Abstract

A robust, asymmetric, multidepth, three-dimensional object encryption scheme based on a computer-generated Fresnel hologram in the cascaded fractional Fourier domain is proposed. A layer-based Fresnel transform is used to generate a computer-generated hologram, which is then decomposed into two phase-only masks with a random phase distribution using matrix composition and decomposition methods. Encryption is implemented by using the created phase-only masks in two cascaded fractional Fourier transform domains, and a pair of private keys is generated in the encryption process. The cryptosystem is asymmetric and possesses high resistance against various potential attacks, such as brute-force, chosen-plaintext, known-plaintext, and ciphertext-only attacks. The simulation results and cryptanalysis confirmed the feasibility and effectiveness of the proposed encryption scheme.

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

Full Article  |  PDF Article
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References

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2018 (3)

L. Ma and W. Jin, “Symmetric and asymmetric hybrid cryptosystem based on compressive sensing and computer generated holography,” Opt. Commun. 407, 51–56 (2018).
[Crossref]

Z. Liu, H. Chen, W. Blondel, Z. Shen, and S. Liu, “Image security based on iterative random phase encoding in expanded fractional Fourier domains,” Opt. Lasers Eng. 105, 1–5 (2018).
[Crossref]

M.-L. Piao, Z.-X. Liu, Y.-L. Piao, H.-Y. Wu, Y. Zhao, and N. Kim, “Multi-depth three-dimensional image encryption based on the phase retrieval algorithm in the Fresnel and fractional Fourier transform domains,” Appl. Opt. 57, 7609–7617 (2018).
[Crossref]

2017 (4)

2016 (6)

D. Kong, X. Shen, L. Cao, H. Zhang, S. Zong, and G. Jin, “Three-dimensional information hierarchical encryption based on computer-generated holograms,” Opt. Commun. 380, 387–393 (2016).
[Crossref]

A. Sinha, “Nonlinear optical cryptosystem resistant to standard and hybrid attacks,” Opt. Lasers Eng. 81, 79–86 (2016).
[Crossref]

W. Chen, “Optical cryptosystem based on single-pixel encoding using the modified Gerchberg-Saxton algorithm with a cascaded structure,” J. Opt. Soc. Am. A. 33, 2305–2311 (2016).
[Crossref]

W. N. Li, C. X. Shi, M. L. Piao, and N. Kim, “Multiple-3D-object secure information system based on phase shifting method and single interference,” Appl. Opt. 55, 4052–4059 (2016).
[Crossref]

A. Fatima and N. K. Nishchal, “Discussion on comparative analysis and a new attack on optical asymmetric cryptosystem,” J. Opt. Soc. Am. A 33, 2034–2040 (2016).
[Crossref]

D. Kong, L. Cao, G. Jin, and B. Javidi, “Three-dimensional scene encryption and display based on computer-generated holograms,” Appl. Opt. 55, 8296–8300 (2016).
[Crossref]

2015 (2)

I. Muniraj, C. L. Guo, B. G. Lee, and J. T. Sheridan, “Interferometry based multispectral photon-limited 2D and 3D integral image encryption employing the Hartley transform,” Opt. Express 23, 15907–15920 (2015).
[Crossref]

N. Rawat, I. C. Hwang, Y. Shi, and B. G. Lee, “Optical image encryption via photon-counting imaging and compressive sensing based ptychography,” J. Opt. 17, 065704 (2015).
[Crossref]

2014 (2)

X. Wang, C. Dai, and J. Chen, “Optical image encryption via reverse engineering of a modified amplitude-phase retrieval-based attack,” Opt. Commun. 328, 67–72 (2014).
[Crossref]

X. Wang, Y. Chen, C. Dai, and D. Zhao, “Discussion and a new attack of the optical asymmetric cryptosystem based on phase-truncated Fourier transform,” Appl. Opt. 53, 208–213 (2014).
[Crossref]

2013 (3)

2011 (1)

2010 (1)

2009 (1)

2007 (2)

Y.-Y. Wang, Y.-R. Wang, Y. Wang, H.-J. Li, and W.-J. Sun, “Optical image encryption based on binary Fourier transform computer-generated hologram and pixel scrambling technology,” Opt. Lasers Eng. 45, 761–765 (2007).
[Crossref]

Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15, 10253–10265 (2007).
[Crossref]

2006 (2)

2005 (1)

2004 (1)

G. Sito and J. Zhang, “A lensless optical security system based on computer-generated phase only mask,” Opt. Commun. 232, 115–122 (2004).
[Crossref]

2002 (1)

2000 (2)

1995 (1)

Arcos, S.

Blondel, W.

Z. Liu, H. Chen, W. Blondel, Z. Shen, and S. Liu, “Image security based on iterative random phase encoding in expanded fractional Fourier domains,” Opt. Lasers Eng. 105, 1–5 (2018).
[Crossref]

Cao, L.

Carnicer, A.

Castro, A.

Chen, H.

Z. Liu, H. Chen, W. Blondel, Z. Shen, and S. Liu, “Image security based on iterative random phase encoding in expanded fractional Fourier domains,” Opt. Lasers Eng. 105, 1–5 (2018).
[Crossref]

Chen, J.

X. Wang, C. Dai, and J. Chen, “Optical image encryption via reverse engineering of a modified amplitude-phase retrieval-based attack,” Opt. Commun. 328, 67–72 (2014).
[Crossref]

Chen, W.

W. Chen, “Optical cryptosystem based on single-pixel encoding using the modified Gerchberg-Saxton algorithm with a cascaded structure,” J. Opt. Soc. Am. A. 33, 2305–2311 (2016).
[Crossref]

Chen, Y.

Cheung, K. W. K.

P. Tsang, K. W. K. Cheung, and T.-C. Poon, “Fast numerical generation and hybrid encryption of a computer-generated Fresnel holographic video sequence,” Chinese Opt. Lett. 11, 020901 (2013).
[Crossref]

P. W. M. Tsang, T.-C. Poon, and K. W. K. Cheung, “Fast numerical generation and encryption of computer-generated Fresnel holograms,” Appl. Opt. 50, B46–B52 (2011).
[Crossref]

Cho, M.

Coëtmellec, S.

Dai, C.

X. Wang, C. Dai, and J. Chen, “Optical image encryption via reverse engineering of a modified amplitude-phase retrieval-based attack,” Opt. Commun. 328, 67–72 (2014).
[Crossref]

X. Wang, Y. Chen, C. Dai, and D. Zhao, “Discussion and a new attack of the optical asymmetric cryptosystem based on phase-truncated Fourier transform,” Appl. Opt. 53, 208–213 (2014).
[Crossref]

Fatima, A.

Frauel, Y.

Gopinathan, U.

Guo, C. L.

Healy, J. J.

Huang, S.

Hwang, I. C.

N. Rawat, I. C. Hwang, Y. Shi, and B. G. Lee, “Optical image encryption via photon-counting imaging and compressive sensing based ptychography,” J. Opt. 17, 065704 (2015).
[Crossref]

Javidi, B.

Jin, G.

Jin, W.

L. Ma and W. Jin, “Symmetric and asymmetric hybrid cryptosystem based on compressive sensing and computer generated holography,” Opt. Commun. 407, 51–56 (2018).
[Crossref]

Joseph, J.

Juvells, I.

Kim, N.

Kong, D.

D. Kong, L. Cao, G. Jin, and B. Javidi, “Three-dimensional scene encryption and display based on computer-generated holograms,” Appl. Opt. 55, 8296–8300 (2016).
[Crossref]

D. Kong, X. Shen, L. Cao, H. Zhang, S. Zong, and G. Jin, “Three-dimensional information hierarchical encryption based on computer-generated holograms,” Opt. Commun. 380, 387–393 (2016).
[Crossref]

Kwon, K.-C.

Y. Zhao, C.-X. Shi, K.-C. Kwon, Y.-L. Piao, M.-L. Piao, and N. Kim, “Fast calculation method of computer-generated hologram using a depth camera with point cloud gridding,” Opt. Commun. 411, 166–169 (2017).
[Crossref]

Lebrun, D.

Lee, B. G.

Li, C.

Li, H.-J.

Y.-Y. Wang, Y.-R. Wang, Y. Wang, H.-J. Li, and W.-J. Sun, “Optical image encryption based on binary Fourier transform computer-generated hologram and pixel scrambling technology,” Opt. Lasers Eng. 45, 761–765 (2007).
[Crossref]

Li, W. N.

Liu, S.

Z. Liu, H. Chen, W. Blondel, Z. Shen, and S. Liu, “Image security based on iterative random phase encoding in expanded fractional Fourier domains,” Opt. Lasers Eng. 105, 1–5 (2018).
[Crossref]

S. Liu, J. Wu, and C. Li, “Cascading the multiple stages of the optical fractional Fourier transforms under different variable scale,” Opt. Lett. 20, 1415–1417 (1995).
[Crossref]

Liu, Z.

Z. Liu, H. Chen, W. Blondel, Z. Shen, and S. Liu, “Image security based on iterative random phase encoding in expanded fractional Fourier domains,” Opt. Lasers Eng. 105, 1–5 (2018).
[Crossref]

Liu, Z.-X.

lto, T.

Ma, L.

L. Ma and W. Jin, “Symmetric and asymmetric hybrid cryptosystem based on compressive sensing and computer generated holography,” Opt. Commun. 407, 51–56 (2018).
[Crossref]

Masuda, N.

Monaghan, D. S.

Montes-Usategui, M.

Muniraj, I.

Naughton, T. J.

Nishchal, N. K.

Özkul, C.

Peng, X.

Piao, M. L.

Piao, M.-L.

M.-L. Piao, Z.-X. Liu, Y.-L. Piao, H.-Y. Wu, Y. Zhao, and N. Kim, “Multi-depth three-dimensional image encryption based on the phase retrieval algorithm in the Fresnel and fractional Fourier transform domains,” Appl. Opt. 57, 7609–7617 (2018).
[Crossref]

Y. Zhao, C.-X. Shi, K.-C. Kwon, Y.-L. Piao, M.-L. Piao, and N. Kim, “Fast calculation method of computer-generated hologram using a depth camera with point cloud gridding,” Opt. Commun. 411, 166–169 (2017).
[Crossref]

Piao, Y.-L.

M.-L. Piao, Z.-X. Liu, Y.-L. Piao, H.-Y. Wu, Y. Zhao, and N. Kim, “Multi-depth three-dimensional image encryption based on the phase retrieval algorithm in the Fresnel and fractional Fourier transform domains,” Appl. Opt. 57, 7609–7617 (2018).
[Crossref]

Y. Zhao, C.-X. Shi, K.-C. Kwon, Y.-L. Piao, M.-L. Piao, and N. Kim, “Fast calculation method of computer-generated hologram using a depth camera with point cloud gridding,” Opt. Commun. 411, 166–169 (2017).
[Crossref]

Poon, T.-C.

P. Tsang, K. W. K. Cheung, and T.-C. Poon, “Fast numerical generation and hybrid encryption of a computer-generated Fresnel holographic video sequence,” Chinese Opt. Lett. 11, 020901 (2013).
[Crossref]

P. W. M. Tsang, T.-C. Poon, and K. W. K. Cheung, “Fast numerical generation and encryption of computer-generated Fresnel holograms,” Appl. Opt. 50, B46–B52 (2011).
[Crossref]

Qin, W.

Ra’ed, M.

Rajput, S. K.

Rawat, N.

N. Rawat, I. C. Hwang, Y. Shi, and B. G. Lee, “Optical image encryption via photon-counting imaging and compressive sensing based ptychography,” J. Opt. 17, 065704 (2015).
[Crossref]

Ryle, J. P.

Shen, X.

D. Kong, X. Shen, L. Cao, H. Zhang, S. Zong, and G. Jin, “Three-dimensional information hierarchical encryption based on computer-generated holograms,” Opt. Commun. 380, 387–393 (2016).
[Crossref]

Shen, Z.

Z. Liu, H. Chen, W. Blondel, Z. Shen, and S. Liu, “Image security based on iterative random phase encoding in expanded fractional Fourier domains,” Opt. Lasers Eng. 105, 1–5 (2018).
[Crossref]

Sheridan, J. T.

Shi, C. X.

Shi, C.-X.

Y. Zhao, C.-X. Shi, K.-C. Kwon, Y.-L. Piao, M.-L. Piao, and N. Kim, “Fast calculation method of computer-generated hologram using a depth camera with point cloud gridding,” Opt. Commun. 411, 166–169 (2017).
[Crossref]

Shi, Y.

N. Rawat, I. C. Hwang, Y. Shi, and B. G. Lee, “Optical image encryption via photon-counting imaging and compressive sensing based ptychography,” J. Opt. 17, 065704 (2015).
[Crossref]

Shimobaba, T.

Singh, K.

Sinha, A.

A. Sinha, “Nonlinear optical cryptosystem resistant to standard and hybrid attacks,” Opt. Lasers Eng. 81, 79–86 (2016).
[Crossref]

Sito, G.

G. Sito and J. Zhang, “A lensless optical security system based on computer-generated phase only mask,” Opt. Commun. 232, 115–122 (2004).
[Crossref]

Song, L.

Sun, W.-J.

Y.-Y. Wang, Y.-R. Wang, Y. Wang, H.-J. Li, and W.-J. Sun, “Optical image encryption based on binary Fourier transform computer-generated hologram and pixel scrambling technology,” Opt. Lasers Eng. 45, 761–765 (2007).
[Crossref]

Tajahuerce, E.

Tsang, P.

P. Tsang, K. W. K. Cheung, and T.-C. Poon, “Fast numerical generation and hybrid encryption of a computer-generated Fresnel holographic video sequence,” Chinese Opt. Lett. 11, 020901 (2013).
[Crossref]

Tsang, P. W. M.

Unnikrishnan, G.

Wang, H.

Wang, X.

Wang, Y.

Y.-Y. Wang, Y.-R. Wang, Y. Wang, H.-J. Li, and W.-J. Sun, “Optical image encryption based on binary Fourier transform computer-generated hologram and pixel scrambling technology,” Opt. Lasers Eng. 45, 761–765 (2007).
[Crossref]

Wang, Y.-R.

Y.-Y. Wang, Y.-R. Wang, Y. Wang, H.-J. Li, and W.-J. Sun, “Optical image encryption based on binary Fourier transform computer-generated hologram and pixel scrambling technology,” Opt. Lasers Eng. 45, 761–765 (2007).
[Crossref]

Wang, Y.-Y.

Y.-Y. Wang, Y.-R. Wang, Y. Wang, H.-J. Li, and W.-J. Sun, “Optical image encryption based on binary Fourier transform computer-generated hologram and pixel scrambling technology,” Opt. Lasers Eng. 45, 761–765 (2007).
[Crossref]

Wei, H.

Wu, H.-Y.

Wu, J.

Xi, S.

Yu, N.

Zhang, H.

H. Zhang, L. Cao, and G. Jin, “Computer-generated hologram with occlusion effect using layer-based processing,” Appl. Opt. 56, F138–F143 (2017).
[Crossref]

D. Kong, X. Shen, L. Cao, H. Zhang, S. Zong, and G. Jin, “Three-dimensional information hierarchical encryption based on computer-generated holograms,” Opt. Commun. 380, 387–393 (2016).
[Crossref]

Zhang, J.

G. Sito and J. Zhang, “A lensless optical security system based on computer-generated phase only mask,” Opt. Commun. 232, 115–122 (2004).
[Crossref]

Zhang, P.

Zhao, D.

Zhao, Y.

M.-L. Piao, Z.-X. Liu, Y.-L. Piao, H.-Y. Wu, Y. Zhao, and N. Kim, “Multi-depth three-dimensional image encryption based on the phase retrieval algorithm in the Fresnel and fractional Fourier transform domains,” Appl. Opt. 57, 7609–7617 (2018).
[Crossref]

Y. Zhao, C.-X. Shi, K.-C. Kwon, Y.-L. Piao, M.-L. Piao, and N. Kim, “Fast calculation method of computer-generated hologram using a depth camera with point cloud gridding,” Opt. Commun. 411, 166–169 (2017).
[Crossref]

Zhu, B.

Zhu, Z.

Zong, S.

D. Kong, X. Shen, L. Cao, H. Zhang, S. Zong, and G. Jin, “Three-dimensional information hierarchical encryption based on computer-generated holograms,” Opt. Commun. 380, 387–393 (2016).
[Crossref]

Appl. Opt. (9)

E. Tajahuerce and B. Javidi, “Encryption three-dimensional information with digital holography,” Appl. Opt. 39, 6595–6601 (2000).
[Crossref]

W. N. Li, C. X. Shi, M. L. Piao, and N. Kim, “Multiple-3D-object secure information system based on phase shifting method and single interference,” Appl. Opt. 55, 4052–4059 (2016).
[Crossref]

P. W. M. Tsang, T.-C. Poon, and K. W. K. Cheung, “Fast numerical generation and encryption of computer-generated Fresnel holograms,” Appl. Opt. 50, B46–B52 (2011).
[Crossref]

X. Wang, Y. Chen, C. Dai, and D. Zhao, “Discussion and a new attack of the optical asymmetric cryptosystem based on phase-truncated Fourier transform,” Appl. Opt. 53, 208–213 (2014).
[Crossref]

S. K. Rajput and N. K. Nishchal, “Known-plaintext attack-based optical cryptosystem using phase-truncated Fresnel transform,” Appl. Opt. 52, 871–878 (2013).
[Crossref]

D. Kong, L. Cao, G. Jin, and B. Javidi, “Three-dimensional scene encryption and display based on computer-generated holograms,” Appl. Opt. 55, 8296–8300 (2016).
[Crossref]

H. Zhang, L. Cao, and G. Jin, “Computer-generated hologram with occlusion effect using layer-based processing,” Appl. Opt. 56, F138–F143 (2017).
[Crossref]

M.-L. Piao, Z.-X. Liu, Y.-L. Piao, H.-Y. Wu, Y. Zhao, and N. Kim, “Multi-depth three-dimensional image encryption based on the phase retrieval algorithm in the Fresnel and fractional Fourier transform domains,” Appl. Opt. 57, 7609–7617 (2018).
[Crossref]

S. Coëtmellec, D. Lebrun, and C. Özkul, “Characterization of diffraction patterns directly from in-line holograms with the fractional Fourier transform,” Appl. Opt. 41, 312–319 (2002).
[Crossref]

Chinese Opt. Lett. (1)

P. Tsang, K. W. K. Cheung, and T.-C. Poon, “Fast numerical generation and hybrid encryption of a computer-generated Fresnel holographic video sequence,” Chinese Opt. Lett. 11, 020901 (2013).
[Crossref]

J. Opt. (1)

N. Rawat, I. C. Hwang, Y. Shi, and B. G. Lee, “Optical image encryption via photon-counting imaging and compressive sensing based ptychography,” J. Opt. 17, 065704 (2015).
[Crossref]

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

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

W. Chen, “Optical cryptosystem based on single-pixel encoding using the modified Gerchberg-Saxton algorithm with a cascaded structure,” J. Opt. Soc. Am. A. 33, 2305–2311 (2016).
[Crossref]

Opt. Commun. (5)

L. Ma and W. Jin, “Symmetric and asymmetric hybrid cryptosystem based on compressive sensing and computer generated holography,” Opt. Commun. 407, 51–56 (2018).
[Crossref]

X. Wang, C. Dai, and J. Chen, “Optical image encryption via reverse engineering of a modified amplitude-phase retrieval-based attack,” Opt. Commun. 328, 67–72 (2014).
[Crossref]

D. Kong, X. Shen, L. Cao, H. Zhang, S. Zong, and G. Jin, “Three-dimensional information hierarchical encryption based on computer-generated holograms,” Opt. Commun. 380, 387–393 (2016).
[Crossref]

Y. Zhao, C.-X. Shi, K.-C. Kwon, Y.-L. Piao, M.-L. Piao, and N. Kim, “Fast calculation method of computer-generated hologram using a depth camera with point cloud gridding,” Opt. Commun. 411, 166–169 (2017).
[Crossref]

G. Sito and J. Zhang, “A lensless optical security system based on computer-generated phase only mask,” Opt. Commun. 232, 115–122 (2004).
[Crossref]

Opt. Express (4)

Opt. Lasers Eng. (3)

A. Sinha, “Nonlinear optical cryptosystem resistant to standard and hybrid attacks,” Opt. Lasers Eng. 81, 79–86 (2016).
[Crossref]

Z. Liu, H. Chen, W. Blondel, Z. Shen, and S. Liu, “Image security based on iterative random phase encoding in expanded fractional Fourier domains,” Opt. Lasers Eng. 105, 1–5 (2018).
[Crossref]

Y.-Y. Wang, Y.-R. Wang, Y. Wang, H.-J. Li, and W.-J. Sun, “Optical image encryption based on binary Fourier transform computer-generated hologram and pixel scrambling technology,” Opt. Lasers Eng. 45, 761–765 (2007).
[Crossref]

Opt. Lett. (8)

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Figures (16)

Fig. 1.
Fig. 1. Fresnel hologram calculation for an L-layer 3D scene.
Fig. 2.
Fig. 2. POM generation process using the matrix superposition and decomposition algorithm.
Fig. 3.
Fig. 3. Multidepth object encryption process.
Fig. 4.
Fig. 4. Decryption process.
Fig. 5.
Fig. 5. Schematic of the optoelectronics used for decryption ($ {{\rm SLM}_1} $, $ {{\rm SLM}_2} $, $ {{\rm SLM}_3} $, and $ {{\rm SLM}_4} $: spatial light modulator schemes 1–4).
Fig. 6.
Fig. 6. (a) Original image, (b) amplitude distribution, and (c) phase distribution of the complex amplitude $ H(u,v) $, (d) encryption keys, and (e) ciphertext image.
Fig. 7.
Fig. 7. Decryption keys and recovered results: (a) decryption keys, (b) decrypted image with all the correct keys in the depth plane, and (c) recovered images with all the correct keys in the out-of-depth plane.
Fig. 8.
Fig. 8. Decrypted results obtained after the use of incorrect keys: (a) incorrect fractional orders $ \alpha $ and $ \beta $, (b) incorrect phase function of decryption keys $ {\psi _\alpha }({\xi _1},{\eta _1}) $ and $ {\psi _\beta }({\xi _2},{\eta _2}) $, and (c) incorrect optical wavelength and Fresnel diffraction distance.
Fig. 9.
Fig. 9. Simulation results for a multidepth gray-scale image. (a) Multidepth image used for encryption, (b) the ciphertext image, and (c) multidepth image decrypted using the proposed scheme.
Fig. 10.
Fig. 10. Relation between MSE of the decrypted image and dislocation.
Fig. 11.
Fig. 11. Normalized MSE versus (a) deviation in the fractional order $ \alpha $ and (b) deviation in the fractional order $ \beta $.
Fig. 12.
Fig. 12. Decrypted multidepth images with (a) private phase functions identical to zero and (b) two arbitrarily selected private phase functions.
Fig. 13.
Fig. 13. MSE variation associated with the decryption image when using an erroneous key (a) $ {\psi _\alpha }^{\prime}({\xi _1},{\eta _1}) $ and (b) $ {\psi _\beta }^{\prime}({\xi _2},{\eta _2}) $ with different parameters $ d $.
Fig. 14.
Fig. 14. (a) Fake plaintext multidepth binary images, (b) plaintext recovered using the CPA scheme with the proposed cryptosystem, and (c) plaintext recovered by using the CPA with the cryptosystem based on PTFrFT.
Fig. 15.
Fig. 15. (a) Retrieved outcomes with the NPA in the proposed scheme, (b) retrieved outcomes with NPA in the PTFrFT-based scheme, and (c) MSEs of phase key retrieval versus the number of iterations for both encryption schemes.
Fig. 16.
Fig. 16. (a) Retrieved outcomes with a COA using the proposed scheme, (b) retrieved outcomes with a COA in the PTFrFT-based scheme, and (c) MSEs of phase key retrieval versus the number of iterations for both encryption schemes.

Tables (1)

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Table 1. Encryption Times Taken (s)

Equations (18)

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O i ( x , y ) = O i ( x , y ) exp [ j θ ( x , y ) ] .
H i ( u , v , 0 ) = exp ( j k z i ) j λ z i O i ( x , y , z i ) × exp { j π λ z i [ ( u x ) 2 + ( v y ) 2 ] } d x d y ,
H ( u , v ) = i = 1 L H i ( u , v ) ,
E α ( ξ 1 , η 1 ) = K ϕ α ( u , v ; ξ 1 , η 1 ) H A ( u , v ) × exp [ j ϕ ( u , v ) ] d u d v , = | E α ( ξ 1 , η 1 ) | × exp [ j ψ α ( ξ 1 , η 1 ) ] ,
K ϕ α ( u , v ; ξ 1 , η 1 ) = exp [ j ( π / 4 ) s i g n ( sin ϕ α ) + j ϕ α / 2 ] | λ f 1 sin ϕ α | × exp { j π [ u 2 + v 2 + ξ 1 2 + η 1 2 λ f 1 tan ϕ α 2 u v + ξ 1 η 1 λ f 1 sin ϕ α ] }
E β ( ξ 2 , η 2 ) = K ϕ β ( ξ 1 , η 1 ; ξ 2 , η 2 ) × { | E α ( ξ 1 , η 1 ) | × exp [ j φ ( ξ 1 , η 1 ) ] } d ξ 1 d η 1 = | E β ( ξ 2 , η 2 ) | × exp [ j ψ β ( ξ 2 , η 2 ) ] ,
D 1 ( ξ 1 , η 1 ) = F r F T β { | E β ( ξ 2 , η 2 ) | × exp [ j ψ β ( ξ 2 , η 2 ) ] } = | D 1 ( ξ 1 , η 1 ) | × exp [ j φ ( ξ 1 , η 1 ) ] ,
D ( u , v ) = F r F T α { | D 1 ( ξ 1 , η 1 ) | × exp [ j ψ α ( ξ 1 , η 1 ) ] } = | D ( u , v ) | × exp [ j ϕ ( u , v ) ] ,
G ( x , y , z i ) = F r T z i λ { | D ( u , v ) | × exp [ j H P ( u , v ) ] } ,
M S E = 1 N × M x = 1 N y = 1 M | O ( x , y ) G ( x , y ) | 2 ,
ψ α ( ξ 1 , η 1 ) = ψ α ( ξ 1 , η 1 ) + d Δ ψ ,
ψ β ( ξ 2 , η 2 ) = ψ β ( ξ 2 , η 2 ) + d Δ ψ ,
G α ( ξ 1 , η 1 ) = P T [ F r F T α { H A ( u , v ) × R α ( u , v ) } ] ,
G β ( ξ 2 , η 2 ) = P T [ F r F T β { G α ( ξ 1 , η 1 ) × R β ( ξ , η ) } ] .
K α ( ξ 1 , η 1 ) = P R [ F r F T α { H A ( u , v ) × R α ( u , v ) } ] ,
K β ( ξ 2 , η 2 ) = P R [ F r F T β { G α ( ξ 1 , η 1 ) × R β ( ξ , η ) } ] ,
H A ( u , v ) = F r F T α { | F r F T β { | E β ( ξ 2 , η 2 ) | × ψ β ( ξ 2 , η 2 ) } | ψ ( ξ 1 , η 1 ) } ,
O n ( x , y , z ) = F r T z i λ { H P ( u , v ) H P ( u , v ) } .

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