Abstract

We perform a numerical analysis of the double random phase encryption–decryption technique to determine how, in the case of both amplitude and phase encoding, the two decryption keys (the image- and Fourier-plane keys) affect the output gray-scale image when they are in error. We perform perfect encryption and imperfect decryption. We introduce errors into the decrypting keys that correspond to the use of random distributions of incorrect pixel values. We quantify the effects that increasing amounts of error in the image-plane key, the Fourier-plane key, and both keys simultaneously have on the decrypted image. Quantization effects are also examined.

© 2008 Optical Society of America

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References

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2007 (2)

2006 (2)

2004 (1)

B. M. Hennelly and J. T. Sheridan, “Random phase and jigsaw encryption in the Fresnel domain,” Opt. Eng. 43, 2239-2249(2004).
[CrossRef]

2003 (1)

B. M. Hennelly and J. T. Sheridan, “Image encryption and the fractional Fourier transform,” Optik (Jena) 114, 251-265(2003).
[CrossRef]

2000 (4)

1998 (1)

1997 (1)

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

1995 (1)

1994 (1)

1967 (1)

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

Bollaro, F.

Castro, A.

Frauel, Y.

Goodman, J. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

Gopinathan, U.

Goudail, F.

Guibert, L.

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

Hennelly, B. M.

B. M. Hennelly and J. T. Sheridan, “Random phase and jigsaw encryption in the Fresnel domain,” Opt. Eng. 43, 2239-2249(2004).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Image encryption and the fractional Fourier transform,” Optik (Jena) 114, 251-265(2003).
[CrossRef]

Javidi, B.

Joseph, J.

Juptner, W.

Lawrence, R. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

Maghzi, N.

Monaghan, D. S.

Naughton, T. J.

Pohit, M.

G. Unnikrishnan, M. Pohit, and K. Singh, “A polarization encoded optical encryption system using ferroelectric spatial light modulator,” Opt. Commun. 185, 25-31 (2000).
[CrossRef]

Pratt, W. K.

W. K. Pratt, Digital Image Processing, 4th ed. (Wiley-Interscience, 2007).
[CrossRef]

Réfrégier, P.

Schnars, U.

Sergent, A.

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

Sheridan, J. T.

Singh, K.

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]

G. Unnikrishnan, M. Pohit, and K. Singh, “A polarization encoded optical encryption system using ferroelectric spatial light modulator,” Opt. Commun. 185, 25-31 (2000).
[CrossRef]

Situ, G.

Tajahuerce, E.

Towghi, N.

Unnikrishnan, G.

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]

G. Unnikrishnan, M. Pohit, and K. Singh, “A polarization encoded optical encryption system using ferroelectric spatial light modulator,” Opt. Commun. 185, 25-31 (2000).
[CrossRef]

Verrall, S. C.

Zhang, G. S.

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

Appl. Opt. (5)

Appl. Phys. Lett. (1)

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

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

Opt. Commun. (1)

G. Unnikrishnan, M. Pohit, and K. Singh, “A polarization encoded optical encryption system using ferroelectric spatial light modulator,” Opt. Commun. 185, 25-31 (2000).
[CrossRef]

Opt. Eng. (2)

B. M. Hennelly and J. T. Sheridan, “Random phase and jigsaw encryption in the Fresnel domain,” Opt. Eng. 43, 2239-2249(2004).
[CrossRef]

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

Opt. Express (2)

Opt. Lett. (2)

Optik (Jena) (1)

B. M. Hennelly and J. T. Sheridan, “Image encryption and the fractional Fourier transform,” Optik (Jena) 114, 251-265(2003).
[CrossRef]

Other (3)

“Lena test image,” http://sipi.usc.edu/database/.

Matlab 7.0.1, http://www.mathworks.com/.

W. K. Pratt, Digital Image Processing, 4th ed. (Wiley-Interscience, 2007).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

Possible optical implementation of DRPE. OFT, optical fourier transform. f A ( x , y ) is the input image, a ( x , y ) and b ( u , v ) are the phase keys, and Ψ ( x , y ) is the encrypted output image.

Fig. 2
Fig. 2

Block diagram of the similar encryption processes for AE and PE and the different decryption processes that lead to an NRMS value for a decrypted image.

Fig. 3
Fig. 3

AE compared with PE. The horizontal axis shows the percentage of pixels that are in error in the Fourier-plane key, b ( u , v ) . The vertical axis shows the NRMS error as described in Section 2. With NRMS = 0 the decryption is perfect.

Fig. 4
Fig. 4

The curve marked by the triangle represents AE. The other three curves look at PE for the cases when the phase keys, a ( x , y ) and b ( u , v ) , are in error separately and at the same time. The dashed lines refer to Figs. 6, 7, 8, which show examples of output images at these error levels.

Fig. 5
Fig. 5

Enlarged graph of a portion of Fig. 4; note the scale on the axes. Superimposed on the AE case is the curve that shows the results for ten runs by using different phase keys and is then averaged (grey dots).

Fig. 6
Fig. 6

Results when 2.59% of phase key pixels are in error. (a) AE case with R 2 in error and NRMS 0.33 . (b)–(d) PE case with a NRMS of (b) 0.17, (c) 0.20, and (d) 0.17. In (b)  R 2 is in error. In (c)  R 1 is in error, and in (d) both R 1 and R 2 are in error.

Fig. 7
Fig. 7

As in Fig. 6 but with 10.07% of the key pixels in error. (a) AE case NRMS of 0.57. (b)–(d) PE case with a NRMS of (a) 0.41, (b) 0.39, and (c) 0.37.

Fig. 8
Fig. 8

As in Fig. 6, 7 but with 30.52% of the key pixels in error. (a) AE case NRMS of 0.71. (b)–(d) PE case with a NRMS of (a) 0.73, (b) 0.64, and (c) 0.67.

Fig. 9
Fig. 9

The AE case and the three PE cases where the decrypting phase-keys are requantized at increasingly lower levels, plotted against the resultant NRMS error.

Equations (6)

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NRMS = ( i = 1 N j = 1 N | I d ( i , j ) I ( i , j ) | 2 ) / ( i = 1 N j = 1 N | I ( i , j ) | 2 ) .
Ψ ( x , y ) = J { J { f A ( x , y ) exp [ + i 2 π a ( x , y ) ] } exp [ + i 2 π b ( u , v ) ] } .
f A ( x , y ) = J 1 { J 1 { Ψ ( x , y ) } exp [ i 2 π b ( u , v ) ] } exp [ i 2 π a ( x , y ) ] .
f P ( x , y ) = exp [ + i 2 π f A ( x , y ) ] ,
Ψ ( x , y ) = J { J { exp [ + i 2 π f A ( x , y ) ] exp [ + i 2 π a ( x , y ) ] } exp [ + i 2 π b ( u , v ) ] } ,
| f P ( x , y ) | = | Arg ( J 1 { J 1 { Ψ ( x , y ) } exp [ i 2 π b ( x , y ) ] } exp [ 2 π a ( u , v ) ] ) | ,

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