Abstract

The security of the encryption and verification techniques with significant output images is examined by a known-plaintext attack. We introduce an iterative phase-retrieval algorithm based on multiple intensity measurements to heuristically estimate the phase key in the Fourier domain by several plaintext-cyphertext pairs. We obtain correlation output images with very low error by correlating the estimated key with corresponding random phase masks. Our studies show that the convergence behavior of this algorithm sensitively depends on the starting point. We also demonstrate that this algorithm can be used to attack the double random phase encoding technique.

© 2007 Optical Society of America

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References

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  1. B. Javidi, ed., Optical and Digital Techniques for Information Security (Springer Verlag, 2005).
  2. P. Réfrégier and B. Javidi, "Optical image encryption based on input plane and Fourier plane random encoding," Opt. Lett. 20, 767-769 (1995).
  3. R. K. Wang, L. A. Watson, and C. Chatwin, "Random phase encoding for optical security," Opt. Eng. 35, 2464-2469 (1996).
    [CrossRef]
  4. P. C. Mogensen and J. Glückstad, "Phase-only optical encryption," Opt. Lett. 25, 566-568 (2000).
  5. G. Unnikrishnan, J. Joseph, and K. Singh, "Optical encryption by double-random phase encoding in the fractional Fourier domain," Opt. Lett. 25, 887-889 (2000).
  6. B. Hennelly and J. T. Sheridan, "Fractional Fourier transform-based image encryption: phase retrieval algorithm," Opt. Commun. 226, 61-80 (2003).
    [CrossRef]
  7. O. Matoba and B. Javidi, "Encrypted optical memory system using three-dimensional keys in the Fresnel domain," Opt. Lett. 24, 762-764 (1999).
  8. G. Situ and J. Zhang, "A lensless optical security system based on computer-generated phase only masks," Opt. Commun. 232, 115-122 (2004).
    [CrossRef]
  9. G. Situ and J. Zhang, "Double random-phase encoding in the Fresnel domain," Opt. Lett. 29, 1584-1586 (2004).
    [CrossRef]
  10. L. Chen and D. Zhao, "Optical image encryption with Hartley transforms," Opt. Lett. 31, 3438-3440 (2006).
    [CrossRef]
  11. J. Rosen and B. Javidi, "Hidden images in halftone pictures," Appl. Opt. 40, 3346-3353 (2001).
  12. J. J. Kim, J. H. Kim, and E. S. Kim, "Optodigital implementation of multiple information hiding and extraction system," Opt. Eng. 43, 113-125 (2004).
    [CrossRef]
  13. Y. Hayasaki, Y. Matsuba, A. Nagaoka, H. Yamamoto, and N. Nishida, "Hiding an image with a light-scattering medium and use of a contrast-discrimination method for readout," Appl. Opt. 43, 1552-1558 (2004).
    [CrossRef]
  14. G. Situ and J. Zhang, "Image hiding with computer-generated phase codes for optical authentication," Opt. Commun. 245, 55-65 (2005).
    [CrossRef]
  15. N. Takai and Y. Mifune, "Digital watermarking by a holographic technique," Appl. Opt. 43, 3078-3084 (2002).
  16. S. Kishk and B. Javidi, "3D object watermarking by a 3D hidden object," Opt. Express 11, 874-888 (2003).
  17. Y. Li, K. Kreske, and J. Rosen, "Security and encryption optical systems based on a correlator with significant output images," Appl. Opt. 39, 5295-5301 (2000).
  18. D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, "Security optical systems based on a joint transform correlator with significant output images," Opt. Eng. 40, 1584-1589 (2001).
    [CrossRef]
  19. M. Yamazaki and J. Ohtsubo, "Optimization of encrypted holograms in optical security systems," Opt. Eng. 40, 132-137 (2001).
    [CrossRef]
  20. D. Abookasis, A. Batikoff, H. Famini, and J. Rosen, "Performance comparison of iterative algorithms for generating digital correlation holograms used in optical security systems," Appl. Opt. 45, 4617-4624 (2006).
    [CrossRef]
  21. H. T. Chang, W. C. Lu, and C. J. Kuo, "Multiple-phase retrieval for optical security systems by use of random-phase encoding," Appl. Opt. 41, 4815-4834 (2002).
  22. G. Situ and J. Zhang, "A cascaded iterative Fourier transform algorithm for optical security applications," Optik 114, 473-477 (2003).
  23. X. F. Meng, L. Z Cai, X. L. Yang, X. X. Shen, and G. Y. Dong, "Information security system by iterative multiple-phase retrieval and pixel random permutation," Appl. Opt. 45, 3289-3297 (2006).
    [CrossRef]
  24. G. Situ and J. Zhang, "Phase-retrieval algorithms applied in a 4-f system for optical image encryption--a comparison," in Information Optics and Photonics Technology, G. Mu, F. T. S. Yu, and S. Jutamulia, eds., Proc. SPIE 5642, 108-115 (2005).
    [CrossRef]
  25. A. Carnicer, M. Montes-Usategui, S. Arcos, and I. Juvells, "Vulnerability to chosen-cyphertext attacks of optical encryption schemes based on double random phase keys," Opt. Lett. 30, 1644-1646 (2005).
    [CrossRef]
  26. Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, "Security analysis of optical encryption," in Optics and Photonics for Defence and Security, E. M. Carapezza, ed., Proc. SPIE 5986, 25-34 (2005).
  27. U. Gopinathan, D. S. Monaghan, T. J. Naughton, and J. T. Sheridan, "A known-plaintext heuristic attack on the Fourier plane encryption algorithm," Opt. Express 14, 3181-3186 (2006).
    [CrossRef]
  28. X. Peng, P. Zhang, H. Wei, and B. Yu, "Known-plaintext attack on optical encryption based on double random phase keys," Opt. Lett. 31, 1044-1046 (2006).
    [CrossRef]
  29. D. C. Youla and H. Webb, "Image restoration by the method of convex projections: part 1--theory," IEEE Trans. Med. Imaging MI-1, 81-94 (1982).
  30. A. Levi and H. Stark, "Image restoration by the method of generalized projections with application to restoration from magnitude," J. Opt. Soc. Am. A 1, 932-943 (1984).
  31. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction-plane pictures," Optik 35, 237-246 (1972).
  32. J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758-2769 (1982).
  33. H. P. Herzig, "Design of refractive and diffractive micro-optics," in Micro-Optics, H. P. Herzig, ed. (Taylor & Francis, 1996), 1-29.
  34. H. Stark, Y. Yang, and D. Gurkan, "Factors affecting convergence in the design of diffractive optics by iterative vector-space methods," J. Opt. Soc. Am. A 16, 149-159 (1999).
  35. W. Stallings, Cryptography and Network Security, 3rd ed. (Prentice Hall, 2004).

2006 (5)

2005 (5)

B. Javidi, ed., Optical and Digital Techniques for Information Security (Springer Verlag, 2005).

G. Situ and J. Zhang, "Image hiding with computer-generated phase codes for optical authentication," Opt. Commun. 245, 55-65 (2005).
[CrossRef]

G. Situ and J. Zhang, "Phase-retrieval algorithms applied in a 4-f system for optical image encryption--a comparison," in Information Optics and Photonics Technology, G. Mu, F. T. S. Yu, and S. Jutamulia, eds., Proc. SPIE 5642, 108-115 (2005).
[CrossRef]

Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, "Security analysis of optical encryption," in Optics and Photonics for Defence and Security, E. M. Carapezza, ed., Proc. SPIE 5986, 25-34 (2005).

A. Carnicer, M. Montes-Usategui, S. Arcos, and I. Juvells, "Vulnerability to chosen-cyphertext attacks of optical encryption schemes based on double random phase keys," Opt. Lett. 30, 1644-1646 (2005).
[CrossRef]

2004 (5)

W. Stallings, Cryptography and Network Security, 3rd ed. (Prentice Hall, 2004).

G. Situ and J. Zhang, "A lensless optical security system based on computer-generated phase only masks," Opt. Commun. 232, 115-122 (2004).
[CrossRef]

J. J. Kim, J. H. Kim, and E. S. Kim, "Optodigital implementation of multiple information hiding and extraction system," Opt. Eng. 43, 113-125 (2004).
[CrossRef]

Y. Hayasaki, Y. Matsuba, A. Nagaoka, H. Yamamoto, and N. Nishida, "Hiding an image with a light-scattering medium and use of a contrast-discrimination method for readout," Appl. Opt. 43, 1552-1558 (2004).
[CrossRef]

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

2003 (3)

S. Kishk and B. Javidi, "3D object watermarking by a 3D hidden object," Opt. Express 11, 874-888 (2003).

G. Situ and J. Zhang, "A cascaded iterative Fourier transform algorithm for optical security applications," Optik 114, 473-477 (2003).

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

2002 (2)

H. T. Chang, W. C. Lu, and C. J. Kuo, "Multiple-phase retrieval for optical security systems by use of random-phase encoding," Appl. Opt. 41, 4815-4834 (2002).

N. Takai and Y. Mifune, "Digital watermarking by a holographic technique," Appl. Opt. 43, 3078-3084 (2002).

2001 (3)

J. Rosen and B. Javidi, "Hidden images in halftone pictures," Appl. Opt. 40, 3346-3353 (2001).

D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, "Security optical systems based on a joint transform correlator with significant output images," Opt. Eng. 40, 1584-1589 (2001).
[CrossRef]

M. Yamazaki and J. Ohtsubo, "Optimization of encrypted holograms in optical security systems," Opt. Eng. 40, 132-137 (2001).
[CrossRef]

2000 (3)

1999 (2)

1996 (2)

H. P. Herzig, "Design of refractive and diffractive micro-optics," in Micro-Optics, H. P. Herzig, ed. (Taylor & Francis, 1996), 1-29.

R. K. Wang, L. A. Watson, and C. Chatwin, "Random phase encoding for optical security," Opt. Eng. 35, 2464-2469 (1996).
[CrossRef]

1995 (1)

1984 (1)

1982 (2)

D. C. Youla and H. Webb, "Image restoration by the method of convex projections: part 1--theory," IEEE Trans. Med. Imaging MI-1, 81-94 (1982).

J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758-2769 (1982).

1972 (1)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction-plane pictures," Optik 35, 237-246 (1972).

Abookasis, D.

D. Abookasis, A. Batikoff, H. Famini, and J. Rosen, "Performance comparison of iterative algorithms for generating digital correlation holograms used in optical security systems," Appl. Opt. 45, 4617-4624 (2006).
[CrossRef]

D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, "Security optical systems based on a joint transform correlator with significant output images," Opt. Eng. 40, 1584-1589 (2001).
[CrossRef]

Arazi, O.

D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, "Security optical systems based on a joint transform correlator with significant output images," Opt. Eng. 40, 1584-1589 (2001).
[CrossRef]

Arcos, S.

Batikoff, A.

Cai, L. Z

Carnicer, A.

Castro, A.

Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, "Security analysis of optical encryption," in Optics and Photonics for Defence and Security, E. M. Carapezza, ed., Proc. SPIE 5986, 25-34 (2005).

Chang, H. T.

H. T. Chang, W. C. Lu, and C. J. Kuo, "Multiple-phase retrieval for optical security systems by use of random-phase encoding," Appl. Opt. 41, 4815-4834 (2002).

Chatwin, C.

R. K. Wang, L. A. Watson, and C. Chatwin, "Random phase encoding for optical security," Opt. Eng. 35, 2464-2469 (1996).
[CrossRef]

Chen, L.

Dong, G. Y.

Famini, H.

Fienup, J. R.

Frauel, Y.

Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, "Security analysis of optical encryption," in Optics and Photonics for Defence and Security, E. M. Carapezza, ed., Proc. SPIE 5986, 25-34 (2005).

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction-plane pictures," Optik 35, 237-246 (1972).

Glückstad, J.

Gopinathan, U.

Gurkan, D.

Hayasaki, Y.

Hennelly, B.

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

Herzig, H. P.

H. P. Herzig, "Design of refractive and diffractive micro-optics," in Micro-Optics, H. P. Herzig, ed. (Taylor & Francis, 1996), 1-29.

Javidi, B.

B. Javidi, ed., Optical and Digital Techniques for Information Security (Springer Verlag, 2005).

Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, "Security analysis of optical encryption," in Optics and Photonics for Defence and Security, E. M. Carapezza, ed., Proc. SPIE 5986, 25-34 (2005).

S. Kishk and B. Javidi, "3D object watermarking by a 3D hidden object," Opt. Express 11, 874-888 (2003).

D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, "Security optical systems based on a joint transform correlator with significant output images," Opt. Eng. 40, 1584-1589 (2001).
[CrossRef]

J. Rosen and B. Javidi, "Hidden images in halftone pictures," Appl. Opt. 40, 3346-3353 (2001).

O. Matoba and B. Javidi, "Encrypted optical memory system using three-dimensional keys in the Fresnel domain," Opt. Lett. 24, 762-764 (1999).

P. Réfrégier and B. Javidi, "Optical image encryption based on input plane and Fourier plane random encoding," Opt. Lett. 20, 767-769 (1995).

Joseph, J.

Juvells, I.

Kim, E. S.

J. J. Kim, J. H. Kim, and E. S. Kim, "Optodigital implementation of multiple information hiding and extraction system," Opt. Eng. 43, 113-125 (2004).
[CrossRef]

Kim, J. H.

J. J. Kim, J. H. Kim, and E. S. Kim, "Optodigital implementation of multiple information hiding and extraction system," Opt. Eng. 43, 113-125 (2004).
[CrossRef]

Kim, J. J.

J. J. Kim, J. H. Kim, and E. S. Kim, "Optodigital implementation of multiple information hiding and extraction system," Opt. Eng. 43, 113-125 (2004).
[CrossRef]

Kishk, S.

Kreske, K.

Kuo, C. J.

H. T. Chang, W. C. Lu, and C. J. Kuo, "Multiple-phase retrieval for optical security systems by use of random-phase encoding," Appl. Opt. 41, 4815-4834 (2002).

Levi, A.

Li, Y.

Lu, W. C.

H. T. Chang, W. C. Lu, and C. J. Kuo, "Multiple-phase retrieval for optical security systems by use of random-phase encoding," Appl. Opt. 41, 4815-4834 (2002).

Matoba, O.

Matsuba, Y.

Meng, X. F.

Mifune, Y.

Mogensen, P. C.

Monaghan, D. S.

Montes-Usategui, M.

Nagaoka, A.

Naughton, T. J.

U. Gopinathan, D. S. Monaghan, T. J. Naughton, and J. T. Sheridan, "A known-plaintext heuristic attack on the Fourier plane encryption algorithm," Opt. Express 14, 3181-3186 (2006).
[CrossRef]

Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, "Security analysis of optical encryption," in Optics and Photonics for Defence and Security, E. M. Carapezza, ed., Proc. SPIE 5986, 25-34 (2005).

Nishida, N.

Ohtsubo, J.

M. Yamazaki and J. Ohtsubo, "Optimization of encrypted holograms in optical security systems," Opt. Eng. 40, 132-137 (2001).
[CrossRef]

Peng, X.

Réfrégier, P.

Rosen, J.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction-plane pictures," Optik 35, 237-246 (1972).

Shen, X. X.

Sheridan, J. T.

U. Gopinathan, D. S. Monaghan, T. J. Naughton, and J. T. Sheridan, "A known-plaintext heuristic attack on the Fourier plane encryption algorithm," Opt. Express 14, 3181-3186 (2006).
[CrossRef]

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

Singh, K.

Situ, G.

G. Situ and J. Zhang, "Phase-retrieval algorithms applied in a 4-f system for optical image encryption--a comparison," in Information Optics and Photonics Technology, G. Mu, F. T. S. Yu, and S. Jutamulia, eds., Proc. SPIE 5642, 108-115 (2005).
[CrossRef]

G. Situ and J. Zhang, "Image hiding with computer-generated phase codes for optical authentication," Opt. Commun. 245, 55-65 (2005).
[CrossRef]

G. Situ and J. Zhang, "A lensless optical security system based on computer-generated phase only masks," Opt. Commun. 232, 115-122 (2004).
[CrossRef]

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

G. Situ and J. Zhang, "A cascaded iterative Fourier transform algorithm for optical security applications," Optik 114, 473-477 (2003).

Stallings, W.

W. Stallings, Cryptography and Network Security, 3rd ed. (Prentice Hall, 2004).

Stark, H.

Takai, N.

Unnikrishnan, G.

Wang, R. K.

R. K. Wang, L. A. Watson, and C. Chatwin, "Random phase encoding for optical security," Opt. Eng. 35, 2464-2469 (1996).
[CrossRef]

Watson, L. A.

R. K. Wang, L. A. Watson, and C. Chatwin, "Random phase encoding for optical security," Opt. Eng. 35, 2464-2469 (1996).
[CrossRef]

Webb, H.

D. C. Youla and H. Webb, "Image restoration by the method of convex projections: part 1--theory," IEEE Trans. Med. Imaging MI-1, 81-94 (1982).

Wei, H.

Yamamoto, H.

Yamazaki, M.

M. Yamazaki and J. Ohtsubo, "Optimization of encrypted holograms in optical security systems," Opt. Eng. 40, 132-137 (2001).
[CrossRef]

Yang, X. L.

Yang, Y.

Youla, D. C.

D. C. Youla and H. Webb, "Image restoration by the method of convex projections: part 1--theory," IEEE Trans. Med. Imaging MI-1, 81-94 (1982).

Yu, B.

Zhang, J.

G. Situ and J. Zhang, "Image hiding with computer-generated phase codes for optical authentication," Opt. Commun. 245, 55-65 (2005).
[CrossRef]

G. Situ and J. Zhang, "Phase-retrieval algorithms applied in a 4-f system for optical image encryption--a comparison," in Information Optics and Photonics Technology, G. Mu, F. T. S. Yu, and S. Jutamulia, eds., Proc. SPIE 5642, 108-115 (2005).
[CrossRef]

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

G. Situ and J. Zhang, "A lensless optical security system based on computer-generated phase only masks," Opt. Commun. 232, 115-122 (2004).
[CrossRef]

G. Situ and J. Zhang, "A cascaded iterative Fourier transform algorithm for optical security applications," Optik 114, 473-477 (2003).

Zhang, P.

Zhao, D.

Appl. Opt. (8)

IEEE Trans. Med. Imaging (1)

D. C. Youla and H. Webb, "Image restoration by the method of convex projections: part 1--theory," IEEE Trans. Med. Imaging MI-1, 81-94 (1982).

J. Opt. Soc. Am. A (2)

Opt. Commun. (3)

G. Situ and J. Zhang, "Image hiding with computer-generated phase codes for optical authentication," Opt. Commun. 245, 55-65 (2005).
[CrossRef]

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

G. Situ and J. Zhang, "A lensless optical security system based on computer-generated phase only masks," Opt. Commun. 232, 115-122 (2004).
[CrossRef]

Opt. Eng. (4)

J. J. Kim, J. H. Kim, and E. S. Kim, "Optodigital implementation of multiple information hiding and extraction system," Opt. Eng. 43, 113-125 (2004).
[CrossRef]

R. K. Wang, L. A. Watson, and C. Chatwin, "Random phase encoding for optical security," Opt. Eng. 35, 2464-2469 (1996).
[CrossRef]

D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, "Security optical systems based on a joint transform correlator with significant output images," Opt. Eng. 40, 1584-1589 (2001).
[CrossRef]

M. Yamazaki and J. Ohtsubo, "Optimization of encrypted holograms in optical security systems," Opt. Eng. 40, 132-137 (2001).
[CrossRef]

Opt. Express (2)

Opt. Lett. (8)

Optik (2)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction-plane pictures," Optik 35, 237-246 (1972).

G. Situ and J. Zhang, "A cascaded iterative Fourier transform algorithm for optical security applications," Optik 114, 473-477 (2003).

Proc. SPIE (2)

G. Situ and J. Zhang, "Phase-retrieval algorithms applied in a 4-f system for optical image encryption--a comparison," in Information Optics and Photonics Technology, G. Mu, F. T. S. Yu, and S. Jutamulia, eds., Proc. SPIE 5642, 108-115 (2005).
[CrossRef]

Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, "Security analysis of optical encryption," in Optics and Photonics for Defence and Security, E. M. Carapezza, ed., Proc. SPIE 5986, 25-34 (2005).

Other (3)

B. Javidi, ed., Optical and Digital Techniques for Information Security (Springer Verlag, 2005).

H. P. Herzig, "Design of refractive and diffractive micro-optics," in Micro-Optics, H. P. Herzig, ed. (Taylor & Francis, 1996), 1-29.

W. Stallings, Cryptography and Network Security, 3rd ed. (Prentice Hall, 2004).

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Figures (8)

Fig. 1
Fig. 1

Flow chart of the attacker algorithm.

Fig. 2
Fig. 2

Original phase key (a) and the difference between the estimated and original 256-phase levels given (b) 4, (c) 8, and (d) 16 arbitrary input–output pairs.

Fig. 3
Fig. 3

Histograms of the distributions in Figs. 2(b)–2(d).

Fig. 4
Fig. 4

Decoded images with (a) K = 4 , (b) K = 4 , (c) K = 16 , and (d) Li and Rosen's algorithm [17].

Fig. 5
Fig. 5

Demonstration of using the cracked phase to obtain other correlation significant images previously encoded with the same ψ.

Fig. 6
Fig. 6

(a) Original binary phase key and the difference between the recovered and the original binary phase keys from (b) 4, (c) 8, and (d) 16 intensity measurements.

Fig. 7
Fig. 7

(Color online) Convergence behavior of the algorithm: phase of (a) binary and (b) 256 gray scale.

Fig. 8
Fig. 8

Decrypted images with the extracted phase key.

Equations (14)

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

g ( x 0 , y 0 ) = | g ( x 0 , y 0 ) | exp [ j α ( x 0 , y 0 ) ] = P 1 { | G ( x 1 , y 1 ) | exp [ j β ( x 1 , y 1 ) ] } ,
| c ( x , y ) | = | F 1 { F { exp [ j ϕ ( x , y ) ] } exp [ j ψ ( u , v ) ] } | ,
| c k ( x , y ) | = | 1 { { exp [ j ϕ k ( x , y ) ] } exp [ j ψ ( u , v ) ] } | ,
k = 1 , 2 ,   . . .   ,   K .
min { k = 1 K P { ϕ k , ψ } | c k | } ,
s k ( u , v ) = | s k ( u , v ) | exp [ j β k ( u , v ) ] = { exp [ j ϕ k ( x , y ) ] } ,
k = 1 , 2 ,   . . .   ,   K .
c ^ k ( n ) ( x , y ) = | c ^ k ( n ) ( x , y ) | exp [ j φ k ( n ) ( x , y ) ] = 1 { s k ( u , v ) exp [ j ψ k ( n ) ( u , v ) ] } .
s ^ k ( n ) ( u , v ) = | s ^ k ( n ) ( u , v ) | exp [ j β ^ k ( n ) ( u , v ) ] = { | c k ( x , y ) | exp [ j φ k ( n ) ( x , y ) ] } ,
ψ k + 1 ( n ) = β ^ k ( n ) β k .
ψ 1 ( n + 1 ) = ψ K + 1 ( n ) .
ψ ^ = ψ K + 1 ( N ) .
NMSE = l = 1 L m = 1 M [ c ^ ( m , l ) c ( m , l ) ] 2 l = 1 L m = 1 M [ c ( m , l ) ] 2
ψ ( n + 1 ) = P 1 P 2 . . . P I ψ ( n ) ,

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