. Hinton, and S. Osindero, “Learning sparse topographic representations with products of Student-t distributions,” in Advances in Neural Information Processing Systems (MIT Press, 2003).

Hinton, G. E.

G. E. Hinton, “Training products of experts by minimizing contrastive divergence,” Neural Comput. 14, 1771–1800 (2002).
[CrossRef] [PubMed]

Horisaki, R.

John, R.

Jolliffe, I. T.

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

Jun-Oh, S.

Katkovnik, V.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef] [PubMed]

Ke, J.

J. Ke, A. Ashok, and M. A. Neifeld, “Block-wise motion detection using compressive imaging system,” Opt. Commun. 284, 1170–1180 (2011).
[CrossRef]

J. Ke, A. Ashok, and M. A. Neifeld, “Object reconstruction from adaptive compressive measurements in feature-specific imaging,” Appl. Opt. 49, H27–H39 (2010).
[CrossRef] [PubMed]

J. Ke, P. Shankar, and M. A. Neifeld, “Distributed imaging using an array of compressive cameras,” Opt. Commun. 282, 185–197 (2009).
[CrossRef]

M. A. Neifeld and J. Ke, “Optical architectures for compressive imaging,” Appl. Opt. 46, 5293–5303 (2007).
[CrossRef] [PubMed]

J. Ke, M. D. Stenner, and M. A. Neifeld, “Minimum reconstruction error in feature-specific imaging,” in Visual Information Processing XIV, Proc. SPIE 5817,7–12(2005).

J. Ke and E. Y. Lam, “Nonlinear image reconstruction in block-based compressive imaging,” in IEEE International Symposium on Circuits and Systems, (IEEE, 2012), pp. 2917–2920.

Kim, T.

Kim, Y. S.

Lam, E. Y.

H. Di, K. Zheng, X. Zhang, E. Y. Lam, T. Kim, Y. S. Kim, T.-C. Poon, and C. Zhou, “Multiple-image encryption by compressive holography,” Appl. Opt. 51, 1000–1009 (2012).
[CrossRef] [PubMed]

Z. Xu and E. Y. Lam, “Image reconstruction using spectroscopic and hyperspectral information for compressive terahertz imaging,” J. Opt. Soc. Am. A 27, 1638–1646 (2010).
[CrossRef]

X. Zhang and E. Y. Lam, “Edge-preserving sectional image reconstruction in optical scanning holography,” J. Opt. Soc. Am. A 27, 1630–1637 (2010).
[CrossRef]

J. Ke and E. Y. Lam, “Nonlinear image reconstruction in block-based compressive imaging,” in IEEE International Symposium on Circuits and Systems, (IEEE, 2012), pp. 2917–2920.

X. Zhang and E. Y. Lam, “Sectional image reconstruction in optical scanning holography usingcompressed sensing,” in IEEE International Conference on Image Processing, (IEEE, 2010), pp. 3349–3352.
[CrossRef]

Lee, S.-H.

Lei, M.

L. Sun, X. Wen, M. Lei, H. Xu, J. Zhu, and Y. Wei, “Signal reconstruction based on block compressed sensing,” Artificial Intelligence and Computational Intelligence, Lecture Notes in Computer Science pp. 312–319 (2011).

Lim, S.

Liu, J. S.

J. S. Liu, Monte Carlo Strategies in Scientific Computing (Springer, 2003).

Loan, C. F. V.

G. H. Golub and C. F. V. Loan, Matrix Computations (The Johns Hopkins University Press, 1996).

Lustig, M.

M. Lustig, D. L. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef] [PubMed]

Marks, D. L.

Needell, D.

D. Needell and J. A. Tropp, “CoSaMP: Iterative signal recovery from incomplete and inaccurate samples,” Appl. Comput. Harmon. Anal. 26, 301–321 (2009).
[CrossRef]

Neifeld, M. A.

J. Ke, A. Ashok, and M. A. Neifeld, “Block-wise motion detection using compressive imaging system,” Opt. Commun. 284, 1170–1180 (2011).
[CrossRef]

J. Ke, A. Ashok, and M. A. Neifeld, “Object reconstruction from adaptive compressive measurements in feature-specific imaging,” Appl. Opt. 49, H27–H39 (2010).
[CrossRef] [PubMed]

P. K. Baheti and M. A. Neifeld, “Recognition using information-optimal adaptive feature-specific imaging,” J. Opt. Soc. Am. A 26, 1055–1070 (2009).
[CrossRef]

J. Ke, P. Shankar, and M. A. Neifeld, “Distributed imaging using an array of compressive cameras,” Opt. Commun. 282, 185–197 (2009).
[CrossRef]

M. A. Neifeld and J. Ke, “Optical architectures for compressive imaging,” Appl. Opt. 46, 5293–5303 (2007).
[CrossRef] [PubMed]

J. Ke, M. D. Stenner, and M. A. Neifeld, “Minimum reconstruction error in feature-specific imaging,” in Visual Information Processing XIV, Proc. SPIE 5817,7–12(2005).

Osher, S.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

Osindero, S.

M. Welling, G. Hinton, and S. Osindero, “Learning sparse topographic representations with products of Student-t distributions,” in Advances in Neural Information Processing Systems (MIT Press, 2003).

Park, H.

Pauly, J. M.

M. Lustig, D. L. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef] [PubMed]

Poon, T.-C.

Roth, S.

S. Roth and M. J. Black, “Fields of experts,” Int. J. Comput. Vis. 82, 205–229 (2009).
[CrossRef]

Rudin, L. I.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

Schulz, T. J.

Shankar, P.

J. Ke, P. Shankar, and M. A. Neifeld, “Distributed imaging using an array of compressive cameras,” Opt. Commun. 282, 185–197 (2009).
[CrossRef]

Son, J.-H.

Stenner, M. D.

J. Ke, M. D. Stenner, and M. A. Neifeld, “Minimum reconstruction error in feature-specific imaging,” in Visual Information Processing XIV, Proc. SPIE 5817,7–12(2005).

Sun, L.

L. Sun, X. Wen, M. Lei, H. Xu, J. Zhu, and Y. Wei, “Signal reconstruction based on block compressed sensing,” Artificial Intelligence and Computational Intelligence, Lecture Notes in Computer Science pp. 312–319 (2011).

Tropp, J. A.

D. Needell and J. A. Tropp, “CoSaMP: Iterative signal recovery from incomplete and inaccurate samples,” Appl. Comput. Harmon. Anal. 26, 301–321 (2009).
[CrossRef]

J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
[CrossRef]

Wei, Y.

L. Sun, X. Wen, M. Lei, H. Xu, J. Zhu, and Y. Wei, “Signal reconstruction based on block compressed sensing,” Artificial Intelligence and Computational Intelligence, Lecture Notes in Computer Science pp. 312–319 (2011).

Welling, M.

M. Welling, G. Hinton, and S. Osindero, “Learning sparse topographic representations with products of Student-t distributions,” in Advances in Neural Information Processing Systems (MIT Press, 2003).

Wen, X.

L. Sun, X. Wen, M. Lei, H. Xu, J. Zhu, and Y. Wei, “Signal reconstruction based on block compressed sensing,” Artificial Intelligence and Computational Intelligence, Lecture Notes in Computer Science pp. 312–319 (2011).

Willett, R. M.

Xu, H.

L. Sun, X. Wen, M. Lei, H. Xu, J. Zhu, and Y. Wei, “Signal reconstruction based on block compressed sensing,” Artificial Intelligence and Computational Intelligence, Lecture Notes in Computer Science pp. 312–319 (2011).

Xu, Z.

Zhang, X.

Zheng, K.

Zhou, C.

Zhu, J.

L. Sun, X. Wen, M. Lei, H. Xu, J. Zhu, and Y. Wei, “Signal reconstruction based on block compressed sensing,” Artificial Intelligence and Computational Intelligence, Lecture Notes in Computer Science pp. 312–319 (2011).

Appl. Comput. Harmon. Anal. (1)

D. Needell and J. A. Tropp, “CoSaMP: Iterative signal recovery from incomplete and inaccurate samples,” Appl. Comput. Harmon. Anal. 26, 301–321 (2009).
[CrossRef]

Appl. Opt. (3)

IEEE Trans. Image Process. (1)

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory (2)

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

J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53, 4655–4666 (2007).
[CrossRef]

Int. J. Comput. Vis. (1)

S. Roth and M. J. Black, “Fields of experts,” Int. J. Comput. Vis. 82, 205–229 (2009).
[CrossRef]

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

Magn. Reson. Med. (1)

M. Lustig, D. L. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef] [PubMed]

Neural Comput. (1)

G. E. Hinton, “Training products of experts by minimizing contrastive divergence,” Neural Comput. 14, 1771–1800 (2002).
[CrossRef] [PubMed]

Opt. Commun. (2)

J. Ke, P. Shankar, and M. A. Neifeld, “Distributed imaging using an array of compressive cameras,” Opt. Commun. 282, 185–197 (2009).
[CrossRef]

J. Ke, A. Ashok, and M. A. Neifeld, “Block-wise motion detection using compressive imaging system,” Opt. Commun. 284, 1170–1180 (2011).
[CrossRef]

Opt. Express (3)

Physica D (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[CrossRef]

Proc. SPIE (1)

J. Ke, M. D. Stenner, and M. A. Neifeld, “Minimum reconstruction error in feature-specific imaging,” in Visual Information Processing XIV, Proc. SPIE 5817,7–12(2005).

Other (11)

National Optical Astronomy Observatory/Association of Universities for Research in Astronomy/National Science Foundation, http://www.noao.edu/image_gallery/ .

J. S. Liu, Monte Carlo Strategies in Scientific Computing (Springer, 2003).

G. H. Golub and C. F. V. Loan, Matrix Computations (The Johns Hopkins University Press, 1996).

The Berkeley Segmentation Dataset and Benchmark, http://www.eecs.berkeley.edu/Research/Projects/CS/vision/bsds/ .

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

M. Welling, G. Hinton, and S. Osindero, “Learning sparse topographic representations with products of Student-t distributions,” in Advances in Neural Information Processing Systems (MIT Press, 2003).

L. Gan, “Block compressed sensing of natural images,” in 2007 15th International Conference on Digital Signal Processing, (IEEE, 2007), pp. 403–406.
[CrossRef]

L. Sun, X. Wen, M. Lei, H. Xu, J. Zhu, and Y. Wei, “Signal reconstruction based on block compressed sensing,” Artificial Intelligence and Computational Intelligence, Lecture Notes in Computer Science pp. 312–319 (2011).

J. Ke and E. Y. Lam, “Nonlinear image reconstruction in block-based compressive imaging,” in IEEE International Symposium on Circuits and Systems, (IEEE, 2012), pp. 2917–2920.

E. J. Candès, “Compressive sampling,” in Proceedings of the International Congress of Mathematicians, (European Mathematical Society, 2006), pp. 1433–1452.

X. Zhang and E. Y. Lam, “Sectional image reconstruction in optical scanning holography usingcompressed sensing,” in IEEE International Conference on Image Processing, (IEEE, 2010), pp. 3349–3352.
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

The BCI system diagram.

Fig. 2
Fig. 2

The diagram to define an object, a block, an image, and an image patch.

Fig. 3
Fig. 3

Three original objects with size 1000×778, 1743×1222, and 1786×1191, respectively.

Fig. 4
Fig. 4

(a) 10 × 10 equal to 100 prewhitened training samples for FoE model, (b) the 24 FoE filters of size 5 × 5 with the corresponding αi labeled.

Fig. 5
Fig. 5

Linear and nonlinear reconstructions using the Wiener operator and the FOE-based method.

Fig. 6
Fig. 6

(a) Reconstruction using the Wiener operator; (b) reconstruction using the FoE-based method.

Fig. 7
Fig. 7

Reconstruction for column 60 using (a) the Wiener operator & (b) the FoE-base method; Reconstruction for column 265 using (c) the Wiener operator & (d) the FoE-base method.

Fig. 8
Fig. 8

(a)&(c) Reconstruction using the Wiener operator; (b)&(d) reconstruction using the FoE-based method.

Fig. 9
Fig. 9

(a) The averaged RMSE vs compression ratio r using the Wiener operator; (b) rmin versus N for RMSE = 0.035; (c) RMSE versus N for compression ratio r = 0.6.

Fig. 10
Fig. 10

Examples of (a) astronomy images [31], (b) nature images [29].

Fig. 11
Fig. 11

(a) The eigenvalues for two kinds of objects (st–star; nat–nature). (b) N versus rmin with RMSE = 0.095 for the astronomy images. (c) N versus rmin with RMSE = 0.06 for the natural images.

Fig. 12
Fig. 12

(a) r versus RMSE for 3 block sizes; (b) Compression ratio r versus N for RMSE = 0.1.

Tables (1)

Tables Icon

Table 1 Contrast ratios for areas pointed out in Fig. 6.

Equations (20)

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

Y = HX + N ,
W = R x H T ( HR x H T + σ 2 I ) 1 ,
p ( X ˜ ) = 1 z ( Θ ) k ˜ = 1 K ˜ i = 1 L ϕ i ( g i T x ˜ ( k ˜ ) ; α i ) , Θ = { θ 1 , , θ L } ,
ϕ i ( g i T x ˜ ( k ˜ ) ; α i ) = [ 1 + 1 2 ( g i T x ˜ ( k ˜ ) ) 2 ] α i .
p ( X ˜ ) = 1 z ( Θ ) exp ( E FoE ( X ˜ , Θ ) )
E FoE ( X ˜ , Θ ) = k ˜ = 1 K ˜ i = 1 L log ϕ i ( g i T x ˜ ( k ˜ ) ; α i ) .
δ θ i = η [ E FoE θ i p j E FoE θ i p 0 ] ,
p ( Y ^ | X ^ ) i = 1 K N exp ( 1 2 σ 2 ( y ^ i x ^ i ) 2 ) ,
X ^ log p ( Y ^ | X ^ ) = 1 σ 2 ( Y ^ | X ^ )
X ^ log p ( X ^ ) = i = 1 L G i * ψ i ( G i * X ^ )
X ^ est ( t + 1 ) = X ^ est ( t ) + η [ i = 1 L G i * ψ i ( G i * X ^ est ( t ) ) + λ σ 2 ( Y ^ X ^ est ( t ) ) ] .
X ^ est ( t + 1 ) = X ^ est ( t ) + η [ i = 1 L G i * ψ i ( G i * X ^ est ( t ) ) + λ σ 2 𝒪 1 { H T Y H T H 𝒪 { X ^ est ( t ) } } ] ,
ε = E { x est x 2 } = Tr { R x } Tr { HR x 2 H T ( HR x 2 H T + σ 0 2 I ) 1 } = i = M + 1 N d i + i = 1 M d i ( 1 1 1 + M 3 σ 0 2 / d i ) ,
ε = E { x est x 2 } = E { W y x 2 } = E { W ( H x + n ) x 2 } = E { ( WH I ) x + W n 2 } .
ε = E { [ ( WH I ) x + W n ] T [ ( WH I ) x + W n ] } = E { [ ( WH I ) x ] T [ ( WH I ) x ] + ( W n ) T ( W n ) } = E { Tr { [ ( WH I ) x ] [ ( WH I ) x ] T } + Tr { ( W n ) ( W n ) T } } .
ε = Tr { ( WH I ) R x ( WH I ) T } + Tr { WR n W T } ,
ε = Tr { R x } Tr { HR x 2 H T ( HR x 2 H T + σ 2 I ) 1 } .
R x = [ q 1 q 2 q N ] [ d 1 0 0 0 d 2 0 0 0 d N ] [ q 1 T q 2 T q N T ]
H = [ q 1 T q 2 T q M T ] T ,
ε = i = 1 N d i i = 1 M d i 2 d i + M 3 σ 0 2 = i = M + 1 N d i + i = 1 M d i ( 1 1 1 + M 3 σ 0 2 / d i ) .

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