Abstract

We present a simple technique to encrypt a digital hologram of a three-dimensional (3-D) object into a stationary white noise by use of virtual optics and then to decrypt it digitally. In this technique the digital hologram is encrypted by our attaching a computer-generated random phase key to it and then forcing them to Fresnel propagate to an arbitrary plane with an illuminating plane wave of a given wavelength. It is shown in experiments that the proposed system is robust to blind decryptions without knowing the correct propagation distance, wavelength, and phase key used in the encryption. Signal-to-noise ratio (SNR) and mean-square-error (MSE) of the reconstructed 3-D object are calculated for various decryption distances and wavelengths, and partial use of the correct phase key.

© 2004 Optical Society of America

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References

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Appl. Opt. (5)

Opt. Commun. (5)

L. Yu, X. Peng, and L. Cai, �??Parameterized multi-dimensional data encryption by digital optics,�?? Opt. Commun. 203, 67-77 (2002).
[CrossRef]

X. Peng, Z. Cui, and T. Tan, �??Information encryption with virtual-optics imaging system,�?? Opt. Commun. 212, 235-245 (2002).
[CrossRef]

L. Yu and L. Cai, �??Multidimensional data encryption with digital holography,�?? Opt. Commun. 215, 271- 284 (2003).
[CrossRef]

X. Peng, L. Yu, and L. Cai, �??Digital watermarking in three-dimensional space with a virtual-optics imaging modality,�?? Opt. Commun. 226, 155-165 (2003
[CrossRef]

S. Lai and M. A. Neifeld, �??Digital wavefront reconstruction and its application to image encryption,�?? Opt. Commun. 178, 283-289 (2000).
[CrossRef]

Opt. Lett. (7)

Optik (1)

B. Zhu, H. Zhao, and S. Liu, �??Image encryption based on pure intensity random coding and digital holography technique,�?? Optik 114, 95-99 (2003).
[CrossRef]

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Figures (10)

Fig. 1.
Fig. 1.

Phase-shifting digital holography. BS’s, Beamsplitter; M’s, Mirror.

Fig. 2.
Fig. 2.

Encryption of digital hologram of a 3-D object by virtual optics. Digital hologram is multiplied by a computer-generated random phase key at CCD plane and then Fresnel propagation of this distorted hologram over a distance de is calculated on a computer.

Fig. 3.
Fig. 3.

Decryption of encrypted digital hologram and reconstruction of 3-D object. Both the decryption and the reconstruction are performed digitally on a computer.

Fig. 4.
Fig. 4.

Digital hologram of a 3-D object. (a) Real part, (b) imaginary part.

Fig. 5.
Fig. 5.

Reconstructed 3-D objects at (a) dr = 250 cm and (b) dr = 150 cm.

Fig. 6.
Fig. 6.

(a) Real part, (b) imaginary part of the encrypted digital hologram; (c) the autocorrelation of the encrypted digital hologram; histograms of (d) real part, (e) imaginary part of the encrypted digital hologram.

Fig. 7.
Fig. 7.

(a) SNR and MSE versus reconstruction distance in blind decryptions without knowing the correct distances, wavelengths, and phase key. (b) SNR and MSE versus percentage of correct phase key used in the decryption, where the correct distances and wavelengths were known.

Fig. 8.
Fig. 8.

Reconstructed 3-D objects when the portion of the correct phase key used in the decryption was (a) 1%, (b) 5%, and (c) 10%.

Fig. 9.
Fig. 9.

Reconstructed 3-D objects when the distance-wavelength product error was (a) 0.01%, (b) 0.05%, and (c) 0.2% in the decryption.

Fig. 10.
Fig. 10.

SNR and MSE versus (a) percent error of λede and (b) percent error of λrdo .

Equations (28)

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U H x y = d o Δ 2 d o + Δ 2 { exp [ j 2 πz λ r ] ( j λ r z ) } exp [ ( x 2 + y 2 ) λ r z ] ×
U o ( x o , y o ; z ) exp [ ( x o 2 + y o 2 ) λ r z ] exp [ j 2 π ( xx o + yy o ) λ r z ] dz dx o dy o
FrT { U o ( x o , y o ; z ) } z = d o ± Δ 2 ,
A H x y exp [ H x y ] .
R ( x , y ; α ) = A R exp [ j ( ϕ R + α ) ] ,
I ( x , y ; α ) = U H ( x , y ) + R ( x , y ; α ) 2
= A H 2 ( x , y ) + A R 2 + 2 A H ( x , y ) A R cos [ ϕ H x y ϕ R α ] .
U H x y = 1 4 [ I ( x , y ; 0 ) I ( x , y ; π ) + j { I ( x , y ; 3 π 2 ) I ( x , y ; π 2 ) } ] .
U E ( x e , y e ) = { exp [ j 2 πd e λ e ] ( e d e ) } exp [ ( x e 2 + y e 2 ) λ e d e ] ×
∫∫ U H x y exp [ E x y ] exp [ ( x 2 + y 2 ) λ e d e ] exp [ j 2 π ( xx e + yy e ) λ e d e ] dxdy ,
E { U E * ( x e , y e ) U E ( x e + p , y e + q ) } = [ 1 ( λ e d e ) 2 ξ = 0 N x 1 η = 0 N y 1 U H ( ξ , η ) 2 ] δ p q ,
U R ( x r , y r ; z = d r 2 ) = IFrT { IFrT { U E ( x e , y e ) } z = d r 1 × exp [ D ( x , y ) ] } z = d r 2 ,
SNR = x = 0 N x 1 y = 0 N y 1 U O ( x , y ; z = d o ) 2 x = 0 N x 1 y = 0 N y 1 [ U O ( x , y ; z = d o ) U R ( x , y ; z = d r 2 ) ] 2 ,
MSE = x = 0 N x 1 y = 0 N y 1 [ U O ( x , y ; z = d o ) U R ( x , y ; z = d r 2 ) ] 2 N x × N y ,
U E ( x e , y e ) = ξ = 0 N x 1 η = 0 N y 1 U H ξ η exp [ j ϕ E ξ η ] h ( x e ξ , y e η ) ,
h x y = exp [ j 2 π d e λ e ] j λ e d e exp [ ( x 2 + y 2 ) λ e d e ] .
E { U E * ( x e , y e ) U E ( x e + p , y e + q ) } = ξ = 0 N x 1 η = 0 N y 1 α = 0 N x 1 β = 0 N y 1 U H * ξ η U H α β
× E { exp [ j ϕ E α β j ϕ E ξ η ] } h * ( x e ξ , y e η ) h ( x e + p α , y e + q β ) .
E { U E * ( x e , y e ) U E ( x e + p , y e + q ) } = ξ = 0 N x 1 η = 0 N y 1 U H * ξ η U H ξ η
× h * ( x e ξ , y e η ) h ( x e + p ξ , y e + q η ) .
h * ( x e ξ , y e η ) h ( x e + p ξ , y e + q η ) = exp [ ( p 2 + q 2 ) λ e d e ] ( λ e d e ) 2
× exp [ j 2 π { ( x e ξ ) p + ( y e η ) q } λ e d e ] .
E { U E * ( x e , y e ) U E ( x e + p , y e + q ) } = exp [ ( p 2 + q 2 ) λ e d e ] ( λ e d e ) 2 exp [ j 2 π ( x e p + y e q ) λ e d e ]
× ξ = 0 N x 1 η = 0 N y 1 U H ξ η 2 exp [ j 2 π ( ξp ηq ) λ e d e ] .
E { U E * ( x e , y e ) U E ( x e + p , y e + q ) } = exp [ ( p 2 + q 2 ) λ e d e ] ( λ e d e ) 2 exp [ j 2 π ( x e p + y e q ) λ e d e ]
× [ ξ = 0 N x 1 η = 0 N y 1 U H ( ξ , η ) 2 ] δ ( p , q )
= [ 1 ( λ e d e ) 2 ξ = 0 N x 1 η = 0 N y 1 U H ξ η 2 ] δ p q .
[ 1 ( λ e d e ) 2 ξ = 0 N x 1 ξ = 0 N y 1 U H ( ξ , η ) 2 ] .

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