Abstract

To obtain a high imaging frame rate, a computational ghost imaging system scheme is proposed based on optical fiber phased array (OFPA). Through high-speed electro-optic modulators, the randomly modulated OFPA can provide much faster speckle projection, which can be precomputed according to the geometry of the fiber array and the known phases for modulation. Receiving the signal light with a low-pixel APD array can effectively decrease the requirement on sampling quantity and computation complexity owing to the reduced data dimensionality while avoiding the image aliasing due to the spatial periodicity of the speckles. The results of analysis and simulation show that the frame rate of the proposed imaging system can be significantly improved compared with traditional systems.

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

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2018 (1)

2017 (7)

M. Le, G. Wang, H. Zheng, J. Liu, Y. Zhou, and Z. Xu, “Underwater computational ghost imaging,” Opt. Express 25(19), 22859–22868 (2017).
[Crossref] [PubMed]

J. Huang and D. Shi, “Multispectral computational ghost imaging with multiplexed illumination,” J. Opt. 19(7), 07570 (2017).
[Crossref]

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3(4), e1601782 (2017).
[Crossref] [PubMed]

M.-J. Sun, W. Chen, T.-F. Liu, and L.-J. Li, “Image Retrieval in Spatial and Temporal Domains With a Quadrant Detector,” IEEE Photonics J. 9(5), 3601206 (2017).
[Crossref]

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. Zafari, S. Ahmadi-Kandjani, and R. Kheradmand, “Noise reduction in selective computational ghost imaging using genetic algorithm,” Opt. Commun. 387, 182–187 (2017).
[Crossref]

H. C. Liu, B. Yang, Q. Guo, J. Shi, C. Guan, G. Zheng, H. Mühlenbernd, G. Li, T. Zentgraf, and S. Zhang, “Single-pixel computational ghost imaging with helicity-dependent metasurface hologram,” Sci. Adv. 3(9), e1701477 (2017).
[Crossref] [PubMed]

2016 (10)

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]

D. B. Phillips, R. He, Q. Chen, G. M. Gibson, and M. J. Padgett, “Non-diffractive computational ghost imaging,” Opt. Express 24(13), 14172–14182 (2016).
[Crossref] [PubMed]

F. Devaux, P. A. Moreau, S. Denis, and E. Lantz, “Computational temporal ghost imaging,” Optica 3(7), 698 (2016).
[Crossref]

Q. Li, Z. Duan, H. Lin, S. Gao, S. Sun, and W. Liu, “Coprime-frequencied sinusoidal modulation for improving the speed of computational ghost imaging with a spatial light modulator,” Chin. Opt. Lett. 14(11), 111103 (2016).
[Crossref]

Y. Liu, J. Shi, and G. Zeng, “Single-photon-counting polarization ghost imaging,” Appl. Opt. 55(36), 10347–10351 (2016).
[Crossref] [PubMed]

H. Wu, X. Zhang, J. Gan, and C. Luo, “High-Quality Computational Ghost Imaging Using an Optimum Distance Search Method,” IEEE Photonics J. 8, 1–9 (2016).

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]

X. Yang, Y. Zhang, C. Yang, L. Xu, Q. Wang, and Y. Zhao, “Heterodyne 3D ghost imaging,” Opt. Commun. 368, 1–6 (2016).
[Crossref]

C. Luo and L. Zhuo, “High-resolution computational ghost imaging and ghost diffraction through turbulence via a beam-shaping method,” Laser Phys. Lett. 14(1), 015201 (2016).
[Crossref]

S. Sun, W. T. Liu, H. Z. Lin, E. F. Zhang, J. Y. Liu, Q. Li, and P. X. Chen, “Multi-scale Adaptive Computational Ghost Imaging,” Sci. Rep. 6(1), 37013 (2016).
[Crossref] [PubMed]

2015 (4)

C. Luo, J. Cheng, A. Chen, and Z.-M. Liu, “Computational ghost imaging with higher-order cosh-Gaussian modulated incoherent sources in atmospheric turbulence,” Opt. Commun. 352, 155–160 (2015).
[Crossref]

H. Ghanbari-Ghalehjoughi, S. Ahmadi-Kandjani, and M. Eslami, “High quality computational ghost imaging using multi-fluorescent screen,” J. Opt. Soc. Am. A 32(2), 323–328 (2015).
[Crossref] [PubMed]

Y. Zhu, J. Shi, Y. Yang, and G. Zeng, “Polarization difference ghost imaging,” Appl. Opt. 54(6), 1279–1284 (2015).
[Crossref] [PubMed]

C. Liu, F. Lu, C. Wu, and X. Han, “Spatial correlation properties of coherent array beams modulated by space-time random phase,” Opt. Commun. 346, 26–33 (2015).
[Crossref]

2014 (1)

M. Zafari, R. Kheradmand, and S. Ahmadi-Kandjani, “Optical encryption with selective computational ghost imaging,” J. Opt. 16(10), 105405 (2014).
[Crossref]

2013 (3)

N. D. Hardy and J. H. Shapiro, “Computational ghost imaging versus imaging laser radar for three-dimensional imaging,” Phys. Rev. A 87(2), 023820 (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]

M. Amann and M. Bayer, “Compressive adaptive computational ghost imaging,” Sci. Rep. 3(1), 1545 (2013).
[Crossref] [PubMed]

2012 (3)

2010 (3)

2009 (2)

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 053840 (2009).
[Crossref]

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95(13), 131110 (2009).
[Crossref]

2008 (2)

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), R061802 (2008).
[Crossref]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. Kelly, and B. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

2007 (1)

2005 (2)

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]

Y. Cai and S.-Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005).
[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 (1)

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]

Ahmadi-Kandjani, S.

M. Zafari, S. Ahmadi-Kandjani, and R. Kheradmand, “Noise reduction in selective computational ghost imaging using genetic algorithm,” Opt. Commun. 387, 182–187 (2017).
[Crossref]

H. Ghanbari-Ghalehjoughi, S. Ahmadi-Kandjani, and M. Eslami, “High quality computational ghost imaging using multi-fluorescent screen,” J. Opt. Soc. Am. A 32(2), 323–328 (2015).
[Crossref] [PubMed]

M. Zafari, R. Kheradmand, and S. Ahmadi-Kandjani, “Optical encryption with selective computational ghost imaging,” J. Opt. 16(10), 105405 (2014).
[Crossref]

Amann, M.

M. Amann and M. Bayer, “Compressive adaptive computational ghost imaging,” Sci. Rep. 3(1), 1545 (2013).
[Crossref] [PubMed]

Astola, J.

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]

Baraniuk, B. G.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. Kelly, and B. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Barnett, S. M.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3(4), e1601782 (2017).
[Crossref] [PubMed]

Basano, L.

Bayer, M.

M. Amann and M. Bayer, “Compressive adaptive computational ghost imaging,” Sci. Rep. 3(1), 1545 (2013).
[Crossref] [PubMed]

Bennink, R. S.

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

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.

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]

Boyd, R. W.

K. W. Chan, M. N. O’Sullivan, and R. W. Boyd, “Optimization of thermal ghost imaging: high-order correlations vs. background subtraction,” Opt. Express 18(6), 5562–5573 (2010).
[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]

Bromberg, Y.

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 053840 (2009).
[Crossref]

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95(13), 131110 (2009).
[Crossref]

Cai, Y.

Y. Cai and S.-Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005).
[Crossref] [PubMed]

Chan, K. W.

Chen, A.

C. Luo, J. Cheng, A. Chen, and Z.-M. Liu, “Computational ghost imaging with higher-order cosh-Gaussian modulated incoherent sources in atmospheric turbulence,” Opt. Commun. 352, 155–160 (2015).
[Crossref]

Chen, P. X.

S. Sun, W. T. Liu, H. Z. Lin, E. F. Zhang, J. Y. Liu, Q. Li, and P. X. Chen, “Multi-scale Adaptive Computational Ghost Imaging,” Sci. Rep. 6(1), 37013 (2016).
[Crossref] [PubMed]

Chen, Q.

Chen, W.

Z.-H. Xu, W. Chen, J. Penuelas, M. Padgett, and M.-J. Sun, “1000 fps computational ghost imaging using LED-based structured illumination,” Opt. Express 26(3), 2427–2434 (2018).
[Crossref] [PubMed]

M.-J. Sun, W. Chen, T.-F. Liu, and L.-J. Li, “Image Retrieval in Spatial and Temporal Domains With a Quadrant Detector,” IEEE Photonics J. 9(5), 3601206 (2017).
[Crossref]

Cheng, J.

C. Luo, J. Cheng, A. Chen, and Z.-M. Liu, “Computational ghost imaging with higher-order cosh-Gaussian modulated incoherent sources in atmospheric turbulence,” Opt. Commun. 352, 155–160 (2015).
[Crossref]

Clemente, P.

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]

Davenport, M. A.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. Kelly, and B. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Denis, S.

Devaux, F.

Duan, Z.

Duarte, M. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. Kelly, and B. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Durán, V.

Edgar, M. P.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3(4), e1601782 (2017).
[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.-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]

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]

Erkmen, B. I.

Eslami, M.

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]

Gan, J.

H. Wu, X. Zhang, J. Gan, and C. Luo, “High-Quality Computational Ghost Imaging Using an Optimum Distance Search Method,” IEEE Photonics J. 8, 1–9 (2016).

Gao, S.

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]

Ghanbari-Ghalehjoughi, H.

Gibson, G. M.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3(4), e1601782 (2017).
[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.-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]

D. B. Phillips, R. He, Q. Chen, G. M. Gibson, and M. J. Padgett, “Non-diffractive computational ghost imaging,” Opt. Express 24(13), 14172–14182 (2016).
[Crossref] [PubMed]

Gong, W.

W. Gong and S. Han, “Experimental investigation of the quality of lensless super-resolution ghost imaging via sparsity constraints,” Phys. Lett. A 376(17), 1519–1522 (2012).
[Crossref]

Guan, C.

H. C. Liu, B. Yang, Q. Guo, J. Shi, C. Guan, G. Zheng, H. Mühlenbernd, G. Li, T. Zentgraf, and S. Zhang, “Single-pixel computational ghost imaging with helicity-dependent metasurface hologram,” Sci. Adv. 3(9), e1701477 (2017).
[Crossref] [PubMed]

Guo, Q.

H. C. Liu, B. Yang, Q. Guo, J. Shi, C. Guan, G. Zheng, H. Mühlenbernd, G. Li, T. Zentgraf, and S. Zhang, “Single-pixel computational ghost imaging with helicity-dependent metasurface hologram,” Sci. Adv. 3(9), e1701477 (2017).
[Crossref] [PubMed]

Han, S.

W. Gong and S. Han, “Experimental investigation of the quality of lensless super-resolution ghost imaging via sparsity constraints,” Phys. Lett. A 376(17), 1519–1522 (2012).
[Crossref]

Han, X.

C. Liu, F. Lu, C. Wu, and X. Han, “Spatial correlation properties of coherent array beams modulated by space-time random phase,” Opt. Commun. 346, 26–33 (2015).
[Crossref]

Hardy, N. D.

N. D. Hardy and J. H. Shapiro, “Computational ghost imaging versus imaging laser radar for three-dimensional imaging,” Phys. Rev. A 87(2), 023820 (2013).
[Crossref]

He, R.

Huang, J.

J. Huang and D. Shi, “Multispectral computational ghost imaging with multiplexed illumination,” J. Opt. 19(7), 07570 (2017).
[Crossref]

Katkovnik, V.

Katz, O.

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95(13), 131110 (2009).
[Crossref]

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 053840 (2009).
[Crossref]

Kelly, K.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. Kelly, and B. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Kheradmand, R.

M. Zafari, S. Ahmadi-Kandjani, and R. Kheradmand, “Noise reduction in selective computational ghost imaging using genetic algorithm,” Opt. Commun. 387, 182–187 (2017).
[Crossref]

M. Zafari, R. Kheradmand, and S. Ahmadi-Kandjani, “Optical encryption with selective computational ghost imaging,” J. Opt. 16(10), 105405 (2014).
[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]

Lancis, J.

Lantz, E.

Laska, J. N.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. Kelly, and B. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Le, M.

Li, G.

H. C. Liu, B. Yang, Q. Guo, J. Shi, C. Guan, G. Zheng, H. Mühlenbernd, G. Li, T. Zentgraf, and S. Zhang, “Single-pixel computational ghost imaging with helicity-dependent metasurface hologram,” Sci. Adv. 3(9), e1701477 (2017).
[Crossref] [PubMed]

Li, L.-J.

M.-J. Sun, W. Chen, T.-F. Liu, and L.-J. Li, “Image Retrieval in Spatial and Temporal Domains With a Quadrant Detector,” IEEE Photonics J. 9(5), 3601206 (2017).
[Crossref]

Li, Q.

Lin, H.

Lin, H. Z.

S. Sun, W. T. Liu, H. Z. Lin, E. F. Zhang, J. Y. Liu, Q. Li, and P. X. Chen, “Multi-scale Adaptive Computational Ghost Imaging,” Sci. Rep. 6(1), 37013 (2016).
[Crossref] [PubMed]

Liu, C.

C. Liu, F. Lu, C. Wu, and X. Han, “Spatial correlation properties of coherent array beams modulated by space-time random phase,” Opt. Commun. 346, 26–33 (2015).
[Crossref]

Liu, H. C.

H. C. Liu, B. Yang, Q. Guo, J. Shi, C. Guan, G. Zheng, H. Mühlenbernd, G. Li, T. Zentgraf, and S. Zhang, “Single-pixel computational ghost imaging with helicity-dependent metasurface hologram,” Sci. Adv. 3(9), e1701477 (2017).
[Crossref] [PubMed]

Liu, J.

Liu, J. Y.

S. Sun, W. T. Liu, H. Z. Lin, E. F. Zhang, J. Y. Liu, Q. Li, and P. X. Chen, “Multi-scale Adaptive Computational Ghost Imaging,” Sci. Rep. 6(1), 37013 (2016).
[Crossref] [PubMed]

Liu, T.-F.

M.-J. Sun, W. Chen, T.-F. Liu, and L.-J. Li, “Image Retrieval in Spatial and Temporal Domains With a Quadrant Detector,” IEEE Photonics J. 9(5), 3601206 (2017).
[Crossref]

Liu, W.

Liu, W. T.

S. Sun, W. T. Liu, H. Z. Lin, E. F. Zhang, J. Y. Liu, Q. Li, and P. X. Chen, “Multi-scale Adaptive Computational Ghost Imaging,” Sci. Rep. 6(1), 37013 (2016).
[Crossref] [PubMed]

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]

Y. Liu, J. Shi, and G. Zeng, “Single-photon-counting polarization ghost imaging,” Appl. Opt. 55(36), 10347–10351 (2016).
[Crossref] [PubMed]

Liu, Z.-M.

C. Luo, J. Cheng, A. Chen, and Z.-M. Liu, “Computational ghost imaging with higher-order cosh-Gaussian modulated incoherent sources in atmospheric turbulence,” Opt. Commun. 352, 155–160 (2015).
[Crossref]

Lu, F.

C. Liu, F. Lu, C. Wu, and X. Han, “Spatial correlation properties of coherent array beams modulated by space-time random phase,” Opt. Commun. 346, 26–33 (2015).
[Crossref]

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]

Luo, C.

C. Luo and L. Zhuo, “High-resolution computational ghost imaging and ghost diffraction through turbulence via a beam-shaping method,” Laser Phys. Lett. 14(1), 015201 (2016).
[Crossref]

H. Wu, X. Zhang, J. Gan, and C. Luo, “High-Quality Computational Ghost Imaging Using an Optimum Distance Search Method,” IEEE Photonics J. 8, 1–9 (2016).

C. Luo, J. Cheng, A. Chen, and Z.-M. Liu, “Computational ghost imaging with higher-order cosh-Gaussian modulated incoherent sources in atmospheric turbulence,” Opt. Commun. 352, 155–160 (2015).
[Crossref]

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]

Moreau, P. A.

Mühlenbernd, H.

H. C. Liu, B. Yang, Q. Guo, J. Shi, C. Guan, G. Zheng, H. Mühlenbernd, G. Li, T. Zentgraf, and S. Zhang, “Single-pixel computational ghost imaging with helicity-dependent metasurface hologram,” Sci. Adv. 3(9), e1701477 (2017).
[Crossref] [PubMed]

O’Sullivan, M. N.

Ottonello, P.

Padgett, M.

Padgett, M. J.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3(4), e1601782 (2017).
[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.-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]

D. B. Phillips, R. He, Q. Chen, G. M. Gibson, and M. J. Padgett, “Non-diffractive computational ghost imaging,” Opt. Express 24(13), 14172–14182 (2016).
[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]

Penuelas, J.

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]

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]

Shapiro, J. H.

N. D. Hardy and J. H. Shapiro, “Computational ghost imaging versus imaging laser radar for three-dimensional imaging,” Phys. Rev. A 87(2), 023820 (2013).
[Crossref]

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), R061802 (2008).
[Crossref]

Shi, D.

J. Huang and D. Shi, “Multispectral computational ghost imaging with multiplexed illumination,” J. Opt. 19(7), 07570 (2017).
[Crossref]

Shi, J.

H. C. Liu, B. Yang, Q. Guo, J. Shi, C. Guan, G. Zheng, H. Mühlenbernd, G. Li, T. Zentgraf, and S. Zhang, “Single-pixel computational ghost imaging with helicity-dependent metasurface hologram,” Sci. Adv. 3(9), e1701477 (2017).
[Crossref] [PubMed]

Y. Liu, J. Shi, and G. Zeng, “Single-photon-counting polarization ghost imaging,” Appl. Opt. 55(36), 10347–10351 (2016).
[Crossref] [PubMed]

Y. Zhu, J. Shi, Y. Yang, and G. Zeng, “Polarization difference ghost imaging,” Appl. Opt. 54(6), 1279–1284 (2015).
[Crossref] [PubMed]

Shih, Y. H.

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), 053840 (2009).
[Crossref]

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95(13), 131110 (2009).
[Crossref]

Situ, G.

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]

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]

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]

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.

Z.-H. Xu, W. Chen, J. Penuelas, M. Padgett, and M.-J. Sun, “1000 fps computational ghost imaging using LED-based structured illumination,” Opt. Express 26(3), 2427–2434 (2018).
[Crossref] [PubMed]

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3(4), e1601782 (2017).
[Crossref] [PubMed]

M.-J. Sun, W. Chen, T.-F. Liu, and L.-J. Li, “Image Retrieval in Spatial and Temporal Domains With a Quadrant Detector,” IEEE Photonics J. 9(5), 3601206 (2017).
[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]

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]

Sun, S.

Sun, T.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. Kelly, and B. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

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]

Tajahuerce, E.

Takhar, D.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. Kelly, and B. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Taylor, J. M.

D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3(4), e1601782 (2017).
[Crossref] [PubMed]

Torres-Company, V.

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, G.

Wang, Q.

X. Yang, Y. Zhang, C. Yang, L. Xu, Q. Wang, and Y. Zhao, “Heterodyne 3D ghost imaging,” Opt. Commun. 368, 1–6 (2016).
[Crossref]

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]

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]

Wu, C.

C. Liu, F. Lu, C. Wu, and X. Han, “Spatial correlation properties of coherent array beams modulated by space-time random phase,” Opt. Commun. 346, 26–33 (2015).
[Crossref]

Wu, H.

H. Wu, X. Zhang, J. Gan, and C. Luo, “High-Quality Computational Ghost Imaging Using an Optimum Distance Search Method,” IEEE Photonics J. 8, 1–9 (2016).

Xu, L.

X. Yang, Y. Zhang, C. Yang, L. Xu, Q. Wang, and Y. Zhao, “Heterodyne 3D ghost imaging,” Opt. Commun. 368, 1–6 (2016).
[Crossref]

Xu, Z.

Xu, Z.-H.

Yang, B.

H. C. Liu, B. Yang, Q. Guo, J. Shi, C. Guan, G. Zheng, H. Mühlenbernd, G. Li, T. Zentgraf, and S. Zhang, “Single-pixel computational ghost imaging with helicity-dependent metasurface hologram,” Sci. Adv. 3(9), e1701477 (2017).
[Crossref] [PubMed]

Yang, C.

X. Yang, Y. Zhang, C. Yang, L. Xu, Q. Wang, and Y. Zhao, “Heterodyne 3D ghost imaging,” Opt. Commun. 368, 1–6 (2016).
[Crossref]

Yang, X.

X. Yang, Y. Zhang, C. Yang, L. Xu, Q. Wang, and Y. Zhao, “Heterodyne 3D ghost imaging,” Opt. Commun. 368, 1–6 (2016).
[Crossref]

Yang, Y.

Zafari, M.

M. Zafari, S. Ahmadi-Kandjani, and R. Kheradmand, “Noise reduction in selective computational ghost imaging using genetic algorithm,” Opt. Commun. 387, 182–187 (2017).
[Crossref]

M. Zafari, R. Kheradmand, and S. Ahmadi-Kandjani, “Optical encryption with selective computational ghost imaging,” J. Opt. 16(10), 105405 (2014).
[Crossref]

Zeng, G.

Zentgraf, T.

H. C. Liu, B. Yang, Q. Guo, J. Shi, C. Guan, G. Zheng, H. Mühlenbernd, G. Li, T. Zentgraf, and S. Zhang, “Single-pixel computational ghost imaging with helicity-dependent metasurface hologram,” Sci. Adv. 3(9), e1701477 (2017).
[Crossref] [PubMed]

Zhang, E. F.

S. Sun, W. T. Liu, H. Z. Lin, E. F. Zhang, J. Y. Liu, Q. Li, and P. X. Chen, “Multi-scale Adaptive Computational Ghost Imaging,” Sci. Rep. 6(1), 37013 (2016).
[Crossref] [PubMed]

Zhang, S.

H. C. Liu, B. Yang, Q. Guo, J. Shi, C. Guan, G. Zheng, H. Mühlenbernd, G. Li, T. Zentgraf, and S. Zhang, “Single-pixel computational ghost imaging with helicity-dependent metasurface hologram,” Sci. Adv. 3(9), e1701477 (2017).
[Crossref] [PubMed]

Zhang, X.

H. Wu, X. Zhang, J. Gan, and C. Luo, “High-Quality Computational Ghost Imaging Using an Optimum Distance Search Method,” IEEE Photonics J. 8, 1–9 (2016).

Zhang, Y.

X. Yang, Y. Zhang, C. Yang, L. Xu, Q. Wang, and Y. Zhao, “Heterodyne 3D ghost imaging,” Opt. Commun. 368, 1–6 (2016).
[Crossref]

Zhao, Y.

X. Yang, Y. Zhang, C. Yang, L. Xu, Q. Wang, and Y. Zhao, “Heterodyne 3D ghost imaging,” Opt. Commun. 368, 1–6 (2016).
[Crossref]

Zheng, G.

H. C. Liu, B. Yang, Q. Guo, J. Shi, C. Guan, G. Zheng, H. Mühlenbernd, G. Li, T. Zentgraf, and S. Zhang, “Single-pixel computational ghost imaging with helicity-dependent metasurface hologram,” Sci. Adv. 3(9), e1701477 (2017).
[Crossref] [PubMed]

Zheng, H.

Zhou, Y.

Zhu, S.-Y.

Y. Cai and S.-Y. Zhu, “Ghost imaging with incoherent and partially coherent light radiation,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 71(5), 056607 (2005).
[Crossref] [PubMed]

Zhu, Y.

Zhuo, L.

C. Luo and L. Zhuo, “High-resolution computational ghost imaging and ghost diffraction through turbulence via a beam-shaping method,” Laser Phys. Lett. 14(1), 015201 (2016).
[Crossref]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95(13), 131110 (2009).
[Crossref]

Chin. Opt. Lett. (1)

IEEE Photonics J. (2)

H. Wu, X. Zhang, J. Gan, and C. Luo, “High-Quality Computational Ghost Imaging Using an Optimum Distance Search Method,” IEEE Photonics J. 8, 1–9 (2016).

M.-J. Sun, W. Chen, T.-F. Liu, and L.-J. Li, “Image Retrieval in Spatial and Temporal Domains With a Quadrant Detector,” IEEE Photonics J. 9(5), 3601206 (2017).
[Crossref]

IEEE Signal Process. Mag. (1)

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. Kelly, and B. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

J. Opt. (2)

M. Zafari, R. Kheradmand, and S. Ahmadi-Kandjani, “Optical encryption with selective computational ghost imaging,” J. Opt. 16(10), 105405 (2014).
[Crossref]

J. Huang and D. Shi, “Multispectral computational ghost imaging with multiplexed illumination,” J. Opt. 19(7), 07570 (2017).
[Crossref]

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

Laser Phys. Lett. (1)

C. Luo and L. Zhuo, “High-resolution computational ghost imaging and ghost diffraction through turbulence via a beam-shaping method,” Laser Phys. Lett. 14(1), 015201 (2016).
[Crossref]

Nat. Commun. (1)

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. (4)

X. Yang, Y. Zhang, C. Yang, L. Xu, Q. Wang, and Y. Zhao, “Heterodyne 3D ghost imaging,” Opt. Commun. 368, 1–6 (2016).
[Crossref]

C. Luo, J. Cheng, A. Chen, and Z.-M. Liu, “Computational ghost imaging with higher-order cosh-Gaussian modulated incoherent sources in atmospheric turbulence,” Opt. Commun. 352, 155–160 (2015).
[Crossref]

M. Zafari, S. Ahmadi-Kandjani, and R. Kheradmand, “Noise reduction in selective computational ghost imaging using genetic algorithm,” Opt. Commun. 387, 182–187 (2017).
[Crossref]

C. Liu, F. Lu, C. Wu, and X. Han, “Spatial correlation properties of coherent array beams modulated by space-time random phase,” Opt. Commun. 346, 26–33 (2015).
[Crossref]

Opt. Express (6)

Opt. Lett. (1)

Optica (1)

Phys. Lett. A (1)

W. Gong and S. Han, “Experimental investigation of the quality of lensless super-resolution ghost imaging via sparsity constraints,” Phys. Lett. A 376(17), 1519–1522 (2012).
[Crossref]

Phys. Rev. A (4)

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), R061802 (2008).
[Crossref]

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79(5), 053840 (2009).
[Crossref]

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]

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D. B. Phillips, M.-J. Sun, J. M. Taylor, M. P. Edgar, S. M. Barnett, G. M. Gibson, and M. J. Padgett, “Adaptive foveated single-pixel imaging with dynamic supersampling,” Sci. Adv. 3(4), e1601782 (2017).
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[Crossref]

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[Crossref]

B. Sun, M. Edgar, R. Bowman, L. Vittert, S. Welsh, A. Bowman, and M. Padgett, “Differential Computational Ghost Imaging,” in Imaging and Applied Optics, OSA Technical Digest (online) (Optical Society of America, 2013), paper CTu1C.4.

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Figures (9)

Fig. 1
Fig. 1 The schematic system of the OFPA based speckle source.
Fig. 2
Fig. 2 The spatial arrangement of three types of regular array beams (up), the corresponding intensity snapshots (middle) and the kernel function normalized by N2 (down). The subplots (a), (d) and (g) are for circular array (N = 16), the subplots (b), (e) and (h) are for square array (N = 25), and the subplots (c), (f) and (i) is for hexagonal array (N = 19).
Fig. 3
Fig. 3 The schematic diagram of the light receiving scheme using a 5 × 5 APD array.
Fig. 4
Fig. 4 The schematic diagram of the computational ghost imaging using the OFPA based speckle source and low-pixel APD array. PBS is short for polarized beam splitter.
Fig. 5
Fig. 5 The object model (1#) used for the imaging simulation, four capital letter ‘S’, ‘P’, ‘O’ and ‘E’ with the reflectivity equal to 1.
Fig. 6
Fig. 6 The reconstructed images of the object with different sampling number M. (a) ~(d): the images restructured using the STC algorithm (up) and (e) ~(h): the images using the CS based algorithm (down).
Fig. 7
Fig. 7 The imaging results of 2# and 3# objects versus the sampling number M. The upper: the object 2# and the reconstructed results; the lower: the object 3# and its reconstructed results.
Fig. 8
Fig. 8 The recovered images of the resized 2# object (128 × 128) versus sampling number M.
Fig. 9
Fig. 9 The MSE curves between the original and reconstructed image versus sampling number M. (a), (b) are corresponding for the STC algorithm; and (c), (d) are for the CS algorithm.

Tables (2)

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Table 1 System Parameters for the Imaging Simulation

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Table 2 Time Consumption of Image Recovery and the Corresponding MSE

Equations (17)

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E n ( x , y ; m ) = a n exp ( ( x x n ) 2 + ( y y n ) 2 ω 0 2 ) exp ( j ϕ n , p + j ϕ i n ) .
F ( ξ , η ; p ) = π ω 0 2 exp ( k 2 ω 0 2 ξ 2 + η 2 4 D 2 ) n = 1 N a n exp ( j k D ( x n ξ + y n η ) ) exp ( j ϕ n , p + j ϕ i n ) .
I ( ξ , η ; p ) = I E N ( ξ , η ) m = 1 N n = 1 N a m a n exp ( j k D ( ( x m x n ) ξ + ( y m y n ) η ) ) exp ( j ( ϕ n , p ϕ m , p ) ) .
I E N ( ξ , η ) = π 2 ω 0 4 exp ( k 2 ω 0 2 ξ 2 + η 2 2 D 2 ) .
I S P ( ξ , η ; p ) = m = 1 N n = 1 N a m a n exp ( j k D ( ( x m x n ) ξ + ( y m y n ) η ) ) exp ( j ( ϕ n , p ϕ m , p ) ) .
C X ( I ( ρ 1 ) , I ( ρ 2 ) ) = I ( ρ 1 ; p ) I ( ρ 2 ; p ) I ( ρ 1 ; p ) I ( ρ 2 ; p ) .
C X ( I ( ρ 1 ) , I ( ρ 2 ) ) = ( n = 1 N a n 2 ) 2 [ I K F ( ξ 1 ξ 2 , η 1 η 2 ) n = 1 N a n 4 + ( n = 1 N a n 2 ) 2 ] .
I K F ( ξ , η ) = n = 1 N m = 1 N a n 2 a m 2 exp { j k D [ ( x m x n ) ξ + ( y m y n ) η ] } .
I K F ( ξ , η ) = N 2 sin 2 ( S k d ξ 2 D ) sin 2 ( S k d η 2 D ) sin 2 ( k d ξ 2 D ) sin 2 ( k d η 2 D ) .
I m = [ I 1 , m I ( K + 1 ) , m I K ( K 1 ) + 1 , m I 2 , m I ( K + 2 ) , m I K ( K 1 ) + 2 , m I K , m I 2 K , m I K 2 , m ] K L × K L .
O p = [ O p ( m ) ] M × 1 = [ R p ( ξ , η ) I p ( ξ , η ; m ) d ξ d η ] M × 1 .
G = [ G 1 G ( K + 1 ) G K ( K 1 ) + 1 G 2 G ( K + 2 ) G K ( K 1 ) + 2 G K G 2 K G K 2 ] K L × K L .
G p ( u , v ) = 1 M I p ( u , v ) 2 [ O p O p ] T [ I p ( u , v ) - I p ( u , v ) ] .
G p ( u , v ) = O p T I p ( u , v ) M I p ( u , v ) 2 O p S p S p T I p ( u , v ) M I p ( u , v ) 2 .
arg min vec ( G p ) 1 , subject to O p = M p vec ( G p ) .
M p = [ vec ( I p , 1 ) vec ( I p , 2 ) vec ( I p , M ) ] T .
MSE = 1 N R N C u = 1 N R v = 1 N C [ G ( u , v ) B ( u , v ) ] 2 .

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