Abstract

The approximate message passing algorithm (AMP) is introduced for practical ghost imaging (GI). This paper proposes to use preconditioning to modified the sensing matrix such that it satisfies the assumptions of the original AMP algorithm. Furthermore, this paper points out that the speckles used in practical GI system are spatially correlated, and this will degrade the imaging performance. The parameter estimations of exponential power distribution reveal that the rows of modified sensing matrix are indeed bounded by the Laplace distribution, which is heavy-tailed. Then, the recovery conditions for such sensing matrix are investigated. Finally, the semi-real experiments and real data processing have validated the proposed imaging method and shown that the required number of measurements for correlated speckles are significantly greater than independent and identically distributed (i.i.d.) measurement. The results of this paper can be used to guide the design of practical GI remote sensing systems.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  4. J. Bioucas-Dias and M. Figueiredo, “Multiplicative noise removal using variable splitting and constrained optimization,” IEEE Transactions on Image Process. 19, 1720–1730 (2010).
    [Crossref]
  5. O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
    [Crossref]
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    [Crossref]
  7. L. Jiying, Z. Jubo, L. Chuan, and H. Shisheng, “High-quality quantum-imaging algorithm and experiment based on compressive sensing,” Opt. Lett. 35, 1206–1208 (2010).
    [Crossref] [PubMed]
  8. W. Gong and S. Han, “Multiple-input ghost imaging via sparsity constraints,” J. Opt. Soc. Am. A 29, 1571–1579 (2012).
    [Crossref]
  9. D. L. Donoho, A. Maleki, and A. Montanari, “Message-passing algorithms for compressed sensing,” Proc. Natl. Acad. Sci. 106, 18914–18919 (2009).
    [Crossref] [PubMed]
  10. H. A. Van der Vorst, Iterative Krylov methods for large linear systems(Cambridge University, 2003).
    [Crossref]
  11. M. Padgett, R. Aspden, G. Gibson, M. Edgar, and G. Spalding, “Ghost imaging,” Opt. Photon. News 27, 38–45 (2016).
    [Crossref]
  12. F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010).
    [Crossref] [PubMed]
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    [Crossref]
  14. X. Yao, W. Yu, X. Liu, L. Li, M. Li, L. Wu, and G. Zhai, “Iterative denoising of ghost imaging,” Opt. Express 22, 24268–24275 (2014).
    [Crossref] [PubMed]
  15. S. Rangan, “Generalized approximate message passing for estimation with random linear mixing,” in The 2011 IEEE International Symposium on Information Theory Proceedings (ISIT), (2011), pp. 2168–2172.
    [Crossref]
  16. J. W. Goodman, Speckle phenomena in optics: theory and applications (Roberts and Company, 2007).
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    [Crossref]
  18. A. Ayebo and T. J. Kozubowski, “An asymmetric generalization of gaussian and laplace laws,” J. Probab. Stat. Sci. 1, 187–210 (2003).
  19. Y. Bromberg and H. Cao, “Generating non-rayleigh speckles with tailored intensity statistics,” Phys. Rev. Lett. 112, 213904 (2014).
    [Crossref]
  20. N. Bender, H. Yilmaz, Y. Bromberg, and H. Cao, “Customizing speckle intensity statistics,” Optica 5, 595–600 (2018).
    [Crossref]
  21. D. Donoho, A. Javanmard, and A. Montanari, “Information-theoretically optimal compressed sensing via spatial coupling and approximate message passing,” IEEE Transactions on Inf. Theory 59, 7434–7464 (2013).
    [Crossref]
  22. C. Guo and M. Davies, “Near optimal compressed sensing without priors: Parametric sure approximate message passing,” IEEE Transactions on Signal Process. 63, 2130–2141 (2015).
    [Crossref]
  23. S. Som and P. Schniter, “Compressive imaging using approximate message passing and a markov-tree prior,” IEEE Transactions on Signal Process. 60, 3439–3448 (2012).
    [Crossref]
  24. J. Tan, Y. Ma, and D. Baron, “Compressive imaging via approximate message passing with image denoising,” IEEE Transactions on Signal Process. 63, 2085–2092 (2015).
    [Crossref]
  25. E. van den Berg and M. P. Friedlander, “Probing the pareto frontier for basis pursuit solutions,” SIAM J. on Sci. Comput. 31, 890–912 (2008).
    [Crossref]
  26. E. van den Berg and M. P. Friedlander, “SPGL1: A solver for large-scale sparse reconstruction,” (2007). http://www.cs.ubc.ca/labs/scl/spgl1 .
  27. S. Kotz, T. J. Kozubowski, and K. Podgorski, The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance(Birkhauser, 2001).
    [Crossref]

2018 (1)

2016 (2)

W. Gong, C. Zhao, H. Yu, M. Chen, W. Xu, and S. Han, “Three-dimensional ghost imaging lidar via sparsity constraint,” Sci. Reports. 6, 26133 (2016).
[Crossref]

M. Padgett, R. Aspden, G. Gibson, M. Edgar, and G. Spalding, “Ghost imaging,” Opt. Photon. News 27, 38–45 (2016).
[Crossref]

2015 (2)

C. Guo and M. Davies, “Near optimal compressed sensing without priors: Parametric sure approximate message passing,” IEEE Transactions on Signal Process. 63, 2130–2141 (2015).
[Crossref]

J. Tan, Y. Ma, and D. Baron, “Compressive imaging via approximate message passing with image denoising,” IEEE Transactions on Signal Process. 63, 2085–2092 (2015).
[Crossref]

2014 (2)

Y. Bromberg and H. Cao, “Generating non-rayleigh speckles with tailored intensity statistics,” Phys. Rev. Lett. 112, 213904 (2014).
[Crossref]

X. Yao, W. Yu, X. Liu, L. Li, M. Li, L. Wu, and G. Zhai, “Iterative denoising of ghost imaging,” Opt. Express 22, 24268–24275 (2014).
[Crossref] [PubMed]

2013 (1)

D. Donoho, A. Javanmard, and A. Montanari, “Information-theoretically optimal compressed sensing via spatial coupling and approximate message passing,” IEEE Transactions on Inf. Theory 59, 7434–7464 (2013).
[Crossref]

2012 (5)

2010 (4)

L. Jiying, Z. Jubo, L. Chuan, and H. Shisheng, “High-quality quantum-imaging algorithm and experiment based on compressive sensing,” Opt. Lett. 35, 1206–1208 (2010).
[Crossref] [PubMed]

B. I. Erkmen and J. H. Shapiro, “Ghost imaging: from quantum to classical to computational,” Adv. Opt. Photon. 2, 405–450 (2010).
[Crossref]

J. Bioucas-Dias and M. Figueiredo, “Multiplicative noise removal using variable splitting and constrained optimization,” IEEE Transactions on Image Process. 19, 1720–1730 (2010).
[Crossref]

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010).
[Crossref] [PubMed]

2009 (2)

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
[Crossref]

D. L. Donoho, A. Maleki, and A. Montanari, “Message-passing algorithms for compressed sensing,” Proc. Natl. Acad. Sci. 106, 18914–18919 (2009).
[Crossref] [PubMed]

2008 (1)

E. van den Berg and M. P. Friedlander, “Probing the pareto frontier for basis pursuit solutions,” SIAM J. on Sci. Comput. 31, 890–912 (2008).
[Crossref]

2005 (1)

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94, 063601 (2005).
[Crossref] [PubMed]

2003 (1)

A. Ayebo and T. J. Kozubowski, “An asymmetric generalization of gaussian and laplace laws,” J. Probab. Stat. Sci. 1, 187–210 (2003).

Aspden, R.

M. Padgett, R. Aspden, G. Gibson, M. Edgar, and G. Spalding, “Ghost imaging,” Opt. Photon. News 27, 38–45 (2016).
[Crossref]

Astola, J.

Ayebo, A.

A. Ayebo and T. J. Kozubowski, “An asymmetric generalization of gaussian and laplace laws,” J. Probab. Stat. Sci. 1, 187–210 (2003).

Baron, D.

J. Tan, Y. Ma, and D. Baron, “Compressive imaging via approximate message passing with image denoising,” IEEE Transactions on Signal Process. 63, 2085–2092 (2015).
[Crossref]

Bender, N.

Bioucas-Dias, J.

J. Bioucas-Dias and M. Figueiredo, “Multiplicative noise removal using variable splitting and constrained optimization,” IEEE Transactions on Image Process. 19, 1720–1730 (2010).
[Crossref]

Bromberg, Y.

N. Bender, H. Yilmaz, Y. Bromberg, and H. Cao, “Customizing speckle intensity statistics,” Optica 5, 595–600 (2018).
[Crossref]

Y. Bromberg and H. Cao, “Generating non-rayleigh speckles with tailored intensity statistics,” Phys. Rev. Lett. 112, 213904 (2014).
[Crossref]

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
[Crossref]

Cao, H.

N. Bender, H. Yilmaz, Y. Bromberg, and H. Cao, “Customizing speckle intensity statistics,” Optica 5, 595–600 (2018).
[Crossref]

Y. Bromberg and H. Cao, “Generating non-rayleigh speckles with tailored intensity statistics,” Phys. Rev. Lett. 112, 213904 (2014).
[Crossref]

Chen, M.

W. Gong, C. Zhao, H. Yu, M. Chen, W. Xu, and S. Han, “Three-dimensional ghost imaging lidar via sparsity constraint,” Sci. Reports. 6, 26133 (2016).
[Crossref]

Chuan, L.

D’Angelo, M.

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94, 063601 (2005).
[Crossref] [PubMed]

Davies, M.

C. Guo and M. Davies, “Near optimal compressed sensing without priors: Parametric sure approximate message passing,” IEEE Transactions on Signal Process. 63, 2130–2141 (2015).
[Crossref]

Donoho, D.

D. Donoho, A. Javanmard, and A. Montanari, “Information-theoretically optimal compressed sensing via spatial coupling and approximate message passing,” IEEE Transactions on Inf. Theory 59, 7434–7464 (2013).
[Crossref]

Donoho, D. L.

D. L. Donoho, A. Maleki, and A. Montanari, “Message-passing algorithms for compressed sensing,” Proc. Natl. Acad. Sci. 106, 18914–18919 (2009).
[Crossref] [PubMed]

Edgar, M.

M. Padgett, R. Aspden, G. Gibson, M. Edgar, and G. Spalding, “Ghost imaging,” Opt. Photon. News 27, 38–45 (2016).
[Crossref]

Edgar, M. P.

Erkmen, B. I.

Ferri, F.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010).
[Crossref] [PubMed]

Figueiredo, M.

J. Bioucas-Dias and M. Figueiredo, “Multiplicative noise removal using variable splitting and constrained optimization,” IEEE Transactions on Image Process. 19, 1720–1730 (2010).
[Crossref]

Friedlander, M. P.

E. van den Berg and M. P. Friedlander, “Probing the pareto frontier for basis pursuit solutions,” SIAM J. on Sci. Comput. 31, 890–912 (2008).
[Crossref]

Gatti, A.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010).
[Crossref] [PubMed]

Gibson, G.

M. Padgett, R. Aspden, G. Gibson, M. Edgar, and G. Spalding, “Ghost imaging,” Opt. Photon. News 27, 38–45 (2016).
[Crossref]

Gong, W.

W. Gong, C. Zhao, H. Yu, M. Chen, W. Xu, and S. Han, “Three-dimensional ghost imaging lidar via sparsity constraint,” Sci. Reports. 6, 26133 (2016).
[Crossref]

W. Gong and S. Han, “Multiple-input ghost imaging via sparsity constraints,” J. Opt. Soc. Am. A 29, 1571–1579 (2012).
[Crossref]

Goodman, J. W.

J. W. Goodman, Speckle phenomena in optics: theory and applications (Roberts and Company, 2007).

Guo, C.

C. Guo and M. Davies, “Near optimal compressed sensing without priors: Parametric sure approximate message passing,” IEEE Transactions on Signal Process. 63, 2130–2141 (2015).
[Crossref]

Han, S.

W. Gong, C. Zhao, H. Yu, M. Chen, W. Xu, and S. Han, “Three-dimensional ghost imaging lidar via sparsity constraint,” Sci. Reports. 6, 26133 (2016).
[Crossref]

W. Gong and S. Han, “Multiple-input ghost imaging via sparsity constraints,” J. Opt. Soc. Am. A 29, 1571–1579 (2012).
[Crossref]

Javanmard, A.

D. Donoho, A. Javanmard, and A. Montanari, “Information-theoretically optimal compressed sensing via spatial coupling and approximate message passing,” IEEE Transactions on Inf. Theory 59, 7434–7464 (2013).
[Crossref]

Jiying, L.

Jubo, Z.

Katkovnik, V.

Katz, O.

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
[Crossref]

Kotz, S.

S. Kotz, T. J. Kozubowski, and K. Podgorski, The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance(Birkhauser, 2001).
[Crossref]

Kozubowski, T. J.

A. Ayebo and T. J. Kozubowski, “An asymmetric generalization of gaussian and laplace laws,” J. Probab. Stat. Sci. 1, 187–210 (2003).

S. Kotz, T. J. Kozubowski, and K. Podgorski, The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance(Birkhauser, 2001).
[Crossref]

Li, L.

Li, M.

Liu, X.

Lugiato, L. A.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010).
[Crossref] [PubMed]

Ma, Y.

J. Tan, Y. Ma, and D. Baron, “Compressive imaging via approximate message passing with image denoising,” IEEE Transactions on Signal Process. 63, 2085–2092 (2015).
[Crossref]

Magatti, D.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010).
[Crossref] [PubMed]

Maleki, A.

D. L. Donoho, A. Maleki, and A. Montanari, “Message-passing algorithms for compressed sensing,” Proc. Natl. Acad. Sci. 106, 18914–18919 (2009).
[Crossref] [PubMed]

Montanari, A.

D. Donoho, A. Javanmard, and A. Montanari, “Information-theoretically optimal compressed sensing via spatial coupling and approximate message passing,” IEEE Transactions on Inf. Theory 59, 7434–7464 (2013).
[Crossref]

D. L. Donoho, A. Maleki, and A. Montanari, “Message-passing algorithms for compressed sensing,” Proc. Natl. Acad. Sci. 106, 18914–18919 (2009).
[Crossref] [PubMed]

Padgett, M.

M. Padgett, R. Aspden, G. Gibson, M. Edgar, and G. Spalding, “Ghost imaging,” Opt. Photon. News 27, 38–45 (2016).
[Crossref]

Padgett, M. J.

Podgorski, K.

S. Kotz, T. J. Kozubowski, and K. Podgorski, The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance(Birkhauser, 2001).
[Crossref]

Rangan, S.

S. Rangan, “Generalized approximate message passing for estimation with random linear mixing,” in The 2011 IEEE International Symposium on Information Theory Proceedings (ISIT), (2011), pp. 2168–2172.
[Crossref]

Scarcelli, G.

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94, 063601 (2005).
[Crossref] [PubMed]

Schniter, P.

S. Som and P. Schniter, “Compressive imaging using approximate message passing and a markov-tree prior,” IEEE Transactions on Signal Process. 60, 3439–3448 (2012).
[Crossref]

Shapiro, J. H.

Shih, Y.

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94, 063601 (2005).
[Crossref] [PubMed]

Shisheng, H.

Silberberg, Y.

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
[Crossref]

Som, S.

S. Som and P. Schniter, “Compressive imaging using approximate message passing and a markov-tree prior,” IEEE Transactions on Signal Process. 60, 3439–3448 (2012).
[Crossref]

Spalding, G.

M. Padgett, R. Aspden, G. Gibson, M. Edgar, and G. Spalding, “Ghost imaging,” Opt. Photon. News 27, 38–45 (2016).
[Crossref]

Sun, B.

Tan, J.

J. Tan, Y. Ma, and D. Baron, “Compressive imaging via approximate message passing with image denoising,” IEEE Transactions on Signal Process. 63, 2085–2092 (2015).
[Crossref]

Valencia, A.

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94, 063601 (2005).
[Crossref] [PubMed]

van den Berg, E.

E. van den Berg and M. P. Friedlander, “Probing the pareto frontier for basis pursuit solutions,” SIAM J. on Sci. Comput. 31, 890–912 (2008).
[Crossref]

Van der Vorst, H. A.

H. A. Van der Vorst, Iterative Krylov methods for large linear systems(Cambridge University, 2003).
[Crossref]

Welsh, S. S.

Wu, L.

Xu, W.

W. Gong, C. Zhao, H. Yu, M. Chen, W. Xu, and S. Han, “Three-dimensional ghost imaging lidar via sparsity constraint,” Sci. Reports. 6, 26133 (2016).
[Crossref]

Yao, X.

Yilmaz, H.

Yu, H.

W. Gong, C. Zhao, H. Yu, M. Chen, W. Xu, and S. Han, “Three-dimensional ghost imaging lidar via sparsity constraint,” Sci. Reports. 6, 26133 (2016).
[Crossref]

Yu, W.

Zhai, G.

Zhao, C.

W. Gong, C. Zhao, H. Yu, M. Chen, W. Xu, and S. Han, “Three-dimensional ghost imaging lidar via sparsity constraint,” Sci. Reports. 6, 26133 (2016).
[Crossref]

Adv. Opt. Photon. (1)

Appl. Phys. Lett. (1)

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95, 131110 (2009).
[Crossref]

IEEE Transactions on Image Process. (1)

J. Bioucas-Dias and M. Figueiredo, “Multiplicative noise removal using variable splitting and constrained optimization,” IEEE Transactions on Image Process. 19, 1720–1730 (2010).
[Crossref]

IEEE Transactions on Inf. Theory (1)

D. Donoho, A. Javanmard, and A. Montanari, “Information-theoretically optimal compressed sensing via spatial coupling and approximate message passing,” IEEE Transactions on Inf. Theory 59, 7434–7464 (2013).
[Crossref]

IEEE Transactions on Signal Process. (3)

C. Guo and M. Davies, “Near optimal compressed sensing without priors: Parametric sure approximate message passing,” IEEE Transactions on Signal Process. 63, 2130–2141 (2015).
[Crossref]

S. Som and P. Schniter, “Compressive imaging using approximate message passing and a markov-tree prior,” IEEE Transactions on Signal Process. 60, 3439–3448 (2012).
[Crossref]

J. Tan, Y. Ma, and D. Baron, “Compressive imaging via approximate message passing with image denoising,” IEEE Transactions on Signal Process. 63, 2085–2092 (2015).
[Crossref]

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

J. Probab. Stat. Sci. (1)

A. Ayebo and T. J. Kozubowski, “An asymmetric generalization of gaussian and laplace laws,” J. Probab. Stat. Sci. 1, 187–210 (2003).

Opt. Express (2)

Opt. Lett. (1)

Opt. Photon. News (1)

M. Padgett, R. Aspden, G. Gibson, M. Edgar, and G. Spalding, “Ghost imaging,” Opt. Photon. News 27, 38–45 (2016).
[Crossref]

Optica (1)

Phys. Rev. Lett. (3)

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104, 253603 (2010).
[Crossref] [PubMed]

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94, 063601 (2005).
[Crossref] [PubMed]

Y. Bromberg and H. Cao, “Generating non-rayleigh speckles with tailored intensity statistics,” Phys. Rev. Lett. 112, 213904 (2014).
[Crossref]

Proc. Natl. Acad. Sci. (1)

D. L. Donoho, A. Maleki, and A. Montanari, “Message-passing algorithms for compressed sensing,” Proc. Natl. Acad. Sci. 106, 18914–18919 (2009).
[Crossref] [PubMed]

Sci. Reports. (1)

W. Gong, C. Zhao, H. Yu, M. Chen, W. Xu, and S. Han, “Three-dimensional ghost imaging lidar via sparsity constraint,” Sci. Reports. 6, 26133 (2016).
[Crossref]

SIAM J. on Sci. Comput. (1)

E. van den Berg and M. P. Friedlander, “Probing the pareto frontier for basis pursuit solutions,” SIAM J. on Sci. Comput. 31, 890–912 (2008).
[Crossref]

Other (5)

E. van den Berg and M. P. Friedlander, “SPGL1: A solver for large-scale sparse reconstruction,” (2007). http://www.cs.ubc.ca/labs/scl/spgl1 .

S. Kotz, T. J. Kozubowski, and K. Podgorski, The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance(Birkhauser, 2001).
[Crossref]

H. A. Van der Vorst, Iterative Krylov methods for large linear systems(Cambridge University, 2003).
[Crossref]

S. Rangan, “Generalized approximate message passing for estimation with random linear mixing,” in The 2011 IEEE International Symposium on Information Theory Proceedings (ISIT), (2011), pp. 2168–2172.
[Crossref]

J. W. Goodman, Speckle phenomena in optics: theory and applications (Roberts and Company, 2007).

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Figures (10)

Fig. 1
Fig. 1 Two temporally successive speckles extracted from a practical GI system.
Fig. 2
Fig. 2 The histogram of a certain row of the matrix A ˜ and the corresponding distribution fittings.
Fig. 3
Fig. 3 The histogram of a certain column of the matrix A ˜ and the corresponding distribution fittings
Fig. 4
Fig. 4 The reconstructed images from semi-real experiments. From top to bottom (a–d), they are i.i.d. Gaussian measurements with wavelet preconditioning, semi-real measurements with wavelet preconditioning, i.i.d. Gaussian measurements with Hadamard preconditioning and semi-real measurements with Hadamard preconditioning, respectively. From left to right (1–5), they are 8000, 7000, 6000, 5000, 4000 measurements, respectively.
Fig. 5
Fig. 5 The PSNR of reconstructions with SPGL1 and AMP algorithms, respectively. The 6000 measurements are i.i.d. Gaussian and the true image is 128*128 House image.
Fig. 6
Fig. 6 The whole images constructed by all the slices. Panel (b) and (c) are reconstructed by Haar wavelet and Hadamard matrix preconditioning, respectively. Panel (a) shows the corresponding optical image of the buildings.
Fig. 7
Fig. 7 Four extracted slices, which are constructed by 8000 measurements. From top to bottom (a–c), they correspond to different methods: traditional correlation-based imaging, preconditioning with Haar wavelet and Hadamard matrix, respectively. From left to right (1–4), they are buildings located at different distances.
Fig. 8
Fig. 8 Four extracted slices, which are constructed by 4000 measurements. From top to bottom (a–c), they correspond to different methods: traditional correlation-based imaging, preconditioning with Haar wavelet and Hadamard matrix, respectively. From left to right (1–4), they are buildings located at different distances.
Fig. 9
Fig. 9 Four extracted slices, which are constructed by 2000 measurements. From top to bottom (a–c), they correspond to different methods: traditional correlation-based imaging, preconditioning with Haar wavelet and Hadamard matrix, respectively. From left to right (1–4), they are buildings located at different distances.
Fig. 10
Fig. 10 Four extracted slices, which are constructed by 1000 measurements. From top to bottom (a–c), they correspond to different methods: traditional correlation-based imaging, preconditioning with Haar wavelet and Hadamard matrix, respectively. From left to right (1–4), they are buildings located at different distances.

Tables (2)

Tables Icon

Table 1 Estimated parameters of the EP distribution

Tables Icon

Table 2 PSNR of the Reconstructed Images in the Simulated and Semi-real Experiments

Equations (19)

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

y = Ax + n
x ^ c ( n ) = 1 M m = 1 M y m A ( m , n ) 1 M m = 1 M y m 1 M m = 1 M A ( m , n )
1 2 y Ax 2 2 + λ x 1
x t + 1 = η ( x t + A T r t ; t h ) r t = y A x t + 1 M x 0 r t 1
η ( x ; t h ) = sign ( x ) max ( | x | t h , 0 )
m = 1 M A ( m , n ) = 0 and m = 1 M A ( m , n ) 2 = 1 ,
A ( m , n ) 2 1 M .
y = AH 1 N H T x + n = A ˜ x ˜ + n
σ ^ ( α ) = ( α ( x ¯ α + x ¯ α ) α 2 ( α + 1 ) ( ( x ¯ α + ) 1 α + 1 + ( x ¯ α ) 1 α + 1 ) ) 1 α , κ ^ ( α ) = ( x ¯ α x ¯ α + ) 1 2 ( α + 1 ) ,
L ( α ) = α Γ ( 1 α ) κ ^ ( α ) 1 + κ ^ 2 ( α ) 1 σ ^ ( α ) exp ( 1 α ) ;
M C ( B ) δ 2 K log 3 K log N ,
E [ δ K ( A ˜ ) ] δ .
ψ ( t ) = exp ( i θ t ) ( 1 1 + i 2 2 σ κ t ) τ ( 1 1 i 2 2 κ σ t ) τ
h ( x ) = 2 exp ( 2 2 σ ( 1 / κ κ ) ( x θ ) ) Γ ( τ ) π σ τ + 1 / 2 ( 2 | x θ | κ + 1 / κ ) τ 1 / 2 × K τ 1 / 2 ( 2 2 σ ( 1 / κ + κ ) | x θ | ) , x 0
X = θ + σ 2 ( 1 κ G 1 κ G 2 )
g ( x ) = 1 Γ ( τ ) x τ 1 e x , τ > 0 , x 0
f ( x ) = α σ Γ ( 1 α ) κ 1 + κ 2 × exp ( κ α σ α [ ( x θ ) + ] α 1 σ α κ α [ ( x θ ) ] α )
u + = { u , u 0 , 0 , u < 0 , and u = { u , u 0 , 0 , u > 0 .
A ˜ ( m , n ) = j = 1 N A ( m , j ) H ( j , n ) = j Ω + A ( m , j ) j Ω A ( m , j )

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