Abstract

The singular value decomposition ghost imaging (SVDGI) is proposed to enhance the fidelity of computational ghost imaging (GI) by constructing a measurement matrix using singular value decomposition (SVD) transform. After SVD transform on a random matrix, the non-zero elements of singular value matrix are all made equal to 1.0, then the measurement matrix is acquired by inverse SVD transform. Eventually, the original objects can be reconstructed by multiplying the transposition of the matrix by a series of collected intensity. SVDGI enables the reconstruction of an N-pixel image using much less than N measurements, and perfectly reconstructs original object with N measurements. Both the simulated and the optical experimental results show that SVDGI always costs less time to accomplish better works. Firstly, it is at least ten times faster than GI and differential ghost imaging (DGI), and several orders of magnitude faster than pseudo-inverse ghost imaging (PGI). Secondly, in comparison with GI, the clarity of SVDGI can get sharply improved, and it is more robust than the other three methods so that it yields a clearer image in the noisy environment.

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

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References

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2018 (1)

K. Xu, “Monolithically integrated Si gate-controlled light-emitting device: science and properties,” J. Opt. 20(2), 024014 (2018).
[Crossref]

2017 (6)

B. Liu, Z. Yang, X. Liu, and L. Wu, “Coloured computational imaging with single-pixel detectors based on a 2D discrete cosine transform,” J. Mod. Opt. 64(3), 259–264 (2017).
[Crossref]

M. Li, R. He, Q. Chen, G. Gu, and W. Zhang, “Research on ghost imaging method based on wavelet transform,” J. Opt. 19(9), 095202 (2017).
[Crossref]

M. Lyu, W. Wang, H. Wang, H. Wang, G. Li, N. Chen, and G. Situ, “Deep-learning-based ghost imaging,” Sci. Rep. 7(1), 17865 (2017).
[Crossref] [PubMed]

Y. Wang, Y. Liu, J. Suo, G. Situ, C. Qiao, and Q. Dai, “High speed computational ghost imaging via spatial sweeping,” Sci. Rep. 7, 45325 (2017).
[Crossref] [PubMed]

M. Alemohammad, J. R. Stroud, B. T. Bosworth, and M. A. Foster, “High-speed all-optical Haar wavelet transform for real-time image compression,” Opt. Express 25(9), 9802–9811 (2017).
[Crossref] [PubMed]

Z. Zhang, X. Wang, G. Zheng, and J. Zhong, “Hadamard single-pixel imaging versus Fourier single-pixel imaging,” Opt. Express 25(16), 19619–19639 (2017).
[Crossref] [PubMed]

2016 (8)

L. Bian, J. Suo, G. Situ, Z. Li, J. Fan, F. Chen, and Q. Dai, “Multispectral imaging using a single bucket detector,” Sci. Rep. 6(1), 24752 (2016).
[Crossref] [PubMed]

K. Xu, “Integrated silicon directly modulated light source using p-well in standard CMOS technology,” IEEE Sens. J. 16(16), 6184–6191 (2016).
[Crossref]

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

L. Bian, J. Suo, X. Hu, F. Chen, and Q. Dai, “Efficient single pixel imaging in Fourier space,” J. Opt. 18(8), 085704 (2016).
[Crossref]

M. J. Sun, M. P. Edgar, D. B. Phillips, G. M. Gibson, and M. J. Padgett, “Improving the signal-to-noise ratio of single-pixel imaging using digital microscanning,” Opt. Express 24(10), 10476–10485 (2016).
[Crossref] [PubMed]

Z. Zhang and J. Zhong, “Three-dimensional single-pixel imaging with far fewer measurements than effective image pixels,” Opt. Lett. 41(11), 2497–2500 (2016).
[Crossref] [PubMed]

L. Wang and S. Zhao, “Fast reconstructed and high-quality ghost imaging with fast Walsh-Hadamard transform,” Photon. Res. 4(6), 240–244 (2016).
[Crossref]

W. Chen, “Optical cryptosystem based on single-pixel encoding using the modified Gerchberg-Saxton algorithm with a cascaded structure,” J. Opt. Soc. Am. A 33(12), 2305–2311 (2016).
[Crossref] [PubMed]

2015 (5)

Z. Zhang, X. Ma, and J. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6(1), 6225 (2015).
[Crossref] [PubMed]

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5(1), 10669 (2015).
[Crossref] [PubMed]

S. S. Welsh, M. P. Edgar, R. Bowman, B. Sun, and M. J. Padgett, “Near video-rate linear Stokes imaging with single-pixel detectors,” J. Opt. 17(2), 025705 (2015).
[Crossref]

W. Gong, “High-resolution pseudo-inverse ghost imaging,” Photon. Res. 3(5), 234–237 (2015).
[Crossref]

W. Wang, X. Hu, J. Liu, S. Zhang, J. Suo, and G. Situ, “Gerchberg-Saxton-like ghost imaging,” Opt. Express 23(22), 28416–28422 (2015).
[Crossref] [PubMed]

2014 (1)

2013 (4)

D. S. Fan, X. F. Meng, Y. R. Wang, X. L. Yang, X. Peng, W. Q. He, G. Y. Dong, and H. Y. Chen, “Optical information encoding and image watermarking scheme based on phase-shifting interferometry and singular value decomposition,” J. Mod. Opt. 60(9), 749–756 (2013).
[Crossref]

L. Chen, D. Zhao, and F. Ge, “Image encryption based on singular value decomposition and arnold transform in fractional domain,” Opt. Commun. 291(291), 98–103 (2013).
[Crossref]

W. Chen and X. Chen, “Ghost imaging for three-dimensional optical security,” Appl. Phys. Lett. 103(22), 221106 (2013).
[Crossref]

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3d computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013).
[Crossref] [PubMed]

2012 (1)

J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” Quantum Inform. Process. 11(4), 949–993 (2012).
[Crossref]

2011 (1)

C. C. Lai, “An improved SVD-based watermarking scheme using human visual characteristics,” Opt. Commun. 284(4), 938–944 (2011).
[Crossref]

2010 (2)

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104(25), 253603 (2010).
[Crossref] [PubMed]

W. Gong and S. Han, “A method to improve the visibility of ghost images obtained by thermal light,” Phys. Lett. A 374(8), 1005–1008 (2010).
[Crossref]

2009 (1)

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), 1–2 (2009).

2008 (1)

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 1744–1747 (2008).

2005 (3)

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94(18), 183602 (2005).
[Crossref] [PubMed]

M. D’Angelo and Y. H. Shih, “Quantum imaging,” Laser Phys. Lett. 2(12), 567–596 (2005).
[Crossref]

D. Zhang, Y. H. Zhai, L. A. Wu, and X. H. Chen, “Correlated two-photon imaging with true thermal light,” Opt. Lett. 30(18), 2354–2356 (2005).
[Crossref] [PubMed]

2004 (3)

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A 70(1), 013802 (2004).
[Crossref]

R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92(3), 033601 (2004).
[Crossref] [PubMed]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93(9), 093602 (2004).
[Crossref] [PubMed]

2002 (1)

R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Two-Photon coincidence imaging with a classical source,” Phys. Rev. Lett. 89(11), 113601 (2002).
[Crossref] [PubMed]

1995 (2)

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52(5), R3429–R3432 (1995).
[Crossref] [PubMed]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74(18), 3600–3603 (1995).
[Crossref] [PubMed]

Alemohammad, M.

Bache, M.

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94(18), 183602 (2005).
[Crossref] [PubMed]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A 70(1), 013802 (2004).
[Crossref]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93(9), 093602 (2004).
[Crossref] [PubMed]

Bennink, R. S.

R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92(3), 033601 (2004).
[Crossref] [PubMed]

R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Two-Photon coincidence imaging with a classical source,” Phys. Rev. Lett. 89(11), 113601 (2002).
[Crossref] [PubMed]

Bentley, S. J.

R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92(3), 033601 (2004).
[Crossref] [PubMed]

R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Two-Photon coincidence imaging with a classical source,” Phys. Rev. Lett. 89(11), 113601 (2002).
[Crossref] [PubMed]

Bian, L.

L. Bian, J. Suo, G. Situ, Z. Li, J. Fan, F. Chen, and Q. Dai, “Multispectral imaging using a single bucket detector,” Sci. Rep. 6(1), 24752 (2016).
[Crossref] [PubMed]

L. Bian, J. Suo, X. Hu, F. Chen, and Q. Dai, “Efficient single pixel imaging in Fourier space,” J. Opt. 18(8), 085704 (2016).
[Crossref]

Bosworth, B. T.

Bowman, A.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3d computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013).
[Crossref] [PubMed]

Bowman, R.

S. S. Welsh, M. P. Edgar, R. Bowman, B. Sun, and M. J. Padgett, “Near video-rate linear Stokes imaging with single-pixel detectors,” J. Opt. 17(2), 025705 (2015).
[Crossref]

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3d computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013).
[Crossref] [PubMed]

Bowman, R. W.

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5(1), 10669 (2015).
[Crossref] [PubMed]

Boyd, R. W.

J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” Quantum Inform. Process. 11(4), 949–993 (2012).
[Crossref]

R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92(3), 033601 (2004).
[Crossref] [PubMed]

R. S. Bennink, S. J. Bentley, and R. W. Boyd, “Two-Photon coincidence imaging with a classical source,” Phys. Rev. Lett. 89(11), 113601 (2002).
[Crossref] [PubMed]

Brambilla, E.

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94(18), 183602 (2005).
[Crossref] [PubMed]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93(9), 093602 (2004).
[Crossref] [PubMed]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A 70(1), 013802 (2004).
[Crossref]

Bromberg, Y.

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 1744–1747 (2008).

Cao, J.

Chen, F.

L. Bian, J. Suo, X. Hu, F. Chen, and Q. Dai, “Efficient single pixel imaging in Fourier space,” J. Opt. 18(8), 085704 (2016).
[Crossref]

L. Bian, J. Suo, G. Situ, Z. Li, J. Fan, F. Chen, and Q. Dai, “Multispectral imaging using a single bucket detector,” Sci. Rep. 6(1), 24752 (2016).
[Crossref] [PubMed]

Chen, H. Y.

D. S. Fan, X. F. Meng, Y. R. Wang, X. L. Yang, X. Peng, W. Q. He, G. Y. Dong, and H. Y. Chen, “Optical information encoding and image watermarking scheme based on phase-shifting interferometry and singular value decomposition,” J. Mod. Opt. 60(9), 749–756 (2013).
[Crossref]

Chen, L.

L. Chen, D. Zhao, and F. Ge, “Image encryption based on singular value decomposition and arnold transform in fractional domain,” Opt. Commun. 291(291), 98–103 (2013).
[Crossref]

Chen, N.

M. Lyu, W. Wang, H. Wang, H. Wang, G. Li, N. Chen, and G. Situ, “Deep-learning-based ghost imaging,” Sci. Rep. 7(1), 17865 (2017).
[Crossref] [PubMed]

Chen, Q.

M. Li, R. He, Q. Chen, G. Gu, and W. Zhang, “Research on ghost imaging method based on wavelet transform,” J. Opt. 19(9), 095202 (2017).
[Crossref]

Chen, W.

Chen, X.

W. Chen and X. Chen, “Ghost imaging for three-dimensional optical security,” Appl. Phys. Lett. 103(22), 221106 (2013).
[Crossref]

Chen, X. H.

D’Angelo, M.

M. D’Angelo and Y. H. Shih, “Quantum imaging,” Laser Phys. Lett. 2(12), 567–596 (2005).
[Crossref]

Dai, Q.

Y. Wang, Y. Liu, J. Suo, G. Situ, C. Qiao, and Q. Dai, “High speed computational ghost imaging via spatial sweeping,” Sci. Rep. 7, 45325 (2017).
[Crossref] [PubMed]

L. Bian, J. Suo, G. Situ, Z. Li, J. Fan, F. Chen, and Q. Dai, “Multispectral imaging using a single bucket detector,” Sci. Rep. 6(1), 24752 (2016).
[Crossref] [PubMed]

L. Bian, J. Suo, X. Hu, F. Chen, and Q. Dai, “Efficient single pixel imaging in Fourier space,” J. Opt. 18(8), 085704 (2016).
[Crossref]

Dong, G. Y.

D. S. Fan, X. F. Meng, Y. R. Wang, X. L. Yang, X. Peng, W. Q. He, G. Y. Dong, and H. Y. Chen, “Optical information encoding and image watermarking scheme based on phase-shifting interferometry and singular value decomposition,” J. Mod. Opt. 60(9), 749–756 (2013).
[Crossref]

Edgar, M. P.

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

M. J. Sun, M. P. Edgar, D. B. Phillips, G. M. Gibson, and M. J. Padgett, “Improving the signal-to-noise ratio of single-pixel imaging using digital microscanning,” Opt. Express 24(10), 10476–10485 (2016).
[Crossref] [PubMed]

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5(1), 10669 (2015).
[Crossref] [PubMed]

S. S. Welsh, M. P. Edgar, R. Bowman, B. Sun, and M. J. Padgett, “Near video-rate linear Stokes imaging with single-pixel detectors,” J. Opt. 17(2), 025705 (2015).
[Crossref]

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3d computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013).
[Crossref] [PubMed]

Fan, D. S.

D. S. Fan, X. F. Meng, Y. R. Wang, X. L. Yang, X. Peng, W. Q. He, G. Y. Dong, and H. Y. Chen, “Optical information encoding and image watermarking scheme based on phase-shifting interferometry and singular value decomposition,” J. Mod. Opt. 60(9), 749–756 (2013).
[Crossref]

Fan, J.

L. Bian, J. Suo, G. Situ, Z. Li, J. Fan, F. Chen, and Q. Dai, “Multispectral imaging using a single bucket detector,” Sci. Rep. 6(1), 24752 (2016).
[Crossref] [PubMed]

Ferri, F.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104(25), 253603 (2010).
[Crossref] [PubMed]

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94(18), 183602 (2005).
[Crossref] [PubMed]

Foster, M. A.

Gao, F.

Gatti, A.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104(25), 253603 (2010).
[Crossref] [PubMed]

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94(18), 183602 (2005).
[Crossref] [PubMed]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93(9), 093602 (2004).
[Crossref] [PubMed]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A 70(1), 013802 (2004).
[Crossref]

Ge, F.

L. Chen, D. Zhao, and F. Ge, “Image encryption based on singular value decomposition and arnold transform in fractional domain,” Opt. Commun. 291(291), 98–103 (2013).
[Crossref]

Gibson, G. M.

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

M. J. Sun, M. P. Edgar, D. B. Phillips, G. M. Gibson, and M. J. Padgett, “Improving the signal-to-noise ratio of single-pixel imaging using digital microscanning,” Opt. Express 24(10), 10476–10485 (2016).
[Crossref] [PubMed]

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5(1), 10669 (2015).
[Crossref] [PubMed]

Gong, W.

W. Gong, “High-resolution pseudo-inverse ghost imaging,” Photon. Res. 3(5), 234–237 (2015).
[Crossref]

W. Gong and S. Han, “A method to improve the visibility of ghost images obtained by thermal light,” Phys. Lett. A 374(8), 1005–1008 (2010).
[Crossref]

Gu, G.

M. Li, R. He, Q. Chen, G. Gu, and W. Zhang, “Research on ghost imaging method based on wavelet transform,” J. Opt. 19(9), 095202 (2017).
[Crossref]

Guan, J.

Guo, S.

Han, S.

W. Gong and S. Han, “A method to improve the visibility of ghost images obtained by thermal light,” Phys. Lett. A 374(8), 1005–1008 (2010).
[Crossref]

He, R.

M. Li, R. He, Q. Chen, G. Gu, and W. Zhang, “Research on ghost imaging method based on wavelet transform,” J. Opt. 19(9), 095202 (2017).
[Crossref]

He, W. Q.

D. S. Fan, X. F. Meng, Y. R. Wang, X. L. Yang, X. Peng, W. Q. He, G. Y. Dong, and H. Y. Chen, “Optical information encoding and image watermarking scheme based on phase-shifting interferometry and singular value decomposition,” J. Mod. Opt. 60(9), 749–756 (2013).
[Crossref]

Howell, J. C.

R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92(3), 033601 (2004).
[Crossref] [PubMed]

Hu, X.

L. Bian, J. Suo, X. Hu, F. Chen, and Q. Dai, “Efficient single pixel imaging in Fourier space,” J. Opt. 18(8), 085704 (2016).
[Crossref]

W. Wang, X. Hu, J. Liu, S. Zhang, J. Suo, and G. Situ, “Gerchberg-Saxton-like ghost imaging,” Opt. Express 23(22), 28416–28422 (2015).
[Crossref] [PubMed]

Katz, O.

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 1744–1747 (2008).

Klyshko, D. N.

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74(18), 3600–3603 (1995).
[Crossref] [PubMed]

Lai, C. C.

C. C. Lai, “An improved SVD-based watermarking scheme using human visual characteristics,” Opt. Commun. 284(4), 938–944 (2011).
[Crossref]

Lamb, R.

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

Li, G.

M. Lyu, W. Wang, H. Wang, H. Wang, G. Li, N. Chen, and G. Situ, “Deep-learning-based ghost imaging,” Sci. Rep. 7(1), 17865 (2017).
[Crossref] [PubMed]

Li, M.

M. Li, R. He, Q. Chen, G. Gu, and W. Zhang, “Research on ghost imaging method based on wavelet transform,” J. Opt. 19(9), 095202 (2017).
[Crossref]

Li, Z.

L. Bian, J. Suo, G. Situ, Z. Li, J. Fan, F. Chen, and Q. Dai, “Multispectral imaging using a single bucket detector,” Sci. Rep. 6(1), 24752 (2016).
[Crossref] [PubMed]

Liu, B.

B. Liu, Z. Yang, X. Liu, and L. Wu, “Coloured computational imaging with single-pixel detectors based on a 2D discrete cosine transform,” J. Mod. Opt. 64(3), 259–264 (2017).
[Crossref]

Liu, J.

Liu, X.

B. Liu, Z. Yang, X. Liu, and L. Wu, “Coloured computational imaging with single-pixel detectors based on a 2D discrete cosine transform,” J. Mod. Opt. 64(3), 259–264 (2017).
[Crossref]

Liu, Y.

Y. Wang, Y. Liu, J. Suo, G. Situ, C. Qiao, and Q. Dai, “High speed computational ghost imaging via spatial sweeping,” Sci. Rep. 7, 45325 (2017).
[Crossref] [PubMed]

Lugiato, L. A.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104(25), 253603 (2010).
[Crossref] [PubMed]

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94(18), 183602 (2005).
[Crossref] [PubMed]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A 70(1), 013802 (2004).
[Crossref]

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93(9), 093602 (2004).
[Crossref] [PubMed]

Lyu, M.

M. Lyu, W. Wang, H. Wang, H. Wang, G. Li, N. Chen, and G. Situ, “Deep-learning-based ghost imaging,” Sci. Rep. 7(1), 17865 (2017).
[Crossref] [PubMed]

Ma, X.

Z. Zhang, X. Ma, and J. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6(1), 6225 (2015).
[Crossref] [PubMed]

Magatti, D.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104(25), 253603 (2010).
[Crossref] [PubMed]

F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94(18), 183602 (2005).
[Crossref] [PubMed]

Meng, X. F.

D. S. Fan, X. F. Meng, Y. R. Wang, X. L. Yang, X. Peng, W. Q. He, G. Y. Dong, and H. Y. Chen, “Optical information encoding and image watermarking scheme based on phase-shifting interferometry and singular value decomposition,” J. Mod. Opt. 60(9), 749–756 (2013).
[Crossref]

Mitchell, K. J.

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5(1), 10669 (2015).
[Crossref] [PubMed]

Padgett, M. J.

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

M. J. Sun, M. P. Edgar, D. B. Phillips, G. M. Gibson, and M. J. Padgett, “Improving the signal-to-noise ratio of single-pixel imaging using digital microscanning,” Opt. Express 24(10), 10476–10485 (2016).
[Crossref] [PubMed]

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5(1), 10669 (2015).
[Crossref] [PubMed]

S. S. Welsh, M. P. Edgar, R. Bowman, B. Sun, and M. J. Padgett, “Near video-rate linear Stokes imaging with single-pixel detectors,” J. Opt. 17(2), 025705 (2015).
[Crossref]

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3d computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013).
[Crossref] [PubMed]

Peng, X.

D. S. Fan, X. F. Meng, Y. R. Wang, X. L. Yang, X. Peng, W. Q. He, G. Y. Dong, and H. Y. Chen, “Optical information encoding and image watermarking scheme based on phase-shifting interferometry and singular value decomposition,” J. Mod. Opt. 60(9), 749–756 (2013).
[Crossref]

Phillips, D. B.

Pittman, T. B.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52(5), R3429–R3432 (1995).
[Crossref] [PubMed]

Qiao, C.

Y. Wang, Y. Liu, J. Suo, G. Situ, C. Qiao, and Q. Dai, “High speed computational ghost imaging via spatial sweeping,” Sci. Rep. 7, 45325 (2017).
[Crossref] [PubMed]

Radwell, N.

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5(1), 10669 (2015).
[Crossref] [PubMed]

Sergienko, A. V.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52(5), R3429–R3432 (1995).
[Crossref] [PubMed]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74(18), 3600–3603 (1995).
[Crossref] [PubMed]

Shapiro, J. H.

J. H. Shapiro and R. W. Boyd, “The physics of ghost imaging,” Quantum Inform. Process. 11(4), 949–993 (2012).
[Crossref]

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), 1–2 (2009).

Shih, Y. H.

M. D’Angelo and Y. H. Shih, “Quantum imaging,” Laser Phys. Lett. 2(12), 567–596 (2005).
[Crossref]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74(18), 3600–3603 (1995).
[Crossref] [PubMed]

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52(5), R3429–R3432 (1995).
[Crossref] [PubMed]

Silberberg, Y.

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 1744–1747 (2008).

Situ, G.

M. Lyu, W. Wang, H. Wang, H. Wang, G. Li, N. Chen, and G. Situ, “Deep-learning-based ghost imaging,” Sci. Rep. 7(1), 17865 (2017).
[Crossref] [PubMed]

Y. Wang, Y. Liu, J. Suo, G. Situ, C. Qiao, and Q. Dai, “High speed computational ghost imaging via spatial sweeping,” Sci. Rep. 7, 45325 (2017).
[Crossref] [PubMed]

L. Bian, J. Suo, G. Situ, Z. Li, J. Fan, F. Chen, and Q. Dai, “Multispectral imaging using a single bucket detector,” Sci. Rep. 6(1), 24752 (2016).
[Crossref] [PubMed]

W. Wang, X. Hu, J. Liu, S. Zhang, J. Suo, and G. Situ, “Gerchberg-Saxton-like ghost imaging,” Opt. Express 23(22), 28416–28422 (2015).
[Crossref] [PubMed]

Strekalov, D. V.

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52(5), R3429–R3432 (1995).
[Crossref] [PubMed]

D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon “ghost” interference and diffraction,” Phys. Rev. Lett. 74(18), 3600–3603 (1995).
[Crossref] [PubMed]

Stroud, J. R.

Sun, B.

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

S. S. Welsh, M. P. Edgar, R. Bowman, B. Sun, and M. J. Padgett, “Near video-rate linear Stokes imaging with single-pixel detectors,” J. Opt. 17(2), 025705 (2015).
[Crossref]

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5(1), 10669 (2015).
[Crossref] [PubMed]

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3d computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013).
[Crossref] [PubMed]

Sun, M. J.

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

M. J. Sun, M. P. Edgar, D. B. Phillips, G. M. Gibson, and M. J. Padgett, “Improving the signal-to-noise ratio of single-pixel imaging using digital microscanning,” Opt. Express 24(10), 10476–10485 (2016).
[Crossref] [PubMed]

Suo, J.

Y. Wang, Y. Liu, J. Suo, G. Situ, C. Qiao, and Q. Dai, “High speed computational ghost imaging via spatial sweeping,” Sci. Rep. 7, 45325 (2017).
[Crossref] [PubMed]

L. Bian, J. Suo, G. Situ, Z. Li, J. Fan, F. Chen, and Q. Dai, “Multispectral imaging using a single bucket detector,” Sci. Rep. 6(1), 24752 (2016).
[Crossref] [PubMed]

L. Bian, J. Suo, X. Hu, F. Chen, and Q. Dai, “Efficient single pixel imaging in Fourier space,” J. Opt. 18(8), 085704 (2016).
[Crossref]

W. Wang, X. Hu, J. Liu, S. Zhang, J. Suo, and G. Situ, “Gerchberg-Saxton-like ghost imaging,” Opt. Express 23(22), 28416–28422 (2015).
[Crossref] [PubMed]

Vittert, L. E.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3d computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013).
[Crossref] [PubMed]

Wang, H.

M. Lyu, W. Wang, H. Wang, H. Wang, G. Li, N. Chen, and G. Situ, “Deep-learning-based ghost imaging,” Sci. Rep. 7(1), 17865 (2017).
[Crossref] [PubMed]

M. Lyu, W. Wang, H. Wang, H. Wang, G. Li, N. Chen, and G. Situ, “Deep-learning-based ghost imaging,” Sci. Rep. 7(1), 17865 (2017).
[Crossref] [PubMed]

Wang, L.

Wang, W.

M. Lyu, W. Wang, H. Wang, H. Wang, G. Li, N. Chen, and G. Situ, “Deep-learning-based ghost imaging,” Sci. Rep. 7(1), 17865 (2017).
[Crossref] [PubMed]

W. Wang, X. Hu, J. Liu, S. Zhang, J. Suo, and G. Situ, “Gerchberg-Saxton-like ghost imaging,” Opt. Express 23(22), 28416–28422 (2015).
[Crossref] [PubMed]

Wang, X.

Wang, Y.

Y. Wang, Y. Liu, J. Suo, G. Situ, C. Qiao, and Q. Dai, “High speed computational ghost imaging via spatial sweeping,” Sci. Rep. 7, 45325 (2017).
[Crossref] [PubMed]

Wang, Y. R.

D. S. Fan, X. F. Meng, Y. R. Wang, X. L. Yang, X. Peng, W. Q. He, G. Y. Dong, and H. Y. Chen, “Optical information encoding and image watermarking scheme based on phase-shifting interferometry and singular value decomposition,” J. Mod. Opt. 60(9), 749–756 (2013).
[Crossref]

Welsh, S.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3d computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013).
[Crossref] [PubMed]

Welsh, S. S.

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5(1), 10669 (2015).
[Crossref] [PubMed]

S. S. Welsh, M. P. Edgar, R. Bowman, B. Sun, and M. J. Padgett, “Near video-rate linear Stokes imaging with single-pixel detectors,” J. Opt. 17(2), 025705 (2015).
[Crossref]

Wu, L.

B. Liu, Z. Yang, X. Liu, and L. Wu, “Coloured computational imaging with single-pixel detectors based on a 2D discrete cosine transform,” J. Mod. Opt. 64(3), 259–264 (2017).
[Crossref]

Wu, L. A.

Xu, K.

K. Xu, “Monolithically integrated Si gate-controlled light-emitting device: science and properties,” J. Opt. 20(2), 024014 (2018).
[Crossref]

K. Xu, “Integrated silicon directly modulated light source using p-well in standard CMOS technology,” IEEE Sens. J. 16(16), 6184–6191 (2016).
[Crossref]

Yang, X. L.

D. S. Fan, X. F. Meng, Y. R. Wang, X. L. Yang, X. Peng, W. Q. He, G. Y. Dong, and H. Y. Chen, “Optical information encoding and image watermarking scheme based on phase-shifting interferometry and singular value decomposition,” J. Mod. Opt. 60(9), 749–756 (2013).
[Crossref]

Yang, Z.

B. Liu, Z. Yang, X. Liu, and L. Wu, “Coloured computational imaging with single-pixel detectors based on a 2D discrete cosine transform,” J. Mod. Opt. 64(3), 259–264 (2017).
[Crossref]

Zhai, Y. H.

Zhang, C.

Zhang, D.

Zhang, S.

Zhang, W.

M. Li, R. He, Q. Chen, G. Gu, and W. Zhang, “Research on ghost imaging method based on wavelet transform,” J. Opt. 19(9), 095202 (2017).
[Crossref]

Zhang, Z.

Zhao, D.

L. Chen, D. Zhao, and F. Ge, “Image encryption based on singular value decomposition and arnold transform in fractional domain,” Opt. Commun. 291(291), 98–103 (2013).
[Crossref]

Zhao, S.

Zheng, G.

Zhong, J.

Appl. Phys. Lett. (1)

W. Chen and X. Chen, “Ghost imaging for three-dimensional optical security,” Appl. Phys. Lett. 103(22), 221106 (2013).
[Crossref]

IEEE Sens. J. (1)

K. Xu, “Integrated silicon directly modulated light source using p-well in standard CMOS technology,” IEEE Sens. J. 16(16), 6184–6191 (2016).
[Crossref]

J. Mod. Opt. (2)

B. Liu, Z. Yang, X. Liu, and L. Wu, “Coloured computational imaging with single-pixel detectors based on a 2D discrete cosine transform,” J. Mod. Opt. 64(3), 259–264 (2017).
[Crossref]

D. S. Fan, X. F. Meng, Y. R. Wang, X. L. Yang, X. Peng, W. Q. He, G. Y. Dong, and H. Y. Chen, “Optical information encoding and image watermarking scheme based on phase-shifting interferometry and singular value decomposition,” J. Mod. Opt. 60(9), 749–756 (2013).
[Crossref]

J. Opt. (4)

M. Li, R. He, Q. Chen, G. Gu, and W. Zhang, “Research on ghost imaging method based on wavelet transform,” J. Opt. 19(9), 095202 (2017).
[Crossref]

K. Xu, “Monolithically integrated Si gate-controlled light-emitting device: science and properties,” J. Opt. 20(2), 024014 (2018).
[Crossref]

L. Bian, J. Suo, X. Hu, F. Chen, and Q. Dai, “Efficient single pixel imaging in Fourier space,” J. Opt. 18(8), 085704 (2016).
[Crossref]

S. S. Welsh, M. P. Edgar, R. Bowman, B. Sun, and M. J. Padgett, “Near video-rate linear Stokes imaging with single-pixel detectors,” J. Opt. 17(2), 025705 (2015).
[Crossref]

J. Opt. Soc. Am. A (1)

Laser Phys. Lett. (1)

M. D’Angelo and Y. H. Shih, “Quantum imaging,” Laser Phys. Lett. 2(12), 567–596 (2005).
[Crossref]

Nat. Commun. (2)

Z. Zhang, X. Ma, and J. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6(1), 6225 (2015).
[Crossref] [PubMed]

M. J. Sun, M. P. Edgar, G. M. Gibson, B. Sun, N. Radwell, R. Lamb, and M. J. Padgett, “Single-pixel three-dimensional imaging with time-based depth resolution,” Nat. Commun. 7, 12010 (2016).
[Crossref] [PubMed]

Opt. Commun. (2)

L. Chen, D. Zhao, and F. Ge, “Image encryption based on singular value decomposition and arnold transform in fractional domain,” Opt. Commun. 291(291), 98–103 (2013).
[Crossref]

C. C. Lai, “An improved SVD-based watermarking scheme using human visual characteristics,” Opt. Commun. 284(4), 938–944 (2011).
[Crossref]

Opt. Express (5)

Opt. Lett. (2)

Photon. Res. (2)

Phys. Lett. A (1)

W. Gong and S. Han, “A method to improve the visibility of ghost images obtained by thermal light,” Phys. Lett. A 374(8), 1005–1008 (2010).
[Crossref]

Phys. Rev. A (4)

T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A 52(5), R3429–R3432 (1995).
[Crossref] [PubMed]

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), 1–2 (2009).

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 1744–1747 (2008).

A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A 70(1), 013802 (2004).
[Crossref]

Phys. Rev. Lett. (6)

R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. 92(3), 033601 (2004).
[Crossref] [PubMed]

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[Crossref] [PubMed]

Quantum Inform. Process. (1)

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[Crossref]

Sci. Rep. (4)

M. P. Edgar, G. M. Gibson, R. W. Bowman, B. Sun, N. Radwell, K. J. Mitchell, S. S. Welsh, and M. J. Padgett, “Simultaneous real-time visible and infrared video with single-pixel detectors,” Sci. Rep. 5(1), 10669 (2015).
[Crossref] [PubMed]

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[Crossref] [PubMed]

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[Crossref] [PubMed]

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[Crossref] [PubMed]

Science (1)

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3d computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013).
[Crossref] [PubMed]

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Figures (7)

Fig. 1
Fig. 1 Schematic diagram of the experimental setup. A cup represents the object which can be arbitrarily replaced by anything.
Fig. 2
Fig. 2 Product comparison. (a) ФT Ф; (b) ФTsvd Фsvd ; (c) two sectional views.
Fig. 3
Fig. 3 Simulated results comparison when η = 0.8.
Fig. 4
Fig. 4 Relationship between CC and the compression ratio η. Four results are restored by (a): SVDGI; (b): PGI; (c): DGI; (d): GI.
Fig. 5
Fig. 5 The relationship between CC and the SNR of B. Four results are restored by (a): SVDGI; (b): PGI; (c): DGI; (d): GI.
Fig. 6
Fig. 6 Reconstruction time comparison.
Fig. 7
Fig. 7 Comparison of the experimental results for binary and 3D objects.

Equations (21)

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Φ=[ I 1 (1,1) I 2 (1,1) I M (1,1) I 1 (1,2) I 2 (1,2) I M (1,2) I 1 (p,p) I 2 (p,p) I M (p,p) ],
[ B 1 B 2 B M ]=Φ[ O(1,1) O(1,2) O(p,p) ].
O ^ (x,y)= 1 M i=1 M ( B i B) I i (x,y),
O ^ (x,y)= 1 M [ I 1 (1,1) I 1 (1,2) I 1 (p,p) I 2 (1,1) I 2 (1,2) I 2 (p,p) I M (1,1) I M (1,2) I M (p,p) ][ B 1 B 2 B M ]B[ I(1,1) I(1,2) I(p,p) ],
O ^ (x,y)= 1 M Φ T Φ[ O(1,1) O(1,2) O(p,p) ],
Φ svd = U M×M [ Λ M×M 0] M×N V T N×N ,
O ^ (x,y)= 1 M V [ Λ M×M 0 0 0 ] N×N V T O(x,y),
Φ T Φ= E A S 1 = E O S 1 ,
Φ T svd Φ Svd = E A S 2 = E O S 2 ,
SNR B ' GP = i=1 M | B GP | 2 i=1 M | F 1 | 2 = O T Φ T ΦO F 1 T F 1 ,
SNR B ' S = i=1 M | B S | 2 i=1 M | F 2 | 2 = O T Φ T svd Φ svd O F 2 T F 2 ,
F 1 T F 1 >> F 2 T F 2 = O T Φ svd T Φ svd O F 1 T F 1 O T Φ T ΦO ,
F 2 T F 2 = O T V[ Λ 0 0 0 ] V T O F 1 T F 1 O T V[ λ 1 2 0 λ m 2 0 0 ] V T O ,
Φ=U[ λ 1 0 λ m ] V T ,
Φ =V[ λ 1 1 λ m 1 0 ] U T ,
SNR PGI = i=1 M | Φ B GP | 2 i=1 M | Φ F 1 | 2 = ( Φ ΦO) T Φ ΦO ( Φ F 1 ) T Φ F 1 = O T V[ Λ 0 0 0 ] V T O F 1 T U[ λ 1 2 λ m 2 ] U T F 1 ,
SNR SVDGI = i=1 M | Φ T svd B S | 2 i=1 M | Φ T svd F 2 | 2 = ( Φ T svd Φ svd O) T Φ T svd Φ svd O ( Φ T svd F 2 ) T Φ T svd F 2 = O T V[ Λ 0 0 0 ] V T O F 2 T F 2 ,
r= SNR PGI SNR SVDGI = F 2 T F 2 F 1 T U[ λ 1 2 λ m 2 ] U T F 1 ,
r= i=1 M j=1 M Ψ i 2 Ω j 2 i=1 M j=1 M ( λ i λ j ) 2 Ψ i 2 Ω j 2 ,
r i=1 M ji M Ψ i 2 Ω j 2 i=1 M ji M ( λ i λ j ) 2 Ψ i 2 Ω j 2 <1.0.
η= M N ,

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