Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Encryption of digital hologram of 3-D object by virtual optics

Hyun Kim, Do-Hyung Kim, and Yeon H. Lee

Open Access Open Access

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

Full Article  |  PDF Article
More Like This
Optimal watermarking of digital hologram of 3-D object

Hyun Kim and Yeon H. Lee
Opt. Express 13(8) 2881-2886 (2005)

Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics

Xiaogang Wang, Daomu Zhao, Feng Jing, and Xiaofeng Wei
Opt. Express 14(4) 1476-1486 (2006)

Encrypting three-dimensional information with digital holography

Enrique Tajahuerce and Bahram Javidi
Appl. Opt. 39(35) 6595-6601 (2000)

View More...

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)
Fig. 1.
Fig. 1. Phase-shifting digital holography. BS’s, Beamsplitter; M’s, Mirror.

Download Full Size | PDF

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.

Download Full Size | PDF

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.

Download Full Size | PDF

Fig. 4.
Fig. 4. Digital hologram of a 3-D object. (a) Real part, (b) imaginary part.

Download Full Size | PDF

Fig. 5.
Fig. 5. Reconstructed 3-D objects at (a) dr = 250 cm and (b) dr = 150 cm.

Download Full Size | PDF

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.

Download Full Size | PDF

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.

Download Full Size | PDF

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%.

Download Full Size | PDF

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.

Download Full Size | PDF

Fig. 10.
Fig. 10. SNR and MSE versus (a) percent error of λede and (b) percent error of λrdo .

Download Full Size | PDF

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

U H x y = ∫ ∫ ∫ d o − Δ 2 d o + Δ 2 { exp [ j 2 πz λ r ] ( j λ r z ) } exp [ jπ ( x 2 + y 2 ) λ r z ] ×
U o ( x o , y o ; z ) exp [ jπ ( 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 [ jϕ 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 ] ( jλ e d e ) } exp [ jπ ( x e 2 + y e 2 ) λ e d e ] ×
∫∫ ∑ U H x y exp [ jϕ E x y ] exp [ jπ ( 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 [ − jϕ 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 [ jπ ( 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 [ jπ ( 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 [ jπ ( 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 [ jπ ( 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 ] .
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use