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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 4 — Feb. 20, 2006
  • pp: 1476–1486
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Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics

Xiaogang Wang, Daomu Zhao, Feng Jing, and Xiaofeng Wei  »View Author Affiliations


Optics Express, Vol. 14, Issue 4, pp. 1476-1486 (2006)
http://dx.doi.org/10.1364/OE.14.001476


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Abstract

A new method is proposed in this paper for the synthesis and encryption of information with digital holography technique and virtual optics. By using a three-step phase-shifting interferometry, the fused or subtracted digital hologram can be calculated from different interference patterns. To protect the digital data that can be transmitted through communication channel, an encryption approach based on virtual optics is also proposed. The encryption method proposed is based on extended fractional Fourier transforms. Both the encryption and decryption processes are performed in all-digital manner. The encrypted data and the synthesized data reconstructed numerically also can be stored and transmitted in the conventional communication channel. Numerical simulation results are given to verify the proposed idea.

© 2006 Optical Society of America

1. Introduction

With the rapid development of internet and computer technique, information security is becoming more and more important. How to protect the synthesized data from unauthorized use and counterfeiting also became an important subject for us. Since Refregier and Javidi proposed the double-random phase encoding technique [7

7. P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995). [CrossRef] [PubMed]

], optical encryption techniques have been attracted increasing interests for their high processing speed and high security level, etc. A number of optical encryption methods have been proposed [8–18

8. 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]

]. Because of physical limitations imposed by optical or electronic hardware, such as the complexity of their optical hardware, lack of flexibility, lack of low-cost optoelectronics devices, etc., most of the encryption methods proposed are limited to image encryption and difficult to be used in real world application. To overcome these problems, several encryption methods based on the concept of virtual optics have also been proposed [12

12. L. Yu and L. Cai, “Multidimensional data encryption with digital holography,” Opt. Commun. 215, 271–284 (2003). [CrossRef]

, 13

13. 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]

, 16

16. H. Kim, D. H. Kim, and Y. H. Lee, “Encryption of digital hologram of 3-D object by virtual optics,” Opt. Exp. 12, 4912–4921 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4912. [CrossRef]

].

In this paper we proposed a simple method to realize information synthesis and data encryption based on digital holography and virtual optics. No special devices such as grating, photosensitive crystal film and no need of filtering operation in the synthesis process. Digital recording of holograms enables us to store, transmit and reconstruct data digitally and has advantages over traditional methods in that no chemical or physical developing is needed after exposure [9

9. E. Tajahuerce, O. Matoba, S. C. Verrall, and B. Javidi, “Optoelectronic information encryption with phase-shifting interferometry,” Appl. Opt. 39, 2313–2320 (2000). [CrossRef]

, 10

10. B. Javidi and T. Nomural, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25, 887–889 (2000). [CrossRef]

, 15

15. N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235, 253–259 (2004). [CrossRef]

]. To eliminate the inconvenience of calibrating phase shifter and increase the flexibility, we use a three-step phase-shifting interferometry [PSI] with arbitrary phase shifts [19

19. L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28, 1808–1810 (2003). [CrossRef] [PubMed]

] to record the synthesized digital hologram. For the purpose of information security, the digital data is to be encrypted by a virtual optical encryption system based on extended fractional Fourier transform [FRT], which was introduced by Hua et al. [20

20. J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14, 3316–3322 (1997). [CrossRef]

] with more parameters compared with the conventional FRT. As a result of the concept of “virtual optics”, there is no need to manufacture costly optical elements, no physical limitations and no difficulty in optical alignment. In this virtual system, we encrypt the input data by use of extended FRTs and computer generated random phase masks. Both the encryption and decryption processes are performed digitally.

2. Principle of information synthesis with digital holography technique

The proposed method for the synthesis of information by complex amplitude addition and subtraction is based on phase-shift digital holography technique. For clarity and simplicity sake, only two 2-D images are used to be synthesized in the following discuss. As shown in Fig. 1, A Fresnel diffraction of the first image can be recorded in CCD plane by use of an interference with an on-axis reference wave. The complex distribution of the object wave U 1(x,y) in CCD plane after Fresnel diffraction can be written as

U1(x,y)=exp(i2πd0λ)d0U0(x0,y0)exp[iπλd0[(xx0)2+(yy0)2]dx0dy0,
(1)

where x 0, y 0 and x, y are the coordinates of object plane and CCD plane, respectively. The function U 0 (x 0, y 0) represents the original object complex wave in object plane. Let the on-axis reference wave in CCD plane at the jth (j = 1, 2, 3) exposure be expressed as

Rj(x,y)=Arexp[i(φr+δj)].
(2)

where δj is the phase shift introduced by phase retarders at the jth exposure, and Ar, φr are the constant amplitude and the phase of reference wave, respectively.

In this three-frame case we have

I1(x,y)=A02(x,y)+Ar2+2A0(x,y)Arcos[φ0(x,y)φrδ1],
(3)
I2(x,y)=A02(x,y)+Ar2+2A0(x,y)Arcos[φ0(x,y)φrδ2],
(4)
I3(x,y)=A02(x,y)+Ar2+2A0(x,y)Arcos[φ0(x,y)φrδ3],
(5)

where we denote U 1(x,y) = A 0(x,y)exp[φ 0(x,y)]. The digital hologram of the first image obtained from the three interference patterns can be expressed as [19

19. L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28, 1808–1810 (2003). [CrossRef] [PubMed]

]

U1(x,y)=14sin[(δ3δ2)2]×{exp[i(δ1+δ2)2]sin[(δ3δ1)2](I1I3)exp[i(δ1+δ3)2]sin[(δ2δ1)2](I1I2)},
(6)

where the constants Ar, φr are replaced with 1 and 0, respectively. In the same way, the corresponding digital hologram of the second image can be expressed as

U2(x,y)=14sin[(δ3δ2)2]×{exp[i(δ1+δ2)2]sin[(δ3δ1)2](I1I3)exp[i(δ1+δ3)2]sin[(δ2δ1)2](I1I2)},
(7)

where I′ 1 , I′ 2, I′ 3 are the intensity distributions of the 1st, 2nd , 3rd interferograms respectively. Calculated from Eq. (6) and Eq. (7), a fused digital hologram can be written as

Uf(x,y)=U1(x,y)+U2(x,y)=14sin[(δ3δ2)2]×{exp[i(δ1+δ2)2]sin[(δ3δ1)2](I1+I1I3I3)exp[i(δ1+δ3)2]sin[(δ2δ1)2](I1+I1I2I2)},
(8)

where the superscript f means the data is obtained during fusion. Similarly, a subtracted digital hologram can also be obtained from Eq. (6) and Eq. (7), which is given by

Us(x,y)=U1(x,y)U2(x,y)=14sin[(δ3δ2)2]×{exp[i(δ1+δ2)2]sin[(δ3δ1)2](I1I3I1+I3)exp[i(δ1+δ3)2]sin[(δ2δ1)2](I1I2I1+I2)},
(9)

where the superscript S means the data is obtained during subtraction process. To obtain the synthesized image, a Fresnel inverse transform of the digital hologram has to be performed digitally. With digital holography technique, one can realize image synthesis by recording digital holograms Uf (x, y) and Us (x, y) . When a half-wave plate and a quarter-wave plate are placed in the reference beam arm to generate phase shifts of 0 , π2 ,π [9

9. E. Tajahuerce, O. Matoba, S. C. Verrall, and B. Javidi, “Optoelectronic information encryption with phase-shifting interferometry,” Appl. Opt. 39, 2313–2320 (2000). [CrossRef]

], the fused digital hologram obtained from Eq. (8

8. 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]

) can be written as

Uf(x,y)=14{(I1+I1I3I3)+i[(2I2I1I3)+(2I2I1I3)]}.
(10)

And the subtracted digital hologram obtained from Eq. (9) can be written as

Us(x,y)=14{(I1I3I1+I3)+i[(2I2I1I3)(2I2I1I3)]}.
(11)
Fig. 1. Phase-shifting digital holography with three-step PSI.

Though the real part and imaginary part of the fused or subtracted digital hologram can be stored and transmitted in communication channel, the fused digital hologram and subtracted digital hologram obtained from the different intensity patterns still need to be encoded for the security of the digital data [16

16. H. Kim, D. H. Kim, and Y. H. Lee, “Encryption of digital hologram of 3-D object by virtual optics,” Opt. Exp. 12, 4912–4921 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4912. [CrossRef]

]. In the following section, the recorded digital hologram will be divided into segments and each segment is to be encrypted independently.

3. Encryption of digital hologram based on the concept of “virtual optics”

Several encryption systems based on extended FRT have been proposed [8

8. 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]

, 14

14. N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase-encrypted memory using cascaded extended fractional Fourier transform,” Opt. Lasers Eng. 42, 141–151 (2004). [CrossRef]

, 18

18. X. Wang, D. Zhao, and L. Chen, “Image encryption based on extended fractional Fourier transform and digital holography technique,” Opt. Commun. (in press). [PubMed]

] to provide enhanced security by increasing the key size compared with those methods based on conventional FRT. In this paper, the encryption system is based on the concept of “virtual optics” and the key size will be enlarged further. As shown in Fig. 2(a), the recorded digital hologram is separated into independent n segments {S 1,S 2,S 3,⋯,Sn } digitally and each segment Sk (k = 1,2,3,⋯n) is extended fractional Fourier transformed two times with a random phase mask placed at the output plane of the first extended FRT. The set of n encoded segments {S1e,S2e,S3e,⋯,Sne } form the encrypted hologram. For clarity and convenience, the encryption process of the kth (k = 1,2,3 ⋯n) segment is used to illustrate this method and a one-dimensional representation is followed in this section. As shown as in Fig. 2(b), let x, x 1 and x 2 denote the coordinates of the input plane and the output plane of the first and second extended FRT, respectively. The complex distribution in output plane can be written as [8

8. 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]

, 14

14. N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase-encrypted memory using cascaded extended fractional Fourier transform,” Opt. Lasers Eng. 42, 141–151 (2004). [CrossRef]

, 18

18. X. Wang, D. Zhao, and L. Chen, “Image encryption based on extended fractional Fourier transform and digital holography technique,” Opt. Commun. (in press). [PubMed]

]

Ske(x2)=gk(x1)exp[iϕk(x1)]exp[iπ(ak2x12+bk2x22)tanφki2πakbksinφkx1x2]dx1,
(12)

gk(x1)=KSk(x)exp[(x)]exp[iπ(ak2x2+bk2x12)tanφki2πakbksinφkxx1]dx.
(13)

ak2=1λk(fk1dk2)12(fk1dk1)121[fk12(fk1dk1)(fk1dk2)]12,
(14)
φk=arccos[(fk1dk1)12(fk1dk2)12fk1],
(15)
bk2=1λk(fk1dk1)12(fk1dk2)121[fk12(fk1dk1)(fk1dk2)]12,
(16)

Fig. 2. (a) The schematic of the digital hologram encryption; (b) the virtual optical setup for encryption of the kth segment based on extended FRT.

4. Decryption of digital hologram

The schematic of decryption system and the virtual optical setup for decryption are as shown in Fig. 3(a) and Fig. 3(b), respectively. The decryption is a reverse operation of the encryption process by taking the conjugate of the encrypted data. The resultant distribution function of decryption is given by

hkd(x)=exp[(x)]×gkd(x1)exp[iϕk(x1)]exp[iπ(bk2x12+ak2x2)tanφki2πbkaksinφkx1x]dx1,
(17)

where the superscript d means the signal is obtained during decryption and gkd (x1) denotes the distribution in the output plane of the first extended FRT during the decryption process that can be given by

gkd(x1)=Sk(x2)exp[iπ(bk2x22+ak2x12)tanφki2πbkaksinφkx2x1]dx2.
(18)

where S′k (x2) = [Ske (x2)]* . By substituting Ske (x 2) from Eq. (12) into Eq. (18) and gkd (x1) from Eq. (18) into Eq. (17), one can show the decrypted function hkd (x′) is equal to Sk (x). With all the correct decrypted digital segments, the fused or subtracted digital hologram can be retrieved.

Fig. 3. (a) The schematic of the digital hologram decryption; (b) the virtual optical setup for decryption of the kth segment.

5. Numerical simulation results

Numerical simulations have been performed to verify the validity of the proposed idea in this paper. Two images with 256×256 pixels are chosen to be synthesized as shown in Fig. 4(a) and Fig. 4(b). The real part and imaginary part of the fused digital hologram are as shown in Fig. 5(a) and Fig. 5(b), respectively. In our numerical simulations, the digital hologram is split into four segments with equality in dimensions. Figure 5(c) and Fig. 5(d) show the real part and imaginary part of one of the four segments respectively. For the encryption of this segment, we set λ1 =520nm, f 1 =17cm, d 1 = 16cm, f 2 = 21cm, d 2=22cm, d 3=24cm, and d 4 =29cm. The parameters of the extended FRT encryption system as calculated from Eqs. (14)–(16) are a 1 =35.4111 + 35.4111i , b 1 =15.8363-15.8363i , φ 1=1.5708-0.1312i, a 2 =39.2154 , b 2 =24.0144 and φ 2 =1.8063 . The real part and imaginary part of the encrypted results are as shown in Fig. 5(e) and Fig. 5(f), respectively. With all the correct keys, the real part and imaginary part of decrypted results of the digital hologram segment are shown in Fig. 5(g) and Fig. 5(h), respectively. Figure 5(i) and Fig. 5(j) show the real part and imaginary part of the decrypted results with correct random phase codes but incorrect parameters (a 1 = 32.2990 + 32.2990i , b 1=16.1495-16.1495i , φ 1=1.5708-0.1174i , a 2 =36.7115 , b 2 = 27.3631 and φ 2 =1.9316 corresponding to the case of λ 1 =560nm, f 1 = 17cm, d 1 = 16cm, f 2 = 19cm, d 2 = 16cm, d 3 =24cm, and d 4 = 28cm). Figure 5(k) and Fig. 5(l) show the real part and imaginary part of the decrypted results with incorrect random phase codes but correct parameters. It can be shown that digital hologram segments cannot be retrieved without knowledge of the respective random phase codes and parameters.

Fig. 4. (a) and (b) The images to be synthesized and encrypted.
Fig. 5. (a) Real part, (b) imaginary part of the recorded fused digital hologram; (c) real part, (d) imaginary part of one of digital hologram segments; (e) real part, (f) imaginary part of one of the encrypted digital hologram segments; decrypted results of one of the encrypted digital hologram segments with all correct keys (g) real part, (h) imaginary part; (i) real part, (j) imaginary part of decrypted results of the digital hologram segment with incorrect parameters but correct random phase codes; (k) real part, (l) imaginary part of decrypted results of the digital hologram segment with correct parameters but incorrect random phase codes.

With all the segments are decrypted correctly, the real part and imaginary part of decrypted digital hologram can be shown in Fig. 6(a) and Fig. 6(b), respectively. Figure 6(c) shows the reconstructed results from the right decrypted data. The corresponding reconstructed image from the wrong decrypted data is as shown in Fig. 6(d).

Fig. 6. (a) Real part, (b) imaginary part of the decrypted digital hologram with correct codes; reconstructed results from (c) correct decrypted digital hologram, (d) incorrect decrypted digital hologram.

The mean square error (MSE) between the decrypted image and the input image is defined as

MSE(I1,I2)=1N×Ni=1Nj=1NI2(i,j)I1(i,j)2
(19)

where I 1(i,j) and I 2(i,j) denote the values of the original image and the decrypted image at the pixel (i, j) , respectively. The MSE between the real part of decrypted digital hologram as shown in Fig. 5(a) and the real part of input digital hologram as shown in Fig. 6(a) is about 3.54×10-12. The corresponding MSE between the imaginary parts as shown in Fig. 5(b) and Fig. 6(b) is about 3.56×10-12.

Figure 7(a) and Fig. 7(b) show the real part and imaginary part of the encrypted results of the subtracted digital hologram. From the right decrypted image of the digital hologram, the subtracted image can be reconstructed digitally as shown in Fig. 7(c).

Fig. 7. (a) Real part, (b) imaginary part of the encrypted results of the subtracted digital hologram; (c) reconstructed results from correct decrypted digital hologram.

Three images, Figs. 8(a), 8(b) and 8(c), are chosen to be synthesis and encrypted. The real part and imaginary part of the encrypted fused digital hologram and the encrypted subtracted digital hologram are as shown in Figs. 8(d), 8(e), 8(f) and 8(g), respectively. In the same way, the reconstructed result from the correct decrypted fused digital hologram can be shown in Fig. 8(h). Figure 8(i) shows the corresponding synthesized image by subtracting Figs. 8(b) and 8(c) from Fig. 8(a).

Fig. 8. (a), (b) and (c) The images to be synthesized and encrypted; (d) real part, (e) imaginary part of encrypted fused digital hologram; (f) real part, (g) imaginary part of encrypted subtracted digital hologram; synthesized results reconstructed from (h) right decrypted fused digital hologram, (i) right decrypted subtracted digital hologram.

6. Conclusions

Acknowledgment

This work was supported by the joint foundation of National Natural Science Foundation of China and the Chinese Academy of Engineering Physics under grant 10276034.

References and links

1.

D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm, “Optical image synthesis (complex amplitude addition and subtraction) by holographic Fourier transformation,” Phys. Lett. 18, 116–118 (1965). [CrossRef]

2.

J. F. Ebersole, “Optical image subtraction,” Opt. Eng. 14, 436–447 (1975).

3.

F. T. S. Yu and A. Tai, “Image subtraction with an encoded extended incoherent source,” Appl. Opt. 20, 4082–4088 (1981). [CrossRef] [PubMed]

4.

A. E. Chiou and P. Yeh, “Parallel image subtraction using a phase-conjugate Michelson interferometer,” Opt. Lett. 11, 306–308 (1986) [CrossRef] [PubMed]

5.

S. Zhivkova and M. Miteva, “Image subtraction using fixed holograms in photorefractive Bi12TiO20 crystals,” Opt. Lett. 16, 750–751 (1991). [CrossRef] [PubMed]

6.

M. Y. Shih, A. Shishido, and I. C. Khoo, “All-optical image processing by means of a photosensitive nonlinear liquid-crystal film: edge enhancement and image addition subtraction,” Opt. Lett. 26, 1140–1142 (2001). [CrossRef]

7.

P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995). [CrossRef] [PubMed]

8.

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]

9.

E. Tajahuerce, O. Matoba, S. C. Verrall, and B. Javidi, “Optoelectronic information encryption with phase-shifting interferometry,” Appl. Opt. 39, 2313–2320 (2000). [CrossRef]

10.

B. Javidi and T. Nomural, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25, 887–889 (2000). [CrossRef]

11.

N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption using a localized fractional Fourier transform,” Opt. Eng. 42, 3566–3571 (2003). [CrossRef]

12.

L. Yu and L. Cai, “Multidimensional data encryption with digital holography,” Opt. Commun. 215, 271–284 (2003). [CrossRef]

13.

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]

14.

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase-encrypted memory using cascaded extended fractional Fourier transform,” Opt. Lasers Eng. 42, 141–151 (2004). [CrossRef]

15.

N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235, 253–259 (2004). [CrossRef]

16.

H. Kim, D. H. Kim, and Y. H. Lee, “Encryption of digital hologram of 3-D object by virtual optics,” Opt. Exp. 12, 4912–4921 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4912. [CrossRef]

17.

L. Chen and D. Zhao, “Optical image encryption based on fractional wavelet transform,” Opt. Commun. 254, 361–367 (2005). [CrossRef]

18.

X. Wang, D. Zhao, and L. Chen, “Image encryption based on extended fractional Fourier transform and digital holography technique,” Opt. Commun. (in press). [PubMed]

19.

L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28, 1808–1810 (2003). [CrossRef] [PubMed]

20.

J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14, 3316–3322 (1997). [CrossRef]

OCIS Codes
(070.6020) Fourier optics and signal processing : Continuous optical signal processing
(090.0090) Holography : Holography
(100.2000) Image processing : Digital image processing

ToC Category:
Holography

History
Original Manuscript: November 15, 2005
Revised Manuscript: January 10, 2006
Manuscript Accepted: February 3, 2006
Published: February 20, 2006

Citation
Xiaogang Wang, Daomu Zhao, Feng Jing, and Xiaofeng Wei, "Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics," Opt. Express 14, 1476-1486 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-4-1476


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References

  1. D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm, "Optical image synthesis (complex amplitude addition and subtraction) by holographic Fourier transformation," Phys. Lett. 18, 116-118 (1965). [CrossRef]
  2. J. F. Ebersole, "Optical image subtraction," Opt. Eng. 14, 436-447 (1975).
  3. F. T. S. Yu and A. Tai, "Image subtraction with an encoded extended incoherent source," Appl. Opt. 20, 4082-4088 (1981). [CrossRef] [PubMed]
  4. A. E. Chiou and P. Yeh, "Parallel image subtraction using a phase-conjugate Michelson interferometer," Opt. Lett. 11, 306-308 (1986) [CrossRef] [PubMed]
  5. S. Zhivkova and M. Miteva, "Image subtraction using fixed holograms in photorefractive Bi12TiO20 crystals," Opt. Lett. 16, 750-751 (1991). [CrossRef] [PubMed]
  6. M. Y. Shih, A. Shishido, and I. C. Khoo, "All-optical image processing by means of a photosensitive nonlinear liquid-crystal film: edge enhancement and image addition subtraction," Opt. Lett. 26, 1140-1142 (2001). [CrossRef]
  7. P. Refregier and B. Javidi, "Optical image encryption based on input plane and Fourier plane random encoding," Opt. Lett. 20, 767-769 (1995). [CrossRef] [PubMed]
  8. 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]
  9. E. Tajahuerce, O. Matoba, S. C. Verrall, and B. Javidi, "Optoelectronic information encryption with phase-shifting interferometry," Appl. Opt. 39, 2313-2320 (2000). [CrossRef]
  10. B. Javidi and T. Nomural, "Optical encryption by double-random phase encoding in the fractional Fourier domain," Opt. Lett. 25, 887-889 (2000). [CrossRef]
  11. N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, "Optical encryption using a localized fractional Fourier transform," Opt. Eng. 42, 3566-3571 (2003). [CrossRef]
  12. L. Yu and L. Cai, "Multidimensional data encryption with digital holography," Opt. Commun. 215, 271-284 (2003). [CrossRef]
  13. 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]
  14. N. K. Nishchal, J. Joseph, and K. Singh, "Fully phase-encrypted memory using cascaded extended fractional Fourier transform," Opt. Lasers Eng. 42, 141-151 (2004). [CrossRef]
  15. N. K. Nishchal, J. Joseph, and K. Singh, "Securing information using fractional Fourier transform in digital holography," Opt. Commun. 235, 253-259 (2004). [CrossRef]
  16. H. Kim, D. H. Kim, and Y. H. Lee, "Encryption of digital hologram of 3-D object by virtual optics," Opt. Express 12, 4912-4921 (2004). [CrossRef]
  17. L. Chen and D. Zhao, "Optical image encryption based on fractional wavelet transform," Opt. Commun. 254, 361-367 (2005). [CrossRef]
  18. X. Wang, D. Zhao, and L. Chen, "Image encryption based on extended fractional Fourier transform and digital holography technique," Opt. Commun. (in press). [PubMed]
  19. L. Z. Cai, Q. Liu, and X. L. Yang, "Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps," Opt. Lett. 28, 1808-1810 (2003). [CrossRef] [PubMed]
  20. J. Hua, L. Liu, and G. Li, "Extended fractional Fourier transforms," J. Opt. Soc. Am. A 14, 3316-3322 (1997). [CrossRef]

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