Abstract

Ghost imaging technologies acquire images through intensity correlation of reference patterns and bucket values. Among them, an interesting method named correspondence imaging can generate positive-negative images by only conditionally averaging reference patterns, but still requires full/over sampling to treat the ensemble average of bucket values as a selection criteria, causing a long acquisition time. Here, we propose a sequential-deviation ghost imaging approach, which can realize real-time reconstructions of positive-negative images with a high image quality close to that of differential ghost imaging. Since it is no longer necessary to compare with the ensemble average, this method can improve the real-time performance. An explanation of its essence is also given here. Both simulation and experimental results have demonstrated the feasibility of this technique. This work may complement the theory of ghost imaging.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (4)

M. P. Edgar, G. M. Gibson, and M. J. Padgett, “Principles and prospects for single-pixel imaging,” Nat. Photonics 13(1), 13–20 (2019).
[Crossref]

G. Musarra, A. Lyons, E. Conca, Y. Altmann, F. Villa, F. Zappa, M. J. Padgett, and D. Faccio, “Non-line-of-sight three-dimensional imaging with a single-pixel camera,” Phys. Rev. Appl. 12(1), 011002 (2019).
[Crossref]

A. M. Paniagua-Diaz, I. Starshynov, N. Fayard, A. Goetschy, R. Pierrat, R. Carminati, and J. Bertolotti, “Blind ghost imaging,” Optica 6(4), 460–464 (2019).
[Crossref]

K. M. Czajkowski, A. Pastuszczak, and R. Kotyński, “Single-pixel imaging with sampling distributed over simplex vertices,” Opt. Lett. 44(5), 1241–1244 (2019).
[Crossref]

2018 (1)

2017 (4)

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]

K. Shibuya, T. Minamikawa, Y. Mizutani, H. Yamamoto, K. Minoshima, T. Yasui, and T. Iwata, “Scan-less hyperspectral dual-comb single-pixel-imaging in both amplitude and phase,” Opt. Express 25(18), 21947–21957 (2017).
[Crossref]

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]

W.-K. Yu, A.-D. Xiong, X.-R. Yao, G.-J. Zhai, and Q. Zhao, “Efficient phase retrieval based on dark fringe extraction and phase pattern construction with a good anti-noise capability,” Opt. Commun. 402, 413–421 (2017).
[Crossref]

2016 (3)

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard X rays,” Phys. Rev. Lett. 117(11), 113901 (2016).
[Crossref]

N. Huynh, E. Zhang, M. Betcke, S. Arridge, P. Beard, and B. Cox, “Single-pixel optical camera for video rate ultrasonic imaging,” Optica 3(1), 26–29 (2016).
[Crossref]

W.-K. Yu, X.-R. Yao, X.-F. Liu, R.-M. Lan, L.-A. Wu, and G.-J. Zhai, “Compressive microscopic imaging with “positive-negative” light modulation,” Opt. Commun. 371, 105–111 (2016).
[Crossref]

2015 (6)

W.-K. Yu, X.-R. Yao, X.-F. Liu, L.-Z. Li, and G.-J. Zhai, “Three-dimensional single-pixel compressive reflectivity imaging based on complementary modulation,” Appl. Opt. 54(3), 363–367 (2015).
[Crossref]

W.-K. Yu, X.-R. Yao, X.-F. Liu, L.-Z. Li, and G.-J. Zhai, “Ghost imaging based on Pearson correlation coefficients,” Chin. Phys. B 24(5), 054203 (2015).
[Crossref]

M.-J. Sun, M.-F. Li, and L.-A. Wu, “Nonlocal imaging of a reflective object using positive and negative correlations,” Appl. Opt. 54(25), 7494–7499 (2015).
[Crossref]

W.-K. Yu, X.-F. Liu, X.-R. Yao, C. Wang, Y. Zhai, and G.-J. Zhai, “Complementary compressive imaging for the telescopic system,” Sci. Rep. 4(1), 5834 (2015).
[Crossref]

W.-K. Yu, X.-R. Yao, X.-F. Liu, L.-Z. Li, and G.-J. Zhai, “Compressive moving target tracking with thermal light based on complementary sampling,” Appl. Opt. 54(13), 4249–4254 (2015).
[Crossref]

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

2014 (2)

2012 (2)

K.-H. Luo, B.-Q. Huang, W.-M. Zheng, and L.-A. Wu, “Nonlocal imaging by conditional averaging of random reference measurements,” Chin. Phys. Lett. 29(7), 074216 (2012).
[Crossref]

J. Wen, “Forming positive-negative images using conditioned partial measurements from reference arm in ghost imaging,” J. Opt. Soc. Am. A 29(9), 1906–1911 (2012).
[Crossref]

2011 (1)

L.-A. Wu and K.-H. Luo, “Two-photon imaging with entangled and thermal light,” AIP Conf. Proc. 1384(1), 223–228 (2011).
[Crossref]

2010 (1)

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

2009 (2)

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]

2008 (2)

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), 061802 (2008).
[Crossref]

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

2005 (3)

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

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]

J. Xiong, D.-Z. Cao, F. Huang, H.-G. Li, X.-J. Sun, and K. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. 94(17), 173601 (2005).
[Crossref]

2004 (1)

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]

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]

1970 (1)

Altmann, Y.

G. Musarra, A. Lyons, E. Conca, Y. Altmann, F. Villa, F. Zappa, M. J. Padgett, and D. Faccio, “Non-line-of-sight three-dimensional imaging with a single-pixel camera,” Phys. Rev. Appl. 12(1), 011002 (2019).
[Crossref]

Arridge, S.

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]

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]

Baraniuk, R. G.

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

Beard, P.

Bertolotti, J.

Betcke, M.

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]

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]

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]

Cao, D.-Z.

J. Xiong, D.-Z. Cao, F. Huang, H.-G. Li, X.-J. Sun, and K. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. 94(17), 173601 (2005).
[Crossref]

Carminati, R.

Chen, L.-M.

Conca, E.

G. Musarra, A. Lyons, E. Conca, Y. Altmann, F. Villa, F. Zappa, M. J. Padgett, and D. Faccio, “Non-line-of-sight three-dimensional imaging with a single-pixel camera,” Phys. Rev. Appl. 12(1), 011002 (2019).
[Crossref]

Cox, B.

Czajkowski, K. M.

Davenport, M. A.

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

Decker, J. A.

Du, G.

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard X rays,” Phys. Rev. Lett. 117(11), 113901 (2016).
[Crossref]

Duarte, M. F.

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

Edgar, M. P.

M. P. Edgar, G. M. Gibson, and M. J. Padgett, “Principles and prospects for single-pixel imaging,” Nat. Photonics 13(1), 13–20 (2019).
[Crossref]

Faccio, D.

G. Musarra, A. Lyons, E. Conca, Y. Altmann, F. Villa, F. Zappa, M. J. Padgett, and D. Faccio, “Non-line-of-sight three-dimensional imaging with a single-pixel camera,” Phys. Rev. Appl. 12(1), 011002 (2019).
[Crossref]

Fayard, N.

Ferri, F.

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

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]

Gatti, A.

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

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]

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]

Gibson, G. M.

M. P. Edgar, G. M. Gibson, and M. J. Padgett, “Principles and prospects for single-pixel imaging,” Nat. Photonics 13(1), 13–20 (2019).
[Crossref]

Goetschy, A.

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]

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]

Han, S.

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard X rays,” Phys. Rev. Lett. 117(11), 113901 (2016).
[Crossref]

He, Y.-H.

Huang, B.-Q.

K.-H. Luo, B.-Q. Huang, W.-M. Zheng, and L.-A. Wu, “Nonlocal imaging by conditional averaging of random reference measurements,” Chin. Phys. Lett. 29(7), 074216 (2012).
[Crossref]

Huang, F.

J. Xiong, D.-Z. Cao, F. Huang, H.-G. Li, X.-J. Sun, and K. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. 94(17), 173601 (2005).
[Crossref]

Huynh, N.

Iwata, T.

Katz, O.

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]

Kelly, K. F.

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

Kotynski, R.

Lan, R.-M.

W.-K. Yu, X.-R. Yao, X.-F. Liu, R.-M. Lan, L.-A. Wu, and G.-J. Zhai, “Compressive microscopic imaging with “positive-negative” light modulation,” Opt. Commun. 371, 105–111 (2016).
[Crossref]

Laska, J. N.

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

Li, C. B.

C. B. Li, “An efficient algorithm for total variation regularization with applications to the single pixel camera and compressive sensing,” M.Sc thesis, (Rice University, 2010).

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]

Li, H.-G.

J. Xiong, D.-Z. Cao, F. Huang, H.-G. Li, X.-J. Sun, and K. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. 94(17), 173601 (2005).
[Crossref]

Li, L.-Z.

Li, M.-F.

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]

Liu, X.-F.

Lu, R.

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard X rays,” Phys. Rev. Lett. 117(11), 113901 (2016).
[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]

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]

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]

Luo, K.-H.

K.-H. Luo, B.-Q. Huang, W.-M. Zheng, and L.-A. Wu, “Nonlocal imaging by conditional averaging of random reference measurements,” Chin. Phys. Lett. 29(7), 074216 (2012).
[Crossref]

L.-A. Wu and K.-H. Luo, “Two-photon imaging with entangled and thermal light,” AIP Conf. Proc. 1384(1), 223–228 (2011).
[Crossref]

Lyons, A.

G. Musarra, A. Lyons, E. Conca, Y. Altmann, F. Villa, F. Zappa, M. J. Padgett, and D. Faccio, “Non-line-of-sight three-dimensional imaging with a single-pixel camera,” Phys. Rev. Appl. 12(1), 011002 (2019).
[Crossref]

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]

Magatti, D.

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

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]

Minamikawa, T.

Minoshima, K.

Mizutani, Y.

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]

Musarra, G.

G. Musarra, A. Lyons, E. Conca, Y. Altmann, F. Villa, F. Zappa, M. J. Padgett, and D. Faccio, “Non-line-of-sight three-dimensional imaging with a single-pixel camera,” Phys. Rev. Appl. 12(1), 011002 (2019).
[Crossref]

Padgett, M. J.

M. P. Edgar, G. M. Gibson, and M. J. Padgett, “Principles and prospects for single-pixel imaging,” Nat. Photonics 13(1), 13–20 (2019).
[Crossref]

G. Musarra, A. Lyons, E. Conca, Y. Altmann, F. Villa, F. Zappa, M. J. Padgett, and D. Faccio, “Non-line-of-sight three-dimensional imaging with a single-pixel camera,” Phys. Rev. Appl. 12(1), 011002 (2019).
[Crossref]

Paniagua-Diaz, A. M.

Pastuszczak, A.

Pierrat, R.

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]

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]

Shapiro, J. H.

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), 061802 (2008).
[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]

Shibuya, K.

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]

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]

Starshynov, I.

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]

Sun, M.-J.

Sun, T.

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

Sun, X.-J.

J. Xiong, D.-Z. Cao, F. Huang, H.-G. Li, X.-J. Sun, and K. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. 94(17), 173601 (2005).
[Crossref]

Takhar, D.

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

Villa, F.

G. Musarra, A. Lyons, E. Conca, Y. Altmann, F. Villa, F. Zappa, M. J. Padgett, and D. Faccio, “Non-line-of-sight three-dimensional imaging with a single-pixel camera,” Phys. Rev. Appl. 12(1), 011002 (2019).
[Crossref]

Wang, B.-B.

Wang, C.

W.-K. Yu, X.-F. Liu, X.-R. Yao, C. Wang, Y. Zhai, and G.-J. Zhai, “Complementary compressive imaging for the telescopic system,” Sci. Rep. 4(1), 5834 (2015).
[Crossref]

Wang, K.

J. Xiong, D.-Z. Cao, F. Huang, H.-G. Li, X.-J. Sun, and K. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. 94(17), 173601 (2005).
[Crossref]

Wang, X.

Wen, J.

Wu, L.-A.

Xiao, T.

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard X rays,” Phys. Rev. Lett. 117(11), 113901 (2016).
[Crossref]

Xie, H.

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard X rays,” Phys. Rev. Lett. 117(11), 113901 (2016).
[Crossref]

Xiong, A.-D.

W.-K. Yu, A.-D. Xiong, X.-R. Yao, G.-J. Zhai, and Q. Zhao, “Efficient phase retrieval based on dark fringe extraction and phase pattern construction with a good anti-noise capability,” Opt. Commun. 402, 413–421 (2017).
[Crossref]

Xiong, J.

J. Xiong, D.-Z. Cao, F. Huang, H.-G. Li, X.-J. Sun, and K. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. 94(17), 173601 (2005).
[Crossref]

Yamamoto, 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]

Yao, X.-R.

W.-K. Yu, A.-D. Xiong, X.-R. Yao, G.-J. Zhai, and Q. Zhao, “Efficient phase retrieval based on dark fringe extraction and phase pattern construction with a good anti-noise capability,” Opt. Commun. 402, 413–421 (2017).
[Crossref]

W.-K. Yu, X.-R. Yao, X.-F. Liu, R.-M. Lan, L.-A. Wu, and G.-J. Zhai, “Compressive microscopic imaging with “positive-negative” light modulation,” Opt. Commun. 371, 105–111 (2016).
[Crossref]

W.-K. Yu, X.-R. Yao, X.-F. Liu, L.-Z. Li, and G.-J. Zhai, “Three-dimensional single-pixel compressive reflectivity imaging based on complementary modulation,” Appl. Opt. 54(3), 363–367 (2015).
[Crossref]

W.-K. Yu, X.-R. Yao, X.-F. Liu, L.-Z. Li, and G.-J. Zhai, “Ghost imaging based on Pearson correlation coefficients,” Chin. Phys. B 24(5), 054203 (2015).
[Crossref]

W.-K. Yu, X.-F. Liu, X.-R. Yao, C. Wang, Y. Zhai, and G.-J. Zhai, “Complementary compressive imaging for the telescopic system,” Sci. Rep. 4(1), 5834 (2015).
[Crossref]

W.-K. Yu, X.-R. Yao, X.-F. Liu, L.-Z. Li, and G.-J. Zhai, “Compressive moving target tracking with thermal light based on complementary sampling,” Appl. Opt. 54(13), 4249–4254 (2015).
[Crossref]

W.-K. Yu, M.-F. Li, X.-R. Yao, X.-F. Liu, L.-A. Wu, and G.-J. Zhai, “Adaptive compressive ghost imaging based on wavelet trees and sparse representation,” Opt. Express 22(6), 7133–7144 (2014).
[Crossref]

X.-R. Yao, W.-K. Yu, X.-F. Liu, L.-Z. Li, M.-F. Li, L.-A. Wu, and G.-J. Zhai, “Iterative denoising of ghost imaging,” Opt. Express 22(20), 24268–24275 (2014).
[Crossref]

Yasui, T.

Yu, H.

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard X rays,” Phys. Rev. Lett. 117(11), 113901 (2016).
[Crossref]

Yu, W.-K.

W.-K. Yu, A.-D. Xiong, X.-R. Yao, G.-J. Zhai, and Q. Zhao, “Efficient phase retrieval based on dark fringe extraction and phase pattern construction with a good anti-noise capability,” Opt. Commun. 402, 413–421 (2017).
[Crossref]

W.-K. Yu, X.-R. Yao, X.-F. Liu, R.-M. Lan, L.-A. Wu, and G.-J. Zhai, “Compressive microscopic imaging with “positive-negative” light modulation,” Opt. Commun. 371, 105–111 (2016).
[Crossref]

W.-K. Yu, X.-R. Yao, X.-F. Liu, L.-Z. Li, and G.-J. Zhai, “Three-dimensional single-pixel compressive reflectivity imaging based on complementary modulation,” Appl. Opt. 54(3), 363–367 (2015).
[Crossref]

W.-K. Yu, X.-R. Yao, X.-F. Liu, L.-Z. Li, and G.-J. Zhai, “Compressive moving target tracking with thermal light based on complementary sampling,” Appl. Opt. 54(13), 4249–4254 (2015).
[Crossref]

W.-K. Yu, X.-R. Yao, X.-F. Liu, L.-Z. Li, and G.-J. Zhai, “Ghost imaging based on Pearson correlation coefficients,” Chin. Phys. B 24(5), 054203 (2015).
[Crossref]

W.-K. Yu, X.-F. Liu, X.-R. Yao, C. Wang, Y. Zhai, and G.-J. Zhai, “Complementary compressive imaging for the telescopic system,” Sci. Rep. 4(1), 5834 (2015).
[Crossref]

X.-R. Yao, W.-K. Yu, X.-F. Liu, L.-Z. Li, M.-F. Li, L.-A. Wu, and G.-J. Zhai, “Iterative denoising of ghost imaging,” Opt. Express 22(20), 24268–24275 (2014).
[Crossref]

W.-K. Yu, M.-F. Li, X.-R. Yao, X.-F. Liu, L.-A. Wu, and G.-J. Zhai, “Adaptive compressive ghost imaging based on wavelet trees and sparse representation,” Opt. Express 22(6), 7133–7144 (2014).
[Crossref]

Zappa, F.

G. Musarra, A. Lyons, E. Conca, Y. Altmann, F. Villa, F. Zappa, M. J. Padgett, and D. Faccio, “Non-line-of-sight three-dimensional imaging with a single-pixel camera,” Phys. Rev. Appl. 12(1), 011002 (2019).
[Crossref]

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]

Zhai, G.-J.

W.-K. Yu, A.-D. Xiong, X.-R. Yao, G.-J. Zhai, and Q. Zhao, “Efficient phase retrieval based on dark fringe extraction and phase pattern construction with a good anti-noise capability,” Opt. Commun. 402, 413–421 (2017).
[Crossref]

W.-K. Yu, X.-R. Yao, X.-F. Liu, R.-M. Lan, L.-A. Wu, and G.-J. Zhai, “Compressive microscopic imaging with “positive-negative” light modulation,” Opt. Commun. 371, 105–111 (2016).
[Crossref]

W.-K. Yu, X.-R. Yao, X.-F. Liu, L.-Z. Li, and G.-J. Zhai, “Three-dimensional single-pixel compressive reflectivity imaging based on complementary modulation,” Appl. Opt. 54(3), 363–367 (2015).
[Crossref]

W.-K. Yu, X.-R. Yao, X.-F. Liu, L.-Z. Li, and G.-J. Zhai, “Compressive moving target tracking with thermal light based on complementary sampling,” Appl. Opt. 54(13), 4249–4254 (2015).
[Crossref]

W.-K. Yu, X.-R. Yao, X.-F. Liu, L.-Z. Li, and G.-J. Zhai, “Ghost imaging based on Pearson correlation coefficients,” Chin. Phys. B 24(5), 054203 (2015).
[Crossref]

W.-K. Yu, X.-F. Liu, X.-R. Yao, C. Wang, Y. Zhai, and G.-J. Zhai, “Complementary compressive imaging for the telescopic system,” Sci. Rep. 4(1), 5834 (2015).
[Crossref]

X.-R. Yao, W.-K. Yu, X.-F. Liu, L.-Z. Li, M.-F. Li, L.-A. Wu, and G.-J. Zhai, “Iterative denoising of ghost imaging,” Opt. Express 22(20), 24268–24275 (2014).
[Crossref]

W.-K. Yu, M.-F. Li, X.-R. Yao, X.-F. Liu, L.-A. Wu, and G.-J. Zhai, “Adaptive compressive ghost imaging based on wavelet trees and sparse representation,” Opt. Express 22(6), 7133–7144 (2014).
[Crossref]

Zhai, Y.

W.-K. Yu, X.-F. Liu, X.-R. Yao, C. Wang, Y. Zhai, and G.-J. Zhai, “Complementary compressive imaging for the telescopic system,” Sci. Rep. 4(1), 5834 (2015).
[Crossref]

Zhai, Y.-H.

Zhang, A.-X.

Zhang, D.

Zhang, E.

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]

Zhang, Z.

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]

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

Zhao, Q.

W.-K. Yu, A.-D. Xiong, X.-R. Yao, G.-J. Zhai, and Q. Zhao, “Efficient phase retrieval based on dark fringe extraction and phase pattern construction with a good anti-noise capability,” Opt. Commun. 402, 413–421 (2017).
[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]

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]

Zheng, W.-M.

K.-H. Luo, B.-Q. Huang, W.-M. Zheng, and L.-A. Wu, “Nonlocal imaging by conditional averaging of random reference measurements,” Chin. Phys. Lett. 29(7), 074216 (2012).
[Crossref]

Zhong, J.

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]

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

Zhu, D.

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard X rays,” Phys. Rev. Lett. 117(11), 113901 (2016).
[Crossref]

AIP Conf. Proc. (1)

L.-A. Wu and K.-H. Luo, “Two-photon imaging with entangled and thermal light,” AIP Conf. Proc. 1384(1), 223–228 (2011).
[Crossref]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

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

Chin. Phys. B (1)

W.-K. Yu, X.-R. Yao, X.-F. Liu, L.-Z. Li, and G.-J. Zhai, “Ghost imaging based on Pearson correlation coefficients,” Chin. Phys. B 24(5), 054203 (2015).
[Crossref]

Chin. Phys. Lett. (1)

K.-H. Luo, B.-Q. Huang, W.-M. Zheng, and L.-A. Wu, “Nonlocal imaging by conditional averaging of random reference measurements,” Chin. Phys. Lett. 29(7), 074216 (2012).
[Crossref]

IEEE Signal Process. Mag. (1)

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

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

Nat. Commun. (1)

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

Nat. Photonics (1)

M. P. Edgar, G. M. Gibson, and M. J. Padgett, “Principles and prospects for single-pixel imaging,” Nat. Photonics 13(1), 13–20 (2019).
[Crossref]

Opt. Commun. (2)

W.-K. Yu, A.-D. Xiong, X.-R. Yao, G.-J. Zhai, and Q. Zhao, “Efficient phase retrieval based on dark fringe extraction and phase pattern construction with a good anti-noise capability,” Opt. Commun. 402, 413–421 (2017).
[Crossref]

W.-K. Yu, X.-R. Yao, X.-F. Liu, R.-M. Lan, L.-A. Wu, and G.-J. Zhai, “Compressive microscopic imaging with “positive-negative” light modulation,” Opt. Commun. 371, 105–111 (2016).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

Optica (3)

Phys. Rev. A (3)

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78(6), 061802 (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]

Phys. Rev. Appl. (1)

G. Musarra, A. Lyons, E. Conca, Y. Altmann, F. Villa, F. Zappa, M. J. Padgett, and D. Faccio, “Non-line-of-sight three-dimensional imaging with a single-pixel camera,” Phys. Rev. Appl. 12(1), 011002 (2019).
[Crossref]

Phys. Rev. Lett. (5)

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard X rays,” Phys. Rev. Lett. 117(11), 113901 (2016).
[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]

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

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]

J. Xiong, D.-Z. Cao, F. Huang, H.-G. Li, X.-J. Sun, and K. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. 94(17), 173601 (2005).
[Crossref]

Sci. Adv. (1)

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]

Sci. Rep. (1)

W.-K. Yu, X.-F. Liu, X.-R. Yao, C. Wang, Y. Zhai, and G.-J. Zhai, “Complementary compressive imaging for the telescopic system,” Sci. Rep. 4(1), 5834 (2015).
[Crossref]

Other (1)

C. B. Li, “An efficient algorithm for total variation regularization with applications to the single pixel camera and compressive sensing,” M.Sc thesis, (Rice University, 2010).

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Figures (8)

Fig. 1.
Fig. 1. Simulation results of $\Delta G^{(2)}$, DGI, CI, SGI mode-1, mode-2 and mode-3. (a) is the original picture of $128\times 128$ pixels. (b)–(d) are the recovered results of $\Delta G^{(2)}$, DGI and SGI mode-1. (e)–(f) are CI positive-negative images, i.e., $G_{CI_+}$ and $G_{CI_-}$. (g)–(h) are retrieved positive-negative images of SGI mode-2, i.e., $G^{(2)}_{B_+}$ and $G^{(2)}_{B_-}$, and (i)–(j) are restored positive-negative images of SGI mode-3, i.e., $G^{(2)}_{R_+}$ and $G^{(2)}_{R_-}$.
Fig. 2.
Fig. 2. Schematic diagram of the double-arm lensless GI setup.
Fig. 3.
Fig. 3. Experimental results. (a) is the light field distribution on the photosensitive surface of the bucket detector in the object arm, i.e., the interaction result of one speckle pattern and the light field of the object. (b)–(d) are the recovered results of $\Delta G^{(2)}$, DGI and SGI mode-1; (e)–(f), (g)–(h) and (i)–(j) are the positive and negative ghost images of CI, SGI mode-2 and mode-3, respectively; (k) gives a cross section plot of the gray-scale images in (c) and (d) to make the image quality easier to be compared, by choosing the same row of the recovered images indicated by a red line. (l) is the cross section plot of retrieved gray-scale images (e), (g) and (i) with the same row, while (m) is the intensity profile of reconstructed gray-scale images (f), (h) and (j). The horizontal coordinate $x$ of (k)–(m) denotes the column pixel index of the restored images.
Fig. 4.
Fig. 4. Comparisons of experimental $\Delta$GI (a)–(j) and SGI mode-1 (k)–(t) reconstructions, using 300 to 30,000 measurements. (u) and (v) are the variance curves of $\Delta$GI and SGI mode-1 recovered images as a function of the measurement number (from 500 to 5,500), corresponding to the background part (BP) and the object part (OP), respectively.
Fig. 5.
Fig. 5. Reconstructed results under different kinds of temperature drift of the light source. (a) shows three different kinds of temperature drifts (TD-1, TD-2 and TD-3). (b)–(d) and (e)–(g) are the restored images of SGI mode-1 and $G_{CI_+}-G_{CI_-}$ under three kinds of temperature drifts of the light source, respectively.
Fig. 6.
Fig. 6. Probability density function of recovered pixel values of $G^{(2)}$, SGI mode-1, mode-2 and mode-3. (a) is an original binary image; (b)–(g) are the probability density distributions and the Gaussian theoretical curves of recovered pixel values falling in the object part (1) and the background part (0), of $G^{(2)}$, SGI mode-1, mode-2 $G_{B_+}^{(2)}$, mode-2 $G_{B_-}^{(2)}$, mode-3 $G_{R_+}^{(2)}$ and mode-3 $G_{R_+}^{(2)}$ respectively. The asterisks and circles represent the simulation data (SD), and the solid lines stand for the theoretical curves (TCs).
Fig. 7.
Fig. 7. Data analysis charts of SGI mode-2 (a) and mode-3 (b) reconstructions, where the abscissa represents the gray-scale value of the original object (here we use a binary object image with its gray-scale value being either 0 or 1), and the ordinate indicates the reconstructed pixel mean. The asterisks and circles denote the reconstructed pixel mean values calculated from the object part and the background part, respectively.
Fig. 8.
Fig. 8. Reconstructions using SCS methods. (a) and (b) are the recovered images of conventional compressive GI and SCS mode-1. (c)–(d) are the positive and negative images restored by SCS mode-2. (e)–(f) are the reconstructed results of SCS mode-3, applying the CS algorithm to the data processed by SGI mode-3 $G^{(2)}_{R_+}$ and $G^{(2)}_{R_-}$.

Tables (1)

Tables Icon

Table 1. Memory consumption of different methods

Equations (18)

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

G ( 2 ) = S B I R ( x R ) ,
Δ G ( 2 ) = ( S B S B ) ( I R ( x R ) I R ( x R ) ) = S B I R ( x R ) S B I R ( x R ) .
G D G I ( 2 ) = ( S B S B S R S R ) ( I R ( x R ) I R ( x R ) = S B I R ( x R ) S B S R S R I R ( x R ) ,
{ G C I + = I R + ,  for  { S B + S B S B 0 } , G C I = I R ,  for  { S B S B S B < 0 } .
mode-1:  G b o t h ( 2 ) = ( S B k + 1 S B k ) ( I R k + 1 I R k ) ;
mode-2:  { G B + ( 2 ) = ( S B k + 1 S B k ) I R k + 1 , G B ( 2 ) = ( S B k + 1 S B k ) I R k ;
mode-3:  { G R + ( 2 ) = S B k + 1 ( I R k + 1 I R k ) , G R ( 2 ) = S B k ( I R k + 1 I R k ) .
PSNR = 10 log ( 255 2 / MSE ) ,
E ( S B k I R j , k ) = E [ ( d j I R j , k + i j d i I R i , k ) I R j , k ] = E ( d j I R j , k I R j , k + i j d i I R i , k I R j , k ) = E ( d j I R j , k I R j , k ) + E ( i j d i I R i , k I R j , k ) = d j E ( I 2 ) + i j d i E ( I ) 2 = d j [ E ( I 2 ) E ( I ) 2 ] + d i E ( I ) 2 = c o n s t 1 d j + c o n s t 2 = μ 1 ,
D ( S B k I R j , k ) = E ( S B k 2 I R j , k 2 ) E ( S B k I R j , k ) 2 = σ 1 2 .
E [ ( S B k + 1 S B k ) ( I R j , k + 1 I R j , k ) ] = E ( S B k + 1 I R j , k + 1 ) + E ( S B k I R j , k ) E ( S B k + 1 I R j , k ) E ( S B k I R j , k + 1 ) = 2 [ E ( S B k I R j , k ) E ( S B k ) E ( I R j , k ) ] = 2 d j ( E ( I 2 ) E ( I ) 2 ) = c o n s t 3 d j = μ 2 ,
D [ ( S B k + 1 S B k ) ( I R j , k + 1 I R j , k ) ] = D [ S B k + 1 I R j , k + 1 + S B k I R j , k S B k + 1 I R j , k S B k I R j , k + 1 ] = 2 E ( S B k 2 I R j , k 2 ) + 2 E ( S B k 2 ) E ( I R j , k 2 ) 4 E ( S B k ) 2 E ( I R j , k ) 2 4 E ( S B k ) E ( S B k I R j , k 2 ) + 8 E ( S B k I R j , k ) E ( S B k ) E ( I R j , k ) 4 E ( S B k 2 I R j , k ) E ( I R j , k ) = σ 2 2 .
E [ ( S B k + 1 S B k ) I R j , k + 1 ] = E [ S B k + 1 ( I R j , k + 1 I R j , k ) ] = E ( S B k + 1 I R j , k + 1 ) E ( S B ) E ( I ) = d j [ E ( I 2 ) E ( I ) 2 ] ,
E [ ( S B k + 1 S B k ) I R j , k ] = E [ S B k ( I R j , k + 1 I R j , k ) ] = E ( S B ) E ( I ) E ( S B k I R j , k ) = d j [ E ( I 2 ) E ( I ) 2 ] .
D [ ( S B k + 1 S B k ) I R j , k + 1 ] = D ( S B k + 1 I R j , k + 1 ) + D ( S B k I R j , k + 1 ) 2 C o v ( S B k + 1 I R j , k + 1 , S B k I R j , k + 1 ) = D ( S B k + 1 I R j , k + 1 ) + E ( S B 2 ) E ( I 2 ) E ( S B ) 2 E ( I ) 2 2 [ E ( S B k + 1 I R j , k + 1 2 ) E ( S B ) E ( S B k + 1 I R j , k + 1 ) E ( S B ) E ( I ) ] ,
D [ ( S B k + 1 S B k ) I R j , k ] = D ( S B k + 1 I R j , k ) + D ( S B k I R j , k ) 2 C o v ( S B k + 1 I R j , k , S B k I R j , k ) = E ( S B 2 ) E ( I 2 ) E ( S B ) 2 E ( I ) 2 + D ( S B k I R j , k ) 2 [ E ( S B k I R j , k 2 ) E ( S B ) E ( S B k I R j , k ) E ( S B ) E ( I ) ] ,
D [ S B k + 1 ( I R j , k + 1 I R j , k ) ] = D ( S B k + 1 I R j , k + 1 ) + D ( S B k + 1 I R j , k ) 2 C o v ( S B k + 1 I R j , k + 1 , S B k + 1 I R j , k ) = D ( S B k + 1 I R j , k + 1 ) + E ( S B 2 ) E ( I 2 ) E ( S B ) 2 E ( I ) 2 2 [ E ( S B k + 1 2 I R j , k + 1 ) E ( I ) E ( S B k + 1 I R j , k + 1 ) E ( S B ) E ( I ) ] ,
D [ S B k ( I R j , k + 1 I R j , k ) ] = D ( S B k I R j , k + 1 ) + D ( S B k I R j , k ) 2 C o v ( S B k I R j , k + 1 , S B k I R j , k ) = E ( S B 2 ) E ( I 2 ) E ( S B ) 2 E ( I ) 2 + D ( S B k I R j , k ) 2 [ E ( S B k 2 I R j , k ) E ( I ) E ( S B k I R j , k ) E ( S B ) E ( I ) ] .

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