Abstract

A novel approach for silhouette-free image encryption based on interference is proposed using discrete multiple-parameter fractional Fourier transform (DMPFrFT), which generalizes from fractional Fourier transform. An original image is firstly applied by chaotic pixel scrambling (CPS) and then encoded into the real part of a complex signal. Using interference principle, the complex signal generates three phase-only masks in DMPFrFT domain. The silhouette of the original image cannot be extracted using one or two of the three phase-only masks. The parameters of both CPS and DMPFrFT can also serve as encryption keys to extend the key space, which further enhance the level of cryptosystem security. Numerical simulations are demonstrated to show the feasibility and validity of this approach.

© 2017 Optical Society of America

1. Introduction

As the greater requirement of image sharing in the contemporary society, image security has caused intense interest in different walks of life. Since Javidi et al. [1] employed double random phases encoding (DRPE) to encrypt an image into white stationary noise, optical encryption methods have been playing increasing roles in application of image security due to their parallelism, high speed and wide degrees of freedom [2]. To improve the security of the cryptosystems, DRPE-based configurations have been extended from Fourier transform (FT) domain to various transform domain, including fractional Fourier transform (FrFT) [3], Fresnel transform (FRT) [4], gyrator transform [5], fractional angular transform [6], fractional random transform [7] and discrete multiple-parameter fractional Fourier transform (DMPFrFT) [8,9]. However, these DRPE-based schemes usually require holographic system to record the complex valued encoded ciphertext, which lead the information storage and transmission to be difficult. Besides, the DRPE-based schemes are also demonstrated with low endurance to several attacks because of inherent linearity [10–13]. The first problem can be solved using phase retrieval methods to iteratively encode the original image [14,15], but they come at the expense of time-consuming. To decrease the linearity, permutation-diffusion processes have been employed using different chaotic maps [16–18], but they complicate the optical cryptosystems. Phase-truncated Fourier transforms (PTFTs) can eliminate the linearity of DRPE completely, but the initial scheme is also vulnerable to a specific attack [19]. Much work has also been done to improve the security of PTFTs, but they also make the encoding scheme more complicated [20]. Recently, Zhang and Wang proposed an encryption approach using interference to encode a plain image into two phase only masks (POMs), which benefits from simplicity and non-iterative operation [21]. Much work has been done on interference-based encryption methods [22–25]. However, the silhouette appears even if only one of the two POMs is utilized during the verification process. Different approaches have been developed to overcome the silhouette problem [26–29], one of which encodes the original image into three POMs with one generated randomly and the other two generated analytically [30]. However, despite significant progress, higher security remains a grand challenge in the image optical encryption using interference.

In this paper, we present a new interference-based image encryption method using CPS and DMPFrFT to generate three POMs. Any information relative to the original image cannot be observed even one or two of the POMs are used for decryption. Furthermore, the key space are improved significantly due to the introduction of CPS and DMPFrFT compared with the traditional optical encryption using interference [21–23].

2. Principle

In an encryption system using interference, an original image o(u, v) to be encrypted is always transferred to a complex value image o′(u, v) [21] firstly as follows.

o(u,v)=o(u,v)exp[iM(u,v)]
where o(u, v) is a nonnegative distribution, and M(u, v) is a random phase distribution in the range of [0, 2π].

To further improve the security of the system, an operation P{a0, λ, t} [8] of chaotic pixel scrambling (CPS) is firstly applied to the original image, and Eq. (1) can then be rewritten as

o(u,v)=P{a0,λ,t}[o(u,v)]exp[iM(u,v)]
where a0 is the initial value located in the range of [0, 1], λ is the coefficient of the map located in the range of [3.57,4], and t is truncated position.

To overcome the inherent characteristics of the silhouette problem in the interference-based encryption system, the encrypted image can be encrypted into three POMs [30] of M1(x, y), M2(x, y) and M3(x, y), where M1(x, y) and M2(x, y) are generated analytically, and M3(x, y) is generated randomly in the range of [0, 2π]. But unlike the previous methods are done in the domain of FRT [21], FT [22] or FrFT [23], our interference is done in the DMPFrFT domain. In terms of the FRT, FT or FrFT, the DMPFrFT has more parameters to provide larger key space and more diversified signal expression, which has been shown better performance and security in image encryption in conventional encryption method [8,9]. When the three POMs are illuminated by a plane wave simultaneously, interference in the image plane can be expressed as

o(u,v)=F(ML,MR)(αL,αR)(n'L,n'R){exp[iM1(x,y)]}+F(ML,MR)(αL,αR)(n'L,n'R){exp[iM2(x,y)]}+F(ML,MR)(αL,αR)(n'L,n'R){exp[iM3(x,y)]}
where F(•) stands for DMPFrFT operation with parameters of (ML,MR; αL, αR; mL, nL; mR, nR), (αL, αR) stands for fractional order, (ML, MR) stands for periodicity, (nL, nR) stands for vector parameter, and n′ is defined as
nk=(kmk+Mmknk+nk)k=0,1,2,,(M1)
and m = (m0, m1, m(M-1))∊ℤM, n = (n0, n1, n(M-1))∊ℤM, and M is arbitrary integer of > 2.

Let us rewrite Eq. (3) so that to obtain a relationship defining M2(x, y) by M1(x, y), M3(x, y) and o′(m, n) as follows

exp[iM2(x,y)]=F-1(ML,MR)(αL,αR)(nL,nR)[o(u,v)]exp[iM3(x,y)]exp[iM1(x,y)]
Since M2(x, y) is phase-only elements, we can obtain
1=|F-1(ML,MR)(αL,αR)(nL,nR)[o'(u,v)]exp[iM3(x,y)]exp[iM1(x,y)]|2={F-1(ML,MR)(αL,αR)(nL,nR)[o'(u,v)]exp[iM3(x,y)]exp[iM1(x,y)]}{F-1(ML,MR)(αL,αR)(nL,nR)[o'(u,v)]exp[iM3(x,y)]exp[iM1(x,y)]}
where * represents the conjugate operation.

Combining Eqs. (3) and (6), we can achieve the phase distributions of M1(x, y) and M2(x, y) as

M1(x,y)=arg{F-1(ML,MR)(αL,αR)(nL,nR)[o(u,v)]exp[iM3(x,y)]}arccos{abs{[F-1(ML,MR)(αL,αR)(nL,nR)[o(u,v)]exp[iM3(x,y)]]/2}}
M2(x,y)=arg{F-1(ML,MR)(αL,αR)(nL,nR)[o(u,v)]exp[iM3(x,y)]exp[iM1(x,y)]}
where arg[•] represents the phase angles of the complex function and abs[•] represents the complex modulus.

In our proposed optical encryption system, the encryption is processed digitally in a computer, while the decryption can be performed digitally or optically. Figure 1 shows the optoelectronic setup for the proposed decryption procedure in DMPFrFT domain. In the setup, SLM1 is employed to generate the summation of the three POMs. SLM2 and Lens are employed to carry out the DMPFrFT [9]. A laser is collimated and expanded by a beam collimator & expander (BCE) to generate coherent parallel light beams. The light is modulated by SLM1, and then transformed by SLM2 and Lens. The scrambled image can be recorded by CCD and transferred to a computer. The decrypted image can be finally obtained after the inverse CPS is digitally implemented in the computer.

 

Fig. 1 Optoelectronic schematic for decryption. BCE, beam collimator and expander; SLM1 and SLM2, spatial light modulator.

Download Full Size | PPT Slide | PDF

In our algorithm, the three POMs can be sent to three different holders to achieve high security. Even if the attackers illegally obtain one or two of the POMs, they cannot retrieve any information about the original image, including its silhouette. Especially, the parameters of both CPS such as initial value a0, coefficient λ, truncated position t, and DMPFrFT such as the periodicities (ML, MR), transform orders (αL, αR) and vector parameters (mL, nL; mR, nR) can also be used as the encryption keys. When the three POMs are illuminated by the plane wave with correct parameters, the plaintext can be achieved in the output plane. To assess the quality of the decrypted image, we use the normalized mean square error (NMSE) [9] as follows

NMSE=y=1px=1q[ID(x,y)IO(x,y)]2/y=1px=1q[IO(x,y)]2
where Io(x, y) is the values of the original image and ID(x, y) is that of the decryption image, and p and q are the size of the images.

3. Results

Numerical simulations have been done to validate the proposed method. Figure 2(a) shows an original image with 256 × 256 pixels size and 256 Gy levels, and Figs. 2(b) and 2(c) show its corresponding POMs of M1 and M2 generated using the proposed encryption approach. The results indicate that the image can be completely encoded, and no any valuable information related to the image can be recognized. The POM of M3 is also generated using CPS, and its distribution is shown in Fig. 2(d). During the encryption process, the parameters of CPS are set as {a0, λ, t} = {0.241, 3.95, 3500}, and those of DMPFrFT are set as (αL,αR; ML,MR) = (0.35, 0.74; 13, 22). The vector parameters (mL, nL) and (mR, nR) are 1 × 13 and 1 × 22 random vectors that contain independent integer values, respectively. Figure 2(e) shows the scrambled-image recorded in CCD and Fig. 2(f) shows the corresponding decrypted image when the correct keys are applied. The decrypted image can be identified obviously without any doubt, and the corresponding NMSE value is 7.9384 × 10−27. But it should be noted that since DMPFrFT is a periodic function, the image can be retrieved with another periodicity when it is fractional transformed with a particular periodicity.

 

Fig. 2 (a) The original image, (b)-(d) the POMs of M1, M2 and M3, (e) the scrambled-image recorded in CCD, (f) the decrypted image.

Download Full Size | PPT Slide | PDF

Next, we illustrate the performance of the security achieved by our proposed method. Figure 3 show the influence of the deviation of fractional order in DMPFrFT on the decrypted image. The results indicate show that our approach is highly sensitive to the fractional order. Any slightly deviation, almost (1 × 10−7, 0.5 × 10−7) from (αL, αR), can make the decrypted image unrecognized. But for the approach using FrFT [23], the deviation from the fractional orders should be ≥0.08.

 

Fig. 3 The decrypted image with (a) (αL + 1 × 10−7, αR) and (b) (αL, αR + 0.5 × 10−7), (c) the curve of NMSE with incorrect fractional order.

Download Full Size | PPT Slide | PDF

Figures 4 and 5 show the decrypted image extracted by wrong periodicity of (ML, MR) and vector parameters (mL, nL; mR, nR), respectively. The same analyses are also performed on the parameters of CPS, and the extremely sensitive results are shown in Figs. 6-8. It’s no doubt that the information about the original image can hardly be extracted when any error occurred in the parameters of DMPFrFT and/or CPS. All those enable our approach to realize higher security hierarchy in comparison to the conventional approaches.

 

Fig. 4 The decrypted image with (a) ML = 14 and (b) MR = 23

Download Full Size | PPT Slide | PDF

 

Fig. 5 The decrypted image with (a) mL + 1, (b) nL + 1,(c) mR + 1 and (d) nR + 1.

Download Full Size | PPT Slide | PDF

 

Fig. 6 (a) The decrypted image with a0 + 0.7 × 10−16, (b) the curve of NMSE.

Download Full Size | PPT Slide | PDF

 

Fig. 7 (a) The decrypted image with λ + 0.3 × 10−15, (b) the curve of NMSE.

Download Full Size | PPT Slide | PDF

 

Fig. 8 (a) The decrypted image with t + 125, (b) the curve of NMSE.

Download Full Size | PPT Slide | PDF

We further demonstrate the silhouette-free capability of our approach using only one or two of the three POMs during the decryption process, and summarized their results in Fig. 9. The corresponding NMSE values are 0.7843, 0.7809, 0.7816, 0.7839, 0.7792 0.7832 respectively. All the retrieved images show fully noise, and no silhouette information leave. Noted that all the parameters of the DMPFrFT and CPS are selected randomly. We have tried ten different combinations of the parameters for all the above analyses and can obtain the same results.

 

Fig. 9 The decrypted image with (a) M1, (b) M2, (c) M3, (d) M1 and M2, (e) M1 and M3, and (f) M2 and M3.

Download Full Size | PPT Slide | PDF

4. Conclusion

In conclusion, we have developed an interference-based image encryption method using CPS and DMPFrFT. The silhouette problem has been thoroughly removed using the summarization of two POMs generated analytically and one POM generated randomly. In our scheme, the original image is firstly scrambled and encoded into a complex signal, and then encoded into the three POMs using DMPFrFT. Numerical results show that our approach has extremely sensitivity to the keys. Of course, we realize this at the cost of storage space for the use of three POMs. However, in contrast to the approaches used two POMs together with phase retrieval algorithm [31], our approach can reduce computational load and time efficiently. Thus, we hope that our new method will be highest security and most practical in applications of interference-based image encryption methods.

Funding

National Natural Science Foundation of China (61377009); Major National Scientific Instrument and Equipment Development Project of China (2013YQ290489); China Scholarship Council (201506685053); Heilongjiang Science Foundation of China (F201411); and Fundamental Research Funds for the Central Universities of China.

References and links

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

2. W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photonics 6(6), 120–155 (2014). [CrossRef]  

3. S. Liansheng, X. Meiting, and T. Ailing, “Multiple-image encryption based on phase mask multiplexing in fractional Fourier transform domain,” Opt. Lett. 38(11), 1996–1998 (2013). [CrossRef]   [PubMed]  

4. S. K. Rajput and N. K. Nishchal, “Known-plaintext attack-based optical cryptosystem using phase-truncated Fresnel transform,” Appl. Opt. 52(4), 871–878 (2013). [CrossRef]   [PubMed]  

5. J. X. Chen, Z. L. Zhu, Z. Liu, C. Fu, L. B. Zhang, and H. Yu, “A novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains,” Opt. Express 22(6), 7349–7361 (2014). [CrossRef]   [PubMed]  

6. L. Sui, K. Duan, and J. Liang, “Double-image encryption based on discrete multiple-parameter fractional angular transform and two-coupled logistic maps,” Opt. Commun. 343, 140–149 (2015). [CrossRef]  

7. L. Sui, K. Duan, J. Liang, and X. Hei, “Asymmetric double-image encryption based on cascaded discrete fractional random transform and logistic maps,” Opt. Express 22(9), 10605–10621 (2014). [CrossRef]   [PubMed]  

8. M. Shan, J. Chang, Z. Zhong, and B. Hao, “Double image encryption based on discrete multiple-parameter fractional Fourier transform and chaotic maps,” Opt. Commun. 285(21–22), 4227–4234 (2012). [CrossRef]  

9. Z. Zhong, Y. Zhang, M. Shan, Y. Wang, Y. Zhang, and H. Xie, “Optical movie encryption based on a discrete multiple-parameter fractional Fourier transform,” J. Opt. 16(12), 125404 (2014). [CrossRef]  

10. A. Carnicer, M. Montes-Usategui, S. Arcos, and I. Juvells, “Vulnerability to chosen-cyphertext attacks of optical encryption schemes based on double random phase keys,” Opt. Lett. 30(13), 1644–1646 (2005). [CrossRef]   [PubMed]  

11. X. Peng, H. Wei, and P. Zhang, “Chosen-plaintext attack on lensless double-random phase encoding in the Fresnel domain,” Opt. Lett. 31(22), 3261–3263 (2006). [CrossRef]   [PubMed]  

12. X. Peng, P. Zhang, H. Wei, and B. Yu, “Known-plaintext attack on optical encryption based on double random phase keys,” Opt. Lett. 31(8), 1044–1046 (2006). [CrossRef]   [PubMed]  

13. Z. Liu, C. Shen, J. Tan, and S. Liu, “A recovery method of double random phase encoding system with a parallel phase retrieval,” IEEE Photonics 8(1), 1–7 (2016). [CrossRef]  

14. H. Di, Y. Kang, Y. Liu, and X. Zhang, “Multiple image encryption by phase retrieval,” Opt. Eng. 55(7), 73–103 (2016). [CrossRef]  

15. C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Lasers Eng. 89, 2–12 (2016). [CrossRef]  

16. H. Liu and X. Wang, “Color image encryption using spatial bit-level permutation and high-dimension chaotic system,” Opt. Commun. 284(16–17), 3895–3903 (2011). [CrossRef]  

17. J. X. Chen, Z. L. Zhu, C. Fu, H. Yu, and L. B. Zhang, “An efficient image encryption scheme using gray code based permutation approach,” Opt. Lasers Eng. 67, 191–204 (2015). [CrossRef]  

18. X. Wang, L. Liu, and Y. Zhang, “A novel chaotic block image encryption algorithm based on dynamic random growth technique,” Opt. Lasers Eng. 66, 10–18 (2015). [CrossRef]  

19. X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285(6), 1078–1081 (2012). [CrossRef]  

20. W. Qin and X. Peng, “Asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Lett. 35(2), 118–120 (2010). [CrossRef]   [PubMed]  

21. Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008). [CrossRef]   [PubMed]  

22. D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284(10), 2485–2487 (2011). [CrossRef]  

23. W. Chen and X. Chen, “Security-enhanced interference-based optical image encryption,” Opt. Commun. 286(1), 123–129 (2013).

24. W. Liu, Z. Liu, and S. Liu, “Optical security validation using Michelson interferometer,” Appl. Opt. 54(7), 1802–1807 (2015). [CrossRef]  

25. D. Lu, W. He, M. Liao, and X. Peng, “Discussion and a new method of optical cryptosystem based on interference,” Opt. Lasers Eng. 89, 13–21 (2016). [CrossRef]  

26. Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A, Pure Appl. Opt. 11(12), 125406 (2009). [CrossRef]  

27. P. Kumar, J. Joseph, and K. Singh, “Optical image encryption using a jigsaw transform for silhouette removal in interference-based methods and decryption with a single spatial light modulator,” Appl. Opt. 50(13), 1805–1811 (2011). [CrossRef]   [PubMed]  

28. X. Wang and D. Zhao, “Optical image hiding with silhouette removal based on the optical interference principle,” Appl. Opt. 51(6), 686–691 (2012). [CrossRef]   [PubMed]  

29. Q. Gong, Z. Wang, X. Lv, and Y. Qin, “Interference-based image encryption with silhouette removal by aid of compressive sensing,” Opt. Commun. 359, 290–296 (2016). [CrossRef]  

30. Q. Wang, “Optical image encryption with silhouette removal based on interference and phase blend processing,” Opt. Commun. 285(21–22), 4294–4301 (2012). [CrossRef]  

31. W. Chen and X. Chen, “Iterative phase retrieval for simultaneously generating two phase-only masks with silhouette removal in interference-based optical encryption,” Opt. Commun. 331(331), 133–138 (2014). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995).
    [Crossref] [PubMed]
  2. W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photonics 6(6), 120–155 (2014).
    [Crossref]
  3. S. Liansheng, X. Meiting, and T. Ailing, “Multiple-image encryption based on phase mask multiplexing in fractional Fourier transform domain,” Opt. Lett. 38(11), 1996–1998 (2013).
    [Crossref] [PubMed]
  4. S. K. Rajput and N. K. Nishchal, “Known-plaintext attack-based optical cryptosystem using phase-truncated Fresnel transform,” Appl. Opt. 52(4), 871–878 (2013).
    [Crossref] [PubMed]
  5. J. X. Chen, Z. L. Zhu, Z. Liu, C. Fu, L. B. Zhang, and H. Yu, “A novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains,” Opt. Express 22(6), 7349–7361 (2014).
    [Crossref] [PubMed]
  6. L. Sui, K. Duan, and J. Liang, “Double-image encryption based on discrete multiple-parameter fractional angular transform and two-coupled logistic maps,” Opt. Commun. 343, 140–149 (2015).
    [Crossref]
  7. L. Sui, K. Duan, J. Liang, and X. Hei, “Asymmetric double-image encryption based on cascaded discrete fractional random transform and logistic maps,” Opt. Express 22(9), 10605–10621 (2014).
    [Crossref] [PubMed]
  8. M. Shan, J. Chang, Z. Zhong, and B. Hao, “Double image encryption based on discrete multiple-parameter fractional Fourier transform and chaotic maps,” Opt. Commun. 285(21–22), 4227–4234 (2012).
    [Crossref]
  9. Z. Zhong, Y. Zhang, M. Shan, Y. Wang, Y. Zhang, and H. Xie, “Optical movie encryption based on a discrete multiple-parameter fractional Fourier transform,” J. Opt. 16(12), 125404 (2014).
    [Crossref]
  10. A. Carnicer, M. Montes-Usategui, S. Arcos, and I. Juvells, “Vulnerability to chosen-cyphertext attacks of optical encryption schemes based on double random phase keys,” Opt. Lett. 30(13), 1644–1646 (2005).
    [Crossref] [PubMed]
  11. X. Peng, H. Wei, and P. Zhang, “Chosen-plaintext attack on lensless double-random phase encoding in the Fresnel domain,” Opt. Lett. 31(22), 3261–3263 (2006).
    [Crossref] [PubMed]
  12. X. Peng, P. Zhang, H. Wei, and B. Yu, “Known-plaintext attack on optical encryption based on double random phase keys,” Opt. Lett. 31(8), 1044–1046 (2006).
    [Crossref] [PubMed]
  13. Z. Liu, C. Shen, J. Tan, and S. Liu, “A recovery method of double random phase encoding system with a parallel phase retrieval,” IEEE Photonics 8(1), 1–7 (2016).
    [Crossref]
  14. H. Di, Y. Kang, Y. Liu, and X. Zhang, “Multiple image encryption by phase retrieval,” Opt. Eng. 55(7), 73–103 (2016).
    [Crossref]
  15. C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Lasers Eng. 89, 2–12 (2016).
    [Crossref]
  16. H. Liu and X. Wang, “Color image encryption using spatial bit-level permutation and high-dimension chaotic system,” Opt. Commun. 284(16–17), 3895–3903 (2011).
    [Crossref]
  17. J. X. Chen, Z. L. Zhu, C. Fu, H. Yu, and L. B. Zhang, “An efficient image encryption scheme using gray code based permutation approach,” Opt. Lasers Eng. 67, 191–204 (2015).
    [Crossref]
  18. X. Wang, L. Liu, and Y. Zhang, “A novel chaotic block image encryption algorithm based on dynamic random growth technique,” Opt. Lasers Eng. 66, 10–18 (2015).
    [Crossref]
  19. X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285(6), 1078–1081 (2012).
    [Crossref]
  20. W. Qin and X. Peng, “Asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Lett. 35(2), 118–120 (2010).
    [Crossref] [PubMed]
  21. Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008).
    [Crossref] [PubMed]
  22. D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284(10), 2485–2487 (2011).
    [Crossref]
  23. W. Chen and X. Chen, “Security-enhanced interference-based optical image encryption,” Opt. Commun. 286(1), 123–129 (2013).
  24. W. Liu, Z. Liu, and S. Liu, “Optical security validation using Michelson interferometer,” Appl. Opt. 54(7), 1802–1807 (2015).
    [Crossref]
  25. D. Lu, W. He, M. Liao, and X. Peng, “Discussion and a new method of optical cryptosystem based on interference,” Opt. Lasers Eng. 89, 13–21 (2016).
    [Crossref]
  26. Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A, Pure Appl. Opt. 11(12), 125406 (2009).
    [Crossref]
  27. P. Kumar, J. Joseph, and K. Singh, “Optical image encryption using a jigsaw transform for silhouette removal in interference-based methods and decryption with a single spatial light modulator,” Appl. Opt. 50(13), 1805–1811 (2011).
    [Crossref] [PubMed]
  28. X. Wang and D. Zhao, “Optical image hiding with silhouette removal based on the optical interference principle,” Appl. Opt. 51(6), 686–691 (2012).
    [Crossref] [PubMed]
  29. Q. Gong, Z. Wang, X. Lv, and Y. Qin, “Interference-based image encryption with silhouette removal by aid of compressive sensing,” Opt. Commun. 359, 290–296 (2016).
    [Crossref]
  30. Q. Wang, “Optical image encryption with silhouette removal based on interference and phase blend processing,” Opt. Commun. 285(21–22), 4294–4301 (2012).
    [Crossref]
  31. W. Chen and X. Chen, “Iterative phase retrieval for simultaneously generating two phase-only masks with silhouette removal in interference-based optical encryption,” Opt. Commun. 331(331), 133–138 (2014).
    [Crossref]

2016 (5)

Z. Liu, C. Shen, J. Tan, and S. Liu, “A recovery method of double random phase encoding system with a parallel phase retrieval,” IEEE Photonics 8(1), 1–7 (2016).
[Crossref]

H. Di, Y. Kang, Y. Liu, and X. Zhang, “Multiple image encryption by phase retrieval,” Opt. Eng. 55(7), 73–103 (2016).
[Crossref]

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Lasers Eng. 89, 2–12 (2016).
[Crossref]

D. Lu, W. He, M. Liao, and X. Peng, “Discussion and a new method of optical cryptosystem based on interference,” Opt. Lasers Eng. 89, 13–21 (2016).
[Crossref]

Q. Gong, Z. Wang, X. Lv, and Y. Qin, “Interference-based image encryption with silhouette removal by aid of compressive sensing,” Opt. Commun. 359, 290–296 (2016).
[Crossref]

2015 (4)

W. Liu, Z. Liu, and S. Liu, “Optical security validation using Michelson interferometer,” Appl. Opt. 54(7), 1802–1807 (2015).
[Crossref]

J. X. Chen, Z. L. Zhu, C. Fu, H. Yu, and L. B. Zhang, “An efficient image encryption scheme using gray code based permutation approach,” Opt. Lasers Eng. 67, 191–204 (2015).
[Crossref]

X. Wang, L. Liu, and Y. Zhang, “A novel chaotic block image encryption algorithm based on dynamic random growth technique,” Opt. Lasers Eng. 66, 10–18 (2015).
[Crossref]

L. Sui, K. Duan, and J. Liang, “Double-image encryption based on discrete multiple-parameter fractional angular transform and two-coupled logistic maps,” Opt. Commun. 343, 140–149 (2015).
[Crossref]

2014 (5)

L. Sui, K. Duan, J. Liang, and X. Hei, “Asymmetric double-image encryption based on cascaded discrete fractional random transform and logistic maps,” Opt. Express 22(9), 10605–10621 (2014).
[Crossref] [PubMed]

Z. Zhong, Y. Zhang, M. Shan, Y. Wang, Y. Zhang, and H. Xie, “Optical movie encryption based on a discrete multiple-parameter fractional Fourier transform,” J. Opt. 16(12), 125404 (2014).
[Crossref]

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photonics 6(6), 120–155 (2014).
[Crossref]

J. X. Chen, Z. L. Zhu, Z. Liu, C. Fu, L. B. Zhang, and H. Yu, “A novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains,” Opt. Express 22(6), 7349–7361 (2014).
[Crossref] [PubMed]

W. Chen and X. Chen, “Iterative phase retrieval for simultaneously generating two phase-only masks with silhouette removal in interference-based optical encryption,” Opt. Commun. 331(331), 133–138 (2014).
[Crossref]

2013 (3)

2012 (4)

M. Shan, J. Chang, Z. Zhong, and B. Hao, “Double image encryption based on discrete multiple-parameter fractional Fourier transform and chaotic maps,” Opt. Commun. 285(21–22), 4227–4234 (2012).
[Crossref]

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285(6), 1078–1081 (2012).
[Crossref]

X. Wang and D. Zhao, “Optical image hiding with silhouette removal based on the optical interference principle,” Appl. Opt. 51(6), 686–691 (2012).
[Crossref] [PubMed]

Q. Wang, “Optical image encryption with silhouette removal based on interference and phase blend processing,” Opt. Commun. 285(21–22), 4294–4301 (2012).
[Crossref]

2011 (3)

D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284(10), 2485–2487 (2011).
[Crossref]

P. Kumar, J. Joseph, and K. Singh, “Optical image encryption using a jigsaw transform for silhouette removal in interference-based methods and decryption with a single spatial light modulator,” Appl. Opt. 50(13), 1805–1811 (2011).
[Crossref] [PubMed]

H. Liu and X. Wang, “Color image encryption using spatial bit-level permutation and high-dimension chaotic system,” Opt. Commun. 284(16–17), 3895–3903 (2011).
[Crossref]

2010 (1)

2009 (1)

Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A, Pure Appl. Opt. 11(12), 125406 (2009).
[Crossref]

2008 (1)

2006 (2)

2005 (1)

1995 (1)

Ailing, T.

Arcos, S.

Carnicer, A.

Chang, J.

M. Shan, J. Chang, Z. Zhong, and B. Hao, “Double image encryption based on discrete multiple-parameter fractional Fourier transform and chaotic maps,” Opt. Commun. 285(21–22), 4227–4234 (2012).
[Crossref]

Chen, J. X.

J. X. Chen, Z. L. Zhu, C. Fu, H. Yu, and L. B. Zhang, “An efficient image encryption scheme using gray code based permutation approach,” Opt. Lasers Eng. 67, 191–204 (2015).
[Crossref]

J. X. Chen, Z. L. Zhu, Z. Liu, C. Fu, L. B. Zhang, and H. Yu, “A novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains,” Opt. Express 22(6), 7349–7361 (2014).
[Crossref] [PubMed]

Chen, K.

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Lasers Eng. 89, 2–12 (2016).
[Crossref]

Chen, W.

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photonics 6(6), 120–155 (2014).
[Crossref]

W. Chen and X. Chen, “Iterative phase retrieval for simultaneously generating two phase-only masks with silhouette removal in interference-based optical encryption,” Opt. Commun. 331(331), 133–138 (2014).
[Crossref]

W. Chen and X. Chen, “Security-enhanced interference-based optical image encryption,” Opt. Commun. 286(1), 123–129 (2013).

Chen, X.

W. Chen and X. Chen, “Iterative phase retrieval for simultaneously generating two phase-only masks with silhouette removal in interference-based optical encryption,” Opt. Commun. 331(331), 133–138 (2014).
[Crossref]

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photonics 6(6), 120–155 (2014).
[Crossref]

W. Chen and X. Chen, “Security-enhanced interference-based optical image encryption,” Opt. Commun. 286(1), 123–129 (2013).

Di, H.

H. Di, Y. Kang, Y. Liu, and X. Zhang, “Multiple image encryption by phase retrieval,” Opt. Eng. 55(7), 73–103 (2016).
[Crossref]

Dong, Z.

Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A, Pure Appl. Opt. 11(12), 125406 (2009).
[Crossref]

Duan, K.

L. Sui, K. Duan, and J. Liang, “Double-image encryption based on discrete multiple-parameter fractional angular transform and two-coupled logistic maps,” Opt. Commun. 343, 140–149 (2015).
[Crossref]

L. Sui, K. Duan, J. Liang, and X. Hei, “Asymmetric double-image encryption based on cascaded discrete fractional random transform and logistic maps,” Opt. Express 22(9), 10605–10621 (2014).
[Crossref] [PubMed]

Fu, C.

J. X. Chen, Z. L. Zhu, C. Fu, H. Yu, and L. B. Zhang, “An efficient image encryption scheme using gray code based permutation approach,” Opt. Lasers Eng. 67, 191–204 (2015).
[Crossref]

J. X. Chen, Z. L. Zhu, Z. Liu, C. Fu, L. B. Zhang, and H. Yu, “A novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains,” Opt. Express 22(6), 7349–7361 (2014).
[Crossref] [PubMed]

Gong, Q.

Q. Gong, Z. Wang, X. Lv, and Y. Qin, “Interference-based image encryption with silhouette removal by aid of compressive sensing,” Opt. Commun. 359, 290–296 (2016).
[Crossref]

Guo, C.

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Lasers Eng. 89, 2–12 (2016).
[Crossref]

Hao, B.

M. Shan, J. Chang, Z. Zhong, and B. Hao, “Double image encryption based on discrete multiple-parameter fractional Fourier transform and chaotic maps,” Opt. Commun. 285(21–22), 4227–4234 (2012).
[Crossref]

He, W.

D. Lu, W. He, M. Liao, and X. Peng, “Discussion and a new method of optical cryptosystem based on interference,” Opt. Lasers Eng. 89, 13–21 (2016).
[Crossref]

Hei, X.

Javidi, B.

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photonics 6(6), 120–155 (2014).
[Crossref]

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

Joseph, J.

Juvells, I.

Kang, Y.

H. Di, Y. Kang, Y. Liu, and X. Zhang, “Multiple image encryption by phase retrieval,” Opt. Eng. 55(7), 73–103 (2016).
[Crossref]

Kumar, P.

Liang, J.

L. Sui, K. Duan, and J. Liang, “Double-image encryption based on discrete multiple-parameter fractional angular transform and two-coupled logistic maps,” Opt. Commun. 343, 140–149 (2015).
[Crossref]

L. Sui, K. Duan, J. Liang, and X. Hei, “Asymmetric double-image encryption based on cascaded discrete fractional random transform and logistic maps,” Opt. Express 22(9), 10605–10621 (2014).
[Crossref] [PubMed]

Liansheng, S.

Liao, M.

D. Lu, W. He, M. Liao, and X. Peng, “Discussion and a new method of optical cryptosystem based on interference,” Opt. Lasers Eng. 89, 13–21 (2016).
[Crossref]

Liu, H.

H. Liu and X. Wang, “Color image encryption using spatial bit-level permutation and high-dimension chaotic system,” Opt. Commun. 284(16–17), 3895–3903 (2011).
[Crossref]

Liu, J.

D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284(10), 2485–2487 (2011).
[Crossref]

Liu, L.

X. Wang, L. Liu, and Y. Zhang, “A novel chaotic block image encryption algorithm based on dynamic random growth technique,” Opt. Lasers Eng. 66, 10–18 (2015).
[Crossref]

Liu, S.

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Lasers Eng. 89, 2–12 (2016).
[Crossref]

Z. Liu, C. Shen, J. Tan, and S. Liu, “A recovery method of double random phase encoding system with a parallel phase retrieval,” IEEE Photonics 8(1), 1–7 (2016).
[Crossref]

W. Liu, Z. Liu, and S. Liu, “Optical security validation using Michelson interferometer,” Appl. Opt. 54(7), 1802–1807 (2015).
[Crossref]

Liu, W.

Liu, Y.

H. Di, Y. Kang, Y. Liu, and X. Zhang, “Multiple image encryption by phase retrieval,” Opt. Eng. 55(7), 73–103 (2016).
[Crossref]

Liu, Z.

Z. Liu, C. Shen, J. Tan, and S. Liu, “A recovery method of double random phase encoding system with a parallel phase retrieval,” IEEE Photonics 8(1), 1–7 (2016).
[Crossref]

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Lasers Eng. 89, 2–12 (2016).
[Crossref]

W. Liu, Z. Liu, and S. Liu, “Optical security validation using Michelson interferometer,” Appl. Opt. 54(7), 1802–1807 (2015).
[Crossref]

J. X. Chen, Z. L. Zhu, Z. Liu, C. Fu, L. B. Zhang, and H. Yu, “A novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains,” Opt. Express 22(6), 7349–7361 (2014).
[Crossref] [PubMed]

Lu, D.

D. Lu, W. He, M. Liao, and X. Peng, “Discussion and a new method of optical cryptosystem based on interference,” Opt. Lasers Eng. 89, 13–21 (2016).
[Crossref]

Lv, X.

Q. Gong, Z. Wang, X. Lv, and Y. Qin, “Interference-based image encryption with silhouette removal by aid of compressive sensing,” Opt. Commun. 359, 290–296 (2016).
[Crossref]

Meiting, X.

Montes-Usategui, M.

Nishchal, N. K.

Peng, X.

Qin, W.

Qin, Y.

Q. Gong, Z. Wang, X. Lv, and Y. Qin, “Interference-based image encryption with silhouette removal by aid of compressive sensing,” Opt. Commun. 359, 290–296 (2016).
[Crossref]

Rajput, S. K.

Refregier, P.

Shan, M.

Z. Zhong, Y. Zhang, M. Shan, Y. Wang, Y. Zhang, and H. Xie, “Optical movie encryption based on a discrete multiple-parameter fractional Fourier transform,” J. Opt. 16(12), 125404 (2014).
[Crossref]

M. Shan, J. Chang, Z. Zhong, and B. Hao, “Double image encryption based on discrete multiple-parameter fractional Fourier transform and chaotic maps,” Opt. Commun. 285(21–22), 4227–4234 (2012).
[Crossref]

Shen, C.

Z. Liu, C. Shen, J. Tan, and S. Liu, “A recovery method of double random phase encoding system with a parallel phase retrieval,” IEEE Photonics 8(1), 1–7 (2016).
[Crossref]

Singh, K.

Sui, L.

L. Sui, K. Duan, and J. Liang, “Double-image encryption based on discrete multiple-parameter fractional angular transform and two-coupled logistic maps,” Opt. Commun. 343, 140–149 (2015).
[Crossref]

L. Sui, K. Duan, J. Liang, and X. Hei, “Asymmetric double-image encryption based on cascaded discrete fractional random transform and logistic maps,” Opt. Express 22(9), 10605–10621 (2014).
[Crossref] [PubMed]

Tan, J.

Z. Liu, C. Shen, J. Tan, and S. Liu, “A recovery method of double random phase encoding system with a parallel phase retrieval,” IEEE Photonics 8(1), 1–7 (2016).
[Crossref]

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Lasers Eng. 89, 2–12 (2016).
[Crossref]

Wang, B.

Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A, Pure Appl. Opt. 11(12), 125406 (2009).
[Crossref]

Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008).
[Crossref] [PubMed]

Wang, Q.

Q. Wang, “Optical image encryption with silhouette removal based on interference and phase blend processing,” Opt. Commun. 285(21–22), 4294–4301 (2012).
[Crossref]

Wang, X.

X. Wang, L. Liu, and Y. Zhang, “A novel chaotic block image encryption algorithm based on dynamic random growth technique,” Opt. Lasers Eng. 66, 10–18 (2015).
[Crossref]

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285(6), 1078–1081 (2012).
[Crossref]

X. Wang and D. Zhao, “Optical image hiding with silhouette removal based on the optical interference principle,” Appl. Opt. 51(6), 686–691 (2012).
[Crossref] [PubMed]

H. Liu and X. Wang, “Color image encryption using spatial bit-level permutation and high-dimension chaotic system,” Opt. Commun. 284(16–17), 3895–3903 (2011).
[Crossref]

Wang, Y.

Z. Zhong, Y. Zhang, M. Shan, Y. Wang, Y. Zhang, and H. Xie, “Optical movie encryption based on a discrete multiple-parameter fractional Fourier transform,” J. Opt. 16(12), 125404 (2014).
[Crossref]

D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284(10), 2485–2487 (2011).
[Crossref]

Wang, Z.

Q. Gong, Z. Wang, X. Lv, and Y. Qin, “Interference-based image encryption with silhouette removal by aid of compressive sensing,” Opt. Commun. 359, 290–296 (2016).
[Crossref]

Wei, C.

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Lasers Eng. 89, 2–12 (2016).
[Crossref]

Wei, H.

Weng, D.

D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284(10), 2485–2487 (2011).
[Crossref]

Wu, Q.

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Lasers Eng. 89, 2–12 (2016).
[Crossref]

Xie, H.

Z. Zhong, Y. Zhang, M. Shan, Y. Wang, Y. Zhang, and H. Xie, “Optical movie encryption based on a discrete multiple-parameter fractional Fourier transform,” J. Opt. 16(12), 125404 (2014).
[Crossref]

Xie, J.

D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284(10), 2485–2487 (2011).
[Crossref]

Yu, B.

Yu, H.

J. X. Chen, Z. L. Zhu, C. Fu, H. Yu, and L. B. Zhang, “An efficient image encryption scheme using gray code based permutation approach,” Opt. Lasers Eng. 67, 191–204 (2015).
[Crossref]

J. X. Chen, Z. L. Zhu, Z. Liu, C. Fu, L. B. Zhang, and H. Yu, “A novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains,” Opt. Express 22(6), 7349–7361 (2014).
[Crossref] [PubMed]

Zhang, L. B.

J. X. Chen, Z. L. Zhu, C. Fu, H. Yu, and L. B. Zhang, “An efficient image encryption scheme using gray code based permutation approach,” Opt. Lasers Eng. 67, 191–204 (2015).
[Crossref]

J. X. Chen, Z. L. Zhu, Z. Liu, C. Fu, L. B. Zhang, and H. Yu, “A novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains,” Opt. Express 22(6), 7349–7361 (2014).
[Crossref] [PubMed]

Zhang, P.

Zhang, X.

H. Di, Y. Kang, Y. Liu, and X. Zhang, “Multiple image encryption by phase retrieval,” Opt. Eng. 55(7), 73–103 (2016).
[Crossref]

Zhang, Y.

X. Wang, L. Liu, and Y. Zhang, “A novel chaotic block image encryption algorithm based on dynamic random growth technique,” Opt. Lasers Eng. 66, 10–18 (2015).
[Crossref]

Z. Zhong, Y. Zhang, M. Shan, Y. Wang, Y. Zhang, and H. Xie, “Optical movie encryption based on a discrete multiple-parameter fractional Fourier transform,” J. Opt. 16(12), 125404 (2014).
[Crossref]

Z. Zhong, Y. Zhang, M. Shan, Y. Wang, Y. Zhang, and H. Xie, “Optical movie encryption based on a discrete multiple-parameter fractional Fourier transform,” J. Opt. 16(12), 125404 (2014).
[Crossref]

Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A, Pure Appl. Opt. 11(12), 125406 (2009).
[Crossref]

Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008).
[Crossref] [PubMed]

Zhao, D.

X. Wang and D. Zhao, “Optical image hiding with silhouette removal based on the optical interference principle,” Appl. Opt. 51(6), 686–691 (2012).
[Crossref] [PubMed]

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285(6), 1078–1081 (2012).
[Crossref]

Zhong, Z.

Z. Zhong, Y. Zhang, M. Shan, Y. Wang, Y. Zhang, and H. Xie, “Optical movie encryption based on a discrete multiple-parameter fractional Fourier transform,” J. Opt. 16(12), 125404 (2014).
[Crossref]

M. Shan, J. Chang, Z. Zhong, and B. Hao, “Double image encryption based on discrete multiple-parameter fractional Fourier transform and chaotic maps,” Opt. Commun. 285(21–22), 4227–4234 (2012).
[Crossref]

Zhu, N.

D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284(10), 2485–2487 (2011).
[Crossref]

Zhu, Z. L.

J. X. Chen, Z. L. Zhu, C. Fu, H. Yu, and L. B. Zhang, “An efficient image encryption scheme using gray code based permutation approach,” Opt. Lasers Eng. 67, 191–204 (2015).
[Crossref]

J. X. Chen, Z. L. Zhu, Z. Liu, C. Fu, L. B. Zhang, and H. Yu, “A novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains,” Opt. Express 22(6), 7349–7361 (2014).
[Crossref] [PubMed]

Adv. Opt. Photonics (1)

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photonics 6(6), 120–155 (2014).
[Crossref]

Appl. Opt. (4)

IEEE Photonics (1)

Z. Liu, C. Shen, J. Tan, and S. Liu, “A recovery method of double random phase encoding system with a parallel phase retrieval,” IEEE Photonics 8(1), 1–7 (2016).
[Crossref]

J. Opt. (1)

Z. Zhong, Y. Zhang, M. Shan, Y. Wang, Y. Zhang, and H. Xie, “Optical movie encryption based on a discrete multiple-parameter fractional Fourier transform,” J. Opt. 16(12), 125404 (2014).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A, Pure Appl. Opt. 11(12), 125406 (2009).
[Crossref]

Opt. Commun. (9)

Q. Gong, Z. Wang, X. Lv, and Y. Qin, “Interference-based image encryption with silhouette removal by aid of compressive sensing,” Opt. Commun. 359, 290–296 (2016).
[Crossref]

Q. Wang, “Optical image encryption with silhouette removal based on interference and phase blend processing,” Opt. Commun. 285(21–22), 4294–4301 (2012).
[Crossref]

W. Chen and X. Chen, “Iterative phase retrieval for simultaneously generating two phase-only masks with silhouette removal in interference-based optical encryption,” Opt. Commun. 331(331), 133–138 (2014).
[Crossref]

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285(6), 1078–1081 (2012).
[Crossref]

D. Weng, N. Zhu, Y. Wang, J. Xie, and J. Liu, “Experimental verification of optical image encryption based on interference,” Opt. Commun. 284(10), 2485–2487 (2011).
[Crossref]

W. Chen and X. Chen, “Security-enhanced interference-based optical image encryption,” Opt. Commun. 286(1), 123–129 (2013).

H. Liu and X. Wang, “Color image encryption using spatial bit-level permutation and high-dimension chaotic system,” Opt. Commun. 284(16–17), 3895–3903 (2011).
[Crossref]

L. Sui, K. Duan, and J. Liang, “Double-image encryption based on discrete multiple-parameter fractional angular transform and two-coupled logistic maps,” Opt. Commun. 343, 140–149 (2015).
[Crossref]

M. Shan, J. Chang, Z. Zhong, and B. Hao, “Double image encryption based on discrete multiple-parameter fractional Fourier transform and chaotic maps,” Opt. Commun. 285(21–22), 4227–4234 (2012).
[Crossref]

Opt. Eng. (1)

H. Di, Y. Kang, Y. Liu, and X. Zhang, “Multiple image encryption by phase retrieval,” Opt. Eng. 55(7), 73–103 (2016).
[Crossref]

Opt. Express (2)

Opt. Lasers Eng. (4)

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Lasers Eng. 89, 2–12 (2016).
[Crossref]

J. X. Chen, Z. L. Zhu, C. Fu, H. Yu, and L. B. Zhang, “An efficient image encryption scheme using gray code based permutation approach,” Opt. Lasers Eng. 67, 191–204 (2015).
[Crossref]

X. Wang, L. Liu, and Y. Zhang, “A novel chaotic block image encryption algorithm based on dynamic random growth technique,” Opt. Lasers Eng. 66, 10–18 (2015).
[Crossref]

D. Lu, W. He, M. Liao, and X. Peng, “Discussion and a new method of optical cryptosystem based on interference,” Opt. Lasers Eng. 89, 13–21 (2016).
[Crossref]

Opt. Lett. (7)

Cited By

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

Alert me when this article is cited.


Figures (9)

Fig. 1
Fig. 1 Optoelectronic schematic for decryption. BCE, beam collimator and expander; SLM1 and SLM2, spatial light modulator.
Fig. 2
Fig. 2 (a) The original image, (b)-(d) the POMs of M1, M2 and M3, (e) the scrambled-image recorded in CCD, (f) the decrypted image.
Fig. 3
Fig. 3 The decrypted image with (a) (αL + 1 × 10−7, αR) and (b) (αL, αR + 0.5 × 10−7), (c) the curve of NMSE with incorrect fractional order.
Fig. 4
Fig. 4 The decrypted image with (a) ML = 14 and (b) MR = 23
Fig. 5
Fig. 5 The decrypted image with (a) mL + 1, (b) nL + 1,(c) mR + 1 and (d) nR + 1.
Fig. 6
Fig. 6 (a) The decrypted image with a0 + 0.7 × 10−16, (b) the curve of NMSE.
Fig. 7
Fig. 7 (a) The decrypted image with λ + 0.3 × 10−15, (b) the curve of NMSE.
Fig. 8
Fig. 8 (a) The decrypted image with t + 125, (b) the curve of NMSE.
Fig. 9
Fig. 9 The decrypted image with (a) M1, (b) M2, (c) M3, (d) M1 and M2, (e) M1 and M3, and (f) M2 and M3.

Equations (9)

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

o ( u , v ) = o ( u , v ) exp [ i M ( u , v ) ]
o ( u , v ) = P { a 0 , λ , t } [ o ( u , v ) ] exp [ i M ( u , v ) ]
o ( u , v ) = F ( M L , M R ) ( α L , α R ) ( n ' L , n ' R ) { exp [ i M 1 ( x , y ) ] } + F ( M L , M R ) ( α L , α R ) ( n ' L , n ' R ) { exp [ i M 2 ( x , y ) ] } + F ( M L , M R ) ( α L , α R ) ( n ' L , n ' R ) { exp [ i M 3 ( x , y ) ] }
n k = ( k m k + M m k n k + n k ) k = 0 , 1 , 2 , , ( M 1 )
exp [ i M 2 ( x , y ) ] = F - 1 ( M L , M R ) ( α L , α R ) ( n L , n R ) [ o ( u , v ) ] exp [ i M 3 ( x , y ) ] exp [ i M 1 ( x , y ) ]
1 = | F - 1 ( M L , M R ) ( α L , α R ) ( n L , n R ) [ o ' ( u , v ) ] exp [ i M 3 ( x , y ) ] exp [ i M 1 ( x , y ) ] | 2 = { F - 1 ( M L , M R ) ( α L , α R ) ( n L , n R ) [ o ' ( u , v ) ] exp [ i M 3 ( x , y ) ] exp [ i M 1 ( x , y ) ] } { F - 1 ( M L , M R ) ( α L , α R ) ( n L , n R ) [ o ' ( u , v ) ] exp [ i M 3 ( x , y ) ] exp [ i M 1 ( x , y ) ] }
M 1 ( x , y ) = arg { F - 1 ( M L , M R ) ( α L , α R ) ( n L , n R ) [ o ( u , v ) ] exp [ i M 3 ( x , y ) ] } arc cos { abs { [ F - 1 ( M L , M R ) ( α L , α R ) ( n L , n R ) [ o ( u , v ) ] exp [ i M 3 ( x , y ) ] ] / 2 } }
M 2 ( x , y ) = arg { F - 1 ( M L , M R ) ( α L , α R ) ( n L , n R ) [ o ( u , v ) ] exp [ i M 3 ( x , y ) ] exp [ i M 1 ( x , y ) ] }
NMSE = y = 1 p x = 1 q [ I D ( x , y ) I O ( x , y ) ] 2 / y = 1 p x = 1 q [ I O ( x , y ) ] 2

Metrics