Abstract

Recently, we have demonstrated that information encryption in phase space offers security enhancement over the traditional encryption schemes operating in real space. However, there is also an important issue with this technique: increasing the cost for data transmitting and storage. To address this issue, here we investigate the problem of decryption using incomplete cyphertext. We show that the analytic solution under the traditional framework set the lower limit of decryption performance. More importantly, we demonstrate that one just needs a small amount of cyphertext to recover the plaintext signal faithfully using compressive sensing, meaning that the amount of data that needs to transmit and store can be significantly reduced. This leads to multiple information encryption so that we can use the system bandwidth more effectively. We also provide an optical experimental result to demonstrate the plaintext recovered in phase space.

© 2016 Optical Society of America

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References

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  1. J. Liu, X. Xu, Q. Wu, J. T. Sheridan, and G. Situ, “Information encryption in phase space,” Opt. Lett. 40, 859–862 (2015).
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  13. E. J. Candés and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Info. Theory 52, 5406–5425 (2006).
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  14. G. Situ and J. Zhang, “Multiple-image encryption by wavelength multiplexing,” Opt. Lett. 30, 1306–1308 (2005).
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  18. F. Mosso, J. Barrera, M. Tebaldi, and N. Bolognini, “All-optical encrypted movie,” Opt. Express 19, 5706–5712 (2011).
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    [Crossref] [PubMed]
  21. B. Deepan, C. Quan, Y. Wang, and C. J. Tay., “Multiple-image encryption by space multiplexing based on compressive sensing and the double-random phase-encoding technique,” Appl. Opt. 53, 4539–4547 (2014).
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  22. Y. Wan, F. Wu, J. Yang, and T. Man, “Multiple-image encryption based on compressive holography using a multiple-beam interferometer,” Opt. Commun. 342, 95–101 (2015).
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  23. W. Liu, Z. Xie, Z. Liu, Y. Zhang, and S. Liu, “Multiple-image encryption based on optical asymmetric key cryptosystem”, Opt. Commun. 335, 205–211 (2015).
    [Crossref]
  24. M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-space Optics (McGraw-Hill, 2009).
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    [Crossref]
  26. G. Situ, L. Waller, and J. W. Fleische, “Experimental observation of 4D Wigner and Ambiguity distribution functions,” in Digital Holography & 3D Imaging, paper DTu3C.5 (Optical Society of America, 2012).
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    [Crossref]
  29. X. Zhou, S. Yuan, S. Wang, and J. Xie, “Affine cryptosystem of double-random-phase encryption based on the fractional fourier transform,” Appl. Opt. 45, 8434–8439 (2006).
    [Crossref]
  30. J. Lang and J. Zhang., “Optical image cryptosystem using chaotic phase-amplitude masks encoding and least-data-driven decryption by compressive sensing,” Opt. Commun. 338, 45–53 (2014).
    [Crossref]
  31. N. Zhou, H. Li, D. Wang, S. Pan, and Z. Zhou, “Image compression and encryption scheme based on 2D compressive sensing and fractional Mellin transform,” Opt. Commun. 343, 10–21 (2015).
    [Crossref]
  32. X. Liu, W. Mei, and H. Du, “Optical image encryption based on compressive sensing and chaos in the fractional Fourier domain,” J. Mod. Opt. 61, 1570–1577 (2014).
    [Crossref]
  33. J. Tropp and A. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inform. Theory 53, 4655–4666 (2007).
    [Crossref]
  34. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
    [Crossref] [PubMed]
  35. M. Liao, W. He, J. Wu, D. Lu, X. Liu, and X. Peng, “Optical authentication based on moire effect of nonlinear gratings in phase space,” J. Opt. 17, 125704 (2015).
    [Crossref]

2015 (5)

J. Liu, X. Xu, Q. Wu, J. T. Sheridan, and G. Situ, “Information encryption in phase space,” Opt. Lett. 40, 859–862 (2015).
[Crossref] [PubMed]

Y. Wan, F. Wu, J. Yang, and T. Man, “Multiple-image encryption based on compressive holography using a multiple-beam interferometer,” Opt. Commun. 342, 95–101 (2015).
[Crossref]

W. Liu, Z. Xie, Z. Liu, Y. Zhang, and S. Liu, “Multiple-image encryption based on optical asymmetric key cryptosystem”, Opt. Commun. 335, 205–211 (2015).
[Crossref]

N. Zhou, H. Li, D. Wang, S. Pan, and Z. Zhou, “Image compression and encryption scheme based on 2D compressive sensing and fractional Mellin transform,” Opt. Commun. 343, 10–21 (2015).
[Crossref]

M. Liao, W. He, J. Wu, D. Lu, X. Liu, and X. Peng, “Optical authentication based on moire effect of nonlinear gratings in phase space,” J. Opt. 17, 125704 (2015).
[Crossref]

2014 (3)

J. Lang and J. Zhang., “Optical image cryptosystem using chaotic phase-amplitude masks encoding and least-data-driven decryption by compressive sensing,” Opt. Commun. 338, 45–53 (2014).
[Crossref]

X. Liu, W. Mei, and H. Du, “Optical image encryption based on compressive sensing and chaos in the fractional Fourier domain,” J. Mod. Opt. 61, 1570–1577 (2014).
[Crossref]

B. Deepan, C. Quan, Y. Wang, and C. J. Tay., “Multiple-image encryption by space multiplexing based on compressive sensing and the double-random phase-encoding technique,” Appl. Opt. 53, 4539–4547 (2014).
[Crossref] [PubMed]

2012 (3)

J. Barrera, M. Tebaldi, C. Rios, E. Rueda, N. Bolognini, and R. Torroba, “Experimental multiplexing of encrypted movies using a JTC architecture,” Opt.Express 20, 3388–3393 (2012).

H. Di, K. Zheng, X. Zhang, E. Y. Lam, T. Kim, Y. S. Kim, T. C. Poon, and C. Zhou, “Multiple-image encryption by compressive holography,” Appl. Opt. 51, 1000–1009 (2012).
[Crossref] [PubMed]

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nature Photonics 6, 474–479 (2012).
[Crossref]

2011 (1)

2010 (2)

2009 (2)

D. S. Monaghan, U. Gopinathan, D. P. Kelly, T. J. Naughton, and J. T. Sheridan, “Systematic errors of an optical encryption system due to the discrete values of a spatial light modulator,” Opt. Eng. 48, 027001 (2009).
[Crossref]

D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
[Crossref] [PubMed]

2008 (1)

2007 (3)

2006 (5)

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]

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

E. J. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Info. Theory 52, 489–509 (2006)
[Crossref]

E. J. Candés and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Info. Theory 52, 5406–5425 (2006).
[Crossref]

X. Zhou, S. Yuan, S. Wang, and J. Xie, “Affine cryptosystem of double-random-phase encryption based on the fractional fourier transform,” Appl. Opt. 45, 8434–8439 (2006).
[Crossref]

2005 (2)

2003 (1)

2001 (1)

B. Wang and C. C. Sun, “Enhancement of signal-to-noise ratio of a double random phase encoding encryption system,” Opt. Eng. 40, 1502–1506 (2001).
[Crossref]

2000 (1)

1999 (1)

1997 (1)

B. Javidi, A. Sergent, G. Zhang, and L. Guibert, “Fault tolerance properties of a double phase encoding encryption technique,” Opt. Eng. 36, 992–998 (1997).
[Crossref]

1995 (1)

Alfalou, A.

Arcos, S.

Barrera, J.

Bolognini, N.

J. Barrera, M. Tebaldi, C. Rios, E. Rueda, N. Bolognini, and R. Torroba, “Experimental multiplexing of encrypted movies using a JTC architecture,” Opt.Express 20, 3388–3393 (2012).

F. Mosso, J. Barrera, M. Tebaldi, and N. Bolognini, “All-optical encrypted movie,” Opt. Express 19, 5706–5712 (2011).
[PubMed]

Brady, D. J.

Brosseau, C.

Candés, E. J.

E. J. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Info. Theory 52, 489–509 (2006)
[Crossref]

E. J. Candés and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Info. Theory 52, 5406–5425 (2006).
[Crossref]

Carnicer, A.

Castro, A.

Choi, K.

Deepan, B.

Di, H.

Dragoman, D.

Du, H.

X. Liu, W. Mei, and H. Du, “Optical image encryption based on compressive sensing and chaos in the fractional Fourier domain,” J. Mod. Opt. 61, 1570–1577 (2014).
[Crossref]

Fleische, J. W.

G. Situ, L. Waller, and J. W. Fleische, “Experimental observation of 4D Wigner and Ambiguity distribution functions,” in Digital Holography & 3D Imaging, paper DTu3C.5 (Optical Society of America, 2012).
[Crossref]

Fleischer, J. W.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nature Photonics 6, 474–479 (2012).
[Crossref]

Frauel, Y.

Gilbert, A.

J. Tropp and A. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inform. Theory 53, 4655–4666 (2007).
[Crossref]

Gopinathan, U.

D. S. Monaghan, U. Gopinathan, D. P. Kelly, T. J. Naughton, and J. T. Sheridan, “Systematic errors of an optical encryption system due to the discrete values of a spatial light modulator,” Opt. Eng. 48, 027001 (2009).
[Crossref]

G. Situ, U. Gopinathan, D. S. Monaghan, and J. T. Sheridan, “Cryptanalysis of optical security systems with significant output images,” Appl. Opt. 46, 5257–5262 (2007).
[Crossref] [PubMed]

Guibert, L.

B. Javidi, A. Sergent, G. Zhang, and L. Guibert, “Fault tolerance properties of a double phase encoding encryption technique,” Opt. Eng. 36, 992–998 (1997).
[Crossref]

He, W.

M. Liao, W. He, J. Wu, D. Lu, X. Liu, and X. Peng, “Optical authentication based on moire effect of nonlinear gratings in phase space,” J. Opt. 17, 125704 (2015).
[Crossref]

Henao, R.

Hennelly, B.

M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-space Optics (McGraw-Hill, 2009).

Horisaki, R.

Javidi, B.

Juvells, I.

Kelly, D. P.

D. S. Monaghan, U. Gopinathan, D. P. Kelly, T. J. Naughton, and J. T. Sheridan, “Systematic errors of an optical encryption system due to the discrete values of a spatial light modulator,” Opt. Eng. 48, 027001 (2009).
[Crossref]

Kim, T.

Kim, Y. S.

Kuroda, K.

Lam, E. Y.

Lang, J.

J. Lang and J. Zhang., “Optical image cryptosystem using chaotic phase-amplitude masks encoding and least-data-driven decryption by compressive sensing,” Opt. Commun. 338, 45–53 (2014).
[Crossref]

Li, H.

N. Zhou, H. Li, D. Wang, S. Pan, and Z. Zhou, “Image compression and encryption scheme based on 2D compressive sensing and fractional Mellin transform,” Opt. Commun. 343, 10–21 (2015).
[Crossref]

Liao, M.

M. Liao, W. He, J. Wu, D. Lu, X. Liu, and X. Peng, “Optical authentication based on moire effect of nonlinear gratings in phase space,” J. Opt. 17, 125704 (2015).
[Crossref]

Lim, S.

Liu, J.

Liu, S.

W. Liu, Z. Xie, Z. Liu, Y. Zhang, and S. Liu, “Multiple-image encryption based on optical asymmetric key cryptosystem”, Opt. Commun. 335, 205–211 (2015).
[Crossref]

Liu, W.

W. Liu, Z. Xie, Z. Liu, Y. Zhang, and S. Liu, “Multiple-image encryption based on optical asymmetric key cryptosystem”, Opt. Commun. 335, 205–211 (2015).
[Crossref]

Liu, X.

M. Liao, W. He, J. Wu, D. Lu, X. Liu, and X. Peng, “Optical authentication based on moire effect of nonlinear gratings in phase space,” J. Opt. 17, 125704 (2015).
[Crossref]

X. Liu, W. Mei, and H. Du, “Optical image encryption based on compressive sensing and chaos in the fractional Fourier domain,” J. Mod. Opt. 61, 1570–1577 (2014).
[Crossref]

Liu, Z.

W. Liu, Z. Xie, Z. Liu, Y. Zhang, and S. Liu, “Multiple-image encryption based on optical asymmetric key cryptosystem”, Opt. Commun. 335, 205–211 (2015).
[Crossref]

Lu, D.

M. Liao, W. He, J. Wu, D. Lu, X. Liu, and X. Peng, “Optical authentication based on moire effect of nonlinear gratings in phase space,” J. Opt. 17, 125704 (2015).
[Crossref]

Man, T.

Y. Wan, F. Wu, J. Yang, and T. Man, “Multiple-image encryption based on compressive holography using a multiple-beam interferometer,” Opt. Commun. 342, 95–101 (2015).
[Crossref]

Marks, D. L.

Matoba, O.

Mei, W.

X. Liu, W. Mei, and H. Du, “Optical image encryption based on compressive sensing and chaos in the fractional Fourier domain,” J. Mod. Opt. 61, 1570–1577 (2014).
[Crossref]

Monaghan, D. S.

D. S. Monaghan, U. Gopinathan, D. P. Kelly, T. J. Naughton, and J. T. Sheridan, “Systematic errors of an optical encryption system due to the discrete values of a spatial light modulator,” Opt. Eng. 48, 027001 (2009).
[Crossref]

G. Situ, U. Gopinathan, D. S. Monaghan, and J. T. Sheridan, “Cryptanalysis of optical security systems with significant output images,” Appl. Opt. 46, 5257–5262 (2007).
[Crossref] [PubMed]

Montes-Usategui, M.

Mosk, A. P.

Mosso, F.

Naughton, T. J.

D. S. Monaghan, U. Gopinathan, D. P. Kelly, T. J. Naughton, and J. T. Sheridan, “Systematic errors of an optical encryption system due to the discrete values of a spatial light modulator,” Opt. Eng. 48, 027001 (2009).
[Crossref]

Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15, 10253–10265 (2007).
[Crossref] [PubMed]

Ojeda-Castaneda, J.

M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-space Optics (McGraw-Hill, 2009).

Pan, S.

N. Zhou, H. Li, D. Wang, S. Pan, and Z. Zhou, “Image compression and encryption scheme based on 2D compressive sensing and fractional Mellin transform,” Opt. Commun. 343, 10–21 (2015).
[Crossref]

Peng, X.

M. Liao, W. He, J. Wu, D. Lu, X. Liu, and X. Peng, “Optical authentication based on moire effect of nonlinear gratings in phase space,” J. Opt. 17, 125704 (2015).
[Crossref]

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]

Poon, T. C.

Quan, C.

Refregier, P.

Rios, C.

J. Barrera, M. Tebaldi, C. Rios, E. Rueda, N. Bolognini, and R. Torroba, “Experimental multiplexing of encrypted movies using a JTC architecture,” Opt.Express 20, 3388–3393 (2012).

Romberg, J.

E. J. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Info. Theory 52, 489–509 (2006)
[Crossref]

Rueda, E.

J. Barrera, M. Tebaldi, C. Rios, E. Rueda, N. Bolognini, and R. Torroba, “Experimental multiplexing of encrypted movies using a JTC architecture,” Opt.Express 20, 3388–3393 (2012).

R. Henao, E. Rueda, J. Barrera, and R. Torroba, “Noise-free recovery of optodigital encrypted and multiplexed images,” Opt. Lett. 35, 333–335 (2010).
[Crossref] [PubMed]

Sergent, A.

B. Javidi, A. Sergent, G. Zhang, and L. Guibert, “Fault tolerance properties of a double phase encoding encryption technique,” Opt. Eng. 36, 992–998 (1997).
[Crossref]

Sheridan, J. T.

Shimura, T.

Situ, G.

J. Liu, X. Xu, Q. Wu, J. T. Sheridan, and G. Situ, “Information encryption in phase space,” Opt. Lett. 40, 859–862 (2015).
[Crossref] [PubMed]

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nature Photonics 6, 474–479 (2012).
[Crossref]

G. Situ, U. Gopinathan, D. S. Monaghan, and J. T. Sheridan, “Cryptanalysis of optical security systems with significant output images,” Appl. Opt. 46, 5257–5262 (2007).
[Crossref] [PubMed]

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

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

G. Situ, L. Waller, and J. W. Fleische, “Experimental observation of 4D Wigner and Ambiguity distribution functions,” in Digital Holography & 3D Imaging, paper DTu3C.5 (Optical Society of America, 2012).
[Crossref]

Sun, C. C.

B. Wang and C. C. Sun, “Enhancement of signal-to-noise ratio of a double random phase encoding encryption system,” Opt. Eng. 40, 1502–1506 (2001).
[Crossref]

Tan, X.

Tao, T.

E. J. Candés, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Info. Theory 52, 489–509 (2006)
[Crossref]

E. J. Candés and T. Tao, “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Info. Theory 52, 5406–5425 (2006).
[Crossref]

Tay., C. J.

Tebaldi, M.

J. Barrera, M. Tebaldi, C. Rios, E. Rueda, N. Bolognini, and R. Torroba, “Experimental multiplexing of encrypted movies using a JTC architecture,” Opt.Express 20, 3388–3393 (2012).

F. Mosso, J. Barrera, M. Tebaldi, and N. Bolognini, “All-optical encrypted movie,” Opt. Express 19, 5706–5712 (2011).
[PubMed]

Testorf, M.

M. Testorf, B. Hennelly, and J. Ojeda-Castaneda, Phase-space Optics (McGraw-Hill, 2009).

Torroba, R.

J. Barrera, M. Tebaldi, C. Rios, E. Rueda, N. Bolognini, and R. Torroba, “Experimental multiplexing of encrypted movies using a JTC architecture,” Opt.Express 20, 3388–3393 (2012).

R. Henao, E. Rueda, J. Barrera, and R. Torroba, “Noise-free recovery of optodigital encrypted and multiplexed images,” Opt. Lett. 35, 333–335 (2010).
[Crossref] [PubMed]

Tropp, J.

J. Tropp and A. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inform. Theory 53, 4655–4666 (2007).
[Crossref]

van Putten, E. G.

Vellekoop, I. M.

Waller, L.

L. Waller, G. Situ, and J. W. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nature Photonics 6, 474–479 (2012).
[Crossref]

G. Situ, L. Waller, and J. W. Fleische, “Experimental observation of 4D Wigner and Ambiguity distribution functions,” in Digital Holography & 3D Imaging, paper DTu3C.5 (Optical Society of America, 2012).
[Crossref]

Wan, Y.

Y. Wan, F. Wu, J. Yang, and T. Man, “Multiple-image encryption based on compressive holography using a multiple-beam interferometer,” Opt. Commun. 342, 95–101 (2015).
[Crossref]

Wang, B.

B. Wang and C. C. Sun, “Enhancement of signal-to-noise ratio of a double random phase encoding encryption system,” Opt. Eng. 40, 1502–1506 (2001).
[Crossref]

Wang, D.

N. Zhou, H. Li, D. Wang, S. Pan, and Z. Zhou, “Image compression and encryption scheme based on 2D compressive sensing and fractional Mellin transform,” Opt. Commun. 343, 10–21 (2015).
[Crossref]

Wang, S.

Wang, Y.

Wei, H.

Wu, F.

Y. Wan, F. Wu, J. Yang, and T. Man, “Multiple-image encryption based on compressive holography using a multiple-beam interferometer,” Opt. Commun. 342, 95–101 (2015).
[Crossref]

Wu, J.

M. Liao, W. He, J. Wu, D. Lu, X. Liu, and X. Peng, “Optical authentication based on moire effect of nonlinear gratings in phase space,” J. Opt. 17, 125704 (2015).
[Crossref]

Wu, Q.

Xie, J.

Xie, Z.

W. Liu, Z. Xie, Z. Liu, Y. Zhang, and S. Liu, “Multiple-image encryption based on optical asymmetric key cryptosystem”, Opt. Commun. 335, 205–211 (2015).
[Crossref]

Xu, X.

Yang, J.

Y. Wan, F. Wu, J. Yang, and T. Man, “Multiple-image encryption based on compressive holography using a multiple-beam interferometer,” Opt. Commun. 342, 95–101 (2015).
[Crossref]

Yu, B.

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Figures (7)
Fig. 1
Fig. 1 The 1 × 64 signal used as plaintext in our simulations.

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Fig. 2
Fig. 2 Experimental demonstration of signal recovered from phase space.

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Fig. 3
Fig. 3 Decryption using part of the cyphertext. One can see that noise appears when the cyphertext is slightly incomplete, meaning that the performance of traditional decryption approach is very sensitive to the completeness of the cyphertext.

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Fig. 4
Fig. 4 Recovery of the plaintext signal from part of the cyphertext : (a) the cyphertext and the part that used in decryption (approximately half of the left-most part in the red rectangle), (b) the recovered plaintext using the OMP algorithm and its comparison with the original signal.

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Fig. 5
Fig. 5 The NRMS value of the decrypted plaintext as a function of the percentage of the cyphertext that is used for decryption. It is clearly seen that that the performance of CS-based decryption approach is insensitive to the completeness of the cyphertext.

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Fig. 6
Fig. 6 Successful rate of reconstruction. Markers and red line represent the numerical and fitting results, respectively.

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Fig. 7
Fig. 7 NRMS values of the recovered plaintext from the cyphertext contains multiple signals. It is clearly shown that the decryption performance is dependent on both the number of synthesized Ambiguity function, and the number of signals in it.

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Equations (26)

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𝒜 f ( u ¯ , x ¯ ) = ∫ f ( x + x ¯ 2 ) f * ( x − x ¯ 2 ) exp [ − j 2 π u ¯ x ] d x ,
g ( u ¯ , x ¯ ) = ℱ { ℱ { 𝒜 f ( u ¯ , x ¯ ) exp [ j ϕ ( u ¯ , x ¯ ) ] } exp [ j ψ ( ξ , η ) ] } ,
𝒜 f ( u ¯ , x ¯ ) = ℱ { ℱ { g ( u ¯ , x ¯ ) } exp [ − j ψ ( ξ , η ) ] } exp [ − j ϕ ( u ¯ , x ¯ ) ] .
I ^ f ( v ) = 𝒜 f ( u ¯ , x ¯ ) | x ¯ = 0 , u ¯ = v ,
| f ( x ) | 2 = ∫ I ^ f ( v ) exp [ j 2 π v x ] d v .
NRMS = [ ∫ | | f ( x ) | 2 − | f ˜ ( x ) | 2 | 2 d x ∫ | f ( x ) | 4 d x ] 1 2 ,
g ˜ ( u ¯ , x ¯ ) = g ( u ¯ , x ¯ ) rect ( u ¯ Ω , x ¯ Ω ) ,
𝒜 ˜ f ( u ¯ , x ¯ ) = Ω 2 𝒜 f ( u ¯ , x ¯ ) + ℛ ( u ¯ , x ¯ ) ,
| f ˜ ( x ) | 2 = Ω 2 | f ( x ) | 2 + r ( x ) ,
E r = ∫ r ( x ) d x = a E f 2 ( Ω − Ω 2 ) ,
NRMS = ( a − 1 ) Ω 2 − a Ω + 1 .
g m , n = ℱ { ℱ { 𝒜 m ′ , n ′ exp [ j ϕ m ′ , n ′ ] } exp [ j ψ μ , ν ] } ,
E = BPBR A ,
E s = S E = SBPBR A = Φ A ,
A ^ = min ‖ A ‖ 0 ,                   s . t .             E s = Φ A .
𝒜 ( u ¯ , n ) = ∑ n = 1 N 𝒜 f n ( u ¯ , 0 ) ,
C = max { X Y : NRMS ( X , Y ) ≤ τ } ,
ℱ { g ˜ ( u ¯ , x ¯ ) } = ℱ { g ( u ¯ , x ¯ ) rect ( u ¯ Ω , x ¯ Ω ) } = ℱ { 𝒜 f ( u ¯ , x ¯ ) exp [ j ϕ ( u ¯ , x ¯ ) ] } exp [ j ψ ( ξ , η ) ] ⊗ ℱ { rect ( u ¯ Ω , x ¯ Ω ) } = Ω 2 ℱ { 𝒜 f ( u ¯ , x ¯ ) exp [ j ϕ ( u ¯ , x ¯ ) ] } exp [ j ψ ( ξ , η ) ] sinc ( Ω ξ ) sinc ( Ω η ) = ∬ Ω 2 𝒜 f ( u ¯ , x ¯ ) exp [ j ϕ ( u ¯ , x ¯ ) ] d u ¯ d x ¯ ∬ exp [ − j 2 π ( u ¯ ξ + x ¯ η ) ] × exp [ j ψ ( ξ , η ) ] sinc [ Ω ( ξ − μ ) ] sinc [ Ω ( η − ν ) ] d μ d ν .
∬ exp [ − j 2 π ( u ¯ ξ + x ¯ η ) ] exp [ j ψ ( ξ , η ) ] sinc [ Ω ( ξ − μ ) ] sinc [ Ω ( η − ν ) ] d μ d ν = exp [ − j 2 π ( u ¯ ξ + x ¯ η ) ] exp [ j ψ ( ξ , η ) ] + ∫ μ ≠ ξ ∫ ν ≠ η exp [ − j 2 π ( u ¯ ξ + x ¯ η ) ] × exp [ j ψ ( ξ , η ) ] sinc [ Ω ( ξ − μ ) ] sinc [ Ω ( η − ν ) ] d μ d ν .
ℱ { g ˜ ( u ¯ , x ¯ ) } = ℱ { 𝒜 f ( u ¯ , x ¯ ) exp [ j ϕ ( u ¯ , x ¯ ) ] } exp [ j ψ ( ξ , η ) ] + ∬ Ω 2 𝒜 f ( u ¯ , x ¯ ) exp [ j ϕ ( u ¯ , x ¯ ) ] d u ¯ d x ¯ ⋅ exp [ j ψ ( ξ , η ) ] × ∫ μ ≠ ξ ∫ ν ≠ η exp [ − j 2 π ( u ¯ ξ + x ¯ η ) ] sinc [ Ω ( ξ − μ ) ] sinc [ Ω ( η − ν ) ] d μ d ν = ℱ { 𝒜 f ( u ¯ , x ¯ ) exp [ j ϕ ( u ¯ , x ¯ ) ] } exp [ j ψ ( ξ , η ) ] + R ( ξ , η ) .
𝒜 ˜ f ( u ¯ , x ¯ ) = ℱ { ℱ { g ˜ ( u ¯ , x ¯ ) } exp [ − j ψ ( ξ , η ) ] } exp [ − j ϕ ( u ¯ , x ¯ ) ] Ω 2 𝒜 f ( u ¯ , x ¯ ) + ℛ ( u ¯ , x ¯ ) ,
ℛ ( u ¯ , x ¯ ) = ℱ { R ( ξ , η ) exp [ − j ψ ( ξ , η ) ] } exp [ − j ϕ ( u ¯ , x ¯ ) ] .
∫ ( | f ( x ) | 2 − | f ˜ ( x ) | 2 ) 2 d x = ∫ | f ( x ) | 4 + | f ˜ ( x ) | 4 − 2 | f ( x ) | 2 | f ˜ ( x ) | 2 d ( x ) .
| f ( x ) | 4 + | f ˜ ( x ) | 4 − 2 | f ( x ) | 2 | f ˜ ( x ) | 2 = | f ( x ) | 4 + Ω 4 | f ( x ) | 4 + 2 Ω 2 | f ( x ) | 2 r ( x ) + r 2 ( x ) − 2 Ω 2 | f ( x ) | 4 − 2 | f ( x ) | 2 r ( x ) = | f ( x ) | 4 ( 1 − Ω 2 ) 2 + 2 Ω 2 | f ( x ) | 2 r ( x ) + r 2 ( x ) − 2 | f ( x ) | 2 r ( x ) .
∫ ( | f ( x ) | 2 − | f ˜ ( x ) | 2 ) 2 d x = ( 1 − Ω 2 ) 2 ∫ | f ( x ) | 4 d x + 2 a Ω 2 ( Ω − Ω 2 ) ∫ | f ( x ) | 4 d x + a 2 ( Ω − Ω 2 ) 2 ∫ | f ( x ) | 4 d x − 2 a ( Ω − Ω 2 ) ∫ | f ( x ) | 4 d x = [ ( 1 − Ω 2 ) − a ( Ω − Ω 2 ) ] 2 ∫ | f ( x ) | 4 d x .
NRMS = [ ∫ | | f ( x ) | 2 − | f ˜ ( x ) | 2 | 2 d x ∫ | f ( x ) | 4 d x ] 1 2 = ( 1 − Ω 2 ) − a ( Ω − Ω 2 ) = ( a − 1 ) Ω 2 − a Ω + 1 .

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