Abstract

We propose a multi-depth three-dimensional (3D) image cryptosystem by employing the phase retrieval algorithm in the Fresnel and fractional Fourier (Fr-FrF) domains. Encryption was realized by applying the phase retrieval algorithm based on the double-random-phase-encoding architecture in which two encryption keys will be incessantly updated in each iteration loop. The phase-only functions (POFs) are generated in two cascaded Fr-FrF transforms (Fr-FrFT), serving as decryption keys to efficiently reduce the speckle noise and crosstalk between encrypted 3D image depths. The use of Fr-FrFT position parameters and fractional order as decryption keys further extended the key space, enhancing the cryptosystem’s security level. Numerical simulations demonstrated the feasibility and robustness of our proposed scheme.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (2)

W. Chen, “3D Gerchberg-Saxton optical correlation,” IEEE Photon. J. 10, 7800409 (2018).
[Crossref]

W. Chen, “Hierarchically optical double-image correlation using 3D phase retrieval algorithm in fractional Fourier transform domain,” Opt. Commun. 427, 374–381 (2018).
[Crossref]

2017 (4)

2016 (3)

H. Di, Y. Kang, Y. Liu, and X. Zhang, “Multiple image encryption by phase retrieval,” Opt. Eng. 55, 073103 (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]

Z. Liu, C. Shen, J. Tan, and S. Liu, “A recovery method of double random phase encoding system with a parallel phase retrieval,” IEEE Photon. J. 8, 7801807 (2016).
[Crossref]

2015 (3)

2014 (2)

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

S. Liu, C. Guo, and J. T. Sheridan, “A review of optical image encryption techniques,” Opt. Laser Technol. 57, 327–342 (2014).
[Crossref]

2013 (2)

M. R. Abuturab, “Color image security system based on discrete Hartley transform in gyrator transform domain,” Opt. Lasers Eng. 51, 317–324 (2013).
[Crossref]

J. M. Vilardy, M. S. Millán, and E. Pérez-Cabré, “Improved decryption quality and security of a joint-transform correlator-based encryption system,” J. Opt. 15, 025401 (2013).
[Crossref]

2012 (3)

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

S. K. Rajput and N. K. Nishchal, “Image encryption based on interference that uses fractional Fourier domain asymmetric keys,” Appl. Opt. 51, 1446–1452 (2012).
[Crossref]

X. Deng and D. Zhao, “Multiple-image encryption using phase retrieve algorithm and intermodulation in Fourier domain,” Opt. Laser Technol. 44, 374–377 (2012).
[Crossref]

2011 (2)

A. Alfalou, C. Brosseau, N. Abdallah, and M. Jridi, “Simulation fusion, compression, and encryption of multiple images,” Opt. Express 19, 24023–24029 (2011).
[Crossref]

W. Chen and X. Chen, “Optical asymmetric cryptography using a three-dimensional space-based model,” J. Opt. 13, 075404 (2011).
[Crossref]

2009 (2)

2008 (2)

H. Li and Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281, 5745–5749 (2008).
[Crossref]

D. Amaya, M. Tebaldi, R. Torroba, and N. Bolognini, “Multichanneled encryption via a joint transform correlator architecture,” Appl. Opt. 47, 5903–5907 (2008).
[Crossref]

2007 (1)

2006 (2)

2004 (1)

2003 (1)

B. Hennelly and J. T. Sheridan, “Fractional Fourier transform-based on image encryption: phase retrieval algorithm,” Opt. Commun. 226, 61–80 (2003).
[Crossref]

2000 (3)

1995 (2)

1993 (1)

Abdallah, N.

Abuturab, M. R.

M. R. Abuturab, “Color image security system based on discrete Hartley transform in gyrator transform domain,” Opt. Lasers Eng. 51, 317–324 (2013).
[Crossref]

Alfalou, A.

Amaya, D.

Bitran, Y.

Bolognini, N.

Brosseau, C.

Chang, H. T.

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 (2017).
[Crossref]

Chen, W.

W. Chen, “3D Gerchberg-Saxton optical correlation,” IEEE Photon. J. 10, 7800409 (2018).
[Crossref]

W. Chen, “Hierarchically optical double-image correlation using 3D phase retrieval algorithm in fractional Fourier transform domain,” Opt. Commun. 427, 374–381 (2018).
[Crossref]

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

W. Chen and X. Chen, “Optical asymmetric cryptography using a three-dimensional space-based model,” J. Opt. 13, 075404 (2011).
[Crossref]

Chen, X.

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

W. Chen and X. Chen, “Optical asymmetric cryptography using a three-dimensional space-based model,” J. Opt. 13, 075404 (2011).
[Crossref]

Deng, X.

X. Deng and D. Zhao, “Multiple-image encryption using phase retrieve algorithm and intermodulation in Fourier domain,” Opt. Laser Technol. 44, 374–377 (2012).
[Crossref]

Di, H.

H. Di, Y. Kang, Y. Liu, and X. Zhang, “Multiple image encryption by phase retrieval,” Opt. Eng. 55, 073103 (2016).
[Crossref]

Dorsch, R. G.

Gil, S. K.

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 (2017).
[Crossref]

C. Guo, S. Liu, and J. T. Sheridan, “Iterative phase retrieval algorithms. I: Optimization,” Appl. Opt. 54, 4698–4708 (2015).
[Crossref]

S. Liu, C. Guo, and J. T. Sheridan, “A review of optical image encryption techniques,” Opt. Laser Technol. 57, 327–342 (2014).
[Crossref]

Guo, C. L.

Healy, J. J.

Hennelly, B.

B. Hennelly and J. T. Sheridan, “Fractional Fourier transform-based on image encryption: phase retrieval algorithm,” Opt. Commun. 226, 61–80 (2003).
[Crossref]

Hennelly, B. M.

Hwang, H. E.

Javidi, B.

Joseph, J.

Jridi, M.

Kang, Y.

H. Di, Y. Kang, Y. Liu, and X. Zhang, “Multiple image encryption by phase retrieval,” Opt. Eng. 55, 073103 (2016).
[Crossref]

Kelly, D. P.

Kim, N.

Lee, B. G.

Lee, S. M.

Li, H.

H. Li and Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281, 5745–5749 (2008).
[Crossref]

Li, W. N.

Lie, W. N.

Liu, L.

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 (2017).
[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 Photon. J. 8, 7801807 (2016).
[Crossref]

C. Guo, S. Liu, and J. T. Sheridan, “Iterative phase retrieval algorithms. I: Optimization,” Appl. Opt. 54, 4698–4708 (2015).
[Crossref]

S. Liu, C. Guo, and J. T. Sheridan, “A review of optical image encryption techniques,” Opt. Laser Technol. 57, 327–342 (2014).
[Crossref]

B. Zhu, S. Liu, and Q. Ran, “Optical image encryption based on multifractional Fourier transforms,” Opt. Lett. 25, 1159–1161 (2000).
[Crossref]

Liu, Y.

H. Di, Y. Kang, Y. Liu, and X. Zhang, “Multiple image encryption by phase retrieval,” Opt. Eng. 55, 073103 (2016).
[Crossref]

Liu, Z.

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 (2017).
[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 Photon. J. 8, 7801807 (2016).
[Crossref]

Lohmann, A. W.

McDonald, J.

Mendlovic, D.

Millán, M. S.

J. M. Vilardy, M. S. Millán, and E. Pérez-Cabré, “Improved decryption quality and security of a joint-transform correlator-based encryption system,” J. Opt. 15, 025401 (2013).
[Crossref]

Muniraj, I.

Naughton, T. J.

Nishchal, N. K.

Ozaktas, H. M.

Peng, X.

Pérez-Cabré, E.

J. M. Vilardy, M. S. Millán, and E. Pérez-Cabré, “Improved decryption quality and security of a joint-transform correlator-based encryption system,” J. Opt. 15, 025401 (2013).
[Crossref]

Phan, A. H.

Piao, M. L.

Qin, H.

Ra’ed, M.

Rajput, S. K.

Ran, Q.

Refregier, P.

Ryle, J. P.

Shan, M.

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 Photon. J. 8, 7801807 (2016).
[Crossref]

Sheridan, J. T.

Shi, C. X.

Singh, K.

Situ, G.

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

G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29, 1584–1586 (2004).
[Crossref]

Tajahuerce, E.

Tan, J.

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 (2017).
[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 Photon. J. 8, 7801807 (2016).
[Crossref]

Tebaldi, M.

Torroba, R.

Unnikrishnan, G.

Vilardy, J. M.

J. M. Vilardy, M. S. Millán, and E. Pérez-Cabré, “Improved decryption quality and security of a joint-transform correlator-based encryption system,” J. Opt. 15, 025401 (2013).
[Crossref]

Wang, X.

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

Wang, Y.

H. Li and Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281, 5745–5749 (2008).
[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 (2017).
[Crossref]

Wei, H.

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 (2017).
[Crossref]

Zhang, J.

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

G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29, 1584–1586 (2004).
[Crossref]

Zhang, P.

Zhang, X.

H. Di, Y. Kang, Y. Liu, and X. Zhang, “Multiple image encryption by phase retrieval,” Opt. Eng. 55, 073103 (2016).
[Crossref]

Zhang, Y.

Zhao, D.

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

X. Deng and D. Zhao, “Multiple-image encryption using phase retrieve algorithm and intermodulation in Fourier domain,” Opt. Laser Technol. 44, 374–377 (2012).
[Crossref]

Zhong, Z.

Zhu, B.

Adv. Opt. Photon. (1)

Appl. Opt. (8)

IEEE Photon. J. (2)

W. Chen, “3D Gerchberg-Saxton optical correlation,” IEEE Photon. J. 10, 7800409 (2018).
[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 Photon. J. 8, 7801807 (2016).
[Crossref]

J. Opt. (2)

J. M. Vilardy, M. S. Millán, and E. Pérez-Cabré, “Improved decryption quality and security of a joint-transform correlator-based encryption system,” J. Opt. 15, 025401 (2013).
[Crossref]

W. Chen and X. Chen, “Optical asymmetric cryptography using a three-dimensional space-based model,” J. Opt. 13, 075404 (2011).
[Crossref]

J. Opt. A (1)

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

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

Opt. Commun. (4)

H. Li and Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281, 5745–5749 (2008).
[Crossref]

B. Hennelly and J. T. Sheridan, “Fractional Fourier transform-based on image encryption: phase retrieval algorithm,” Opt. Commun. 226, 61–80 (2003).
[Crossref]

W. Chen, “Hierarchically optical double-image correlation using 3D phase retrieval algorithm in fractional Fourier transform domain,” Opt. Commun. 427, 374–381 (2018).
[Crossref]

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

Opt. Eng. (1)

H. Di, Y. Kang, Y. Liu, and X. Zhang, “Multiple image encryption by phase retrieval,” Opt. Eng. 55, 073103 (2016).
[Crossref]

Opt. Express (4)

Opt. Laser Technol. (2)

X. Deng and D. Zhao, “Multiple-image encryption using phase retrieve algorithm and intermodulation in Fourier domain,” Opt. Laser Technol. 44, 374–377 (2012).
[Crossref]

S. Liu, C. Guo, and J. T. Sheridan, “A review of optical image encryption techniques,” Opt. Laser Technol. 57, 327–342 (2014).
[Crossref]

Opt. Lasers Eng. (2)

M. R. Abuturab, “Color image security system based on discrete Hartley transform in gyrator transform domain,” Opt. Lasers Eng. 51, 317–324 (2013).
[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 (2017).
[Crossref]

Opt. Lett. (8)

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Figures (12)

Fig. 1.
Fig. 1. Optical encryption architecture with RPMs in the Fr-FrFT domains.
Fig. 2.
Fig. 2. Flowchart of the n th loop of the iteration.
Fig. 3.
Fig. 3. Flowchart of the decryption process.
Fig. 4.
Fig. 4. Simulation results for the multi-depth 3D image with binary images. (a) Multi-depth 3D image to be encrypted, (b) encrypted target multi-depth 3D image, (c) decrypted image using phase retrieval only one time with all of the matched keys, and (d) decrypted image using phase retrieval 200 times with all of the matched keys.
Fig. 5.
Fig. 5. Convergence based on the number of the iterations.
Fig. 6.
Fig. 6. Decrypted results obtained after using incorrect keys: (a) incorrect POFs, (b) incorrect free-space propagation distance, (c) incorrect optical wavelength, and (d) incorrect fractional order.
Fig. 7.
Fig. 7. Simulation results for a multi-depth 3D gray-scale image. (a) Multi-depth 3D image to be encrypted, (b) the encrypted image, and (c) the image decrypted using the phase retrieval algorithm with one RPM retrieving one POF in the Fresnel domain; (d) the encrypted image and (e) the image decrypted using the phase retrieval algorithm with two RPMs retrieving one POF in the Fresnel domain; and (f) the encrypted image and (g) the image decrypted using the proposed method.
Fig. 8.
Fig. 8. (a) MSE value and iterative loops and (b) CC values and iterative loops for three different phase retrieval algorithms.
Fig. 9.
Fig. 9. (a) Normalized MSE versus the deviation of fractional order α in different depth planes, (b) the decrypted multi-depth 3D image with correct decryption keys but α = 0.493 ( Δ α = 0.008 ) , and (c) the decrypted multi-depth 3D image with correct decryption keys but α = 0.5 ( Δ α = 0.015 ) .
Fig. 10.
Fig. 10. Decrypted images with varying coefficients: (a)  k = 0.2 , (b)  k = 0.4 , (c)  k = 0.8 , and (d)  k = 1.0 .
Fig. 11.
Fig. 11. Robustness against occlusion attack: (a) 25% corner attack and (b) corresponding decrypted 3D image focused on the 8, 8.5, and 9 cm depth planes (from left to right).
Fig. 12.
Fig. 12. Robustness against occlusion attack: (a) 45% center attack and (b) corresponding decrypted 3D image focused on the 8, 8.5, and 9 cm depth planes (from left to right).

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

f i ( x , y ; z i ) = | f i ( x , y ; z i ) | × exp [ j 2 π r ( x , y ) ; z i ] .
F i ( u , v ) = FrT λ z i [ f i ( x , y ) ; z i ] = exp ( j 2 π z i / λ ) j λ z i f i ( x , y ; z i ) × exp { j π λ z i [ ( u x ) 2 + ( v y ) 2 ] } d x d y ,
F ( u , v ) = i = 1 M F i ( u , v ) = | F ( u , v ) | × exp [ j ϕ ( u , v ) ] .
F ( u , v ) = | F ( u , v ) | × exp [ j 2 π R ( u , v ) ] .
E ( ξ , η ) = FrFT α [ F ( u , v ) ] = K [ F ( u , v ) ] × exp { j π [ u 2 + v 2 + ξ 2 + η 2 λ f a tan ( ϕ ) 2 u v + ξ η λ f a sin ( ϕ ) ] } d u d v = | E ( ξ , η ) | × exp [ j ψ ( ξ , η ) ] ,
K = exp [ j ( π / 4 ) sign ( sin ϕ ) + j ϕ / 2 ] | λ f a · sin ϕ | ,
f i _ n ( x , y ; z i ) = | f i ( x , y ; z i ) | × exp [ j r n ( x , y ; z i ) ] .
F n ( u , v ) = i = 1 M FrT z i λ [ f i _ n ( x , y ; z i ) ] = | F n ( u , v ) | × exp [ j ϕ n ( u , v ) ] ,
E n ( ξ , η ) = | E n ( ξ , η ) | × exp [ j ψ n ( ξ , η ) ] ,
F n ( u , v ) = | F n ( u , v ) | × exp [ j ψ n ( u , v ) ] = | F n ( u , v ) | × exp [ j R n + 1 ( u , v ) ] ,
f i _ n ( x , y ; z i ) = | f i ( x , y ; z i ) | × exp [ j ϕ n ( x , y ; z i ) ] ,
f i _ n ( x , y ; z i ) = | f i ( x , y ; z i ) | × exp [ j ϕ n ( x , y ; z i ) ] = | f i ( x , y ; z i ) | × exp [ j r n + 1 ( x , y ; z i ) ] .
CC = E { [ f E [ f ] ] [ | f | E [ | f | ] ] } E { [ f E [ f ] ] 2 } E { [ | f | E [ | f | ] ] 2 } ,
K 1 ( u , v ) = exp [ j R n + 1 ( u , v ) ] × exp [ j ϕ n ( u , v ) ] ,
K 2 ( ξ , η ) = exp [ j ψ n ( ξ , η ) ] .
D ( u , v ) = FrFT α [ | E n ( ξ , η ) | × K 2 ( ξ , η ) ] ,
d i ( x , y ; z i ) = | FrT λ z i [ D ( u , v ) × K 1 ( u , v ) ] | ,
MSE = 1 / ( M × N ) m = 1 M n = 1 N | f ( m , n ) d ( m , n ) | 2 ,
C = C ( 1 + k G ) ,

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