Abstract

A spatial multiplexing reconstruction method has been proposed to improve the sampling efficiency and image quality of Fourier-transform ghost imaging. In this method, the sensing equation of Fourier-transform ghost imaging is established based on recombination and reutilization of the correlated intensity distributions of light fields. It is theoretically proved that the scale of the sensing matrix in the sensing equation can be greatly reduced, and spatial multiplexing combined with this matrix reduction provides the feasibility of ghost imaging with just a few measurements. Experimental results show better visibility and signal-to-noise ratio in the Fourier spectrums reconstructed via spatial multiplexing compared with previous methods. The transmittance of an object is also recovered in spatial domain with better image quality based on its spectrum of spatial multiplexing reconstruction. This method is especially important to x-ray ghost imaging applications due to its potential for reducing radiation damage and achieving high quality images in x-ray microscopy.

© 2018 Optical Society of America

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References

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2016 (4)

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).
[PubMed]

D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganin, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117(11), 113902 (2016).
[PubMed]

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

Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
[PubMed]

2015 (1)

2014 (1)

2012 (2)

H. Wang and S. Han, “Coherent ghost imaging based on sparsity constraint without phase-sensitive detection,” EPL 98(2), 24003 (2012).

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101(14), 141123 (2012).

2011 (3)

R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. 98(11), 111115 (2011).

N. Tian, Q. Guo, A. Wang, D. Xu, and L. Fu, “Fluorescence ghost imaging with pseudothermal light,” Opt. Lett. 36(16), 3302–3304 (2011).
[PubMed]

M. F. Duarte and Y. C. Eldar, “Structured compressed sensing: From theory to applications,” IEEE Trans Sign. Process. 59(9), 4053–4085 (2011).

2010 (2)

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

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

2009 (4)

2008 (1)

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

2007 (3)

M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, J. Cheng, and S. Han, “Lensless Fourier-transform ghost imaging with classical incoherent light,” Phys. Rev. A 75(2), 021803 (2007).

J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53(12), 4655–4666 (2007).

X. H Chen, I. N. Agafonovei, K. H. Luo, Q. Liu, R. Xian, M. V. Chekhova, and L. A. Wu, “LHigh-visibility, high-order lensless ghost imaging with thermal light,” Opt. Lett. 35(8), 1166–1168 (2007).

2005 (3)

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

D. Z. Cao, J. Xiong, and K. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71(1), 013801 (2005).

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94(6), 063601 (2005).
[PubMed]

2004 (3)

J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. 92(9), 093903 (2004).
[PubMed]

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

J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. 92(9), 093903 (2004).
[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).
[PubMed]

1994 (1)

V. Belinskii and D. N. Klyshko, “Two-photon optics: diffraction, holography, and transformation of two-dimensional signals,” J. Exp. Theor. Phys. 78, 259–262 (1994).

1982 (1)

1978 (1)

Agafonovei, I. N.

Bache, M.

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

Belinskii, V.

V. Belinskii and D. N. Klyshko, “Two-photon optics: diffraction, holography, and transformation of two-dimensional signals,” J. Exp. Theor. Phys. 78, 259–262 (1994).

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).
[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).
[PubMed]

Boyd, R. W.

K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “High-order thermal ghost imaging,” Opt. Lett. 34(21), 3343–3345 (2009).
[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).
[PubMed]

Brambilla, E. G.

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

Bromberg, Y.

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

Cantelli, V.

D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganin, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117(11), 113902 (2016).
[PubMed]

Cao, D. Z.

D. Z. Cao, J. Xiong, and K. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71(1), 013801 (2005).

Chan, K. W. C.

Chekhova, M. V.

Chen, M.

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

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101(14), 141123 (2012).

Chen, X. H

Chen, X. H.

Cheng, J.

M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, J. Cheng, and S. Han, “Lensless Fourier-transform ghost imaging with classical incoherent light,” Phys. Rev. A 75(2), 021803 (2007).

J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. 92(9), 093903 (2004).
[PubMed]

J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. 92(9), 093903 (2004).
[PubMed]

Collett, E.

D’Angelo, M.

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94(6), 063601 (2005).
[PubMed]

Deacon, K. S.

R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. 98(11), 111115 (2011).

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).
[PubMed]

Duarte, M. F.

M. F. Duarte and Y. C. Eldar, “Structured compressed sensing: From theory to applications,” IEEE Trans Sign. Process. 59(9), 4053–4085 (2011).

Eldar, Y. C.

M. F. Duarte and Y. C. Eldar, “Structured compressed sensing: From theory to applications,” IEEE Trans Sign. Process. 59(9), 4053–4085 (2011).

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images,” Opt. Express 17(26), 23920–23946 (2009).
[PubMed]

Ferri, F.

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

Fienup, J. R.

Fu, L.

Gatti, A.

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

Gazit, S.

Gilbert, A. C.

J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53(12), 4655–4666 (2007).

Gong, W.

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

H. Yu, E. Li, W. Gong, and S. Han, “Structured image reconstruction for three-dimensional ghost imaging lidar,” Opt. Express 23(11), 14541–14551 (2015).
[PubMed]

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101(14), 141123 (2012).

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

Guo, Q.

Han, S.

Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
[PubMed]

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

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).
[PubMed]

H. Yu, E. Li, W. Gong, and S. Han, “Structured image reconstruction for three-dimensional ghost imaging lidar,” Opt. Express 23(11), 14541–14551 (2015).
[PubMed]

H. Wang and S. Han, “Coherent ghost imaging based on sparsity constraint without phase-sensitive detection,” EPL 98(2), 24003 (2012).

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101(14), 141123 (2012).

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

M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, J. Cheng, and S. Han, “Lensless Fourier-transform ghost imaging with classical incoherent light,” Phys. Rev. A 75(2), 021803 (2007).

J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. 92(9), 093903 (2004).
[PubMed]

J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. 92(9), 093903 (2004).
[PubMed]

Katz, O.

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

Klyshko, D. N.

V. Belinskii and D. N. Klyshko, “Two-photon optics: diffraction, holography, and transformation of two-dimensional signals,” J. Exp. Theor. Phys. 78, 259–262 (1994).

Li, E.

Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
[PubMed]

H. Yu, E. Li, W. Gong, and S. Han, “Structured image reconstruction for three-dimensional ghost imaging lidar,” Opt. Express 23(11), 14541–14551 (2015).
[PubMed]

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101(14), 141123 (2012).

Li, M. F.

Liu, H.

M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, J. Cheng, and S. Han, “Lensless Fourier-transform ghost imaging with classical incoherent light,” Phys. Rev. A 75(2), 021803 (2007).

Liu, Q.

Liu, X. F.

Liu, Y.

M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, J. Cheng, and S. Han, “Lensless Fourier-transform ghost imaging with classical incoherent light,” Phys. Rev. A 75(2), 021803 (2007).

Liu, Z.

Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
[PubMed]

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).
[PubMed]

Lugiato, L. A.

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

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

Luo, K. H.

Magatti, D.

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

Meyers, R. E.

R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. 98(11), 111115 (2011).

O’Sullivan, M. N.

Paganin, D. M.

D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganin, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117(11), 113902 (2016).
[PubMed]

Pelliccia, D.

D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganin, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117(11), 113902 (2016).
[PubMed]

Rack, A.

D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganin, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117(11), 113902 (2016).
[PubMed]

Scarcelli, G.

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94(6), 063601 (2005).
[PubMed]

Scheel, M.

D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganin, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117(11), 113902 (2016).
[PubMed]

Segev, M.

Shapiro, H. J.

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

Shen, X.

Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
[PubMed]

M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, J. Cheng, and S. Han, “Lensless Fourier-transform ghost imaging with classical incoherent light,” Phys. Rev. A 75(2), 021803 (2007).

Shih, Y.

R. E. Meyers, K. S. Deacon, and Y. Shih, “Turbulence-free ghost imaging,” Appl. Phys. Lett. 98(11), 111115 (2011).

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94(6), 063601 (2005).
[PubMed]

Silberberg, Y.

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

Szameit, A.

Tan, S.

Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
[PubMed]

Tian, N.

Tropp, J. A.

J. A. Tropp and A. C. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53(12), 4655–4666 (2007).

Valencia, A.

A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94(6), 063601 (2005).
[PubMed]

Wang, A.

Wang, H.

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101(14), 141123 (2012).

H. Wang and S. Han, “Coherent ghost imaging based on sparsity constraint without phase-sensitive detection,” EPL 98(2), 24003 (2012).

Wang, K.

D. Z. Cao, J. Xiong, and K. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71(1), 013801 (2005).

Wei, Q.

M. Zhang, Q. Wei, X. Shen, Y. Liu, H. Liu, J. Cheng, and S. Han, “Lensless Fourier-transform ghost imaging with classical incoherent light,” Phys. Rev. A 75(2), 021803 (2007).

Wolf, E.

Wu, J.

Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6, 25718 (2016).
[PubMed]

Wu, L. A.

Xian, R.

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).
[PubMed]

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).
[PubMed]

Xiong, J.

D. Z. Cao, J. Xiong, and K. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A 71(1), 013801 (2005).

Xu, D.

Xu, W.

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

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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).
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Figures (4)

Fig. 1
Fig. 1 Scheme of Fourier-transform ghost imaging via sparsity constrains (GISC). BS is a beam splitter, and the information of the object is retrieved from the sensing equation constructed with the intensities recorded by the two CCDs
Fig. 2
Fig. 2 Fourier spectrums of a double slits obtained with different reconstructions. (a) Traditional intensity correlation reconstruction, (b) standard single point reconstruction, (c)spatial multiplexing reconstruction of 40 points, (d) spatial multiplexing reconstruction of 1000 points. The insets in (a)(b)(c)(d) are the reconstructed Fourier spectrums at the CCD plane, and the curves in (a)(b)(c)(d) are the corresponding cross-section distributions. The measurements for (a)(b)(c) is 800, and the measurements for (d) is 40.
Fig. 3
Fig. 3 Fourier spectrums and recovered objects of different reconstruction methods. (a) Fourier spectrum retrieved using spatial multiplexing reconstruction of 100 points, (b) Fourier spectrum retrieved using standard single point reconstruction, (c) object recovered from (a), (d) object recovered from (b).
Fig. 4
Fig. 4 Statistical distributions of the Fourier spectrums. The blue line is the curve for the spatial multiplexing reconstruction of 100 points in Fig. 3(a), the red line is the curve for standard single point reconstruction in Fig. 3(b).

Tables (1)

Tables Icon

Table 1 Visibility and SNR of different reconstructions

Equations (19)

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E 1 ( x 1 , y 1 )= s E 0 ( x 0 , y 0 ) h( x 0 , y 0 ; x 1 , y 1 )d x 0 d y 0 ,
E t (x,y)= ref E 1 ( x 1 , y 1 ) h * ( x 1 , y 1 ;x,y)d x 1 d y 1 ,
E 2 ( x 2 , y 2 )= obj E t (x,y) t(x,y)h(x,y; x 2 , y 2 )dxdy ,
h(x,y;ξ,η)= e ikz iλz exp{ ik 2z [ (xξ) 2 + (yη) 2 ]} ,
E 2 ( x 2 , y 2 )= exp[ ik 2 d 22 ( x 2 2 + y 2 2 )] λ 2 d 22 2 ref E 1 ( x 1 , y 1 ) exp[ ik 2 d 22 ( x 1 2 + y 1 2 )]T( x 1 x 2 λ d 22 , y 1 y 2 λ d 22 )d x 1 d y 1 .
I( x 2 , y 2 )= E 2 ( x 2 , y 2 ) E 2 * ( x 2 , y 2 )               ref,re f E 1 ( x 1 , y 1 ) E 1 * ( x 1 , y 1 ) exp[ ik( x 1 2 + y 1 2 x 1 2 y 1 2 ) 2 d 22 ] T( x 1 x 2 λ d 22 , y 1 y 2 λ d 22 ) T * ( x 1 x 2 λ d 22 , y 1 y 2 λ d 22 )d x 1 d y 1 d x 1 d y 1
g Q (0) ( x 1 x 1 ' , y 1 y 1 ' )exp[ ( x 1 x 1 ' ) 2 + ( y 1 y 1 ' ) 2 8 σ L 2 ] ,
I Q (0) ( x 1 , y 1 )= ( σ L / σ Q ) 2 A L exp( x 1 2 + y 1 2 σ Q 2 ]
g Q (0) ( x 1 , y 1 ; x 1 , y 1 )= g Q (0) ( x 1 x 1 , y 1 y 1 )= E 1 ( x 1 , y 1 ) E 1 * ( x 1 , y 1 ) E 1 ( x 1 , y 1 ) E 1 * ( x 1 , y 1 ) E 1 ( x 1 , y 1 ) E 1 * ( x 1 , y 1 )
I 2 ( x 2 , y 2 )= ref,ref' I 1 ( x 1 , y 1 ) I 1 ( x 1 , y 1 ) exp[ ik( x 1 2 + y 1 2 x 1 2 y 1 2 ) 2 d 22 ]exp[ ( x 1 x 1 ) 2 + ( y 1 y 1 ) 2 8 σ L 2 ]                               ×T( x 1 x 2 λ d 22 , y 1 y 2 λ d 22 )T( x 1 x 2 λ d 22 , y 1 y 2 λ d 22 )d x 1 d y 1 d x 1 d y 1
I 2 ( x 2 , y 2 ) ref I 1 ( x 1 , y 1 ) | T( ( x 1 x 2 ) λ d 22 , ( y 1 y 2 ) λ d 22 ) | 2 d x 1 d y 1 .
I 2 q ( x 2i , y 2i )= j I 1 q ( x 1j , y 1j ) | T( x 1j x 2i λ d 22 , y 1j y 2i λ d 22 ) | 2 Δ x 1j Δ y 1j
Y p ={ I 2 1 I 2 2 I 2 m }= A P X={ I 1 1 I 1 2 I 1 m }X .
I 2 ( x 2 , y 2 ) window I 1 ( x 1 , y 1 ) | T( x 1 x 2 λ d 22 , y 1 y 2 ) λ d 22 ) | 2 d x 1 d y 1 .
Y={ Y P 1 Y p 2 Y p n }={ A p 1 A p 2 A p n }X=AX .
Min X 1 s.t.Y=AX .
μ(A)= max 1ijn | a i , a j | a i 2 a j 2 ,
μ( A s )= max 1ijn | a i , a j |= max 1ijn a i , a j .
μ( A m )= max 1ijn s=1 P a is , a js P s=1 P max 1ijn a is , a js P = s=1 P μ( A s ) P ,

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