Abstract

This paper describes a novel algorithm to encrypt double color images into a single undistinguishable image in quaternion gyrator domain. By using an iterative phase retrieval algorithm, the phase masks used for encryption are obtained. Subsequently, the encrypted image is generated via cascaded quaternion gyrator transforms with different rotation angles. The parameters in quaternion gyrator transforms and phases serve as encryption keys. By knowing these keys, the original color images can be fully restituted. Numerical simulations have demonstrated the validity of the proposed encryption system as well as its robustness against loss of data and additive Gaussian noise.

© 2014 Optical Society of America

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  1. P. Refregier, B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995).
    [CrossRef] [PubMed]
  2. S. T. Liu, L. Yu, B. H. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 187(1–3), 57–63 (2001).
    [CrossRef]
  3. B. Hennelly, J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28(4), 269–271 (2003).
    [CrossRef] [PubMed]
  4. N. K. Nishchal, J. Joseph, K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235(4–6), 253–259 (2004).
    [CrossRef]
  5. J. L. Zang, Z. W. Xie, Y. Zhang, “Optical image encryption with spatially incoherent illumination,” Opt. Lett. 38(8), 1289–1291 (2013).
    [CrossRef] [PubMed]
  6. Y. S. Shi, T. Li, Y. L. Wang, Q. K. Gao, S. G. Zhang, H. F. Li, “Optical image encryption via ptychography,” Opt. Lett. 38(9), 1425–1427 (2013).
    [CrossRef] [PubMed]
  7. W. Liu, Z. J. Liu, S. T. Liu, “Asymmetric cryptosystem using random binary phase modulation based on mixture retrieval type of Yang-Gu algorithm,” Opt. Lett. 38(10), 1651–1653 (2013).
    [CrossRef] [PubMed]
  8. W. Chen, X. D. Chen, A. Stern, B. Javidi, “Phase-modulated optical system with sparse representation for information encoding and authentication,” IEEE Photonics J. 5(2), 6900113 (2013).
    [CrossRef]
  9. M. J. Swain, D. H. Ballard, “Color indexing,” Int. J. Comput. Vis. 7(1), 11–32 (1991).
    [CrossRef]
  10. T. Gevers, A. W. M. Smeulders, “Color-based object recognition,” Pattern Recognit. 32(3), 453–464 (1999).
    [CrossRef]
  11. S. Q. Zhang, M. A. Karim, “Color image encryption using double random phase encoding,” Microw. Opt. Technol. Lett. 21(5), 318–323 (1999).
    [CrossRef]
  12. L. F. Chen, D. M. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Express 14(19), 8552–8560 (2006).
    [CrossRef] [PubMed]
  13. L. F. Chen, D. M. Zhao, “Color information processing (coding and synthesis) with fractional Fourier transforms and digital holography,” Opt. Express 15(24), 16080–16089 (2007).
    [CrossRef] [PubMed]
  14. L. S. Sui, M. T. Xin, A. L. Tian, H. Y. Jin, “Single-channel color image encryption using phase retrieve algorithm in fractional Fourier domain,” Opt. Lasers Eng. 51(12), 1297–1309 (2013).
    [CrossRef]
  15. M. R. Abuturab, “Color image security system using double random-structured phase encoding in gyrator transform domain,” Appl. Opt. 51(15), 3006–3016 (2012).
    [CrossRef] [PubMed]
  16. L. S. Sui, B. Gao, “Color image encryption based on gyrator transform and Arnold transform,” Opt. Laser Technol. 48, 530–538 (2013).
    [CrossRef]
  17. W. R. Hamilton, Elements of Quaternions (Longmans Green, 1866).
  18. X. L. Wang, H. C. Zhai, Z. L. Li, Q. Ge, “Double random-phase encryption based on discrete quaternion Fourier-transforms,” Optik (Stuttg.) 122(20), 1856–1859 (2011).
    [CrossRef]
  19. Z. H. Shao, J. S. Wu, J. L. Coatrieux, G. Coatrieux, and H. Z. Shu, “Quaternion gyrator transform and its application to color image encryption,” in Proceedings of IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2013), pp. 4579–4582.
  20. G. H. Situ, J. J. Zhang, “Multiple-image encryption by wavelength multiplexing,” Opt. Lett. 30(11), 1306–1308 (2005).
    [CrossRef] [PubMed]
  21. Y. S. Shi, G. H. Situ, J. J. Zhang, “Multiple-image hiding in the Fresnel domain,” Opt. Lett. 32(13), 1914–1916 (2007).
    [CrossRef] [PubMed]
  22. H. E. Hwang, H. T. Chang, W. N. Lie, “Multiple-image encryption and multiplexing using a modified Gerchberg-Saxton algorithm and phase modulation in Fresnel-transform domain,” Opt. Lett. 34(24), 3917–3919 (2009).
    [CrossRef] [PubMed]
  23. H. T. Chang, H. Hwang, C. Lee, “Position multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 284(18), 4146–4151 (2011).
    [CrossRef]
  24. H. T. Chang, H. E. Hwang, C. L. Lee, M. T. Lee, “Wavelength multiplexing multiple-image encryption using cascaded phase-only masks in the Fresnel transform domain,” Appl. Opt. 50(5), 710–716 (2011).
    [CrossRef] [PubMed]
  25. A. Alfalou, A. Mansour, “Double random phase encryption scheme to multiplex and simultaneous encode multiple images,” Appl. Opt. 48(31), 5933–5947 (2009).
    [CrossRef] [PubMed]
  26. X. P. Deng, D. M. Zhao, “Multiple-image encryption using phase retrieve algorithm and intermodulation in Fourier domain,” Opt. Laser Technol. 44(2), 374–377 (2012).
    [CrossRef]
  27. X. G. Wang, D. M. Zhao, “Fully phase multiple-image encryption based on superposition principle and the digital holographic technique,” Opt. Commun. 285(21-22), 4280–4284 (2012).
    [CrossRef]
  28. S. Liansheng, X. Meiting, T. Ailing, “Multiple-image encryption based on phase mask multiplexing in fractional Fourier transform domain,” Opt. Lett. 38(11), 1996–1998 (2013).
    [CrossRef] [PubMed]
  29. D. Z. Kong, X. J. Shen, Q. Z. Xu, W. Xin, H. Q. Guo, “Multiple-image encryption scheme based on cascaded fractional Fourier transform,” Appl. Opt. 52(12), 2619–2625 (2013).
    [CrossRef] [PubMed]
  30. R. Tao, Y. Xin, Y. Wang, “Double image encryption based on random phase encoding in the fractional Fourier domain,” Opt. Express 15(24), 16067–16079 (2007).
    [CrossRef] [PubMed]
  31. Z. J. Liu, S. T. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275(2), 324–329 (2007).
    [CrossRef]
  32. Z. J. Liu, J. M. Dai, X. G. Sun, S. T. Liu, “Triple image encryption scheme in fractional Fourier transform domains,” Opt. Commun. 282(4), 518–522 (2009).
    [CrossRef]
  33. J. H. Wu, X. Z. Luo, N. R. Zhou, “Four-image encryption method based on spectrum truncation, chaos and the MODFrFT,” Opt. Laser Technol. 45, 571–577 (2013).
    [CrossRef]
  34. M. Joshi, C. Shakher, K. Singh, “Color image encryption and decryption for twin images in fractional Fourier domain,” Opt. Commun. 281(23), 5713–5720 (2008).
    [CrossRef]
  35. M. Joshi, Chandrashakher, K. Singh, “Fractional Fourier transform based image multiplexing and encryption technique for four-color images using input images as keys,” Opt. Commun. 283(12), 2496–2505 (2010).
    [CrossRef]
  36. Z. J. Liu, Q. Guo, L. Xu, M. A. Ahmad, S. T. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express 18(11), 12033–12043 (2010).
    [CrossRef] [PubMed]
  37. H. J. Li, Y. R. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281(23), 5745–5749 (2008).
    [CrossRef]
  38. J. A. Rodrigo, T. Alieva, M. L. Calvo, “Gyrator transform: properties and applications,” Opt. Express 15(5), 2190–2203 (2007).
    [CrossRef] [PubMed]
  39. J. A. Rodrigo, T. Alieva, M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278(2), 279–284 (2007).
    [CrossRef]
  40. S. Said, N. Le Bihan, S. J. Sangwine, “Fast complexified quaternion Fourier transform,” IEEE Trans. Signal Process. 56(4), 1522–1531 (2008).
    [CrossRef]
  41. Original color images: http://sipi.usc.edu/database/database.php .

2013 (9)

W. Chen, X. D. Chen, A. Stern, B. Javidi, “Phase-modulated optical system with sparse representation for information encoding and authentication,” IEEE Photonics J. 5(2), 6900113 (2013).
[CrossRef]

L. S. Sui, M. T. Xin, A. L. Tian, H. Y. Jin, “Single-channel color image encryption using phase retrieve algorithm in fractional Fourier domain,” Opt. Lasers Eng. 51(12), 1297–1309 (2013).
[CrossRef]

L. S. Sui, B. Gao, “Color image encryption based on gyrator transform and Arnold transform,” Opt. Laser Technol. 48, 530–538 (2013).
[CrossRef]

J. H. Wu, X. Z. Luo, N. R. Zhou, “Four-image encryption method based on spectrum truncation, chaos and the MODFrFT,” Opt. Laser Technol. 45, 571–577 (2013).
[CrossRef]

J. L. Zang, Z. W. Xie, Y. Zhang, “Optical image encryption with spatially incoherent illumination,” Opt. Lett. 38(8), 1289–1291 (2013).
[CrossRef] [PubMed]

D. Z. Kong, X. J. Shen, Q. Z. Xu, W. Xin, H. Q. Guo, “Multiple-image encryption scheme based on cascaded fractional Fourier transform,” Appl. Opt. 52(12), 2619–2625 (2013).
[CrossRef] [PubMed]

Y. S. Shi, T. Li, Y. L. Wang, Q. K. Gao, S. G. Zhang, H. F. Li, “Optical image encryption via ptychography,” Opt. Lett. 38(9), 1425–1427 (2013).
[CrossRef] [PubMed]

W. Liu, Z. J. Liu, S. T. Liu, “Asymmetric cryptosystem using random binary phase modulation based on mixture retrieval type of Yang-Gu algorithm,” Opt. Lett. 38(10), 1651–1653 (2013).
[CrossRef] [PubMed]

S. Liansheng, X. Meiting, T. Ailing, “Multiple-image encryption based on phase mask multiplexing in fractional Fourier transform domain,” Opt. Lett. 38(11), 1996–1998 (2013).
[CrossRef] [PubMed]

2012 (3)

X. P. Deng, D. M. Zhao, “Multiple-image encryption using phase retrieve algorithm and intermodulation in Fourier domain,” Opt. Laser Technol. 44(2), 374–377 (2012).
[CrossRef]

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

M. R. Abuturab, “Color image security system using double random-structured phase encoding in gyrator transform domain,” Appl. Opt. 51(15), 3006–3016 (2012).
[CrossRef] [PubMed]

2011 (3)

H. T. Chang, H. E. Hwang, C. L. Lee, M. T. Lee, “Wavelength multiplexing multiple-image encryption using cascaded phase-only masks in the Fresnel transform domain,” Appl. Opt. 50(5), 710–716 (2011).
[CrossRef] [PubMed]

X. L. Wang, H. C. Zhai, Z. L. Li, Q. Ge, “Double random-phase encryption based on discrete quaternion Fourier-transforms,” Optik (Stuttg.) 122(20), 1856–1859 (2011).
[CrossRef]

H. T. Chang, H. Hwang, C. Lee, “Position multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 284(18), 4146–4151 (2011).
[CrossRef]

2010 (2)

M. Joshi, Chandrashakher, K. Singh, “Fractional Fourier transform based image multiplexing and encryption technique for four-color images using input images as keys,” Opt. Commun. 283(12), 2496–2505 (2010).
[CrossRef]

Z. J. Liu, Q. Guo, L. Xu, M. A. Ahmad, S. T. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express 18(11), 12033–12043 (2010).
[CrossRef] [PubMed]

2009 (3)

2008 (3)

S. Said, N. Le Bihan, S. J. Sangwine, “Fast complexified quaternion Fourier transform,” IEEE Trans. Signal Process. 56(4), 1522–1531 (2008).
[CrossRef]

H. J. Li, Y. R. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281(23), 5745–5749 (2008).
[CrossRef]

M. Joshi, C. Shakher, K. Singh, “Color image encryption and decryption for twin images in fractional Fourier domain,” Opt. Commun. 281(23), 5713–5720 (2008).
[CrossRef]

2007 (6)

2006 (1)

2005 (1)

2004 (1)

N. K. Nishchal, J. Joseph, K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235(4–6), 253–259 (2004).
[CrossRef]

2003 (1)

2001 (1)

S. T. Liu, L. Yu, B. H. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 187(1–3), 57–63 (2001).
[CrossRef]

1999 (2)

T. Gevers, A. W. M. Smeulders, “Color-based object recognition,” Pattern Recognit. 32(3), 453–464 (1999).
[CrossRef]

S. Q. Zhang, M. A. Karim, “Color image encryption using double random phase encoding,” Microw. Opt. Technol. Lett. 21(5), 318–323 (1999).
[CrossRef]

1995 (1)

1991 (1)

M. J. Swain, D. H. Ballard, “Color indexing,” Int. J. Comput. Vis. 7(1), 11–32 (1991).
[CrossRef]

Abuturab, M. R.

Ahmad, M. A.

Ailing, T.

Alfalou, A.

Alieva, T.

J. A. Rodrigo, T. Alieva, M. L. Calvo, “Gyrator transform: properties and applications,” Opt. Express 15(5), 2190–2203 (2007).
[CrossRef] [PubMed]

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

Ballard, D. H.

M. J. Swain, D. H. Ballard, “Color indexing,” Int. J. Comput. Vis. 7(1), 11–32 (1991).
[CrossRef]

Calvo, M. L.

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

J. A. Rodrigo, T. Alieva, M. L. Calvo, “Gyrator transform: properties and applications,” Opt. Express 15(5), 2190–2203 (2007).
[CrossRef] [PubMed]

Chandrashakher,

M. Joshi, Chandrashakher, K. Singh, “Fractional Fourier transform based image multiplexing and encryption technique for four-color images using input images as keys,” Opt. Commun. 283(12), 2496–2505 (2010).
[CrossRef]

Chang, H. T.

Chen, L. F.

Chen, W.

W. Chen, X. D. Chen, A. Stern, B. Javidi, “Phase-modulated optical system with sparse representation for information encoding and authentication,” IEEE Photonics J. 5(2), 6900113 (2013).
[CrossRef]

Chen, X. D.

W. Chen, X. D. Chen, A. Stern, B. Javidi, “Phase-modulated optical system with sparse representation for information encoding and authentication,” IEEE Photonics J. 5(2), 6900113 (2013).
[CrossRef]

Dai, J. M.

Z. J. Liu, J. M. Dai, X. G. Sun, S. T. Liu, “Triple image encryption scheme in fractional Fourier transform domains,” Opt. Commun. 282(4), 518–522 (2009).
[CrossRef]

Deng, X. P.

X. P. Deng, D. M. Zhao, “Multiple-image encryption using phase retrieve algorithm and intermodulation in Fourier domain,” Opt. Laser Technol. 44(2), 374–377 (2012).
[CrossRef]

Gao, B.

L. S. Sui, B. Gao, “Color image encryption based on gyrator transform and Arnold transform,” Opt. Laser Technol. 48, 530–538 (2013).
[CrossRef]

Gao, Q. K.

Ge, Q.

X. L. Wang, H. C. Zhai, Z. L. Li, Q. Ge, “Double random-phase encryption based on discrete quaternion Fourier-transforms,” Optik (Stuttg.) 122(20), 1856–1859 (2011).
[CrossRef]

Gevers, T.

T. Gevers, A. W. M. Smeulders, “Color-based object recognition,” Pattern Recognit. 32(3), 453–464 (1999).
[CrossRef]

Guo, H. Q.

Guo, Q.

Hennelly, B.

Hwang, H.

H. T. Chang, H. Hwang, C. Lee, “Position multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 284(18), 4146–4151 (2011).
[CrossRef]

Hwang, H. E.

Javidi, B.

W. Chen, X. D. Chen, A. Stern, B. Javidi, “Phase-modulated optical system with sparse representation for information encoding and authentication,” IEEE Photonics J. 5(2), 6900113 (2013).
[CrossRef]

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

Jin, H. Y.

L. S. Sui, M. T. Xin, A. L. Tian, H. Y. Jin, “Single-channel color image encryption using phase retrieve algorithm in fractional Fourier domain,” Opt. Lasers Eng. 51(12), 1297–1309 (2013).
[CrossRef]

Joseph, J.

N. K. Nishchal, J. Joseph, K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235(4–6), 253–259 (2004).
[CrossRef]

Joshi, M.

M. Joshi, Chandrashakher, K. Singh, “Fractional Fourier transform based image multiplexing and encryption technique for four-color images using input images as keys,” Opt. Commun. 283(12), 2496–2505 (2010).
[CrossRef]

M. Joshi, C. Shakher, K. Singh, “Color image encryption and decryption for twin images in fractional Fourier domain,” Opt. Commun. 281(23), 5713–5720 (2008).
[CrossRef]

Karim, M. A.

S. Q. Zhang, M. A. Karim, “Color image encryption using double random phase encoding,” Microw. Opt. Technol. Lett. 21(5), 318–323 (1999).
[CrossRef]

Kong, D. Z.

Le Bihan, N.

S. Said, N. Le Bihan, S. J. Sangwine, “Fast complexified quaternion Fourier transform,” IEEE Trans. Signal Process. 56(4), 1522–1531 (2008).
[CrossRef]

Lee, C.

H. T. Chang, H. Hwang, C. Lee, “Position multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 284(18), 4146–4151 (2011).
[CrossRef]

Lee, C. L.

Lee, M. T.

Li, H. F.

Li, H. J.

H. J. Li, Y. R. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281(23), 5745–5749 (2008).
[CrossRef]

Li, T.

Li, Z. L.

X. L. Wang, H. C. Zhai, Z. L. Li, Q. Ge, “Double random-phase encryption based on discrete quaternion Fourier-transforms,” Optik (Stuttg.) 122(20), 1856–1859 (2011).
[CrossRef]

Liansheng, S.

Lie, W. N.

Liu, S. T.

W. Liu, Z. J. Liu, S. T. Liu, “Asymmetric cryptosystem using random binary phase modulation based on mixture retrieval type of Yang-Gu algorithm,” Opt. Lett. 38(10), 1651–1653 (2013).
[CrossRef] [PubMed]

Z. J. Liu, Q. Guo, L. Xu, M. A. Ahmad, S. T. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express 18(11), 12033–12043 (2010).
[CrossRef] [PubMed]

Z. J. Liu, J. M. Dai, X. G. Sun, S. T. Liu, “Triple image encryption scheme in fractional Fourier transform domains,” Opt. Commun. 282(4), 518–522 (2009).
[CrossRef]

Z. J. Liu, S. T. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275(2), 324–329 (2007).
[CrossRef]

S. T. Liu, L. Yu, B. H. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 187(1–3), 57–63 (2001).
[CrossRef]

Liu, W.

Liu, Z. J.

W. Liu, Z. J. Liu, S. T. Liu, “Asymmetric cryptosystem using random binary phase modulation based on mixture retrieval type of Yang-Gu algorithm,” Opt. Lett. 38(10), 1651–1653 (2013).
[CrossRef] [PubMed]

Z. J. Liu, Q. Guo, L. Xu, M. A. Ahmad, S. T. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express 18(11), 12033–12043 (2010).
[CrossRef] [PubMed]

Z. J. Liu, J. M. Dai, X. G. Sun, S. T. Liu, “Triple image encryption scheme in fractional Fourier transform domains,” Opt. Commun. 282(4), 518–522 (2009).
[CrossRef]

Z. J. Liu, S. T. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275(2), 324–329 (2007).
[CrossRef]

Luo, X. Z.

J. H. Wu, X. Z. Luo, N. R. Zhou, “Four-image encryption method based on spectrum truncation, chaos and the MODFrFT,” Opt. Laser Technol. 45, 571–577 (2013).
[CrossRef]

Mansour, A.

Meiting, X.

Nishchal, N. K.

N. K. Nishchal, J. Joseph, K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235(4–6), 253–259 (2004).
[CrossRef]

Refregier, P.

Rodrigo, J. A.

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

J. A. Rodrigo, T. Alieva, M. L. Calvo, “Gyrator transform: properties and applications,” Opt. Express 15(5), 2190–2203 (2007).
[CrossRef] [PubMed]

Said, S.

S. Said, N. Le Bihan, S. J. Sangwine, “Fast complexified quaternion Fourier transform,” IEEE Trans. Signal Process. 56(4), 1522–1531 (2008).
[CrossRef]

Sangwine, S. J.

S. Said, N. Le Bihan, S. J. Sangwine, “Fast complexified quaternion Fourier transform,” IEEE Trans. Signal Process. 56(4), 1522–1531 (2008).
[CrossRef]

Shakher, C.

M. Joshi, C. Shakher, K. Singh, “Color image encryption and decryption for twin images in fractional Fourier domain,” Opt. Commun. 281(23), 5713–5720 (2008).
[CrossRef]

Shen, X. J.

Sheridan, J. T.

Shi, Y. S.

Singh, K.

M. Joshi, Chandrashakher, K. Singh, “Fractional Fourier transform based image multiplexing and encryption technique for four-color images using input images as keys,” Opt. Commun. 283(12), 2496–2505 (2010).
[CrossRef]

M. Joshi, C. Shakher, K. Singh, “Color image encryption and decryption for twin images in fractional Fourier domain,” Opt. Commun. 281(23), 5713–5720 (2008).
[CrossRef]

N. K. Nishchal, J. Joseph, K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235(4–6), 253–259 (2004).
[CrossRef]

Situ, G. H.

Smeulders, A. W. M.

T. Gevers, A. W. M. Smeulders, “Color-based object recognition,” Pattern Recognit. 32(3), 453–464 (1999).
[CrossRef]

Stern, A.

W. Chen, X. D. Chen, A. Stern, B. Javidi, “Phase-modulated optical system with sparse representation for information encoding and authentication,” IEEE Photonics J. 5(2), 6900113 (2013).
[CrossRef]

Sui, L. S.

L. S. Sui, M. T. Xin, A. L. Tian, H. Y. Jin, “Single-channel color image encryption using phase retrieve algorithm in fractional Fourier domain,” Opt. Lasers Eng. 51(12), 1297–1309 (2013).
[CrossRef]

L. S. Sui, B. Gao, “Color image encryption based on gyrator transform and Arnold transform,” Opt. Laser Technol. 48, 530–538 (2013).
[CrossRef]

Sun, X. G.

Z. J. Liu, J. M. Dai, X. G. Sun, S. T. Liu, “Triple image encryption scheme in fractional Fourier transform domains,” Opt. Commun. 282(4), 518–522 (2009).
[CrossRef]

Swain, M. J.

M. J. Swain, D. H. Ballard, “Color indexing,” Int. J. Comput. Vis. 7(1), 11–32 (1991).
[CrossRef]

Tao, R.

Tian, A. L.

L. S. Sui, M. T. Xin, A. L. Tian, H. Y. Jin, “Single-channel color image encryption using phase retrieve algorithm in fractional Fourier domain,” Opt. Lasers Eng. 51(12), 1297–1309 (2013).
[CrossRef]

Wang, X. G.

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

Wang, X. L.

X. L. Wang, H. C. Zhai, Z. L. Li, Q. Ge, “Double random-phase encryption based on discrete quaternion Fourier-transforms,” Optik (Stuttg.) 122(20), 1856–1859 (2011).
[CrossRef]

Wang, Y.

Wang, Y. L.

Wang, Y. R.

H. J. Li, Y. R. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281(23), 5745–5749 (2008).
[CrossRef]

Wu, J. H.

J. H. Wu, X. Z. Luo, N. R. Zhou, “Four-image encryption method based on spectrum truncation, chaos and the MODFrFT,” Opt. Laser Technol. 45, 571–577 (2013).
[CrossRef]

Xie, Z. W.

Xin, M. T.

L. S. Sui, M. T. Xin, A. L. Tian, H. Y. Jin, “Single-channel color image encryption using phase retrieve algorithm in fractional Fourier domain,” Opt. Lasers Eng. 51(12), 1297–1309 (2013).
[CrossRef]

Xin, W.

Xin, Y.

Xu, L.

Xu, Q. Z.

Yu, L.

S. T. Liu, L. Yu, B. H. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 187(1–3), 57–63 (2001).
[CrossRef]

Zang, J. L.

Zhai, H. C.

X. L. Wang, H. C. Zhai, Z. L. Li, Q. Ge, “Double random-phase encryption based on discrete quaternion Fourier-transforms,” Optik (Stuttg.) 122(20), 1856–1859 (2011).
[CrossRef]

Zhang, J. J.

Zhang, S. G.

Zhang, S. Q.

S. Q. Zhang, M. A. Karim, “Color image encryption using double random phase encoding,” Microw. Opt. Technol. Lett. 21(5), 318–323 (1999).
[CrossRef]

Zhang, Y.

Zhao, D. M.

X. G. Wang, D. M. 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. P. Deng, D. M. Zhao, “Multiple-image encryption using phase retrieve algorithm and intermodulation in Fourier domain,” Opt. Laser Technol. 44(2), 374–377 (2012).
[CrossRef]

L. F. Chen, D. M. Zhao, “Color information processing (coding and synthesis) with fractional Fourier transforms and digital holography,” Opt. Express 15(24), 16080–16089 (2007).
[CrossRef] [PubMed]

L. F. Chen, D. M. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Express 14(19), 8552–8560 (2006).
[CrossRef] [PubMed]

Zhou, N. R.

J. H. Wu, X. Z. Luo, N. R. Zhou, “Four-image encryption method based on spectrum truncation, chaos and the MODFrFT,” Opt. Laser Technol. 45, 571–577 (2013).
[CrossRef]

Zhu, B. H.

S. T. Liu, L. Yu, B. H. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 187(1–3), 57–63 (2001).
[CrossRef]

Appl. Opt. (4)

IEEE Photonics J. (1)

W. Chen, X. D. Chen, A. Stern, B. Javidi, “Phase-modulated optical system with sparse representation for information encoding and authentication,” IEEE Photonics J. 5(2), 6900113 (2013).
[CrossRef]

IEEE Trans. Signal Process. (1)

S. Said, N. Le Bihan, S. J. Sangwine, “Fast complexified quaternion Fourier transform,” IEEE Trans. Signal Process. 56(4), 1522–1531 (2008).
[CrossRef]

Int. J. Comput. Vis. (1)

M. J. Swain, D. H. Ballard, “Color indexing,” Int. J. Comput. Vis. 7(1), 11–32 (1991).
[CrossRef]

Microw. Opt. Technol. Lett. (1)

S. Q. Zhang, M. A. Karim, “Color image encryption using double random phase encoding,” Microw. Opt. Technol. Lett. 21(5), 318–323 (1999).
[CrossRef]

Opt. Commun. (10)

S. T. Liu, L. Yu, B. H. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 187(1–3), 57–63 (2001).
[CrossRef]

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

Z. J. Liu, S. T. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275(2), 324–329 (2007).
[CrossRef]

Z. J. Liu, J. M. Dai, X. G. Sun, S. T. Liu, “Triple image encryption scheme in fractional Fourier transform domains,” Opt. Commun. 282(4), 518–522 (2009).
[CrossRef]

M. Joshi, C. Shakher, K. Singh, “Color image encryption and decryption for twin images in fractional Fourier domain,” Opt. Commun. 281(23), 5713–5720 (2008).
[CrossRef]

M. Joshi, Chandrashakher, K. Singh, “Fractional Fourier transform based image multiplexing and encryption technique for four-color images using input images as keys,” Opt. Commun. 283(12), 2496–2505 (2010).
[CrossRef]

H. J. Li, Y. R. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281(23), 5745–5749 (2008).
[CrossRef]

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

H. T. Chang, H. Hwang, C. Lee, “Position multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 284(18), 4146–4151 (2011).
[CrossRef]

N. K. Nishchal, J. Joseph, K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235(4–6), 253–259 (2004).
[CrossRef]

Opt. Express (5)

Opt. Laser Technol. (3)

X. P. Deng, D. M. Zhao, “Multiple-image encryption using phase retrieve algorithm and intermodulation in Fourier domain,” Opt. Laser Technol. 44(2), 374–377 (2012).
[CrossRef]

J. H. Wu, X. Z. Luo, N. R. Zhou, “Four-image encryption method based on spectrum truncation, chaos and the MODFrFT,” Opt. Laser Technol. 45, 571–577 (2013).
[CrossRef]

L. S. Sui, B. Gao, “Color image encryption based on gyrator transform and Arnold transform,” Opt. Laser Technol. 48, 530–538 (2013).
[CrossRef]

Opt. Lasers Eng. (1)

L. S. Sui, M. T. Xin, A. L. Tian, H. Y. Jin, “Single-channel color image encryption using phase retrieve algorithm in fractional Fourier domain,” Opt. Lasers Eng. 51(12), 1297–1309 (2013).
[CrossRef]

Opt. Lett. (9)

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

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

B. Hennelly, J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28(4), 269–271 (2003).
[CrossRef] [PubMed]

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

H. E. Hwang, H. T. Chang, W. N. Lie, “Multiple-image encryption and multiplexing using a modified Gerchberg-Saxton algorithm and phase modulation in Fresnel-transform domain,” Opt. Lett. 34(24), 3917–3919 (2009).
[CrossRef] [PubMed]

J. L. Zang, Z. W. Xie, Y. Zhang, “Optical image encryption with spatially incoherent illumination,” Opt. Lett. 38(8), 1289–1291 (2013).
[CrossRef] [PubMed]

Y. S. Shi, T. Li, Y. L. Wang, Q. K. Gao, S. G. Zhang, H. F. Li, “Optical image encryption via ptychography,” Opt. Lett. 38(9), 1425–1427 (2013).
[CrossRef] [PubMed]

W. Liu, Z. J. Liu, S. T. Liu, “Asymmetric cryptosystem using random binary phase modulation based on mixture retrieval type of Yang-Gu algorithm,” Opt. Lett. 38(10), 1651–1653 (2013).
[CrossRef] [PubMed]

S. Liansheng, X. Meiting, T. Ailing, “Multiple-image encryption based on phase mask multiplexing in fractional Fourier transform domain,” Opt. Lett. 38(11), 1996–1998 (2013).
[CrossRef] [PubMed]

Optik (Stuttg.) (1)

X. L. Wang, H. C. Zhai, Z. L. Li, Q. Ge, “Double random-phase encryption based on discrete quaternion Fourier-transforms,” Optik (Stuttg.) 122(20), 1856–1859 (2011).
[CrossRef]

Pattern Recognit. (1)

T. Gevers, A. W. M. Smeulders, “Color-based object recognition,” Pattern Recognit. 32(3), 453–464 (1999).
[CrossRef]

Other (3)

Z. H. Shao, J. S. Wu, J. L. Coatrieux, G. Coatrieux, and H. Z. Shu, “Quaternion gyrator transform and its application to color image encryption,” in Proceedings of IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 2013), pp. 4579–4582.

W. R. Hamilton, Elements of Quaternions (Longmans Green, 1866).

Original color images: http://sipi.usc.edu/database/database.php .

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Figures (9)

Fig. 1
Fig. 1

Tested color images: each column constitutes a tested group (Primary color images are selected from USC-SIPI image database).

Fig. 2
Fig. 2

The evolution of CC values over the 1000 iteration loops.

Fig. 3
Fig. 3

Examples of encryption and granted decryption for double color images: (a) encrypted image; (b) and (c), decrypted results.

Fig. 4
Fig. 4

Decrypted results with randomly guessed keys: (a) and (b) correspond to Lena, (c) and (d) correspond to Pepper: (a) and (c) correspond to phases randomly guessed, (b) and (d) correspond to angles randomly guessed.

Fig. 5
Fig. 5

The NMSE values of decrypted images when varying one rotation angle while the other keys are set to the right value.

Fig. 6
Fig. 6

The NMSE values of decrypted images with different information loss severities.

Fig. 7
Fig. 7

Decrypted results of encrypted data with 50% information loss.

Fig. 8
Fig. 8

The NMSE values of decrypted images with different Gaussian white noise using different encryption algorithms.

Fig. 9
Fig. 9

Examples of decrypted images from noisy encrypted data with standard deviation 0.05.

Tables (1)

Tables Icon

Table 1 NMSE Values of Decrypted Images with Granted Keys

Equations (16)

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f ( x , y ) = i f R ( x , y ) + j f G ( x , y ) + k f B ( x , y ) .
Q G α [ f ( x , y ) ] ( u , v ) = 1 | sin α | f ( x , y ) exp ( μ 2 π ( u v + x y ) cos α ( u y + v x ) sin α ) d x d y ,
h exp ( Ι θ ) = G α 2 { G α 1 { f exp ( μ 11 ϕ 1 ) } exp ( μ 12 ϕ 2 ) } = G β 2 { G β 1 { g exp ( μ 21 φ 1 ) } exp ( μ 22 φ 2 ) }
g exp ( μ 2 1 φ 1 ) = G β 1 { G β 2 { G α 2 { G α 1 { f exp ( μ 11 ϕ 1 ) } exp ( μ 1 2 ϕ 2 ) } } exp ( μ 22 φ 2 ) } .
g ^ n = G β 1 { G β 2 { G α 2 { G α 1 { f exp ( μ 11 ϕ 1 n ) } exp ( μ 1 2 ϕ 2 n ) } } exp ( μ 22 φ 2 n ) } .
φ 1 n = arg { g ¯ g ^ n }
φ 2 n + 1 = arg { 1 G β 1 { g exp ( μ 2 1 φ 1 n ) } G β 2 { G α 2 { G α 1 { f exp ( μ 11 ϕ 1 n ) } exp ( μ 1 2 ϕ 2 n ) } } }
ϕ 2 n + 1 = arg { 1 G α 1 { f exp ( μ 11 ϕ 1 n ) } G α 2 { G β 2 { G β 1 { g exp ( μ 21 φ 1 n ) } exp ( μ 22 φ 2 n + 1 ) } } }
ϕ 1 n + 1 = arg { f ¯ G α 1 { G α 2 { G β 2 { G β 1 { g exp ( μ 21 φ 1 n ) } exp ( μ 22 φ 2 n + 1 ) } } exp ( μ 12 ϕ 2 n + 1 ) } }
C C = c { R , G , B } E { [ g c E ( g c ) ] [ g c n E ( g c n ) ] } 3 E { [ g c E ( g c ) ] 2 } E { [ g c n E ( g c n ) ] 2 } .
ϕ 1 = ϕ 1 n , ϕ 2 = ϕ 2 n , φ 1 = φ 1 n 1 , φ 2 = φ 2 n .
h = | G α 2 { G α 1 { f exp ( μ 11 ϕ 1 ) } exp ( μ 12 ϕ 2 ) } |
h = | G β 2 { G β 1 { g exp ( μ 21 φ 1 ) } exp ( μ 22 φ 2 ) } |
f = G α 1 { G α 2 { h exp ( Ι θ ) } exp ( μ 12 ϕ 2 ) } exp ( μ 11 ϕ 1 ) .
g = G β 1 { G β 2 { h exp ( Ι θ ) } exp ( μ 22 φ 2 ) } exp ( μ 21 φ 1 ) .
NMSE = x = 0 N 1 y = 0 M 1 | f ( x , y ) f ^ ( x , y ) | 2 x = 0 N 1 y = 0 M 1 | f ( x , y ) | 2 .

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