Abstract

We consider the recently proposed double phase-encoding system [Opt. Lett. 20, 767 (1995)]. We study the robustness of the decoding process, that is, the way in which a perturbation of the coded image modifies the decoded image. We demonstrate that the amplitude signal-to-noise ratio (SNR) in the decoded image is strictly (and not only statistically) equal to the SNR in the coded image for different kinds of coded-image perturbations. In optical implementations the intensity of the decoded image is measured at the output of the decoding system. We show that there exists a simple relation between the intensity SNR of the decoded image and the amplitude SNR of the coded image and that this relation is quasi-independent of the nature of the coded-image perturbation. The results presented could provide a simple and efficient way of determining the precision level of the components of the optical decoding system necessary to reach a predefined quality level of the decoded image.

© 1998 Optical Society of America

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References

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  1. Ph. Réfrégier, B. Javidi, “Optical image encryption using input plane and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995).
    [CrossRef]
  2. B. Javidi, J. L. Horner, “Optical pattern recognition system for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
    [CrossRef]
  3. 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]
  4. L. G. Neto, Y. Sheng, “Optical implementation of image encryption using random phase encoding,” Opt. Eng. 35, 2459–2463 (1996).
    [CrossRef]
  5. E. G. Johnson, J. D. Brasher, “Phase encryption of biometrics in diffractive optical elements,” Opt. Lett. 21, 1271–1273 (1996).
    [CrossRef] [PubMed]
  6. B. Javidi, A. Sergent, G. Zhang, L. Guibert, “Fault tolerance properties of a double phase encoding encryption technique,” Opt. Eng. 36, 992–998 (1997).
    [CrossRef]
  7. H. S. Li, Y. Qiao, D. Psaltis, “Optical network for real-time face recognition,” Appl. Opt. 32, 5026–5035 (1993).
    [CrossRef] [PubMed]
  8. K. H. Fielding, J. L. Horner, C. K. Makekau, “Optical fingerprint identification by binary joint-transform correlation,” Opt. Eng. 30, 1958–1961. (1991).
    [CrossRef]
  9. D. Raj, D. W. Prather, R. A. Athale, J. N. Mait, “Performance analysis of optical shadow-casting correlators,” Appl. Opt. 32, 3108–3112 (1993).
    [CrossRef] [PubMed]
  10. “Special issue on optical security,” Opt. Eng. 35, 2451–2547 (1996).
  11. B. Javidi, A. Sergent, “Fully phase encoded key and biometrics for security verification,” Opt. Eng. 36, 935–942 (1997).
    [CrossRef]
  12. M. H. Hayes, “The unique reconstruction of multidimensional sequences from Fourier transform magnitude or phase,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, San Diego, Calif., 1987), pp. 195–230.
  13. J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, San Diego, Calif., 1987), pp. 231–275.

1997

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

B. Javidi, A. Sergent, “Fully phase encoded key and biometrics for security verification,” Opt. Eng. 36, 935–942 (1997).
[CrossRef]

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]

L. G. Neto, Y. Sheng, “Optical implementation of image encryption using random phase encoding,” Opt. Eng. 35, 2459–2463 (1996).
[CrossRef]

“Special issue on optical security,” Opt. Eng. 35, 2451–2547 (1996).

E. G. Johnson, J. D. Brasher, “Phase encryption of biometrics in diffractive optical elements,” Opt. Lett. 21, 1271–1273 (1996).
[CrossRef] [PubMed]

1995

1994

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

1993

1991

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

Athale, R. A.

Brasher, J. D.

Dainty, J. C.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, San Diego, Calif., 1987), pp. 231–275.

Fielding, K. H.

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

Fienup, J. R.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, San Diego, Calif., 1987), pp. 231–275.

Guibert, L.

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

Hayes, M. H.

M. H. Hayes, “The unique reconstruction of multidimensional sequences from Fourier transform magnitude or phase,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, San Diego, Calif., 1987), pp. 195–230.

Horner, J. L.

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

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

Javidi, B.

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

B. Javidi, A. Sergent, “Fully phase encoded key and biometrics for security verification,” Opt. Eng. 36, 935–942 (1997).
[CrossRef]

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]

Ph. Réfrégier, B. Javidi, “Optical image encryption using input plane and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995).
[CrossRef]

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

Johnson, E. G.

Li, H. S.

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]

Mait, J. N.

Makekau, C. K.

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

Neto, L. G.

L. G. Neto, Y. Sheng, “Optical implementation of image encryption using random phase encoding,” Opt. Eng. 35, 2459–2463 (1996).
[CrossRef]

Prather, D. W.

Psaltis, D.

Qiao, Y.

Raj, D.

Réfrégier, Ph.

Sergent, A.

B. Javidi, A. Sergent, “Fully phase encoded key and biometrics for security verification,” Opt. Eng. 36, 935–942 (1997).
[CrossRef]

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

Sheng, Y.

L. G. Neto, Y. Sheng, “Optical implementation of image encryption using random phase encoding,” Opt. Eng. 35, 2459–2463 (1996).
[CrossRef]

Zhang, G.

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

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.

Opt. Eng.

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

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]

L. G. Neto, Y. Sheng, “Optical implementation of image encryption using random phase encoding,” Opt. Eng. 35, 2459–2463 (1996).
[CrossRef]

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

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

“Special issue on optical security,” Opt. Eng. 35, 2451–2547 (1996).

B. Javidi, A. Sergent, “Fully phase encoded key and biometrics for security verification,” Opt. Eng. 36, 935–942 (1997).
[CrossRef]

Opt. Lett.

Other

M. H. Hayes, “The unique reconstruction of multidimensional sequences from Fourier transform magnitude or phase,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, San Diego, Calif., 1987), pp. 195–230.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, San Diego, Calif., 1987), pp. 231–275.

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Figures (7)

Fig. 1
Fig. 1

Optical implementation of the decoding of a double phase-coded image.

Fig. 2
Fig. 2

(a) Image f to be coded. It is a binary image whose nonnull values are equal to 255. (b) Real part of the coded image Re[ψ]. (c) Real part of the perturbed coded image Re[ϕ]. The perturbation was an occlusion of 30% of the pixels of the coded image. (d) Decoded image g.

Fig. 3
Fig. 3

Characterization of the spatial distribution of ñ. The input image is given in Fig. 2(a). The coded-image perturbation is an occlusion of 30% of the pixels of the coded image (see Fig. 2). (a) Perturbed decoded image g, (b) least-mean-square estimation of ñ, (c) histogram of ñ, (d) cross section of the autocorrelation function of ñ.

Fig. 4
Fig. 4

Plot of Rdec as a function of ρcod for different types of perturbation. The coded image is given in Fig. 2(a). The theoretical curve (solid curve) uses Eq. (39) and (40), and the experimental points are determined from numerical simulations. Each point of the plot corresponds to one realization of the noise. Squares, additive noise; diamonds, multiplicative amplitude noise; triangles, multiplicative phase noise; filled circles, occlusion.

Fig. 5
Fig. 5

Image used in the simulations of Fig. 7.

Fig. 6
Fig. 6

Plot of Rdec as a function of ρcod. The coded image is aversion of Fig. 5 subsampled so that its dimension is 32×32 pixels. The theoretical curve (solid curve) uses Eqs. (39) and (40), and the experimental points are determined from numerical simulations. Each point of the plot corresponds to one realization of the noise. Squares, additive noise; diamonds, multiplicative amplitude noise; triangles, multiplicative phase noise; filled circles, occlusion.

Fig. 7
Fig. 7

Plot of Rdec as a function of ρcod for different types of perturbation. The coded image is given in Fig. 2(a). The theoretical curve (solid curve) uses Eqs. (39) and (40), and the experimental curves are determined from numerical simulations. Each point of the plot corresponds to one realization of the noise. Squares, results obtained after the quantization of the phase and of the modulus of the coded image with the same number of levels (16, 8, and 4). Triangles, results obtained after taking only the real part of the coded image and quantizing it (number of levels: 256, 128, 64, 32, 16, 8, 4, and 2).

Equations (60)

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hˆ(ν)=exp[i2πb(ν)],
ψ(x)=h(x){f(x)exp[i2πa(x)]},
h¯ˆ(ν)=hˆ*(ν)=exp[-i2πb(ν)],
ϕ=ψ+Δψ,
Δψ(x)=αu(x)ψ(x)+ηn(x),
g=H[ϕ],
g(x)=[h¯ϕ](x)exp[-2iπa(x)].
g=H[ψ]+H[Δψ]=f+Δf.
ϕ(x)=ψ(x)+n(x),
Δψ(x)=n(x).
ϕ(x)=ψ(x)[1+n(x)],
Δψ(x)=ψ(x)n(x).
ϕ(x)=ψ(x)exp[2iπn(x)],
Δψ(x)=ψ(x)[exp[2iπn(x)]-1].
ϕ(x)=ψ(x)[1-w(x)],
Δψ(x)=ψ(x)w(x).
f|g=x=0N-1f(x)g*(x).
|f|2=f|f=x=0N-1|f(x)|2.
f|g=ψ|ϕ.
Δf2=Δψ2.
cod=Δψ2,dec=Δf2.
codo=mincϕ-cψ2,
deco=mindg-df2,
c˜=ψϕψ2,d˜=fgf2.
codo=ϕ2-[ψϕ]2ψ2,
deco=g2-[fg]2f2.
c˜=d˜,codo=deco.
ρcod=10 log10 ϕ2codo,ρdec=10 log10 g2deco.
Ecodo=mincΦ-cΨ2,
Edeco=mincG-cF2=G2-[FG]2F2.
Rdec=5 log10 G2Edeco.
g(x)=λf(x)+n(x),
λ˜=arg minλg-λf2,
En=minλg-λf2.
λ˜=fgf2=ψϕψ2,
En=deco=g2-[fg]2f2=ϕ2-[ψϕ]2ψ2.
n˜(x)=g(x)-λ˜f(x).
C(τ)=x=0N-1n˜(x)n˜*(x-τ).
ρcod=10 log10[1+γ],
Rdec=5 log10 γ2+4γ+22γ+2-1,
γ=λ˜2E2f2σ2,=E4f[E2f]2,
Ekf=1Ni=0N-1fik.
f(x)=H[ψ](x)=[h¯ψ](x)exp[-2iπa(x)],
g(x)=H[ϕ](x)=[h¯ϕ](x)exp[-2iπa(x)].
f|g=x=0N-1[h¯ψ](x)[h¯ϕ]*(x).
f|g=ν=0N-1ψˆ(ν)ϕˆ*(ν).
f|g=ψ|ϕ,
ρcod=ρdec=-10 log101-|f|g|2f2g2.
E[|f|g|2]=λ2[E2f]2+2σ2 E2fN,
E[g|g]=λ2E2f+2σ2,
k,Ekf=x=0N-1fk(x).
λ2[E2f]22σ2 E2fN.
E[|f|g|2]λ2[E2f]2.
ρcod=10 log10[1+γ],
γ=λ2E2f2σ2.
Rdec=5 log10G2Edeco=-5 log101-[F|G]2F2G2.
E[|F|G|2]λ4[E4f]2+4λ2σ2E2fE4f+4σ4[E2f]2,
E[G|G]=λ4E4f+8λ2σ2E2f+8σ4.
Rdec=5 log10 γ2+4γ+22γ+2-1,
=E4f[E2f]2.

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