Abstract

A novel and efficient algorithm based on a modified Gerchberg–Saxton algorithm (MGSA) in the Fresnel domain is presented, together with mathematical derivation, and two pure phase-only masks (POMs) are generated. The algorithm’s application to data hiding is demonstrated by a simulation procedure, in which a hidden image/logo is encoded into phase forms. A hidden image/logo can be extracted by the proposed high-performance lensless optical data-hiding system. The reconstructed image shows good quality and the errors are close to zero. In addition, the robustness of our data-hiding technique is illustrated by simulation results. The position coordinates of the POMs as well as the wavelength are used as secure keys that can ensure sufficient information security and robustness. The main advantages of this proposed watermarking system are that it uses fewer iterative processes to produce the masks, and the image-hiding scheme is straightforward.

© 2013 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  24. 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]
  25. 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).
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2009 (2)

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. E. Hwang and P. Han, “Fast algorithm of phase masks for image encryption in the Fresnel domain,” J. Opt. Soc. Am. A 23, 1870–1874 (2006).
[CrossRef]

2005 (2)

2004 (5)

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]

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]

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 (1)

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

2002 (2)

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

S. Sinzinger, “Microoptically integrated correlators for security applications,” Opt. Commun. 209, 69–74 (2002).
[CrossRef]

2001 (2)

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)

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, 275–283 (1971).

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]

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.

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]

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).
[CrossRef]

Chang, H. T.

Chen, N. X.

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, 275–283 (1971).

Glückstad, J.

Gu, B. Y.

Han, P.

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]

Kim, D. H.

Kim, H.

Kreske, K.

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.

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).

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.

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).
[CrossRef]

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).

R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–283 (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]

Sinzinger, S.

S. Sinzinger, “Microoptically integrated correlators for security applications,” Opt. Commun. 209, 69–74 (2002).
[CrossRef]

Situ, G.

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

Tan, X.

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]

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).
[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. (6)

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

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. (3)

S. Sinzinger, “Microoptically integrated correlators for security applications,” Opt. Commun. 209, 69–74 (2002).
[CrossRef]

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]

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. (2)

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]

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

Opt. Express (3)

Opt. Lett. (7)

Optik (4)

R. W. Gerchberg and W. O. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik 34, 275–283 (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).

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

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]

Other (1)

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

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Figures (13)

Fig. 1.
Fig. 1.

Optical setup of the two POMs optical security system.

Fig. 2.
Fig. 2.

Flow chart of the GSA as used for performing phase retrieval if source intensity (in space domain) and target image (in Fourier domain) are known.

Fig. 3.
Fig. 3.

Flow chart of the MGSA used for encoding the target image into a pure phase function.

Fig. 4.
Fig. 4.

(a) Continuous-tone image used as the hidden image/logo for computer simulation. (b) Binary-tone image used as the hidden image/logo for computer simulation.

Fig. 5.
Fig. 5.

Image used as the host image for computer simulation.

Fig. 6.
Fig. 6.

Phase distribution of exp[jh(x2,y2)+jψ2(x2,y2)].

Fig. 7.
Fig. 7.

Phase distribution ψ2(x2,y2) of POM2.

Fig. 8.
Fig. 8.

Phase distribution ψ1(x1,y1) of POM1.

Fig. 9.
Fig. 9.

(a) Decryption of the continuous-tone hidden image/logo with the correct keys and correlation coefficient ρ equal to 0.9997. (b) Decryption of the halftone hidden image/logo with the correct keys and correlation coefficient ρ equal to 0.9969.

Fig. 10.
Fig. 10.

Correlation coefficient between the continuous-tone hidden image/logo and the extracted hidden image/logo as a function of the wavelength difference between the encryption and decryption beams.

Fig. 11.
Fig. 11.

(a) Retrieved continuous-tone hidden image/logo when wavelength difference Δλ=9.0nm and ρ=0.0358. (b) Retrieved binary-tone hidden image/logo when wavelength difference Δλ=9.0nm and ρ=0.0172.

Fig. 12.
Fig. 12.

Correlation coefficient between the continuous-tone hidden image/logo and the recovered hidden image/logo as a function of the axial offset of POM1 from its matched position.

Fig. 13.
Fig. 13.

(a) Retrieved continuous-tone hidden image/logo when wavelength difference Δz=800μm and ρ=0.0750. (b) Retrieved binary-tone hidden image/logo when wavelength difference Δz=800μm and ρ=0.2379.

Equations (16)

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FT{exp[jψ2(x2,y2)];λ;z2}=exp[jψ2(x2,y2)]·exp{j2π(x2x1+y2y1)λz2}dx2dy2=G^(x1,y1)exp[jψG(x1,y1)],
F(x1,y1)=exp(j2πzλ)jλzf(x0,y0)exp{jπλz[(x1x0)2+(y1y0)2]}dx0dy0,
IFT[g(x0,y0);λ;z1]=1λz1g(x0,y0)exp{j2π(x1x0+y1y0)λz1}dx0dy0=G(x1,y1)exp[jψt(x1,y1)],
g^(x0,y0;λ;z1)=FT{G^(x1,y1)·exp[jψt(x1,y1)];λ;z1}.
ψ2(x2,y2)=h(x2,y2)2πz2λ+ψ2(x2,y2)π(x22+y22)λz2.
FrT{exp[jh(x2,y2)+jψ2(x2,y2)];λ;z2}=exp(j2πz2λ)jλz2exp[jj2πz2λ+ψ2(x2,y2)jπ(x22+y22)λz2]×exp[jπλz2(x2x1)2+jπλz2(y2y1)2]dx2dy2=1jλz2exp[jπ(x12+y12)λz2]exp[jψ2(x2,y2)]exp{j2π(x2x1+y2y1)λz2}dx2dy2=exp[jπ(x12+y12)λz2]FT{exp[jψ2(x2,y2)];λ;z2}=exp[jπ(x12+y12)λz2]G^(x1,y1)exp[jψG(x1,y1)].
G^(x1,y1)=FrT{exp[jψ2(x2,y2)];λ;z2}exp[jψG(x1,y1)]·exp[jπ(x12+y12)λz2].
ψ1(x1,y1)=2πz1λ+ψt(x1,y1)ψG(x1,y1)π(x12+y12)λ(1z2+1z1).
|FrT(FrT{exp[jh(x2,y2)+jψ2(x2,y2)];λ;z2}·exp[jψ1(x1,y1)];λ;z1)|=|FrT{exp[jπ(x12+y12)λz2]G^(x1,y1)exp[jψG(x1,y1)]×exp[jψ1(x1,y1)];λ;z1}|=|FrT{exp(j2πz1λ)exp[jπ(x12+y12)λz1]G^(x1,y1)exp[jψt(x1,y1)];λ;z1}|=|exp[j2πz1λ]jλz1exp(j2πz1λ)exp[jπ(x12+y12)λz1]G^(x1,y1)exp[jψt(x1,y1)]×exp[jπλz1(x1x0)2+jπλz1(y1y0)2]dx1dy1|=|1jλz1exp[jπ(x02+y02)λz1]G^(x1,y1)exp[jψt(x1,y1)]×exp[j2π(x1x0+y1y0)λz1]dx1dy1|=|exp[jπ(x02+y02)λz1]FT{G^(x1,y1)exp[jψt(x1,y1)]}|=g^(x0,y0).
ρ=E{[gE[g]][|g^|E[|g^|]]}{E{[gE[g]]2}E{[|g^|E[|g^|]]2}}1/2,
G(x1,y1)exp[jψt(x1,y1)]=IFT[g(x0,y0);λ;z1],
ψ2(x2,y2)=MGSA{[G(x1,y1);λ;z2]},
ψ2(x2,y2)=h(x2,y2)2πz2λ+ψ2(x2,y2)π(x22+y22)λz2,
ψG(x1,y1)=arg(FT{exp[jψ2(x2,y2)];λ;z2}),
ψ1(x1,y1)=2πz1λ+ψt(x1,y1)ψG(x1,y1)π(x12+y12)λ(1z2+1z1).
g^(x0,y0)=|FrT(FrT{exp[jψ2(x2,y2)];λ;z2}·exp[jψ1(x1,y1)];λ;z1)|.

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