Abstract

A novel approach for silhouette-free image encryption based on interference is proposed using discrete multiple-parameter fractional Fourier transform (DMPFrFT), which generalizes from fractional Fourier transform. An original image is firstly applied by chaotic pixel scrambling (CPS) and then encoded into the real part of a complex signal. Using interference principle, the complex signal generates three phase-only masks in DMPFrFT domain. The silhouette of the original image cannot be extracted using one or two of the three phase-only masks. The parameters of both CPS and DMPFrFT can also serve as encryption keys to extend the key space, which further enhance the level of cryptosystem security. Numerical simulations are demonstrated to show the feasibility and validity of this approach.

© 2017 Optical Society of America

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OSA Recommended Articles
Image encryption based on interference that uses fractional Fourier domain asymmetric keys

Sudheesh K. Rajput and Naveen K. Nishchal
Appl. Opt. 51(10) 1446-1452 (2012)

References

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    [Crossref]
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    [Crossref]

2016 (5)

Z. Liu, C. Shen, J. Tan, and S. Liu, “A recovery method of double random phase encoding system with a parallel phase retrieval,” IEEE Photonics 8(1), 1–7 (2016).
[Crossref]

H. Di, Y. Kang, Y. Liu, and X. Zhang, “Multiple image encryption by phase retrieval,” Opt. Eng. 55(7), 73–103 (2016).
[Crossref]

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Lasers Eng. 89, 2–12 (2016).
[Crossref]

D. Lu, W. He, M. Liao, and X. Peng, “Discussion and a new method of optical cryptosystem based on interference,” Opt. Lasers Eng. 89, 13–21 (2016).
[Crossref]

Q. Gong, Z. Wang, X. Lv, and Y. Qin, “Interference-based image encryption with silhouette removal by aid of compressive sensing,” Opt. Commun. 359, 290–296 (2016).
[Crossref]

2015 (4)

W. Liu, Z. Liu, and S. Liu, “Optical security validation using Michelson interferometer,” Appl. Opt. 54(7), 1802–1807 (2015).
[Crossref]

J. X. Chen, Z. L. Zhu, C. Fu, H. Yu, and L. B. Zhang, “An efficient image encryption scheme using gray code based permutation approach,” Opt. Lasers Eng. 67, 191–204 (2015).
[Crossref]

X. Wang, L. Liu, and Y. Zhang, “A novel chaotic block image encryption algorithm based on dynamic random growth technique,” Opt. Lasers Eng. 66, 10–18 (2015).
[Crossref]

L. Sui, K. Duan, and J. Liang, “Double-image encryption based on discrete multiple-parameter fractional angular transform and two-coupled logistic maps,” Opt. Commun. 343, 140–149 (2015).
[Crossref]

2014 (5)

L. Sui, K. Duan, J. Liang, and X. Hei, “Asymmetric double-image encryption based on cascaded discrete fractional random transform and logistic maps,” Opt. Express 22(9), 10605–10621 (2014).
[Crossref] [PubMed]

Z. Zhong, Y. Zhang, M. Shan, Y. Wang, Y. Zhang, and H. Xie, “Optical movie encryption based on a discrete multiple-parameter fractional Fourier transform,” J. Opt. 16(12), 125404 (2014).
[Crossref]

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

J. X. Chen, Z. L. Zhu, Z. Liu, C. Fu, L. B. Zhang, and H. Yu, “A novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains,” Opt. Express 22(6), 7349–7361 (2014).
[Crossref] [PubMed]

W. Chen and X. Chen, “Iterative phase retrieval for simultaneously generating two phase-only masks with silhouette removal in interference-based optical encryption,” Opt. Commun. 331(331), 133–138 (2014).
[Crossref]

2013 (3)

2012 (4)

M. Shan, J. Chang, Z. Zhong, and B. Hao, “Double image encryption based on discrete multiple-parameter fractional Fourier transform and chaotic maps,” Opt. Commun. 285(21–22), 4227–4234 (2012).
[Crossref]

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285(6), 1078–1081 (2012).
[Crossref]

X. Wang and D. Zhao, “Optical image hiding with silhouette removal based on the optical interference principle,” Appl. Opt. 51(6), 686–691 (2012).
[Crossref] [PubMed]

Q. Wang, “Optical image encryption with silhouette removal based on interference and phase blend processing,” Opt. Commun. 285(21–22), 4294–4301 (2012).
[Crossref]

2011 (3)

D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284(10), 2485–2487 (2011).
[Crossref]

P. Kumar, J. Joseph, and K. Singh, “Optical image encryption using a jigsaw transform for silhouette removal in interference-based methods and decryption with a single spatial light modulator,” Appl. Opt. 50(13), 1805–1811 (2011).
[Crossref] [PubMed]

H. Liu and X. Wang, “Color image encryption using spatial bit-level permutation and high-dimension chaotic system,” Opt. Commun. 284(16–17), 3895–3903 (2011).
[Crossref]

2010 (1)

2009 (1)

Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A, Pure Appl. Opt. 11(12), 125406 (2009).
[Crossref]

2008 (1)

2006 (2)

2005 (1)

1995 (1)

Ailing, T.

Arcos, S.

Carnicer, A.

Chang, J.

M. Shan, J. Chang, Z. Zhong, and B. Hao, “Double image encryption based on discrete multiple-parameter fractional Fourier transform and chaotic maps,” Opt. Commun. 285(21–22), 4227–4234 (2012).
[Crossref]

Chen, J. X.

J. X. Chen, Z. L. Zhu, C. Fu, H. Yu, and L. B. Zhang, “An efficient image encryption scheme using gray code based permutation approach,” Opt. Lasers Eng. 67, 191–204 (2015).
[Crossref]

J. X. Chen, Z. L. Zhu, Z. Liu, C. Fu, L. B. Zhang, and H. Yu, “A novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains,” Opt. Express 22(6), 7349–7361 (2014).
[Crossref] [PubMed]

Chen, K.

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Lasers Eng. 89, 2–12 (2016).
[Crossref]

Chen, W.

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

W. Chen and X. Chen, “Iterative phase retrieval for simultaneously generating two phase-only masks with silhouette removal in interference-based optical encryption,” Opt. Commun. 331(331), 133–138 (2014).
[Crossref]

W. Chen and X. Chen, “Security-enhanced interference-based optical image encryption,” Opt. Commun. 286(1), 123–129 (2013).

Chen, X.

W. Chen and X. Chen, “Iterative phase retrieval for simultaneously generating two phase-only masks with silhouette removal in interference-based optical encryption,” Opt. Commun. 331(331), 133–138 (2014).
[Crossref]

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

W. Chen and X. Chen, “Security-enhanced interference-based optical image encryption,” Opt. Commun. 286(1), 123–129 (2013).

Di, H.

H. Di, Y. Kang, Y. Liu, and X. Zhang, “Multiple image encryption by phase retrieval,” Opt. Eng. 55(7), 73–103 (2016).
[Crossref]

Dong, Z.

Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A, Pure Appl. Opt. 11(12), 125406 (2009).
[Crossref]

Duan, K.

L. Sui, K. Duan, and J. Liang, “Double-image encryption based on discrete multiple-parameter fractional angular transform and two-coupled logistic maps,” Opt. Commun. 343, 140–149 (2015).
[Crossref]

L. Sui, K. Duan, J. Liang, and X. Hei, “Asymmetric double-image encryption based on cascaded discrete fractional random transform and logistic maps,” Opt. Express 22(9), 10605–10621 (2014).
[Crossref] [PubMed]

Fu, C.

J. X. Chen, Z. L. Zhu, C. Fu, H. Yu, and L. B. Zhang, “An efficient image encryption scheme using gray code based permutation approach,” Opt. Lasers Eng. 67, 191–204 (2015).
[Crossref]

J. X. Chen, Z. L. Zhu, Z. Liu, C. Fu, L. B. Zhang, and H. Yu, “A novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains,” Opt. Express 22(6), 7349–7361 (2014).
[Crossref] [PubMed]

Gong, Q.

Q. Gong, Z. Wang, X. Lv, and Y. Qin, “Interference-based image encryption with silhouette removal by aid of compressive sensing,” Opt. Commun. 359, 290–296 (2016).
[Crossref]

Guo, C.

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Lasers Eng. 89, 2–12 (2016).
[Crossref]

Hao, B.

M. Shan, J. Chang, Z. Zhong, and B. Hao, “Double image encryption based on discrete multiple-parameter fractional Fourier transform and chaotic maps,” Opt. Commun. 285(21–22), 4227–4234 (2012).
[Crossref]

He, W.

D. Lu, W. He, M. Liao, and X. Peng, “Discussion and a new method of optical cryptosystem based on interference,” Opt. Lasers Eng. 89, 13–21 (2016).
[Crossref]

Hei, X.

Javidi, B.

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

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

Joseph, J.

Juvells, I.

Kang, Y.

H. Di, Y. Kang, Y. Liu, and X. Zhang, “Multiple image encryption by phase retrieval,” Opt. Eng. 55(7), 73–103 (2016).
[Crossref]

Kumar, P.

Liang, J.

L. Sui, K. Duan, and J. Liang, “Double-image encryption based on discrete multiple-parameter fractional angular transform and two-coupled logistic maps,” Opt. Commun. 343, 140–149 (2015).
[Crossref]

L. Sui, K. Duan, J. Liang, and X. Hei, “Asymmetric double-image encryption based on cascaded discrete fractional random transform and logistic maps,” Opt. Express 22(9), 10605–10621 (2014).
[Crossref] [PubMed]

Liansheng, S.

Liao, M.

D. Lu, W. He, M. Liao, and X. Peng, “Discussion and a new method of optical cryptosystem based on interference,” Opt. Lasers Eng. 89, 13–21 (2016).
[Crossref]

Liu, H.

H. Liu and X. Wang, “Color image encryption using spatial bit-level permutation and high-dimension chaotic system,” Opt. Commun. 284(16–17), 3895–3903 (2011).
[Crossref]

Liu, J.

D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284(10), 2485–2487 (2011).
[Crossref]

Liu, L.

X. Wang, L. Liu, and Y. Zhang, “A novel chaotic block image encryption algorithm based on dynamic random growth technique,” Opt. Lasers Eng. 66, 10–18 (2015).
[Crossref]

Liu, S.

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Lasers Eng. 89, 2–12 (2016).
[Crossref]

Z. Liu, C. Shen, J. Tan, and S. Liu, “A recovery method of double random phase encoding system with a parallel phase retrieval,” IEEE Photonics 8(1), 1–7 (2016).
[Crossref]

W. Liu, Z. Liu, and S. Liu, “Optical security validation using Michelson interferometer,” Appl. Opt. 54(7), 1802–1807 (2015).
[Crossref]

Liu, W.

Liu, Y.

H. Di, Y. Kang, Y. Liu, and X. Zhang, “Multiple image encryption by phase retrieval,” Opt. Eng. 55(7), 73–103 (2016).
[Crossref]

Liu, Z.

Z. Liu, C. Shen, J. Tan, and S. Liu, “A recovery method of double random phase encoding system with a parallel phase retrieval,” IEEE Photonics 8(1), 1–7 (2016).
[Crossref]

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Lasers Eng. 89, 2–12 (2016).
[Crossref]

W. Liu, Z. Liu, and S. Liu, “Optical security validation using Michelson interferometer,” Appl. Opt. 54(7), 1802–1807 (2015).
[Crossref]

J. X. Chen, Z. L. Zhu, Z. Liu, C. Fu, L. B. Zhang, and H. Yu, “A novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains,” Opt. Express 22(6), 7349–7361 (2014).
[Crossref] [PubMed]

Lu, D.

D. Lu, W. He, M. Liao, and X. Peng, “Discussion and a new method of optical cryptosystem based on interference,” Opt. Lasers Eng. 89, 13–21 (2016).
[Crossref]

Lv, X.

Q. Gong, Z. Wang, X. Lv, and Y. Qin, “Interference-based image encryption with silhouette removal by aid of compressive sensing,” Opt. Commun. 359, 290–296 (2016).
[Crossref]

Meiting, X.

Montes-Usategui, M.

Nishchal, N. K.

Peng, X.

Qin, W.

Qin, Y.

Q. Gong, Z. Wang, X. Lv, and Y. Qin, “Interference-based image encryption with silhouette removal by aid of compressive sensing,” Opt. Commun. 359, 290–296 (2016).
[Crossref]

Rajput, S. K.

Refregier, P.

Shan, M.

Z. Zhong, Y. Zhang, M. Shan, Y. Wang, Y. Zhang, and H. Xie, “Optical movie encryption based on a discrete multiple-parameter fractional Fourier transform,” J. Opt. 16(12), 125404 (2014).
[Crossref]

M. Shan, J. Chang, Z. Zhong, and B. Hao, “Double image encryption based on discrete multiple-parameter fractional Fourier transform and chaotic maps,” Opt. Commun. 285(21–22), 4227–4234 (2012).
[Crossref]

Shen, C.

Z. Liu, C. Shen, J. Tan, and S. Liu, “A recovery method of double random phase encoding system with a parallel phase retrieval,” IEEE Photonics 8(1), 1–7 (2016).
[Crossref]

Singh, K.

Sui, L.

L. Sui, K. Duan, and J. Liang, “Double-image encryption based on discrete multiple-parameter fractional angular transform and two-coupled logistic maps,” Opt. Commun. 343, 140–149 (2015).
[Crossref]

L. Sui, K. Duan, J. Liang, and X. Hei, “Asymmetric double-image encryption based on cascaded discrete fractional random transform and logistic maps,” Opt. Express 22(9), 10605–10621 (2014).
[Crossref] [PubMed]

Tan, J.

Z. Liu, C. Shen, J. Tan, and S. Liu, “A recovery method of double random phase encoding system with a parallel phase retrieval,” IEEE Photonics 8(1), 1–7 (2016).
[Crossref]

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Lasers Eng. 89, 2–12 (2016).
[Crossref]

Wang, B.

Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A, Pure Appl. Opt. 11(12), 125406 (2009).
[Crossref]

Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008).
[Crossref] [PubMed]

Wang, Q.

Q. Wang, “Optical image encryption with silhouette removal based on interference and phase blend processing,” Opt. Commun. 285(21–22), 4294–4301 (2012).
[Crossref]

Wang, X.

X. Wang, L. Liu, and Y. Zhang, “A novel chaotic block image encryption algorithm based on dynamic random growth technique,” Opt. Lasers Eng. 66, 10–18 (2015).
[Crossref]

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285(6), 1078–1081 (2012).
[Crossref]

X. Wang and D. Zhao, “Optical image hiding with silhouette removal based on the optical interference principle,” Appl. Opt. 51(6), 686–691 (2012).
[Crossref] [PubMed]

H. Liu and X. Wang, “Color image encryption using spatial bit-level permutation and high-dimension chaotic system,” Opt. Commun. 284(16–17), 3895–3903 (2011).
[Crossref]

Wang, Y.

Z. Zhong, Y. Zhang, M. Shan, Y. Wang, Y. Zhang, and H. Xie, “Optical movie encryption based on a discrete multiple-parameter fractional Fourier transform,” J. Opt. 16(12), 125404 (2014).
[Crossref]

D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284(10), 2485–2487 (2011).
[Crossref]

Wang, Z.

Q. Gong, Z. Wang, X. Lv, and Y. Qin, “Interference-based image encryption with silhouette removal by aid of compressive sensing,” Opt. Commun. 359, 290–296 (2016).
[Crossref]

Wei, C.

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Lasers Eng. 89, 2–12 (2016).
[Crossref]

Wei, H.

Weng, D.

D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284(10), 2485–2487 (2011).
[Crossref]

Wu, Q.

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Lasers Eng. 89, 2–12 (2016).
[Crossref]

Xie, H.

Z. Zhong, Y. Zhang, M. Shan, Y. Wang, Y. Zhang, and H. Xie, “Optical movie encryption based on a discrete multiple-parameter fractional Fourier transform,” J. Opt. 16(12), 125404 (2014).
[Crossref]

Xie, J.

D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284(10), 2485–2487 (2011).
[Crossref]

Yu, B.

Yu, H.

J. X. Chen, Z. L. Zhu, C. Fu, H. Yu, and L. B. Zhang, “An efficient image encryption scheme using gray code based permutation approach,” Opt. Lasers Eng. 67, 191–204 (2015).
[Crossref]

J. X. Chen, Z. L. Zhu, Z. Liu, C. Fu, L. B. Zhang, and H. Yu, “A novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains,” Opt. Express 22(6), 7349–7361 (2014).
[Crossref] [PubMed]

Zhang, L. B.

J. X. Chen, Z. L. Zhu, C. Fu, H. Yu, and L. B. Zhang, “An efficient image encryption scheme using gray code based permutation approach,” Opt. Lasers Eng. 67, 191–204 (2015).
[Crossref]

J. X. Chen, Z. L. Zhu, Z. Liu, C. Fu, L. B. Zhang, and H. Yu, “A novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains,” Opt. Express 22(6), 7349–7361 (2014).
[Crossref] [PubMed]

Zhang, P.

Zhang, X.

H. Di, Y. Kang, Y. Liu, and X. Zhang, “Multiple image encryption by phase retrieval,” Opt. Eng. 55(7), 73–103 (2016).
[Crossref]

Zhang, Y.

X. Wang, L. Liu, and Y. Zhang, “A novel chaotic block image encryption algorithm based on dynamic random growth technique,” Opt. Lasers Eng. 66, 10–18 (2015).
[Crossref]

Z. Zhong, Y. Zhang, M. Shan, Y. Wang, Y. Zhang, and H. Xie, “Optical movie encryption based on a discrete multiple-parameter fractional Fourier transform,” J. Opt. 16(12), 125404 (2014).
[Crossref]

Z. Zhong, Y. Zhang, M. Shan, Y. Wang, Y. Zhang, and H. Xie, “Optical movie encryption based on a discrete multiple-parameter fractional Fourier transform,” J. Opt. 16(12), 125404 (2014).
[Crossref]

Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A, Pure Appl. Opt. 11(12), 125406 (2009).
[Crossref]

Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008).
[Crossref] [PubMed]

Zhao, D.

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Figures (9)

Fig. 1
Fig. 1 Optoelectronic schematic for decryption. BCE, beam collimator and expander; SLM1 and SLM2, spatial light modulator.
Fig. 2
Fig. 2 (a) The original image, (b)-(d) the POMs of M1, M2 and M3, (e) the scrambled-image recorded in CCD, (f) the decrypted image.
Fig. 3
Fig. 3 The decrypted image with (a) (αL + 1 × 10−7, αR) and (b) (αL, αR + 0.5 × 10−7), (c) the curve of NMSE with incorrect fractional order.
Fig. 4
Fig. 4 The decrypted image with (a) ML = 14 and (b) MR = 23
Fig. 5
Fig. 5 The decrypted image with (a) mL + 1, (b) nL + 1,(c) mR + 1 and (d) nR + 1.
Fig. 6
Fig. 6 (a) The decrypted image with a0 + 0.7 × 10−16, (b) the curve of NMSE.
Fig. 7
Fig. 7 (a) The decrypted image with λ + 0.3 × 10−15, (b) the curve of NMSE.
Fig. 8
Fig. 8 (a) The decrypted image with t + 125, (b) the curve of NMSE.
Fig. 9
Fig. 9 The decrypted image with (a) M1, (b) M2, (c) M3, (d) M1 and M2, (e) M1 and M3, and (f) M2 and M3.

Equations (9)

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o (u,v)= o(u,v) exp[ iM(u,v) ]
o (u,v)= P { a 0 ,λ,t } [ o(u,v) ]exp[ iM(u,v) ]
o (u,v)= F ( M L , M R ) ( α L , α R ) ( n ' L ,n ' R ){ exp[ i M 1 ( x,y ) ] } + F ( M L , M R ) ( α L , α R ) ( n ' L ,n ' R ){ exp[ i M 2 ( x,y ) ] } + F ( M L , M R ) ( α L , α R ) ( n ' L ,n ' R ){ exp[ i M 3 ( x,y ) ] }
n k =( k m k +M m k n k + n k ) k=0,1,2,,(M1)
exp[ i M 2 ( x,y ) ]= F -1 ( M L , M R ) ( α L , α R ) ( n L , n R )[ o ( u,v ) ] exp[ i M 3 ( x,y ) ]exp[ i M 1 ( x,y ) ]
1= | F -1 ( M L , M R ) ( α L , α R ) ( n L , n R )[ o'( u,v ) ]exp[ i M 3 ( x,y ) ]exp[ i M 1 ( x,y ) ] | 2 ={ F -1 ( M L , M R ) ( α L , α R ) ( n L , n R )[ o'( u,v ) ]exp[ i M 3 ( x,y ) ]exp[ i M 1 ( x,y ) ] } { F -1 ( M L , M R ) ( α L , α R ) ( n L , n R )[ o'( u,v ) ]exp[ i M 3 ( x,y ) ]exp[ i M 1 ( x,y ) ] }
M 1 ( x,y )=arg{ F -1 ( M L , M R ) ( α L , α R ) ( n L , n R )[ o ( u,v ) ]exp[ i M 3 ( x,y ) ] } arccos{ abs{ [ F -1 ( M L , M R ) ( α L , α R ) ( n L , n R )[ o ( u,v ) ] exp[ i M 3 ( x,y ) ] ] /2 } }
M 2 ( x,y )=arg{ F -1 ( M L , M R ) ( α L , α R ) ( n L , n R )[ o ( u,v ) ] exp[ i M 3 ( x,y ) ]exp[ i M 1 ( x,y ) ] }
NMSE= y=1 p x=1 q [ I D (x,y) I O (x,y) ] 2 / y=1 p x=1 q [ I O (x,y) ] 2

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