Abstract

The requirement of a large number of samplings limits the performance of ghost imaging for moving objects. Conventionally, tracking and imaging of the moving objects are done independently; thus, sequential clear images of the moving target during its evolution are required. In this Letter, we propose to obtain the displacement of the object via cross correlation between sequential unclear rough images. Then, a high-quality image of the moving object can be reconstructed gradually during its evolution. Our method works well for translating and rotating objects.

© 2019 Optical Society of America

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References

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2019 (4)

2018 (4)

Z. Xu, W. Chen, J. Penuelas, M. Padgett, and M. Sun, Opt. Express 26, 2427 (2018).
[Crossref]

X. Liu, J. Shi, X. Wu, and G. Zeng, Sci. Rep. 8, 5012 (2018).
[Crossref]

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[Crossref]

A. Zhang, Y. He, L. Wu, L. Chen, and B. Wang, Optica 5, 374 (2018).
[Crossref]

2017 (2)

M. Lyu, W. Wang, H. Wang, H. Wang, G. Li, N. Chen, and G. Situ, Sci. Rep. 7, 17865 (2017).
[Crossref]

D. Phillips, M. Sun, J. Taylor, M. Edgar, S. Barnett, G. Gibson, and M. Padgett, Sci. Adv. 3, e1601728 (2017).
[Crossref]

2016 (6)

Z. Bo, W. Gong, and S. Han, Chin. Opt. Lett. 14, 070301 (2016).

L. Wang and S. Zhao, Photon. Res. 4, 240 (2016).
[Crossref]

S. Sun, W. Liu, H. Lin, J. Liu, Q. Li, and P. Chen, Sci. Rep. 6, 37013 (2016).
[Crossref]

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and T. Zhu, Phys. Rev. Lett. 117, 113901 (2016).
[Crossref]

P. Morris, R. Aspden, J. Bell, R. Boyd, and M. Padgett, Nat. Commun. 6, 5913 (2016).
[Crossref]

W. Gong, C. Zhang, H. Yu, M. Chen, W. Xu, and S. Han, Sci. Rep. 6, 26133 (2016).
[Crossref]

2015 (3)

2014 (1)

E. Li, Z. Bo, M. Chen, W. Gong, and S. S. Han, Appl. Phys. Lett. 104, 251120 (2014).
[Crossref]

2013 (2)

M. Aßmann and M. Bayer, Sci. Rep. 3, 1545 (2013).
[Crossref]

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, Science 340, 844 (2013).
[Crossref]

2012 (1)

2011 (2)

W. Gong and S. Han, Opt. Lett. 36, 394 (2011).
[Crossref]

H. Li, J. Xiong, and G. Zeng, Opt. Eng. 50, 127005 (2011).
[Crossref]

2010 (2)

K. W. Chan, M. N. O’Sullivan, and R. W. Boyd, Opt. Express 18, 5562 (2010).
[Crossref]

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, Phys. Rev. Lett. 104, 253603 (2010).
[Crossref]

2009 (1)

Y. Bromberg, O. Katz, and Y. Silberberg, Phys. Rev. A 79, 053840 (2009).
[Crossref]

2008 (1)

J. Shapiro, Phys. Rev. A 78, 061802 (2008).
[Crossref]

2005 (1)

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, Phys. Rev. Lett. 94, 063601 (2005).
[Crossref]

1995 (1)

T. Pittman, Y. Shih, D. Strekalov, and A. Sergienko, Phys. Rev. A 52, R3429 (1995).
[Crossref]

Aspden, R.

P. Morris, R. Aspden, J. Bell, R. Boyd, and M. Padgett, Nat. Commun. 6, 5913 (2016).
[Crossref]

Aßmann, M.

M. Aßmann and M. Bayer, Sci. Rep. 3, 1545 (2013).
[Crossref]

Bai, Y.

Barnett, S.

D. Phillips, M. Sun, J. Taylor, M. Edgar, S. Barnett, G. Gibson, and M. Padgett, Sci. Adv. 3, e1601728 (2017).
[Crossref]

Bayer, M.

M. Aßmann and M. Bayer, Sci. Rep. 3, 1545 (2013).
[Crossref]

Bell, J.

P. Morris, R. Aspden, J. Bell, R. Boyd, and M. Padgett, Nat. Commun. 6, 5913 (2016).
[Crossref]

Bo, Z.

Z. Bo, W. Gong, and S. Han, Chin. Opt. Lett. 14, 070301 (2016).

E. Li, Z. Bo, M. Chen, W. Gong, and S. S. Han, Appl. Phys. Lett. 104, 251120 (2014).
[Crossref]

Bowman, A.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, Science 340, 844 (2013).
[Crossref]

Bowman, R.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, Science 340, 844 (2013).
[Crossref]

Boyd, R.

J. Zhao, Y. E. K. Williams, X. Zhang, and R. Boyd, Light Sci. Appl. 8, 55 (2019).
[Crossref]

P. Morris, R. Aspden, J. Bell, R. Boyd, and M. Padgett, Nat. Commun. 6, 5913 (2016).
[Crossref]

Boyd, R. W.

Bromberg, Y.

Y. Bromberg, O. Katz, and Y. Silberberg, Phys. Rev. A 79, 053840 (2009).
[Crossref]

Chan, K. W.

Chen, H.

Y. He, G. Wang, G. Dong, S. Zhu, H. Chen, A. Zhang, and Z. Xu, Sci. Rep. 8, 6469 (2018).
[Crossref]

Chen, L.

Chen, M.

W. Gong, C. Zhang, H. Yu, M. Chen, W. Xu, and S. Han, Sci. Rep. 6, 26133 (2016).
[Crossref]

X. Li, C. Deng, M. Chen, W. Gong, and S. Han, Photon. Res. 3, 153 (2015).
[Crossref]

E. Li, Z. Bo, M. Chen, W. Gong, and S. S. Han, Appl. Phys. Lett. 104, 251120 (2014).
[Crossref]

Chen, N.

M. Lyu, W. Wang, H. Wang, H. Wang, G. Li, N. Chen, and G. Situ, Sci. Rep. 7, 17865 (2017).
[Crossref]

Chen, P.

Chen, W.

D’Angelo, M.

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, Phys. Rev. Lett. 94, 063601 (2005).
[Crossref]

Dai, H.

Deng, C.

Dong, G.

Y. He, G. Wang, G. Dong, S. Zhu, H. Chen, A. Zhang, and Z. Xu, Sci. Rep. 8, 6469 (2018).
[Crossref]

Du, G.

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and T. Zhu, Phys. Rev. Lett. 117, 113901 (2016).
[Crossref]

Edgar, M.

D. Phillips, M. Sun, J. Taylor, M. Edgar, S. Barnett, G. Gibson, and M. Padgett, Sci. Adv. 3, e1601728 (2017).
[Crossref]

B. Sun, S. Welsh, M. Edgar, J. Shapiro, and M. Padgett, Opt. Express 20, 16892 (2012).
[Crossref]

Edgar, M. P.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, Science 340, 844 (2013).
[Crossref]

Ferri, F.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, Phys. Rev. Lett. 104, 253603 (2010).
[Crossref]

Fu, X.

Gao, Y.

Gatti, A.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, Phys. Rev. Lett. 104, 253603 (2010).
[Crossref]

Gibson, G.

D. Phillips, M. Sun, J. Taylor, M. Edgar, S. Barnett, G. Gibson, and M. Padgett, Sci. Adv. 3, e1601728 (2017).
[Crossref]

Gong, W.

W. Gong, C. Zhang, H. Yu, M. Chen, W. Xu, and S. Han, Sci. Rep. 6, 26133 (2016).
[Crossref]

Z. Bo, W. Gong, and S. Han, Chin. Opt. Lett. 14, 070301 (2016).

X. Li, C. Deng, M. Chen, W. Gong, and S. Han, Photon. Res. 3, 153 (2015).
[Crossref]

E. Li, Z. Bo, M. Chen, W. Gong, and S. S. Han, Appl. Phys. Lett. 104, 251120 (2014).
[Crossref]

W. Gong and S. Han, Opt. Lett. 36, 394 (2011).
[Crossref]

Han, S.

Z. Bo, W. Gong, and S. Han, Chin. Opt. Lett. 14, 070301 (2016).

W. Gong, C. Zhang, H. Yu, M. Chen, W. Xu, and S. Han, Sci. Rep. 6, 26133 (2016).
[Crossref]

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and T. Zhu, Phys. Rev. Lett. 117, 113901 (2016).
[Crossref]

X. Li, C. Deng, M. Chen, W. Gong, and S. Han, Photon. Res. 3, 153 (2015).
[Crossref]

W. Gong and S. Han, Opt. Lett. 36, 394 (2011).
[Crossref]

Han, S. S.

E. Li, Z. Bo, M. Chen, W. Gong, and S. S. Han, Appl. Phys. Lett. 104, 251120 (2014).
[Crossref]

He, G.

He, Y.

A. Zhang, Y. He, L. Wu, L. Chen, and B. Wang, Optica 5, 374 (2018).
[Crossref]

Y. He, G. Wang, G. Dong, S. Zhu, H. Chen, A. Zhang, and Z. Xu, Sci. Rep. 8, 6469 (2018).
[Crossref]

Huang, X.

Jiang, S.

Jiang, W.

Jiao, S.

Katz, O.

Y. Bromberg, O. Katz, and Y. Silberberg, Phys. Rev. A 79, 053840 (2009).
[Crossref]

Lei, T.

Li, E.

E. Li, Z. Bo, M. Chen, W. Gong, and S. S. Han, Appl. Phys. Lett. 104, 251120 (2014).
[Crossref]

Li, G.

M. Lyu, W. Wang, H. Wang, H. Wang, G. Li, N. Chen, and G. Situ, Sci. Rep. 7, 17865 (2017).
[Crossref]

Li, H.

H. Li, J. Xiong, and G. Zeng, Opt. Eng. 50, 127005 (2011).
[Crossref]

Li, Q.

Li, X.

Lin, H.

S. Sun, W. Liu, H. Lin, J. Liu, Q. Li, and P. Chen, Sci. Rep. 6, 37013 (2016).
[Crossref]

E. Zhang, H. Lin, W. Liu, Q. Li, and P. Chen, Opt. Express 23, 33506 (2015).
[Crossref]

Liu, J.

S. Sun, W. Liu, H. Lin, J. Liu, Q. Li, and P. Chen, Sci. Rep. 6, 37013 (2016).
[Crossref]

Liu, W.

Liu, X.

X. Liu, J. Shi, X. Wu, and G. Zeng, Sci. Rep. 8, 5012 (2018).
[Crossref]

Lu, R.

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and T. Zhu, Phys. Rev. Lett. 117, 113901 (2016).
[Crossref]

Lugiato, L. A.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, Phys. Rev. Lett. 104, 253603 (2010).
[Crossref]

Lyu, M.

M. Lyu, W. Wang, H. Wang, H. Wang, G. Li, N. Chen, and G. Situ, Sci. Rep. 7, 17865 (2017).
[Crossref]

Magatti, D.

F. Ferri, D. Magatti, L. A. Lugiato, and A. Gatti, Phys. Rev. Lett. 104, 253603 (2010).
[Crossref]

Morris, P.

P. Morris, R. Aspden, J. Bell, R. Boyd, and M. Padgett, Nat. Commun. 6, 5913 (2016).
[Crossref]

Nan, S.

O’Sullivan, M. N.

Padgett, M.

Z. Xu, W. Chen, J. Penuelas, M. Padgett, and M. Sun, Opt. Express 26, 2427 (2018).
[Crossref]

D. Phillips, M. Sun, J. Taylor, M. Edgar, S. Barnett, G. Gibson, and M. Padgett, Sci. Adv. 3, e1601728 (2017).
[Crossref]

P. Morris, R. Aspden, J. Bell, R. Boyd, and M. Padgett, Nat. Commun. 6, 5913 (2016).
[Crossref]

B. Sun, S. Welsh, M. Edgar, J. Shapiro, and M. Padgett, Opt. Express 20, 16892 (2012).
[Crossref]

Padgett, M. J.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, Science 340, 844 (2013).
[Crossref]

Penuelas, J.

Phillips, D.

D. Phillips, M. Sun, J. Taylor, M. Edgar, S. Barnett, G. Gibson, and M. Padgett, Sci. Adv. 3, e1601728 (2017).
[Crossref]

Pittman, T.

T. Pittman, Y. Shih, D. Strekalov, and A. Sergienko, Phys. Rev. A 52, R3429 (1995).
[Crossref]

Scarcelli, G.

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, Phys. Rev. Lett. 94, 063601 (2005).
[Crossref]

Sergienko, A.

T. Pittman, Y. Shih, D. Strekalov, and A. Sergienko, Phys. Rev. A 52, R3429 (1995).
[Crossref]

Shapiro, J.

Shi, J.

X. Liu, J. Shi, X. Wu, and G. Zeng, Sci. Rep. 8, 5012 (2018).
[Crossref]

Shih, Y.

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, Phys. Rev. Lett. 94, 063601 (2005).
[Crossref]

T. Pittman, Y. Shih, D. Strekalov, and A. Sergienko, Phys. Rev. A 52, R3429 (1995).
[Crossref]

Silberberg, Y.

Y. Bromberg, O. Katz, and Y. Silberberg, Phys. Rev. A 79, 053840 (2009).
[Crossref]

Situ, G.

M. Lyu, W. Wang, H. Wang, H. Wang, G. Li, N. Chen, and G. Situ, Sci. Rep. 7, 17865 (2017).
[Crossref]

Strekalov, D.

T. Pittman, Y. Shih, D. Strekalov, and A. Sergienko, Phys. Rev. A 52, R3429 (1995).
[Crossref]

Sun, B.

Sun, M.

Sun, S.

S. Sun, W. Liu, H. Lin, J. Liu, Q. Li, and P. Chen, Sci. Rep. 6, 37013 (2016).
[Crossref]

Tan, W.

Taylor, J.

D. Phillips, M. Sun, J. Taylor, M. Edgar, S. Barnett, G. Gibson, and M. Padgett, Sci. Adv. 3, e1601728 (2017).
[Crossref]

Valencia, A.

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, Phys. Rev. Lett. 94, 063601 (2005).
[Crossref]

Vittert, L. E.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, Science 340, 844 (2013).
[Crossref]

Wang, B.

Wang, G.

Y. He, G. Wang, G. Dong, S. Zhu, H. Chen, A. Zhang, and Z. Xu, Sci. Rep. 8, 6469 (2018).
[Crossref]

Wang, H.

M. Lyu, W. Wang, H. Wang, H. Wang, G. Li, N. Chen, and G. Situ, Sci. Rep. 7, 17865 (2017).
[Crossref]

M. Lyu, W. Wang, H. Wang, H. Wang, G. Li, N. Chen, and G. Situ, Sci. Rep. 7, 17865 (2017).
[Crossref]

Wang, L.

Wang, W.

M. Lyu, W. Wang, H. Wang, H. Wang, G. Li, N. Chen, and G. Situ, Sci. Rep. 7, 17865 (2017).
[Crossref]

Wang, Y.

Welsh, S.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, Science 340, 844 (2013).
[Crossref]

B. Sun, S. Welsh, M. Edgar, J. Shapiro, and M. Padgett, Opt. Express 20, 16892 (2012).
[Crossref]

Williams, Y. E. K.

J. Zhao, Y. E. K. Williams, X. Zhang, and R. Boyd, Light Sci. Appl. 8, 55 (2019).
[Crossref]

Wu, L.

Wu, X.

X. Liu, J. Shi, X. Wu, and G. Zeng, Sci. Rep. 8, 5012 (2018).
[Crossref]

Xiao, T.

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and T. Zhu, Phys. Rev. Lett. 117, 113901 (2016).
[Crossref]

Xie, H.

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and T. Zhu, Phys. Rev. Lett. 117, 113901 (2016).
[Crossref]

Xie, Z.

Xiong, J.

H. Li, J. Xiong, and G. Zeng, Opt. Eng. 50, 127005 (2011).
[Crossref]

Xu, W.

W. Gong, C. Zhang, H. Yu, M. Chen, W. Xu, and S. Han, Sci. Rep. 6, 26133 (2016).
[Crossref]

Xu, Y.

Xu, Z.

Z. Xu, W. Chen, J. Penuelas, M. Padgett, and M. Sun, Opt. Express 26, 2427 (2018).
[Crossref]

Y. He, G. Wang, G. Dong, S. Zhu, H. Chen, A. Zhang, and Z. Xu, Sci. Rep. 8, 6469 (2018).
[Crossref]

Yu, H.

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and T. Zhu, Phys. Rev. Lett. 117, 113901 (2016).
[Crossref]

W. Gong, C. Zhang, H. Yu, M. Chen, W. Xu, and S. Han, Sci. Rep. 6, 26133 (2016).
[Crossref]

Yuan, X.

Zeng, G.

X. Liu, J. Shi, X. Wu, and G. Zeng, Sci. Rep. 8, 5012 (2018).
[Crossref]

H. Li, J. Xiong, and G. Zeng, Opt. Eng. 50, 127005 (2011).
[Crossref]

Zhang, A.

Y. He, G. Wang, G. Dong, S. Zhu, H. Chen, A. Zhang, and Z. Xu, Sci. Rep. 8, 6469 (2018).
[Crossref]

A. Zhang, Y. He, L. Wu, L. Chen, and B. Wang, Optica 5, 374 (2018).
[Crossref]

Zhang, C.

W. Gong, C. Zhang, H. Yu, M. Chen, W. Xu, and S. Han, Sci. Rep. 6, 26133 (2016).
[Crossref]

Zhang, E.

Zhang, X.

J. Zhao, Y. E. K. Williams, X. Zhang, and R. Boyd, Light Sci. Appl. 8, 55 (2019).
[Crossref]

Zhang, Z.

Zhao, J.

J. Zhao, Y. E. K. Williams, X. Zhang, and R. Boyd, Light Sci. Appl. 8, 55 (2019).
[Crossref]

Zhao, S.

Zhu, S.

Y. He, G. Wang, G. Dong, S. Zhu, H. Chen, A. Zhang, and Z. Xu, Sci. Rep. 8, 6469 (2018).
[Crossref]

Zhu, T.

H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and T. Zhu, Phys. Rev. Lett. 117, 113901 (2016).
[Crossref]

Appl. Phys. Lett. (1)

E. Li, Z. Bo, M. Chen, W. Gong, and S. S. Han, Appl. Phys. Lett. 104, 251120 (2014).
[Crossref]

Chin. Opt. Lett. (1)

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

Light Sci. Appl. (1)

J. Zhao, Y. E. K. Williams, X. Zhang, and R. Boyd, Light Sci. Appl. 8, 55 (2019).
[Crossref]

Nat. Commun. (1)

P. Morris, R. Aspden, J. Bell, R. Boyd, and M. Padgett, Nat. Commun. 6, 5913 (2016).
[Crossref]

Opt. Eng. (1)

H. Li, J. Xiong, and G. Zeng, Opt. Eng. 50, 127005 (2011).
[Crossref]

Opt. Express (7)

Opt. Lett. (1)

Optica (1)

Photon. Res. (2)

Phys. Rev. A (3)

J. Shapiro, Phys. Rev. A 78, 061802 (2008).
[Crossref]

Y. Bromberg, O. Katz, and Y. Silberberg, Phys. Rev. A 79, 053840 (2009).
[Crossref]

T. Pittman, Y. Shih, D. Strekalov, and A. Sergienko, Phys. Rev. A 52, R3429 (1995).
[Crossref]

Phys. Rev. Lett. (3)

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Supplementary Material (6)

NameDescription
» Visualization 1       Translating car to be imaged.
» Visualization 2       Blurry images of the translating object obtained from ghost imaging.
» Visualization 3       Tracking and imaging of the translating object by crosscorrelation based ghost imaging.
» Visualization 4       Translating and rotating car to be imaged.
» Visualization 5       Blurry images of the translating and rotating obtained from ghost imaging.
» Visualization 6       Tracking and imaging of the translating object by crosscorrelation based ghost imaging.

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Figures (3)

Fig. 1.
Fig. 1. Schematic diagram of the experimental setup. On the Fourier plane of L1 ( f 1 = 30 cm ), there are a CCD1 and a DMD. The moving object is displayed by the DMD, and the light reflected from the DMD is recorded by CCD2. F1 and F2 are narrowband filters (FL532-10).
Fig. 2.
Fig. 2. Reconstructed images and the calculated trajectory with CBGI. (a) The first row is the car at different positions (see Visualization 1). The second row is the blurry images with 400 samplings at each time frame (see Visualization 2). The third row is the reconstructed images with CBGI (see Visualization 3). (b) shows the real trajectory and calculated trajectory with a different number of samplings at each time frame.
Fig. 3.
Fig. 3. Results for a translating and rotating object. The first row is the object at different positions (see Visualization 4). The second row is blurry images from 600 samplings at each time frame (see Visualization 5). The third row is the reconstructed images of the object with CBGI (see Visualization 6).

Equations (14)

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G ( x ) = 1 N i = 0 N I i ( x ) B i 1 N i = 0 N I i ( x ) 1 N i = 0 N B i ,
C ( Δ x ) = x = 1 M G k ( x ) G l ( x Δ x ) .
S G K ( x ) = 1 K k = 1 K G k ( x Δ x k ) ,
C a ( Δ θ ) = x = 1 M G k ( x ; θ ) R ( x 0 ; Δ θ ) G l ( x ; θ ) ,
C s ( Δ x ) = x = 1 M G k ( x ; θ ) S ( Δ x ) { R ( x 0 ; Δ θ l ) G l ( x ; θ ) } ,
S G k ( x ) = 1 K k = 1 K S ( Δ x k ) { R ( x 0 , ; Δ θ k ) G k ( x ; θ ) } .
CNR G = G ( x in ) G ( x out ) 1 2 [ Δ 2 G ( x in ) + Δ 2 G ( x out ) ] ,
G ( x ) = { 1 + 1 T x x in 1 x x out
Δ 2 G ( x ) = { 2 ( T + 3 ) N T 2 x x in 2 N T x x out .
CNR G N 2 T + 3 .
CNR ps = C ( x Δ k ) C ( x Δ k 1 ) 1 2 ( Δ 2 C ( x Δ k ) + Δ 2 C ( x Δ k 1 ) ) ,
C ( x Δ k ) = x x in T G k ( x ) G l ( x ) + x x out M T G k ( x ) G l ( x )
C ( x Δ k 1 ) = x x in T E G k ( x ) G l ( x 1 ) + x x eg E G k ( x ) G l ( x out 1 ) + x x eg E G k ( x out ) G l ( x 1 ) + x x out M ( T + E ) G k ( x out ) G l ( x out 1 ) ,
CNR ps γ N 2 T + 3 ,