Abstract

A robust lensless optical data embedding system using both the computer-generated concealogram and the digital Fresnel hologram (DFH) is proposed. The concealogram and the DFH are cascaded as the input and filter planes, respectively. When the concealogram is illuminated by a plane wave, the hidden data can be extracted at the output plane without using any lenses. The longitudinal positions of the filter and the output planes, as well as the wavelength, can be used as the secret keys to enhance system security. Furthermore, the robustness of the proposed system against noise and distortion is also demonstrated.

© 2011 Optical Society of America

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    [CrossRef]
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2009 (1)

2006 (2)

S. Deng, L. Liu, H. Lang, D. Zhao, and X. Liu, “Cascaded Fresnel digital hologram and its application to watermarking,” Opt. Appl. 36, 413–420 (2006).

H. T. Chang and C. C. Chen, “Fully phase asymmetric image verification system based on joint transform correlator,” Opt. Express 14, 1458–1467 (2006).
[CrossRef] [PubMed]

2005 (3)

2004 (9)

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] [PubMed]

O. Ripoll, V. Kettunen, and H. P. Herzig, “Review of iterative Fourier-transform algorithms for beam shaping applications,” Opt. Eng. 43, 2549–2556 (2004).
[CrossRef]

H. T. Chang and C. T. Chen, “Asymmetric-image verification for security optical system based on joint transform correlator,” Opt. Commun. 239, 43–54 (2004).
[CrossRef]

H. T. Chang and C. T. Chen, “Enhanced optical image verification based on joint transform correlator adopting Fourier hologram,” Opt. Rev. 11, 165–169 (2004).
[CrossRef]

Y. C. Chang, H. T. Chang, and C. J. Kuo, “Hybrid image cryptosystem based on dyadic phase displacement in the Fourier domain,” Opt. Commun. 236, 245–257 (2004).
[CrossRef]

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase encryption using digital holography,” Opt. Eng. 43, 2959–2966(2004).
[CrossRef]

H. Kim, D.-H. Kim, and Y. H. Lee, “Encryption of digital hologram of 3-D object by virtue optics,” Opt. Express 12, 4912–4921(2004).
[CrossRef] [PubMed]

X. Peng, P. Zhang, and L. Cai, “Information security system based on virtual-optics imaging methodology and public key infrastructure,” Optik 115, 420–426 (2004).
[CrossRef]

2003 (2)

G. Situ and J. Zhang, “A cascaded iterative Fourier transform algorithm for optical security applications,” Optik 114, 473–477(2003).
[CrossRef]

G. S. Lin, H. T. Chang, W. N. Lie, and C. H. Chuang, “A public-key-based optical image cryptosystem based on data embedding techniques,” Opt. Eng. 42, 2331–2339 (2003).
[CrossRef]

2002 (3)

2001 (3)

H. T. Chang, “Optical image encryption using separate amplitude-based virtual image and iteratively-retrieved phase information,” Opt. Eng. 40, 2165–2171 (2001).
[CrossRef]

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

J. Rosen and B. Javidi, “Hidden images in halftone pictures,” Appl. Opt. 40, 3346–3353 (2001).
[CrossRef]

2000 (4)

1999 (4)

1997 (1)

1995 (1)

1987 (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).

1971 (1)

R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

1966 (1)

Abookasis, D.

D. Abookasis, O. Montal, O. Abramson, and J. Rosen, “Watermarks encrypted in a concealogram and deciphered by a modified joint transform correlator,” Appl. Opt. 44, 3019–3023 (2005).
[CrossRef] [PubMed]

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

Abramson, O.

D. Abookasis, O. Montal, O. Abramson, and J. Rosen, “Watermarks encrypted in a concealogram and deciphered by a modified joint transform correlator,” Appl. Opt. 44, 3019–3023 (2005).
[CrossRef] [PubMed]

Arazi, O.

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

Bernardo, L. M.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithm for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Brown, B. R.

Cai, L.

X. Peng, P. Zhang, and L. Cai, “Information security system based on virtual-optics imaging methodology and public key infrastructure,” Optik 115, 420–426 (2004).
[CrossRef]

X. Peng, L. Yu, and L. Cai, “Double-lock for image encryption with virtual optical wavelength,” Opt. Express 10, 41–45(2002).
[PubMed]

Chang, H. T.

H. E. Hwang, H. T. Chang, and W. N. Lie, “Fast double-phase retrieval in Fresnel domain using modified Gerchberg-Saxton algorithm for lensless optical security systems,” Opt. Express 17, 13700–13710 (2009).
[CrossRef] [PubMed]

H. T. Chang and C. C. Chen, “Fully phase asymmetric image verification system based on joint transform correlator,” Opt. Express 14, 1458–1467 (2006).
[CrossRef] [PubMed]

H. T. Chang and C. L. Tsan, “Image watermarking by use of digital holography embedded in DCT domain,” Appl. Opt. 44, 6211–6219 (2005).
[CrossRef] [PubMed]

Y. C. Chang, H. T. Chang, and C. J. Kuo, “Hybrid image cryptosystem based on dyadic phase displacement in the Fourier domain,” Opt. Commun. 236, 245–257 (2004).
[CrossRef]

H. T. Chang and C. T. Chen, “Asymmetric-image verification for security optical system based on joint transform correlator,” Opt. Commun. 239, 43–54 (2004).
[CrossRef]

H. T. Chang and C. T. Chen, “Enhanced optical image verification based on joint transform correlator adopting Fourier hologram,” Opt. Rev. 11, 165–169 (2004).
[CrossRef]

G. S. Lin, H. T. Chang, W. N. Lie, and C. H. Chuang, “A public-key-based optical image cryptosystem based on data embedding techniques,” Opt. Eng. 42, 2331–2339 (2003).
[CrossRef]

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, 4825–4834 (2002).
[CrossRef] [PubMed]

H. T. Chang, “Optical image encryption using separate amplitude-based virtual image and iteratively-retrieved phase information,” Opt. Eng. 40, 2165–2171 (2001).
[CrossRef]

Chang, Y. C.

Y. C. Chang, H. T. Chang, and C. J. Kuo, “Hybrid image cryptosystem based on dyadic phase displacement in the Fourier domain,” Opt. Commun. 236, 245–257 (2004).
[CrossRef]

Chen, C. C.

Chen, C. T.

H. T. Chang and C. T. Chen, “Enhanced optical image verification based on joint transform correlator adopting Fourier hologram,” Opt. Rev. 11, 165–169 (2004).
[CrossRef]

H. T. Chang and C. T. Chen, “Asymmetric-image verification for security optical system based on joint transform correlator,” Opt. Commun. 239, 43–54 (2004).
[CrossRef]

Chen, N. X.

Chuang, C. H.

G. S. Lin, H. T. Chang, W. N. Lie, and C. H. Chuang, “A public-key-based optical image cryptosystem based on data embedding techniques,” Opt. Eng. 42, 2331–2339 (2003).
[CrossRef]

Cong, W. X.

Deng, S.

S. Deng, L. Liu, H. Lang, D. Zhao, and X. Liu, “Cascaded Fresnel digital hologram and its application to watermarking,” Opt. Appl. 36, 413–420 (2006).

Ferreira, C.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithm for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Garcia, J.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithm for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

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).

R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

Glückstad, J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 355–358.

Gu, B. Y.

Herzig, H. P.

O. Ripoll, V. Kettunen, and H. P. Herzig, “Review of iterative Fourier-transform algorithms for beam shaping applications,” Opt. Eng. 43, 2549–2556 (2004).
[CrossRef]

Hwang, H. E.

Javidi, B.

Joseph, J.

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase encryption using digital holography,” Opt. Eng. 43, 2959–2966(2004).
[CrossRef]

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).
[CrossRef]

Kettunen, V.

O. Ripoll, V. Kettunen, and H. P. Herzig, “Review of iterative Fourier-transform algorithms for beam shaping applications,” Opt. Eng. 43, 2549–2556 (2004).
[CrossRef]

Kim, D.-H.

Kim, H.

Kishk, S.

Kreske, K.

Kuo, C. J.

Y. C. Chang, H. T. Chang, and C. J. Kuo, “Hybrid image cryptosystem based on dyadic phase displacement in the Fourier domain,” Opt. Commun. 236, 245–257 (2004).
[CrossRef]

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, 4825–4834 (2002).
[CrossRef] [PubMed]

Kuroda, K.

Lang, H.

S. Deng, L. Liu, H. Lang, D. Zhao, and X. Liu, “Cascaded Fresnel digital hologram and its application to watermarking,” Opt. Appl. 36, 413–420 (2006).

Lee, Y. H.

Li, J.

Li, Y.

Lie, W. N.

H. E. Hwang, H. T. Chang, and W. N. Lie, “Fast double-phase retrieval in Fresnel domain using modified Gerchberg-Saxton algorithm for lensless optical security systems,” Opt. Express 17, 13700–13710 (2009).
[CrossRef] [PubMed]

G. S. Lin, H. T. Chang, W. N. Lie, and C. H. Chuang, “A public-key-based optical image cryptosystem based on data embedding techniques,” Opt. Eng. 42, 2331–2339 (2003).
[CrossRef]

Lin, G. S.

G. S. Lin, H. T. Chang, W. N. Lie, and C. H. Chuang, “A public-key-based optical image cryptosystem based on data embedding techniques,” Opt. Eng. 42, 2331–2339 (2003).
[CrossRef]

Liu, L.

S. Deng, L. Liu, H. Lang, D. Zhao, and X. Liu, “Cascaded Fresnel digital hologram and its application to watermarking,” Opt. Appl. 36, 413–420 (2006).

Liu, X.

S. Deng, L. Liu, H. Lang, D. Zhao, and X. Liu, “Cascaded Fresnel digital hologram and its application to watermarking,” Opt. Appl. 36, 413–420 (2006).

Lohmann, A. W.

Lu, W. C.

Marinho, F.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithm for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Mas, D.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithm for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

Matoba, O.

Mogensen, P. C.

Montal, O.

D. Abookasis, O. Montal, O. Abramson, and J. Rosen, “Watermarks encrypted in a concealogram and deciphered by a modified joint transform correlator,” Appl. Opt. 44, 3019–3023 (2005).
[CrossRef] [PubMed]

Nishchal, N. K.

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase encryption using digital holography,” Opt. Eng. 43, 2959–2966(2004).
[CrossRef]

Peng, X.

X. Peng, P. Zhang, and L. Cai, “Information security system based on virtual-optics imaging methodology and public key infrastructure,” Optik 115, 420–426 (2004).
[CrossRef]

X. Peng, L. Yu, and L. Cai, “Double-lock for image encryption with virtual optical wavelength,” Opt. Express 10, 41–45(2002).
[PubMed]

Réfrégier, P.

Ripoll, O.

O. Ripoll, V. Kettunen, and H. P. Herzig, “Review of iterative Fourier-transform algorithms for beam shaping applications,” Opt. Eng. 43, 2549–2556 (2004).
[CrossRef]

Rosen, J.

Rosen, , J.

D. Abookasis, O. Montal, O. Abramson, and J. Rosen, “Watermarks encrypted in a concealogram and deciphered by a modified joint transform correlator,” Appl. Opt. 44, 3019–3023 (2005).
[CrossRef] [PubMed]

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).

R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

Shimura, T.

Singh, K.

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase encryption using digital holography,” Opt. Eng. 43, 2959–2966(2004).
[CrossRef]

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).
[CrossRef]

Situ, G.

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

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] [PubMed]

G. Situ and J. Zhang, “A cascaded iterative Fourier transform algorithm for optical security applications,” Optik 114, 473–477(2003).
[CrossRef]

Tan, X.

Tricoles, G.

Tsan, C. L.

Unnikrishnan, G.

Yu, L.

Zhang, G.

Zhang, J.

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

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] [PubMed]

G. Situ and J. Zhang, “A cascaded iterative Fourier transform algorithm for optical security applications,” Optik 114, 473–477(2003).
[CrossRef]

Zhang, P.

X. Peng, P. Zhang, and L. Cai, “Information security system based on virtual-optics imaging methodology and public key infrastructure,” Optik 115, 420–426 (2004).
[CrossRef]

Zhao, D.

S. Deng, L. Liu, H. Lang, D. Zhao, and X. Liu, “Cascaded Fresnel digital hologram and its application to watermarking,” Opt. Appl. 36, 413–420 (2006).

Appl. Opt. (11)

D. Abookasis, O. Montal, O. Abramson, and J. Rosen, “Watermarks encrypted in a concealogram and deciphered by a modified joint transform correlator,” Appl. Opt. 44, 3019–3023 (2005).
[CrossRef] [PubMed]

B. R. Brown and A. W. Lohmann, “Complex spatial filtering with binary masks,” Appl. Opt. 5, 967–969 (1966).
[CrossRef] [PubMed]

G. Tricoles, “Computer generated holograms: an historical review,” Appl. Opt. 26, 4351–4357 (1987).
[CrossRef] [PubMed]

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

O. Matoba and B. Javidi, “Encrypted optical storage with wavelength-key and random phase codes,” Appl. Opt. 38, 6785–6790 (1999).
[CrossRef]

Y. Li, K. Kreske, and J. Rosen, “Security and encryption optical systems based on a correlator with significant output image,” Appl. Opt. 39, 5295–5301 (2000).
[CrossRef]

X. Tan, O. Matoba, T. Shimura, K. Kuroda, and B. Javidi, “Secure optical storage that uses fully phase encryption,” Appl. Opt. 39, 6689–6694 (2000).
[CrossRef]

J. Rosen and B. Javidi, “Hidden images in halftone pictures,” Appl. Opt. 40, 3346–3353 (2001).
[CrossRef]

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, 4825–4834 (2002).
[CrossRef] [PubMed]

S. Kishk and B. Javidi, “Information hiding technique with double phase encoding,” Appl. Opt. 41, 5462–5470(2002).
[CrossRef] [PubMed]

H. T. Chang and C. L. Tsan, “Image watermarking by use of digital holography embedded in DCT domain,” Appl. Opt. 44, 6211–6219 (2005).
[CrossRef] [PubMed]

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

Opt. Appl. (1)

S. Deng, L. Liu, H. Lang, D. Zhao, and X. Liu, “Cascaded Fresnel digital hologram and its application to watermarking,” Opt. Appl. 36, 413–420 (2006).

Opt. Commun. (4)

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, “Fast algorithm for free-space diffraction patterns calculation,” Opt. Commun. 164, 233–245 (1999).
[CrossRef]

H. T. Chang and C. T. Chen, “Asymmetric-image verification for security optical system based on joint transform correlator,” Opt. Commun. 239, 43–54 (2004).
[CrossRef]

Y. C. Chang, H. T. Chang, and C. J. Kuo, “Hybrid image cryptosystem based on dyadic phase displacement in the Fourier domain,” Opt. Commun. 236, 245–257 (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]

Opt. Eng. (5)

H. T. Chang, “Optical image encryption using separate amplitude-based virtual image and iteratively-retrieved phase information,” Opt. Eng. 40, 2165–2171 (2001).
[CrossRef]

G. S. Lin, H. T. Chang, W. N. Lie, and C. H. Chuang, “A public-key-based optical image cryptosystem based on data embedding techniques,” Opt. Eng. 42, 2331–2339 (2003).
[CrossRef]

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase encryption using digital holography,” Opt. Eng. 43, 2959–2966(2004).
[CrossRef]

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

O. Ripoll, V. Kettunen, and H. P. Herzig, “Review of iterative Fourier-transform algorithms for beam shaping applications,” Opt. Eng. 43, 2549–2556 (2004).
[CrossRef]

Opt. Express (4)

Opt. Lett. (6)

Opt. Rev. (1)

H. T. Chang and C. T. Chen, “Enhanced optical image verification based on joint transform correlator adopting Fourier hologram,” Opt. Rev. 11, 165–169 (2004).
[CrossRef]

Optik (4)

X. Peng, P. Zhang, and L. Cai, “Information security system based on virtual-optics imaging methodology and public key infrastructure,” Optik 115, 420–426 (2004).
[CrossRef]

G. Situ and J. Zhang, “A cascaded iterative Fourier transform algorithm for optical security applications,” Optik 114, 473–477(2003).
[CrossRef]

R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–284 (1971).

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).

Other (2)

H.P.Herzig, ed., Micro-Optics (Taylor & Francis, 1996).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 355–358.

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Figures (10)

Fig. 1
Fig. 1

Concealogram construction based on encoding any complex data into the size and position of a subcell in a cell.

Fig. 2
Fig. 2

(a) Optical setup of a security system based on a CDFH, (b) optical setup of a security system based on a concealogram b ( x 2 , y 2 ) and a digital Fresnel hologram POM 1 .

Fig. 3
Fig. 3

Block diagram of the proposed MGSA.

Fig. 4
Fig. 4

Block diagram of compositing a concealogram based on the proposed method.

Fig. 5
Fig. 5

Simulation results: (a) host image h ( x 2 , y 2 ) , (b) hidden image g ( x 0 , y 0 ) , (c) computed concealogram b ( x 2 , y 2 ) , (d) enlarged portion of the concealogram.

Fig. 6
Fig. 6

(a) Computed G ^ ( x 1 , y 1 ) at λ = 632.8 nm and z 2 = 30 mm , whose correlation coefficient, with respect to the original G ( x 1 , y 1 ) , is 0.9997. (b) Phase distribution ψ 1 ( x 1 , y 1 ) on POM 1 . (c) Extracted image with correct keys (i.e., λ and z 2 ).

Fig. 7
Fig. 7

(a) Correlation coefficient between the target image g ( x 0 , y 0 ) and the recovered image g ^ ( x 0 , y 0 ) as a function of the wavelength deviation Δ λ . (b) Retrieved image at Δ λ = 1.0 nm ( ρ = 0.2121 ). (c) Retrieved image at Δ λ = 1.2 nm ( ρ = 0.1592 ).

Fig. 8
Fig. 8

(a) Correlation coefficient between the target image g ( x 0 , y 0 ) and the recovered image g ^ ( x 0 , y 0 ) as a function of the axial offset Δ z of POM 1 from its correct position. (b) Retrieved image at Δ z = 20 μm ( ρ = 0.3043 ). (c) Retrieved image at wavelength difference Δ z = 80 μm ( ρ = 0.1909 ).

Fig. 9
Fig. 9

(a) Halftone image zeroed by 25% of the area, (b) hidden image recovered from (a) with ρ = 0.8657 . (c) Halftone image with a zeroed area of up to 75%, (d) hidden image recovered from (c) with ρ = 0.8577 .

Fig. 10
Fig. 10

(a) Halftone image with 50.12% of randomly flipped pixels, (b) hidden image recovered from (a) with ρ = 0.8045 . (c) Halftone image with 69.09% of randomly flipped pixels, (d) hidden image recovered from (c) with ρ = 0.7493 .

Equations (34)

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DFT { h ( m δ x 2 , n δ y 2 ) exp [ j ψ h ( m δ x 2 , n δ y 2 ) ] ; λ ; z 2 } = m = M / 2 M / 2 1 n = N / 2 N / 2 1 h ( m δ x 2 , n δ y 2 ) exp [ j ψ h ( m δ x 2 , n δ y 2 ) ] exp { j 2 π ( m x 1 δ x 2 + n y 1 δ y 2 ) λ z 2 } = G ^ ( x 1 , y 1 ) exp [ j ψ G ( x 1 , y 1 ) ] ,
b t ( x 2 , y 2 ) = rect ( x 2 a X W X ) rect ( y 2 a Y W Y ) ,
rect ( x ) = { 1 , for     1 2 x 1 2 0 , other . .
B t ( x 1 , y 1 ) = FT [ b t ( x 2 , y 2 ) ] = W X W Y λ z 2 exp [ j 2 π λ z 2 ( x 1 a X + y 1 a Y ) ] sinc ( W X x 1 λ z 2 ) sinc ( W Y y 1 λ z 2 ) ,
B t ( x 1 , y 1 ) W X W Y λ z 2 exp [ j 2 π λ z 2 ( x 1 a X + y 1 a Y ) ] .
{ a X = m δ x 2 , a Y = n δ y 2 .
B t ( x 1 , y 1 ) W X m , n W Y m , n λ z 2 exp [ j 2 π λ z 2 ( x 1 m δ x 2 + y 1 n δ y 2 ) ] .
B ( x 1 , y 1 ) m = M / 2 M / 2 1 n = N / 2 N / 2 1 W X m , n W Y m , n λ z 2 exp [ j 2 π ( m x 1 δ x 2 + n y 1 δ y 2 ) λ z 2 ] .
{ a X = m δ x 2 + ( Δ x 2 ) m n , a Y = n δ y 2 ,
B ( x 1 , y 1 ) m = M / 2 M / 2 1 n = N / 2 N / 2 1 W X m , n W Y m , n λ z 2 exp { j 2 π [ ( Δ x 2 ) m , n x 1 + m x 1 δ x 2 + n y 1 δ y 2 ] λ z 2 } = m = M / 2 M / 2 1 n = N / 2 N / 2 1 W X m , n W Y m , n λ z 2 exp [ j 2 π ( Δ x 2 ) m , n x 1 λ z 2 ] exp [ j 2 π ( m x 1 δ x 2 + n y 1 δ y 2 ) λ z 2 ] .
B ( x 1 , y 1 ) m = M / 2 M / 2 1 n = N / 2 N / 2 1 W X m , n W Y m , n λ z 2 exp [ j 2 π ( Δ x 2 ) m , n δ x 2 ] exp [ j 2 π ( m x 1 δ x 2 + n y 1 δ y 2 ) λ z 2 ] .
W X m , n W Y m , n λ z 2 = h ( m δ x 2 , n δ y 2 ) ,
exp [ j 2 π ( Δ x 2 ) m , n δ x 2 ] = exp [ j ψ h ( m δ x 2 , n δ y 2 ) ] .
( Δ x 2 ) m , n δ x 2 = ψ h ( m δ x 2 , n δ y 2 ) 2 π .
b ( x 2 , y 2 ) = m = M / 2 M / 2 1 n = N / 2 N / 2 1 rect { x 2 d [ m + ψ h ( m δ x 2 , n δ y 2 ) / 2 π ] d | h ( m δ x 2 , n δ y 2 ) | } rect { y 2 d n d | h ( m δ x 2 , n δ y 2 ) | } .
FT { h ( x 2 , y 2 ) exp [ j ψ h ( x 2 , y 2 ) ] ; λ ; z 2 } = h ( x 2 , y 2 ) exp [ j ψ h ( x 2 , y 2 ) ] exp { j 2 π ( x 2 x 1 + y 2 y 1 ) λ z 2 } d x 2 d y 2 = G ^ ( x 1 , y 1 ) exp [ j ψ G ( x 1 , y 1 ) ] ,
F ( x 1 , y 1 ) = exp ( j 2 π z 2 λ ) j λ z 2 h ( x 2 , y 2 ) exp [ j ψ 2 ( x 2 , y 2 ) ] exp { j π λ z 2 [ ( x 1 x 2 ) 2 + ( y 1 y 2 ) 2 ] } d x 2 d y 2 .
F ( x 1 , y 1 ) = FrT { h ( x 2 , y 2 ) exp [ j ψ 2 ( x 2 , y 2 ) ] ; λ ; z 2 } .
g ^ ( x 0 , y 0 ) = FrT { FrT { h ( x 2 , y 2 ) exp [ j ψ 2 ( x 2 , y 2 ) ] ; λ ; z 2 } exp [ j ψ 1 ( x 1 , y 1 ) ] ; λ ; z 1 } = | g ^ ( x 0 , y 0 ) | exp [ j ψ g ( x 0 , y 0 ) ] .
IFT [ g ( x 0 , y 0 ) ; λ ; z 1 ] = 1 λ z 1 g ( x 0 , y 0 ) exp { j 2 π ( x 1 x 0 + y 1 y 0 ) λ z 1 } d x 0 d y 0 = G ( x 1 , y 1 ) exp [ j ψ t ( x 1 , y 1 ) ] ,
FT { h ( x 2 , y 2 ) exp [ j ψ h ( x 2 , y 2 ) ] ; λ ; z 2 } = h ( x 2 , y 2 ) exp [ j ψ h ( x 2 , y 2 ) ] exp { j 2 π ( x 2 x 1 + y 2 y 1 ) λ z 2 } d x 2 d y 2 = G ^ ( x 1 , y 1 ) exp [ j ψ G ( x 1 , y 1 ) ] ,
g ^ ( x 0 , y 0 ) = FT { G ^ ( x 1 , y 1 ) exp [ j ψ t ( x 1 , y 1 ) ] ; λ ; z 1 } .
ψ 2 ( x 2 , y 2 ) = 2 π z 2 λ + ψ h ( x 2 , y 2 ) π ( x 2 2 + y 2 2 ) λ z 2 .
FrT { h ( x 2 , y 2 ) exp [ j ψ 2 ( x 2 , y 2 ) ] ; λ ; z 2 } = exp [ j π ( x 1 2 + y 1 2 ) λ z 2 ] FT { h ( x 2 , y 2 ) exp [ j ψ h ( x 2 , y 2 ) ] ; λ ; z 2 } ,
FrT { h ( x 2 , y 2 ) exp [ j ψ 2 ( x 2 , y 2 ) ] ; λ ; z 2 } = exp [ j π ( x 1 2 + y 1 2 ) λ z 2 ] G ^ ( x 1 , y 1 ) exp [ j ψ G ( x 1 , y 1 ) ] .
G ^ ( x 1 , y 1 ) = FrT { h ( x 2 , y 2 ) exp [ j ψ 2 ( x 2 , y 2 ) ] ; λ ; z 2 } exp [ j ψ G ( x 1 , y 1 ) ] exp [ j π ( x 1 2 + y 1 2 ) λ z 2 ] .
ψ 1 ( x 1 , y 1 ) = 2 π z 1 λ + ψ t ( x 1 , y 1 ) ψ G ( x 1 , y 1 ) π ( x 1 2 + y 1 2 ) λ ( 1 z 2 + 1 z 1 ) .
| FrT ( FrT { h ( x 2 , y 2 ) exp [ j ψ 2 ( x 2 , y 2 ) ] ; λ ; z 2 } exp [ j ψ 1 ( x 1 , y 1 ) ] ; λ ; z 1 ) | = | exp [ j π ( x 0 2 + y 0 2 ) λ z 1 ] FT { G ^ ( x 1 , y 1 ) exp [ j ψ t ( x 1 , y 1 ) ] } | = g ^ ( x 0 , y 0 ) .
ρ = E { [ g E [ g ] ] [ | g est | E [ | g est | ] ] } { E { [ g E [ g ] ] 2 } E { [ | g est | E [ | g est | ] ] 2 } } 1 / 2 ,
G ( x 1 , y 1 ) exp [ j ψ t ( x 1 , y 1 ) ] as IFT [ g ( x 0 , y 0 ) ; λ ; z 1 ] ,
ψ h ( x 2 , y 2 ) and ψ G ( x 1 , y 1 )   via MGSA { [ h ( x 2 , y 2 ) ; G ( x 1 , y 1 ) ; λ ; z 2 ] } ,
ψ 2 ( x 2 , y 2 ) = 2 π z 2 λ + ψ h ( x 2 , y 2 ) π ( x 2 2 + y 2 2 ) λ z 2 .
ψ 1 ( x 1 , y 1 ) = 2 π z 1 λ + ψ t ( x 1 , y 1 ) ψ G ( x 1 , y 1 ) π ( x 1 2 + y 1 2 ) λ ( 1 z 2 + 1 z 1 ) .
h ( x 2 , y 2 ) exp [ j ψ 2 ( x 2 , y 2 ) ]

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