Abstract

A novel optical image encryption scheme based on interference is proposed. The original image is digitally encoded into one phase-only mask by employing an improved Gerchberg-Saxton phase retrieval algorithm together with another predefined random phase mask which serves as the encryption key. The decryption process can be implemented optically based on Michelson interferometer by using the same key. The scheme can be regarded as a stream-cipher-like encryption system, the encryption and decryption keys are the same, however the operations are different. The position coordinates and light wavelength can also be used as additional keys during the decryption. Numerical simulations have demonstrated the validity and robustness of the proposed method.

© 2011 Optical Society of America

Full Article  |  PDF Article
More Like This
Image encryption by encoding with a nonuniform optical beam in gyrator transform domains

Zhengjun Liu, Lie Xu, Chuang Lin, and Shutian Liu
Appl. Opt. 49(29) 5632-5637 (2010)

Optical color image encryption based on a voice key under the framework of speckle-illuminated Fourier ptychography

Yonggang Su, Wenjun Xu, and Jie Zhao
OSA Continuum 3(11) 3267-3279 (2020)

Triple-image encryption based on phase-truncated Fresnel transform and basic vector operation

Xuemei Pan, Xiangfeng Meng, Xiulun Yang, Yurong Wang, Xiang Peng, Wenqi He, Guoyan Dong, and Hongyi Chen
Appl. Opt. 54(28) 8485-8493 (2015)

View More...

References

  • View by:
  • Article Order
  • Year
  • Author
  • Publication
  1. P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995).
    [Crossref] [PubMed]
  2. D. S. Monaghan, U. Gopinathan, T. J. Naughton, and J. T. Sheridan, “Key-space analysis of double random phase encryption technique,” Appl. Opt. 46, 6641–6647 (2007).
    [Crossref] [PubMed]
  3. B. Javidi, G. Zhang, and J. Li, “Encrypted optical memory using double-random phase encoding,” Opt. Eng. 36, 1054–1058 (1997).
  4. O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain” Opt. Lett. 24, 762–764 (1999).
    [Crossref]
  5. 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] [PubMed]
  6. 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] [PubMed]
  7. S. Liu, Q. Mi, and B. Zhu, “Optical image encryption with multistage and multichannel fractional Fourier-domain filtering,” Opt. Lett. 26, 1242–1244 (2001).
    [Crossref]
  8. Y. Zhang, CH. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202, 277–285 (2002).
    [Crossref]
  9. X. Wang and D. Zhao, “Image encryption based on anamorphic fractional Fourier transform and three-step phase-shifting interferometry,” Opt. Commun. 268, 240–244 (2006).
    [Crossref]
  10. Z. Liu and S. Liu, “Random fractional Fourier transform,” Opt. Lett. 32, 2088–2090 (2007).
    [Crossref] [PubMed]
  11. L. Chen and D. Zhao, “Optical image encryption with Hartley transforms,” Opt. Lett. 31, 3438–3440 (2006).
    [Crossref] [PubMed]
  12. H. Kim, D. H. Kim, and Y. Lee, “Encryption of digital hologram of 3-d object by virtual optics,” Opt. Express 12, 4912–4921 (2004).
    [Crossref] [PubMed]
  13. Z. Liu and S. Liu, “Randomization of the Fourier transform,” Opt. Lett. 32, 478–480 (2007).
    [Crossref] [PubMed]
  14. Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275, 324–329 (2007).
    [Crossref]
  15. G. Situ and J. Zhang, “Multiple-image encryption by wavelength multiplexing,” Opt. Lett. 30, 1306–1308 (2005).
    [Crossref] [PubMed]
  16. H. Li and Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281, 5745–5749 (2008).
    [Crossref]
  17. M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption and watermarking by random phase matching,” Opt. Commun. 247, 29–37 (2005).
    [Crossref]
  18. Z. Liu, J. Dai, X. Sun, and S. Liu, “Triple image encryption scheme in fractional Fourier transform domains,” Opt. Commun. 282, 518–522 (2009).
    [Crossref]
  19. Z. Liu, Q. Li, J. Dai, X. Sun, S. Liu, and M. A. Ahmad, “A new kind of double image encryption by using a cutting spectrum in the 1-D fractional Fourier transform domains,” Opt. Commun. 282, 1536–1540 (2009).
    [Crossref]
  20. Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33, 2443–2445 (2008).
    [Crossref] [PubMed]
  21. M. Madjarova, M. Kakuta, M. Yamaguchi, and N. Ohyama, “Optical implementation of the stream cipher based on the irreversible cellular automata algorithm,” Opt. Lett. 22, 1624–1626 (1997).
    [Crossref]
  22. J. W. Han, C. S. Park, D. H. Ryu, and E. S. Kim, “Optical image encryption based on XOR operations,” Opt. Eng. 38, 47–54 (1999).
    [Crossref]
  23. J. W. Han, S. Lee, and E. S. Kim, “Optical key bit stream generator,” Opt. Eng. 38, 33–38 (1999).
    [Crossref]
  24. S. Zhang and R. Karim, “High-security optical integrated stream ciphers,” Opt. Eng. 38, 20–24 (1999).
    [Crossref]
  25. T. Sasaki, H. Togo, J. Tanida, and Y. Ichioka, “Stream cipher based on pseudorandom number generation with optical affine transformation,” Appl. Opt. 39, 2340–2346 (2000).
    [Crossref]
  26. 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).
  27. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [Crossref] [PubMed]

2009 (2)

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

Z. Liu, Q. Li, J. Dai, X. Sun, S. Liu, and M. A. Ahmad, “A new kind of double image encryption by using a cutting spectrum in the 1-D fractional Fourier transform domains,” Opt. Commun. 282, 1536–1540 (2009).
[Crossref]

2008 (2)

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

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

2007 (4)

2006 (3)

2005 (3)

2004 (1)

2002 (1)

Y. Zhang, CH. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202, 277–285 (2002).
[Crossref]

2001 (1)

2000 (1)

1999 (4)

J. W. Han, C. S. Park, D. H. Ryu, and E. S. Kim, “Optical image encryption based on XOR operations,” Opt. Eng. 38, 47–54 (1999).
[Crossref]

J. W. Han, S. Lee, and E. S. Kim, “Optical key bit stream generator,” Opt. Eng. 38, 33–38 (1999).
[Crossref]

S. Zhang and R. Karim, “High-security optical integrated stream ciphers,” Opt. Eng. 38, 20–24 (1999).
[Crossref]

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

1997 (2)

1995 (1)

1982 (1)

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).

Ahmad, M. A.

Z. Liu, Q. Li, J. Dai, X. Sun, S. Liu, and M. A. Ahmad, “A new kind of double image encryption by using a cutting spectrum in the 1-D fractional Fourier transform domains,” Opt. Commun. 282, 1536–1540 (2009).
[Crossref]

Arcos, S.

Cai, L. Z.

M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption and watermarking by random phase matching,” Opt. Commun. 247, 29–37 (2005).
[Crossref]

Carnicer, A.

Chen, L.

Dai, J.

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

Z. Liu, Q. Li, J. Dai, X. Sun, S. Liu, and M. A. Ahmad, “A new kind of double image encryption by using a cutting spectrum in the 1-D fractional Fourier transform domains,” Opt. Commun. 282, 1536–1540 (2009).
[Crossref]

Fienup, J. R.

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).

Gopinathan, U.

Han, J. W.

J. W. Han, C. S. Park, D. H. Ryu, and E. S. Kim, “Optical image encryption based on XOR operations,” Opt. Eng. 38, 47–54 (1999).
[Crossref]

J. W. Han, S. Lee, and E. S. Kim, “Optical key bit stream generator,” Opt. Eng. 38, 33–38 (1999).
[Crossref]

He, M. Z.

M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption and watermarking by random phase matching,” Opt. Commun. 247, 29–37 (2005).
[Crossref]

Ichioka, Y.

Javidi, B.

Juvells, I.

Kakuta, M.

Karim, R.

S. Zhang and R. Karim, “High-security optical integrated stream ciphers,” Opt. Eng. 38, 20–24 (1999).
[Crossref]

Kim, D. H.

Kim, E. S.

J. W. Han, S. Lee, and E. S. Kim, “Optical key bit stream generator,” Opt. Eng. 38, 33–38 (1999).
[Crossref]

J. W. Han, C. S. Park, D. H. Ryu, and E. S. Kim, “Optical image encryption based on XOR operations,” Opt. Eng. 38, 47–54 (1999).
[Crossref]

Kim, H.

Lee, S.

J. W. Han, S. Lee, and E. S. Kim, “Optical key bit stream generator,” Opt. Eng. 38, 33–38 (1999).
[Crossref]

Lee, Y.

Li, H.

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

Li, J.

B. Javidi, G. Zhang, and J. Li, “Encrypted optical memory using double-random phase encoding,” Opt. Eng. 36, 1054–1058 (1997).

Li, Q.

Z. Liu, Q. Li, J. Dai, X. Sun, S. Liu, and M. A. Ahmad, “A new kind of double image encryption by using a cutting spectrum in the 1-D fractional Fourier transform domains,” Opt. Commun. 282, 1536–1540 (2009).
[Crossref]

Liu, Q.

M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption and watermarking by random phase matching,” Opt. Commun. 247, 29–37 (2005).
[Crossref]

Liu, S.

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

Z. Liu, Q. Li, J. Dai, X. Sun, S. Liu, and M. A. Ahmad, “A new kind of double image encryption by using a cutting spectrum in the 1-D fractional Fourier transform domains,” Opt. Commun. 282, 1536–1540 (2009).
[Crossref]

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

Z. Liu and S. Liu, “Randomization of the Fourier transform,” Opt. Lett. 32, 478–480 (2007).
[Crossref] [PubMed]

Z. Liu and S. Liu, “Random fractional Fourier transform,” Opt. Lett. 32, 2088–2090 (2007).
[Crossref] [PubMed]

S. Liu, Q. Mi, and B. Zhu, “Optical image encryption with multistage and multichannel fractional Fourier-domain filtering,” Opt. Lett. 26, 1242–1244 (2001).
[Crossref]

Liu, Z.

Z. Liu, Q. Li, J. Dai, X. Sun, S. Liu, and M. A. Ahmad, “A new kind of double image encryption by using a cutting spectrum in the 1-D fractional Fourier transform domains,” Opt. Commun. 282, 1536–1540 (2009).
[Crossref]

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

Z. Liu and S. Liu, “Random fractional Fourier transform,” Opt. Lett. 32, 2088–2090 (2007).
[Crossref] [PubMed]

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

Z. Liu and S. Liu, “Randomization of the Fourier transform,” Opt. Lett. 32, 478–480 (2007).
[Crossref] [PubMed]

Madjarova, M.

Matoba, O.

Meng, X. F.

M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption and watermarking by random phase matching,” Opt. Commun. 247, 29–37 (2005).
[Crossref]

Mi, Q.

Monaghan, D. S.

Montes-Usategui, M.

Naughton, T. J.

Ohyama, N.

Park, C. S.

J. W. Han, C. S. Park, D. H. Ryu, and E. S. Kim, “Optical image encryption based on XOR operations,” Opt. Eng. 38, 47–54 (1999).
[Crossref]

Peng, X.

Refregier, P.

Ryu, D. H.

J. W. Han, C. S. Park, D. H. Ryu, and E. S. Kim, “Optical image encryption based on XOR operations,” Opt. Eng. 38, 47–54 (1999).
[Crossref]

Sasaki, T.

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).

Sheridan, J. T.

Situ, G.

Sun, X.

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

Z. Liu, Q. Li, J. Dai, X. Sun, S. Liu, and M. A. Ahmad, “A new kind of double image encryption by using a cutting spectrum in the 1-D fractional Fourier transform domains,” Opt. Commun. 282, 1536–1540 (2009).
[Crossref]

Tanida, J.

Tanno, N.

Y. Zhang, CH. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202, 277–285 (2002).
[Crossref]

Togo, H.

Wang, B.

Wang, X.

X. Wang and D. Zhao, “Image encryption based on anamorphic fractional Fourier transform and three-step phase-shifting interferometry,” Opt. Commun. 268, 240–244 (2006).
[Crossref]

Wang, X. C.

M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption and watermarking by random phase matching,” Opt. Commun. 247, 29–37 (2005).
[Crossref]

Wang, Y.

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

Wei, H.

Yamaguchi, M.

Yu, B.

Zhang, G.

B. Javidi, G. Zhang, and J. Li, “Encrypted optical memory using double-random phase encoding,” Opt. Eng. 36, 1054–1058 (1997).

Zhang, J.

Zhang, P.

Zhang, S.

S. Zhang and R. Karim, “High-security optical integrated stream ciphers,” Opt. Eng. 38, 20–24 (1999).
[Crossref]

Zhang, Y.

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

Y. Zhang, CH. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202, 277–285 (2002).
[Crossref]

Zhao, D.

X. Wang and D. Zhao, “Image encryption based on anamorphic fractional Fourier transform and three-step phase-shifting interferometry,” Opt. Commun. 268, 240–244 (2006).
[Crossref]

L. Chen and D. Zhao, “Optical image encryption with Hartley transforms,” Opt. Lett. 31, 3438–3440 (2006).
[Crossref] [PubMed]

Zheng, CH.

Y. Zhang, CH. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202, 277–285 (2002).
[Crossref]

Zhu, B.

Appl. Opt. (3)

Opt. Commun. (7)

Y. Zhang, CH. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202, 277–285 (2002).
[Crossref]

X. Wang and D. Zhao, “Image encryption based on anamorphic fractional Fourier transform and three-step phase-shifting interferometry,” Opt. Commun. 268, 240–244 (2006).
[Crossref]

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

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

M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption and watermarking by random phase matching,” Opt. Commun. 247, 29–37 (2005).
[Crossref]

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

Z. Liu, Q. Li, J. Dai, X. Sun, S. Liu, and M. A. Ahmad, “A new kind of double image encryption by using a cutting spectrum in the 1-D fractional Fourier transform domains,” Opt. Commun. 282, 1536–1540 (2009).
[Crossref]

Opt. Eng. (4)

B. Javidi, G. Zhang, and J. Li, “Encrypted optical memory using double-random phase encoding,” Opt. Eng. 36, 1054–1058 (1997).

J. W. Han, C. S. Park, D. H. Ryu, and E. S. Kim, “Optical image encryption based on XOR operations,” Opt. Eng. 38, 47–54 (1999).
[Crossref]

J. W. Han, S. Lee, and E. S. Kim, “Optical key bit stream generator,” Opt. Eng. 38, 33–38 (1999).
[Crossref]

S. Zhang and R. Karim, “High-security optical integrated stream ciphers,” Opt. Eng. 38, 20–24 (1999).
[Crossref]

Opt. Express (1)

Opt. Lett. (11)

Z. Liu and S. Liu, “Randomization of the Fourier transform,” Opt. Lett. 32, 478–480 (2007).
[Crossref] [PubMed]

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

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

M. Madjarova, M. Kakuta, M. Yamaguchi, and N. Ohyama, “Optical implementation of the stream cipher based on the irreversible cellular automata algorithm,” Opt. Lett. 22, 1624–1626 (1997).
[Crossref]

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

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] [PubMed]

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] [PubMed]

S. Liu, Q. Mi, and B. Zhu, “Optical image encryption with multistage and multichannel fractional Fourier-domain filtering,” Opt. Lett. 26, 1242–1244 (2001).
[Crossref]

Z. Liu and S. Liu, “Random fractional Fourier transform,” Opt. Lett. 32, 2088–2090 (2007).
[Crossref] [PubMed]

L. Chen and D. Zhao, “Optical image encryption with Hartley transforms,” Opt. Lett. 31, 3438–3440 (2006).
[Crossref] [PubMed]

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

Optik (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).

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)
Fig. 1
Fig. 1 The flowchart of the optical process of our algorithm. Two incident light beams, which are modulated by phase only masks POM1 and POM2 respectively, propagate to the output plane and interfere with each other to construct the original image.

Download Full Size | PPT Slide | PDF

Fig. 2
Fig. 2 Flowchart of the proposed phase retrieval algorithm.

Download Full Size | PPT Slide | PDF

Fig. 3
Fig. 3 Schematic illustration of the optical decryption setup.

Download Full Size | PPT Slide | PDF

Fig. 4
Fig. 4 Encryption results: (a) the primary image for encryption, (b) decryption result with right keys. The phase distributions of (c) POM1 and (d) POM2.

Download Full Size | PPT Slide | PDF

Fig. 5
Fig. 5 The NMSE curves of recovered images with the change of: (a) l2, (b) λ. The attached drawings is retrieved with parameters of a small value shifting when NMSE≈ 0.2 as the red lines indicates.

Download Full Size | PPT Slide | PDF

Fig. 6
Fig. 6 The noise-attack test: (a) dependence of NMSE on the change of noise intensity, the attach image is retrieved from POM2 with noise intensity of 15(dB), (b)dependence of NMSE on the change of missing data, the attach image is retrieved from POM2 when 1/16 of the mask POM2 is blocked.

Download Full Size | PPT Slide | PDF

Tables (1)

Tables Icon

Table 1 Convergence speed of the phase retrieval algorithm

View Table

Equations (9)

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

f ( x , y ) = I ( x , y ) .
f ( x , y ) exp [ i φ ( x , y ) ] = f 1 ( x , y ) exp [ i φ 1 ( x , y ) ] + f 2 ( x , y ) exp [ i φ 2 ( x , y ) ] = exp [ i ϕ 1 ( x , y ) ] ⊗ h ( x , y , l 1 ) + exp [ i ϕ 2 ( x , y ) ] ⊗ h ( x , y , l 2 ) .
h ( x , y , l ) = exp ( 2 i π l / λ ) i λ l exp [ i π λ l ( x 2 + y 2 ) ] ,
exp [ i ϕ 2 ( x 2 , y 2 ) ] = ℱ − 1 { ℱ { f 2 ( x , y ) exp [ i φ 2 ( x , y ) ] } ℱ { h ( x , y , l 2 ) } } ,
f 2 ( k ) exp [ i φ 2 ( k ) ] = f exp [ i φ ( k ) ] − f 1 exp [ i φ 1 ] ,
g ( k ) exp [ i ϕ 2 ( k ) ] = ℱ − 1 { ℱ { f 2 ( k ) exp [ i φ 2 ( k ) ] } ℱ { h ( x , y , l 2 ) } } ,
f 2 ( k + 1 ) exp [ i φ 2 ( k + 1 ) ] = exp [ i ϕ 2 ( k ) ] ⊗ h ( x , y , l 2 ) ,
f ( k + 1 ) exp [ i φ ( k + 1 ) ] = f 1 exp ( i φ 1 ) + f 2 ( k + 1 ) exp [ i φ 2 ( k + 1 ) ] .
NMSE = ∑ m = 1 N ∑ n = 1 N | | f ( k ) ( m , n ) | − | f ( m , n ) | | 2 ∑ m = 1 N ∑ n = 1 N | f ( m , n ) | 2 ,

Metrics