Abstract

A fully phase image encryption technique that uses the double random-phase encoding method is presented. The performance of this fully phase encryption is compared with that of amplitude-based encryption in the presence of noise. Analytic bounds on the mean squared error for the decrypted images are obtained. The accuracy of the theoretical error bounds is confirmed by comparing them with the mean squared errors obtained by using numerical and statistical methods for actual images. Fully phase-based encryption shows better performance than amplitude-based encryption with respect to the mean-squared-error metric.

© 1999 Optical Society of America

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References

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  1. Special issue on optical security, Opt. Eng. 35, 2451–2541 (1996).
  2. B. Javidi, J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
    [CrossRef]
  3. P. Refregier, B. Javidi, “Optical image encryption using input and Fourier plane random phase encoding,” Opt. Lett. 20, 767–769 (1995).
    [CrossRef] [PubMed]
  4. Q. Huang, J. Caulfield, “Wave guide holography and its applications,” in Practical Holography V, S. A. Benton, ed., Proc. SPIE1461, 303–312 (1991).
  5. B. Javidi, G. Zhang, J. Li, “Experimental demonstration of the random phase encoding technique for image encryption and security verification,” Opt. Eng. 35, 2506–2512 (1996).
    [CrossRef]
  6. H. Fielding, J. L. Horner, C. K. Makekau, “Optical fingerprint identification by binary joint transform correlation,” Opt. Eng. 30, 1958–1961 (1991).
    [CrossRef]
  7. H-Y. Li, Y. Qiao, D. Psaltis, “Optical network for real-time face recognition,” Appl. Opt. 32, 5026–5035 (1993).
    [CrossRef] [PubMed]
  8. B. Javidi, J. Li, Q. Tang, “Optical implementation of neural networks for face recognition by the use of nonlinear joint transform correlators,” Appl. Opt. 34, 3950–3962 (1995).
    [CrossRef] [PubMed]
  9. J. Rodolfo, H. Rajbenbach, J.-P. Huignard, “Performance of a photorefractive joint transform correlator for fingerprint identification,” Opt. Eng. 34, 1166–1171 (1995).
    [CrossRef]
  10. C. L. Wilson, C. I. Watson, E. G. Paek, “Combined optical and neural network fingerprint matching,” in Optical Pattern Recognition VIII, D. Casasent, Tien-Hsin Chao, eds., Proc. SPIE3073, 373–382 (1997).
    [CrossRef]
  11. N. Riza, M. Howlader, “Photonics security system using spatial codes and remote coded coherent optical communications,” Opt. Eng. 35, 2487–2498 (1996).
    [CrossRef]
  12. M. Drake, M. Lidd, M. A. Fiddy, “Wave guide hologram fingerprint entry device,” Opt. Eng. 35, 2499–2505 (1996).
    [CrossRef]
  13. F. Goudail, F. Bollaro, B. Javidi, P. Réfrégier, “Influence of a perturbation in a double phase-encoding system,” J. Opt. Soc. Am. A 15, 2629–2638 (1998).
    [CrossRef]

1998 (1)

1996 (4)

N. Riza, M. Howlader, “Photonics security system using spatial codes and remote coded coherent optical communications,” Opt. Eng. 35, 2487–2498 (1996).
[CrossRef]

M. Drake, M. Lidd, M. A. Fiddy, “Wave guide hologram fingerprint entry device,” Opt. Eng. 35, 2499–2505 (1996).
[CrossRef]

Special issue on optical security, Opt. Eng. 35, 2451–2541 (1996).

B. Javidi, G. Zhang, J. Li, “Experimental demonstration of the random phase encoding technique for image encryption and security verification,” Opt. Eng. 35, 2506–2512 (1996).
[CrossRef]

1995 (3)

1994 (1)

B. Javidi, J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
[CrossRef]

1993 (1)

1991 (1)

H. Fielding, J. L. Horner, C. K. Makekau, “Optical fingerprint identification by binary joint transform correlation,” Opt. Eng. 30, 1958–1961 (1991).
[CrossRef]

Bollaro, F.

Caulfield, J.

Q. Huang, J. Caulfield, “Wave guide holography and its applications,” in Practical Holography V, S. A. Benton, ed., Proc. SPIE1461, 303–312 (1991).

Drake, M.

M. Drake, M. Lidd, M. A. Fiddy, “Wave guide hologram fingerprint entry device,” Opt. Eng. 35, 2499–2505 (1996).
[CrossRef]

Fiddy, M. A.

M. Drake, M. Lidd, M. A. Fiddy, “Wave guide hologram fingerprint entry device,” Opt. Eng. 35, 2499–2505 (1996).
[CrossRef]

Fielding, H.

H. Fielding, J. L. Horner, C. K. Makekau, “Optical fingerprint identification by binary joint transform correlation,” Opt. Eng. 30, 1958–1961 (1991).
[CrossRef]

Goudail, F.

Horner, J. L.

B. Javidi, J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
[CrossRef]

H. Fielding, J. L. Horner, C. K. Makekau, “Optical fingerprint identification by binary joint transform correlation,” Opt. Eng. 30, 1958–1961 (1991).
[CrossRef]

Howlader, M.

N. Riza, M. Howlader, “Photonics security system using spatial codes and remote coded coherent optical communications,” Opt. Eng. 35, 2487–2498 (1996).
[CrossRef]

Huang, Q.

Q. Huang, J. Caulfield, “Wave guide holography and its applications,” in Practical Holography V, S. A. Benton, ed., Proc. SPIE1461, 303–312 (1991).

Huignard, J.-P.

J. Rodolfo, H. Rajbenbach, J.-P. Huignard, “Performance of a photorefractive joint transform correlator for fingerprint identification,” Opt. Eng. 34, 1166–1171 (1995).
[CrossRef]

Javidi, B.

Li, H-Y.

Li, J.

B. Javidi, G. Zhang, J. Li, “Experimental demonstration of the random phase encoding technique for image encryption and security verification,” Opt. Eng. 35, 2506–2512 (1996).
[CrossRef]

B. Javidi, J. Li, Q. Tang, “Optical implementation of neural networks for face recognition by the use of nonlinear joint transform correlators,” Appl. Opt. 34, 3950–3962 (1995).
[CrossRef] [PubMed]

Lidd, M.

M. Drake, M. Lidd, M. A. Fiddy, “Wave guide hologram fingerprint entry device,” Opt. Eng. 35, 2499–2505 (1996).
[CrossRef]

Makekau, C. K.

H. Fielding, J. L. Horner, C. K. Makekau, “Optical fingerprint identification by binary joint transform correlation,” Opt. Eng. 30, 1958–1961 (1991).
[CrossRef]

Paek, E. G.

C. L. Wilson, C. I. Watson, E. G. Paek, “Combined optical and neural network fingerprint matching,” in Optical Pattern Recognition VIII, D. Casasent, Tien-Hsin Chao, eds., Proc. SPIE3073, 373–382 (1997).
[CrossRef]

Psaltis, D.

Qiao, Y.

Rajbenbach, H.

J. Rodolfo, H. Rajbenbach, J.-P. Huignard, “Performance of a photorefractive joint transform correlator for fingerprint identification,” Opt. Eng. 34, 1166–1171 (1995).
[CrossRef]

Refregier, P.

Réfrégier, P.

Riza, N.

N. Riza, M. Howlader, “Photonics security system using spatial codes and remote coded coherent optical communications,” Opt. Eng. 35, 2487–2498 (1996).
[CrossRef]

Rodolfo, J.

J. Rodolfo, H. Rajbenbach, J.-P. Huignard, “Performance of a photorefractive joint transform correlator for fingerprint identification,” Opt. Eng. 34, 1166–1171 (1995).
[CrossRef]

Tang, Q.

Watson, C. I.

C. L. Wilson, C. I. Watson, E. G. Paek, “Combined optical and neural network fingerprint matching,” in Optical Pattern Recognition VIII, D. Casasent, Tien-Hsin Chao, eds., Proc. SPIE3073, 373–382 (1997).
[CrossRef]

Wilson, C. L.

C. L. Wilson, C. I. Watson, E. G. Paek, “Combined optical and neural network fingerprint matching,” in Optical Pattern Recognition VIII, D. Casasent, Tien-Hsin Chao, eds., Proc. SPIE3073, 373–382 (1997).
[CrossRef]

Zhang, G.

B. Javidi, G. Zhang, J. Li, “Experimental demonstration of the random phase encoding technique for image encryption and security verification,” Opt. Eng. 35, 2506–2512 (1996).
[CrossRef]

Appl. Opt. (2)

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

Opt. Eng. (7)

B. Javidi, G. Zhang, J. Li, “Experimental demonstration of the random phase encoding technique for image encryption and security verification,” Opt. Eng. 35, 2506–2512 (1996).
[CrossRef]

H. Fielding, J. L. Horner, C. K. Makekau, “Optical fingerprint identification by binary joint transform correlation,” Opt. Eng. 30, 1958–1961 (1991).
[CrossRef]

N. Riza, M. Howlader, “Photonics security system using spatial codes and remote coded coherent optical communications,” Opt. Eng. 35, 2487–2498 (1996).
[CrossRef]

M. Drake, M. Lidd, M. A. Fiddy, “Wave guide hologram fingerprint entry device,” Opt. Eng. 35, 2499–2505 (1996).
[CrossRef]

J. Rodolfo, H. Rajbenbach, J.-P. Huignard, “Performance of a photorefractive joint transform correlator for fingerprint identification,” Opt. Eng. 34, 1166–1171 (1995).
[CrossRef]

Special issue on optical security, Opt. Eng. 35, 2451–2541 (1996).

B. Javidi, J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
[CrossRef]

Opt. Lett. (1)

Other (2)

Q. Huang, J. Caulfield, “Wave guide holography and its applications,” in Practical Holography V, S. A. Benton, ed., Proc. SPIE1461, 303–312 (1991).

C. L. Wilson, C. I. Watson, E. G. Paek, “Combined optical and neural network fingerprint matching,” in Optical Pattern Recognition VIII, D. Casasent, Tien-Hsin Chao, eds., Proc. SPIE3073, 373–382 (1997).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Optical implementation of (a) the fully phase encryption optical processor and (b) the decryption method.

Fig. 2
Fig. 2

Theoretical upper bound mean squared error and experimental mean squared error for fully phase encryption and amplitude-based encryption: (a) for a gray-scale image and (b) for a binary image.

Fig. 3
Fig. 3

Fully phase encryption and amplitude-based encryption for a gray-scale image in additive white noise with standard deviation of σ=0.2: (a) original gray-scale image of Franklin D. Roosevelt, (b) white noise n(x, y), (c) fully phase encrypted image ψP(x, y), (d) amplitude-based encrypted image ψA(x, y), (e) fully phase encrypted image ψP(x, y) plus white noise n(x, y), (f) amplitude-based encrypted image ψA(x, y) plus white noise n(x, y), (g) recovered image from a fully phase encrypted image (MSE=0.0021), (h) recovered image from an amplitude-based encrypted image (MSE=0.0401).

Fig. 4
Fig. 4

Fully phase encryption and amplitude-based encryption for a binary image in additive white noise with standard deviation of σ=0.2: (a) original binary text, (b) white noise n(x, y), (c) fully phase encrypted image ψP(x, y), (d) amplitude-based encrypted image ψA(x, y), (e) fully phase encrypted image ψP(x, y) plus white noise n(x, y), (f) amplitude-based encrypted image ψA(x, y) plus white noise n(x, y), (g) recovered image from a fully phase encrypted image (MSE=0.0021), (h) recovered image from an amplitude-based encrypted image (MSE=0.0404).

Fig. 5
Fig. 5

Fully phase encryption and amplitude-based encryption for a gray-scale image in additive color Gaussian noise with standard deviation of σ=0.2 and bandwidth of 5×5 pixels: (a) original gray-scale image of Franklin D. Roosevelt, (b) white noise n(x, y), (c) fully phase encrypted image ψP(x, y), (d) amplitude-based encrypted image ψA(x, y), (e) fully phase encrypted image ψP(x, y) plus white noise n(x, y), (f) amplitude-based encrypted image ψA(x, y) plus white noise n(x, y), (g) recovered image from a fully phase encrypted image (MSE=0.0026), (h) recovered image from an amplitude-based encrypted image (MSE=0.0488).

Fig. 6
Fig. 6

Mean squared error versus standard deviation in the presence of additive Gaussian color noise with bandwidth of 5×5 pixels: (a) for a gray-scale image and (b) for a binary image.

Equations (72)

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ψP(x, y)={exp[iπf(x, y)]exp[i2πp(x, y)]}*h(x, y),
ψA(x, y)={f(x, y)exp[i2πp(x, y)]}*h(x, y).
ψP(x, y)={exp[iπf(x, y)+i2πp(x, y)]*h(x, y)}+n(x, y).
A(x, y)exp[iπfP(x, y)]=exp[iπf(x, y)]+n0(x, y),
n0(x, y)=FT-1{N(v, w)exp[-i2πb(v, w)]}×exp[-i2πp(x, y)].
n0(x, y)=n0R(x, y)+in0I(x, y),
A(x, y){exp[iπfP(x, y)]}
={cos[πf(x, y)]+n0R(x, y)}+i{sin[πf(x, y)]+n0I(x, y)}.
A(x, y)exp[iπfP(x, y)]=exp[iπf(x, y)].
|fP(x, y)|=|Arg{A exp[iπfP(x, y)]/π}|.
ψA(x, y)={f(x, y)exp[i2πp(x, y)]*h(x, y)}+n(x, y).
fA(x, y)=f(x, y)+FT-1{N(v, w)exp[-i2πb(x, y)]}×exp[-i2πp(x, y)]=f(x, y)+n0(x, y),
Err(|fP|)=E1N×M x=0N-1y=0M-1[|f(x, y)-|fP(x, y)||2],
Err(|fA|)=E1N×M x=0N-1y=0M-1[|f(x, y)-|fA(x, y)2],
Err(|fp|)σ02π2 erf122σ0+6σ02π 1-exp-18σ02+29 erfc122σ0+12 erf22σ0-erf122σ0+σ022π2 erf32σ0-erf22σ0+14 erfc32σ0,
erf(x)=2π 0x exp(-t2)dt,
erfc(x)=2π x- exp(-t2)dt.
|f(x, y)-|fA(x, y)|f(x, y)-fA(x, y)|.
Err(|fA|)E1N×M x=0N-1y=0M-1[|n0(x, y)|2]=σ2.
Ang(z)=tan-1Im(z)Re(z).
Err(|fP|)1π2×N×M×x=0N-1y=0M-1E({Ang[n0(x, y)+1]}2).
Err(|fP|)1π2×N×M×x=0N-1y=0M-1Etan-1n0I(x, y)1+n0R(x, y)2.
Etan-1n0I(x, y)1+n0R(x, y)2.
h=n0R(x, y),g=n0I(x, y).
Etan-1g1+h2
=12πσ02--tan-1g1+h2×exp-(h2+g2)2σ02dhdg
=12πσ02-1/2-tan-1g1+h2
×exp-h22σ02exp-g22σ02dhdg
+12πσ02-2-1/2-tan-1g1+h2×exp-h22σ02exp-g22σ02dhdg
+12πσ02--2-tan-1g1+h2
×exp-h2σ02exp-g2σ02dhdg.
12πσ02-1/2--tan-1g1+h2
×exp-h22σ02exp-g22σ02dhdg
12πσ02-1/2-g1+h2×exp-h22σ02exp-g22σ02dhdg
(σ02)12πσ0-1/211+h2 exp-h22σ02dh,
12πσ0 -g2 exp-g22σ02dg=σ02.
12πσ0-1/211+h2 exp-h22σ02dh,
12πσ0-1/211+h2 exp-h22σ02dh
=12πσ000.5 1(1-h)2 exp-h22σ02dh+12πσ00 1(1+h)2 exp-h22σ02dh
=I+II.
I=12πσ000.5 exp-h22σ02dh+62πσ000.5h exp-h22σ02dh=0.5 erf122σ0+6σ02π1-exp-18σ02.
II12πσ000.5 exp-h22σ02dh+49 12πσ01/2 exp-h22σ02dh=0.5 erf122σ0+29 erfc122σ0.
σ02erf122σ0+6σ02π 1-exp-18σ02+29 erfc122σ0,
erf(x)=2π 0x exp(-t2)dt,
erfc(x)=2π x- exp(-t2)dt.
12πσ02-2-1/2 -tan-1g1+h2
×exp-h22σ02exp-g22σ02dhdg
(π/2)2(2πσ0)2 -2-1/2 -exp-g22σ02dg exp-h22σ02dh.
12πσ0 - exp-g22σ02dg=1,
π22 12πσ0 -2-1/2 exp-h22σ02dh
=π28 erf22σ0-erf122σ0.
12πσ02--2-tan-1g1+h2
×exp-h22σ02exp-g22σ02dhdg
12πσ02--2 -g1+h2×exp-h22σ02exp-g22σ02dhdg(σ02)12πσ0--211+h2 exp-h22σ02dh=(σ02)12πσ0211-h2 exp-h22σ02dh=σ022πσ0 2311-h2 exp-h22σ02dh+σ022πσ0 311-h2 exp-h22σ02dh
σ022πσ0 23 exp-h22σ02dh+14 3 exp-h22σ02dh=σ020.5 erf32σ0-0.5 erf22σ0+18 erfc32σ0.
Errσ02π2 erf122σ0+6σ02π 1-exp-18σ02+29 erfc122σ0+18 erf22σ0-erf122σ0+σ022π2 erf32σ0-erf22σ0+14 erfc32σ0.
Err(|fP|)1π2×N×M x=0N-1y=0M-1Etan-1n0I(x, y)1+n0R(x, y)2.
|fP(x, y)|=|Arg{A exp[iπfP(x, y)]}/π|,
A exp[iπfP(x, y)]={cos[πf(x, y)]+n0R(x, y)}+i{sin[πf(x, y)]+n0I(x, y)}.
n0=n0(x, y)=n0R(x, y)+in0I(x, y)=n0R+in0I.
π2E|f(x, y)-|fP||2E({Ang[n0(x, y)+1]}2).
πf=Arg[exp(iπf )].
π|f-|fP|||Ang[exp(ifπ)+n0]-Ang[exp(ifπ)]|.
I(ω)=1if|n0(ω)|<10if|n0(ω)|1.
π2E(I|f-|fP||2)E{I|Ang[exp(iπ)f+n0]-Ang[exp(ifπ)]|2}.
π2E(I|f-|fP||2)E[I|Ang(n0+1)|2].
n0(x, y)=r(x, y)exp[iω(x, y)]orn0=r exp(iω).
tan(eπ)=r sin(ω-fπ)r cos(ω-fπ)+1.
tan(fπ-eπ)=r sin(fπ-ω)r cos(fπ-ω)-1.
π2E(Ic|f-|fP2)=E[Ic|Ang(n0+1)|2].
E(|f-|fP2)=E(I|f-|fP2)+E(Ic|f-|fP2),
π2E(|f-|fP2)E[|Ang(n0+1)|2].

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