Abstract

Recent developments in 3D computational optical imaging such as digital holographic microscopy has ushered in a new era for biological research. Therefore, efficient and secure storage and retrieval of digital holograms is a challenging task for future cloud computing services. In this study, we propose a novel scheme to securely store and retrieve multiple encrypted digital holograms by using phase encoding multiplexing. In the proposed schemes, an encrypted hologram can only be accessed using a binary phase mask, which is the key to retrieve the image. In addition, it is possible to independently store, retrieve, and manage the encrypted digital holograms without affecting other groups of the encrypted holograms multiplexed using different sets of binary phase masks, due to the orthogonality properties of the Hadamard matrices with high autocorrelation and low cross-correlation. The desired encrypted holograms may also be searched for, removed, and added independently of other groups of the encrypted holograms. More and more 3D images or digital holograms can be securely and efficiently stored, retrieved, and managed.

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

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References

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    [Crossref]

2018 (3)

Z. Xia, Y. Zhu, X. Sun, Z. Qin, and K. Ren, “Towards privacy-preserving content-based image retrieval in cloud computing,” IEEE Trans. Cloud Comput. 6(1), 276–286 (2018).
[Crossref]

N. Wang, J. Fu, B. K. Bhargava, and J. Zeng, “Efficient retrieval over documents encrypted by attributes in cloud computing,” IEEE Trans. Inf. Forensics Security 13(10), 2653–2667 (2018).
[Crossref]

Y. Kim, J. Song, I. Moon, and Y. Lee, “Interference-based multiple-image encryption using binary phase masks,” Opt. Lasers Eng. 107, 281–287 (2018).
[Crossref]

2016 (1)

Z. Xia, X. Wang, L. Zhang, Z. Qin, X. Sun, and K. Ren, “A privacy-preserving and copy-deterrence content-based image retrieval scheme in cloud computing,” IEEE Trans. Inf. Forensics Security 11(11), 2594–2608 (2016).
[Crossref]

2015 (1)

I. Mehra, K. Singh, A. K. Agarwal, U. Gopinathan, and N. K. Nishchal, “Encrypting digital hologram of three-dimensional object using diffractive imaging,” J. Opt. 17(3), 035707 (2015).
[Crossref]

2014 (3)

2013 (1)

2012 (1)

X. Wang and D. Zhao, “Fully phase multiple-image encryption based on superposition principle and the digital holographic technique,” Opt. Commun. 285(21-22), 4280–4284 (2012).
[Crossref]

2011 (2)

N. K. Nishchal and T. J. Naughton, “Flexible optical encryption with multiple users and multiple security levels,” Opt. Commun. 284(3), 735–739 (2011).
[Crossref]

X. Wang and D. Zhao, “Image encoding based on coherent superposition and basic vector operations,” Opt. Commun. 284(4), 945–951 (2011).
[Crossref]

2010 (1)

2009 (3)

2008 (1)

2007 (2)

Y. Shi, G. Situ, and J. Zhang, “Multiple-image hiding in the Fresnel domain,” Opt. Lett. 32(13), 1914–1916 (2007).
[Crossref] [PubMed]

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278(2), 279–284 (2007).
[Crossref]

2006 (1)

G. Situ and J. Zhang, “Position multiplexing for multiple-image encryption,” J. Opt. A, Pure Appl. Opt. 8(5), 391–397 (2006).
[Crossref]

2005 (2)

M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption and watermarking by random phase matching,” Opt. Commun. 247(1-3), 29–37 (2005).
[Crossref]

G. Situ and J. Zhang, “Multiple-image encryption by wavelength multiplexing,” Opt. Lett. 30(11), 1306–1308 (2005).
[Crossref] [PubMed]

2004 (2)

2003 (1)

2002 (1)

U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13(9), R85–R101 (2002).
[Crossref]

2000 (3)

1999 (1)

1998 (1)

C. Denz, K.-O. Müller, T. Heimann, and T. Tschudi, “Volume holographic storage demonstrator based on phase-coded multiplexing,” IEEE J. Sel. Top. Quantum Electron. 4(5), 832–839 (1998).
[Crossref]

1995 (2)

P. Réfrégier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995).
[Crossref] [PubMed]

J. Lembcke, C. Denz, and T. Tschudi, “General formalism for angular and phase-encoding multiplexing in holographic image storage,” Opt. Mater. 4(2-3), 428–432 (1995).
[Crossref]

1994 (1)

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33(6), 1752–1756 (1994).
[Crossref]

1991 (1)

C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, “Volume hologram multiplexing using a deterministic phase encoding method,” Opt. Commun. 85(2-3), 171–176 (1991).
[Crossref]

Abuturab, M. R.

Agarwal, A. K.

I. Mehra, K. Singh, A. K. Agarwal, U. Gopinathan, and N. K. Nishchal, “Encrypting digital hologram of three-dimensional object using diffractive imaging,” J. Opt. 17(3), 035707 (2015).
[Crossref]

Alfalou, A.

Alieva, T.

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278(2), 279–284 (2007).
[Crossref]

Bevilacqua, F.

Bhargava, B. K.

N. Wang, J. Fu, B. K. Bhargava, and J. Zeng, “Efficient retrieval over documents encrypted by attributes in cloud computing,” IEEE Trans. Inf. Forensics Security 13(10), 2653–2667 (2018).
[Crossref]

Cai, L. Z.

M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption and watermarking by random phase matching,” Opt. Commun. 247(1-3), 29–37 (2005).
[Crossref]

Calvo, M. L.

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278(2), 279–284 (2007).
[Crossref]

Chang, H. T.

Chen, W.

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photonics 6(2), 120–155 (2014).
[Crossref]

Chen, X.

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photonics 6(2), 120–155 (2014).
[Crossref]

Cuche, E.

Denz, C.

C. Denz, K.-O. Müller, T. Heimann, and T. Tschudi, “Volume holographic storage demonstrator based on phase-coded multiplexing,” IEEE J. Sel. Top. Quantum Electron. 4(5), 832–839 (1998).
[Crossref]

J. Lembcke, C. Denz, and T. Tschudi, “General formalism for angular and phase-encoding multiplexing in holographic image storage,” Opt. Mater. 4(2-3), 428–432 (1995).
[Crossref]

C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, “Volume hologram multiplexing using a deterministic phase encoding method,” Opt. Commun. 85(2-3), 171–176 (1991).
[Crossref]

Depeursinge, C.

Fu, J.

N. Wang, J. Fu, B. K. Bhargava, and J. Zeng, “Efficient retrieval over documents encrypted by attributes in cloud computing,” IEEE Trans. Inf. Forensics Security 13(10), 2653–2667 (2018).
[Crossref]

Gopinathan, U.

I. Mehra, K. Singh, A. K. Agarwal, U. Gopinathan, and N. K. Nishchal, “Encrypting digital hologram of three-dimensional object using diffractive imaging,” J. Opt. 17(3), 035707 (2015).
[Crossref]

Guo, Q.

He, M. Z.

M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption and watermarking by random phase matching,” Opt. Commun. 247(1-3), 29–37 (2005).
[Crossref]

Heimann, T.

C. Denz, K.-O. Müller, T. Heimann, and T. Tschudi, “Volume holographic storage demonstrator based on phase-coded multiplexing,” IEEE J. Sel. Top. Quantum Electron. 4(5), 832–839 (1998).
[Crossref]

Hennelly, B.

Horner, J. L.

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33(6), 1752–1756 (1994).
[Crossref]

Hwang, H. E.

Javidi, B.

I. Moon, F. Yi, Y. H. Lee, and B. Javidi, “Avalanche and bit independence characteristics of double random phase encoding in the Fourier and Fresnel domains,” J. Opt. Soc. Am. A 31(5), 1104–1111 (2014).
[Crossref] [PubMed]

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photonics 6(2), 120–155 (2014).
[Crossref]

O. Matoba, T. Nomura, E. Pérez-Cabré, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97(6), 1128–1148 (2009).
[Crossref]

B. Javidi and T. Nomura, “Securing information by use of digital holography,” Opt. Lett. 25(1), 28–30 (2000).
[Crossref] [PubMed]

P. Réfrégier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995).
[Crossref] [PubMed]

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33(6), 1752–1756 (1994).
[Crossref]

Joseph, J.

Jüptner, W. P. O.

U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13(9), R85–R101 (2002).
[Crossref]

Kim, H.

Kim, Y.

Y. Kim, J. Song, I. Moon, and Y. Lee, “Interference-based multiple-image encryption using binary phase masks,” Opt. Lasers Eng. 107, 281–287 (2018).
[Crossref]

Lee, Y.

Y. Kim, J. Song, I. Moon, and Y. Lee, “Interference-based multiple-image encryption using binary phase masks,” Opt. Lasers Eng. 107, 281–287 (2018).
[Crossref]

Lee, Y. H.

Lei, L.

Lembcke, J.

J. Lembcke, C. Denz, and T. Tschudi, “General formalism for angular and phase-encoding multiplexing in holographic image storage,” Opt. Mater. 4(2-3), 428–432 (1995).
[Crossref]

Lie, W. N.

Liu, Q.

M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption and watermarking by random phase matching,” Opt. Commun. 247(1-3), 29–37 (2005).
[Crossref]

Mansour, A.

Marquet, P.

Matoba, O.

O. Matoba, T. Nomura, E. Pérez-Cabré, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97(6), 1128–1148 (2009).
[Crossref]

Mehra, I.

I. Mehra, K. Singh, A. K. Agarwal, U. Gopinathan, and N. K. Nishchal, “Encrypting digital hologram of three-dimensional object using diffractive imaging,” J. Opt. 17(3), 035707 (2015).
[Crossref]

Meng, X. F.

M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption and watermarking by random phase matching,” Opt. Commun. 247(1-3), 29–37 (2005).
[Crossref]

Millan, M. S.

O. Matoba, T. Nomura, E. Pérez-Cabré, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97(6), 1128–1148 (2009).
[Crossref]

Moon, I.

Müller, K.-O.

C. Denz, K.-O. Müller, T. Heimann, and T. Tschudi, “Volume holographic storage demonstrator based on phase-coded multiplexing,” IEEE J. Sel. Top. Quantum Electron. 4(5), 832–839 (1998).
[Crossref]

Naughton, T. J.

N. K. Nishchal and T. J. Naughton, “Flexible optical encryption with multiple users and multiple security levels,” Opt. Commun. 284(3), 735–739 (2011).
[Crossref]

Nishchal, N. K.

I. Mehra, K. Singh, A. K. Agarwal, U. Gopinathan, and N. K. Nishchal, “Encrypting digital hologram of three-dimensional object using diffractive imaging,” J. Opt. 17(3), 035707 (2015).
[Crossref]

N. K. Nishchal and T. J. Naughton, “Flexible optical encryption with multiple users and multiple security levels,” Opt. Commun. 284(3), 735–739 (2011).
[Crossref]

Nomura, T.

O. Matoba, T. Nomura, E. Pérez-Cabré, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97(6), 1128–1148 (2009).
[Crossref]

B. Javidi and T. Nomura, “Securing information by use of digital holography,” Opt. Lett. 25(1), 28–30 (2000).
[Crossref] [PubMed]

Pauliat, G.

C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, “Volume hologram multiplexing using a deterministic phase encoding method,” Opt. Commun. 85(2-3), 171–176 (1991).
[Crossref]

Peng, X.

Pérez-Cabré, E.

O. Matoba, T. Nomura, E. Pérez-Cabré, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97(6), 1128–1148 (2009).
[Crossref]

Qin, W.

Qin, Z.

Z. Xia, Y. Zhu, X. Sun, Z. Qin, and K. Ren, “Towards privacy-preserving content-based image retrieval in cloud computing,” IEEE Trans. Cloud Comput. 6(1), 276–286 (2018).
[Crossref]

Z. Xia, X. Wang, L. Zhang, Z. Qin, X. Sun, and K. Ren, “A privacy-preserving and copy-deterrence content-based image retrieval scheme in cloud computing,” IEEE Trans. Inf. Forensics Security 11(11), 2594–2608 (2016).
[Crossref]

Réfrégier, P.

Ren, K.

Z. Xia, Y. Zhu, X. Sun, Z. Qin, and K. Ren, “Towards privacy-preserving content-based image retrieval in cloud computing,” IEEE Trans. Cloud Comput. 6(1), 276–286 (2018).
[Crossref]

Z. Xia, X. Wang, L. Zhang, Z. Qin, X. Sun, and K. Ren, “A privacy-preserving and copy-deterrence content-based image retrieval scheme in cloud computing,” IEEE Trans. Inf. Forensics Security 11(11), 2594–2608 (2016).
[Crossref]

Rodrigo, J. A.

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278(2), 279–284 (2007).
[Crossref]

Roosen, G.

C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, “Volume hologram multiplexing using a deterministic phase encoding method,” Opt. Commun. 85(2-3), 171–176 (1991).
[Crossref]

Schnars, U.

U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13(9), R85–R101 (2002).
[Crossref]

Sheridan, J. T.

Shi, Y.

Singh, K.

I. Mehra, K. Singh, A. K. Agarwal, U. Gopinathan, and N. K. Nishchal, “Encrypting digital hologram of three-dimensional object using diffractive imaging,” J. Opt. 17(3), 035707 (2015).
[Crossref]

G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25(12), 887–889 (2000).
[Crossref] [PubMed]

Situ, G.

Song, J.

Y. Kim, J. Song, I. Moon, and Y. Lee, “Interference-based multiple-image encryption using binary phase masks,” Opt. Lasers Eng. 107, 281–287 (2018).
[Crossref]

Sun, X.

Z. Xia, Y. Zhu, X. Sun, Z. Qin, and K. Ren, “Towards privacy-preserving content-based image retrieval in cloud computing,” IEEE Trans. Cloud Comput. 6(1), 276–286 (2018).
[Crossref]

Z. Xia, X. Wang, L. Zhang, Z. Qin, X. Sun, and K. Ren, “A privacy-preserving and copy-deterrence content-based image retrieval scheme in cloud computing,” IEEE Trans. Inf. Forensics Security 11(11), 2594–2608 (2016).
[Crossref]

Tschudi, T.

C. Denz, K.-O. Müller, T. Heimann, and T. Tschudi, “Volume holographic storage demonstrator based on phase-coded multiplexing,” IEEE J. Sel. Top. Quantum Electron. 4(5), 832–839 (1998).
[Crossref]

J. Lembcke, C. Denz, and T. Tschudi, “General formalism for angular and phase-encoding multiplexing in holographic image storage,” Opt. Mater. 4(2-3), 428–432 (1995).
[Crossref]

C. Denz, G. Pauliat, G. Roosen, and T. Tschudi, “Volume hologram multiplexing using a deterministic phase encoding method,” Opt. Commun. 85(2-3), 171–176 (1991).
[Crossref]

Unnikrishnan, G.

Wang, B.

Wang, N.

N. Wang, J. Fu, B. K. Bhargava, and J. Zeng, “Efficient retrieval over documents encrypted by attributes in cloud computing,” IEEE Trans. Inf. Forensics Security 13(10), 2653–2667 (2018).
[Crossref]

Wang, Q.

Wang, X.

Z. Xia, X. Wang, L. Zhang, Z. Qin, X. Sun, and K. Ren, “A privacy-preserving and copy-deterrence content-based image retrieval scheme in cloud computing,” IEEE Trans. Inf. Forensics Security 11(11), 2594–2608 (2016).
[Crossref]

X. Wang and D. Zhao, “Fully phase multiple-image encryption based on superposition principle and the digital holographic technique,” Opt. Commun. 285(21-22), 4280–4284 (2012).
[Crossref]

X. Wang and D. Zhao, “Image encoding based on coherent superposition and basic vector operations,” Opt. Commun. 284(4), 945–951 (2011).
[Crossref]

Wang, X. C.

M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption and watermarking by random phase matching,” Opt. Commun. 247(1-3), 29–37 (2005).
[Crossref]

Xia, Z.

Z. Xia, Y. Zhu, X. Sun, Z. Qin, and K. Ren, “Towards privacy-preserving content-based image retrieval in cloud computing,” IEEE Trans. Cloud Comput. 6(1), 276–286 (2018).
[Crossref]

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Figures (13)

Fig. 1
Fig. 1 Off-axis configuration in digital holography microscopy.
Fig. 2
Fig. 2 Reconstruction of an off-axis hologram. (a) original hologram, (b) spatial spectrum of the hologram, (c) filtered spectrum of the hologram, (d) filtered hologram, (e) amplitude image, (f) phase contrast image.
Fig. 3
Fig. 3 Schematics for encrypting and multiplexing digital holograms using DRPE and phase encoding multiplexing. O is the filtered hologram, RPM denotes the random phase-only mask, BPM denotes the binary phase mask, FT and FT−1 are the Fourier and inverse Fourier transform operators, E is the encrypted hologram, and M is the multiplexed image.
Fig. 4
Fig. 4 Phase encoding and multiplexing of the (x, y)th pixel in encrypted holograms using phase-encoding multiplexing. (a)-(c) Phase encoding of the first encrypted hologram with the first row of the Hadamard matrix. (d) Phase encoding of the fourth encrypted hologram with the fourth row of the Hadamard matrix. (e) Multiplexing of four phase encoded images. ρi is the phase of a pixel in the ith encrypted hologram and ψij denotes the phase (0 or π radian) of the (i, j)th element of the Hadamard matrix.
Fig. 5
Fig. 5 Schematics for the restoration of digital holograms. M is the multiplexed image, FT and FT−1 are the Fourier and the inverse Fourier transform operators, E is the recovered encrypted hologram, D’ is the decrypted hologram, and * is the complex conjugate operator.
Fig. 6
Fig. 6 Four digital holograms used in numerical simulations for the proposed scheme.
Fig. 7
Fig. 7 (a) Amplitude and (b) phase distributions of the multiplexed image.
Fig. 8
Fig. 8 (a)-(d) Four holograms decrypted by using the correct BPMs and the correct RPMs. (e)-(h) Four holograms decrypted by using the correct RPMs but wrong BPMs.
Fig. 9
Fig. 9 16 digital holograms (four groups of four holograms) used in numerical simulations. (a)-(d) Group I, (e)-(h) group II, (i)-(l) group III, (m)-(p) group IV.
Fig. 10
Fig. 10 Four multiplexed images obtained from four groups of four encrypted holograms. (a)-(d) Amplitude distributions, (e)-(h) phase distributions; (a) and (e) the multiplexed image from group I, (b) and (f) the multiplexed image from group II, (c) and (g) the multiplexed image from group III, (d) and (h) the multiplexed image from group IV.
Fig. 11
Fig. 11 Decrypted holograms obtained from four multiplexed images in Fig. 10. (a)-(d) Holograms decrypted by the 2nd BPM, (e)-(h) holograms decrypted by the 8th BPM, (i)-(l) holograms decrypted by the 9th BPM, (m)-(p) holograms decrypted by the 15th BPM.
Fig. 12
Fig. 12 Amplitude and phase contrast images of correct decrypted holograms in Fig. 11. (a)-(d) Amplitude images, (e)-(h) phase contrast images; (a) and (e) for the 2nd decrypted hologram, (b) and (f) for the 8th decrypted hologram, (c) and (g) for the 9th decrypted hologram, (d) and (h) for the 15th decrypted hologram.
Fig. 13
Fig. 13 Correction coefficients of the decrypted hologram according to the ratio of correct Hadamard matrices in BPMs.

Tables (1)

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Table 1 The correlation coefficients between original holograms and decrypted holograms.

Equations (16)

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I H ( x,y )= | R | 2 + | O | 2 + R * O+R O * .
I R F ( x,y )=F T 1 { SFFT[ I H ( x,y ) ] }=R O * ,
Ψ( m,n )=Aexp[ iπ λd ( m 2 Δ ξ 2 + n 2 Δ η 2 ) ] ×FT{ R D ( k,l ) I R F ( k,l )exp[ iπ λd ( k 2 Δ x 2 + l 2 Δ y 2 ) ] },
R D ( k,l )= A R exp[ i( 2π/λ )( k x kΔx+ k y lΔy ) ],
I( m,n )=Im [ Ψ( m,n ) ] 2 +Re [ Ψ( m,n ) ] 2 .
ϕ( m,n )= tan 1 { Im[ Ψ( m,n ) ] Re[ Ψ( m,n ) ] }.
F i ( ξ,η )=FT[ O i ( x,y ) e j φ i 1 ( x,y ) ]fori=1,2,,n,
E i ( x,y )=F T 1 { F i ( ξ,η ) e j φ i 2 ( ξ,η ) }fori=1,2,,n,
H 1×1 =1, H 2×2 =[ H 1×1 H 1×1 H 1×1 H 1×1 ],, H n×n =[ H ( n/2 )×( n/2 ) H ( n/2 )×( n/2 ) H ( n/2 )×( n/2 ) H ( n/2 )×( n/2 ) ],
H T H=nI,
j=1 n h ij h kj =n δ ik ,
δ ik ={ 1,i=k 0,ik.
M( x,y )= H n×n T [ e 1 ( x,y ) e j ρ 1 ( x,y ) e 2 ( x,y ) e j ρ 2 ( x,y ) e 3 ( x,y ) e j ρ 3 ( x,y ) e n ( x,y ) e j ρ n ( x,y ) ],
E( x,y )= 1 n H n×n H n×n T [ e 1 ( x,y ) e j ρ 1 ( x,y ) e 2 ( x,y ) e j ρ 2 ( x,y ) e 3 ( x,y ) e j ρ 3 ( x,y ) e n ( x,y ) e j ρ n ( x,y ) ].
D i ( x,y )= [ e j φ i 1 ( x,y ) ] * F T 1 { [ e j φ i 2 ( ξ,η ) ] * FT[ E( x,y ) ] }fori=1,2,,n,
CC= cov( D,O ) σ( D )σ( O ) ,

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