Optical Image Encryption Based on Joint Fresnel ... - Semantic Scholar
Optical Image Encryption Based on Joint Fresnel Transform Correlator Hsuan-Ting Chang* and Zong-Yu Wu Photonics and Information Laboratory, Institute of Communications Engineering, National Yunlin University of Science and Technology, Douliu Yunlin, 64002 Taiwan R.O.C.
Abstract. Optical verification systems can make image hide in high-security phase key through the phase retrieval operation Conventionally, optical verification systems based on the joint transform correlator usually use Fourier transform. In this paper we propose the joint Fresnel transform architecture and use the methods of the projection onto constraint set and nonlinear transform to determine the pure phase key. As shown in our simulation results, the advantages of the proposed method include that our system is lensless and the additional wavelength and distance parameters enhance the system security. Keywords: Image encryption, Fresnel transform, phase retrieval, nonlinear transform.
1 Introduction In current communication systems, cryptography [9-11] is an important issue because the information is easily attacked, stolen, or forgery. Encryption for information is indispensable. The developing of systems and technologies in optic can create more developmental and application. Compared to the previous digital encryption, optical encryption is faster, the computing speed equivalent to the light, and having characteristics of parallel processing in image signal. The other, the optical encryption can store in the form of phase and amplitude. In contrast, the optical encryption has a large space for development. The demand on the device, optical system needing more precise of instruments in encrypting and decrypt, so the invasion risk is reduced. Previous joint transform correlators (JTCs) calculate far-field projection by using the optical Fourier transform (FT), then through CCD(Charge-Coupled Device) to receive image intensity, and implement IFT (Inverse Fourier Transform) to get target image. Reference to this framework, we propose FrT (Fresnel Transform) [14] to calculus near-field projection, thus can making lensless and increase distance parameters. In the latter segment of this system, we using POCS (Projection Onto Constraint Set) [2], [8], [12], [13] to iterate the reconstruct information to pure phase mask [3~7]. To make all of the operations based on pure phase, we using nonlinear transform and normalize to convert amplitude to phase. We reference to the iterative encrypt system *
Corresponding author. [email protected]
J.-S. Pan, S.-M. Chen, and N.T. Nguyen (Eds.): ICCCI 2010, Part I, LNAI 6421, pp. 81–89, 2010. © Springer-Verlag Berlin Heidelberg 2010
82
H.-T. Chang and Z.-Y. Wu
for one image and one random phase mask based on FrT, and two types of nonlinear conversion. According to the idea of using near-field projection and the algorithm of pure phase mask, we proposed the joint Fresnel transform system as shown in Fig. 1. Section 2 is divided into optical iterative encryption and nonlinear transform, we will introduce the calculus process of POCS algorithm for enhance the quality of reconstruction, and the use of nonlinear transformation. The experiment of Section 3, we using image example to test this algorithm by MATLAB, and using the results of MSE (Mean square error) and CC (Correlation coefficient) to verify the quality of this algorithm. The final section, we referred to conclusions of this algorithm and the development of future applications.
Fig. 1. The proposed joint Fresnel transform system for optical image encryption
2 Encryption algorithm 2.1 POCS The flow chart of our system divided into the iteration part (red block) and the transform part (green block), which are shown in Fig. 2. In conversion operation of signals, we use the near-field projection for FrT. FrT{Η (x,y);z} =
h
,
, where k
.
Optical Image Encryption Based on Joint Fresnel Transform Correlator
83
In the iteration part, we give an original image g and it is transformed with the FrT, the transform distance is z2 (the block diagram is shown in Fig. 3), such as Eq. (1): t(x2,y2)exp[js(x2,y2)] = IFrT{g(x3,y3);z2}
(1)
t is the part of amplitude and s is the part of phase. We extract s to convert by FrT, such as Eq. (2): g’(x3,y3)exp[ jψ(x3,y3)] = FrT{exp[ js(x2,y2)];z2}
(2)
We make the amplitude g’ to normalize, judge with g for pixel error. If some pixel error is greater than threshold that we enter, the pixel of g’ can replace by the same position pixel of g. After the judge, we using the new image to multiplied by ψ , and continue to execute the next loop by back to Eq. (1), and so on.
Fig. 2. The systematic block diagram of the proposed joint Fresnel transform system
2.2 Nonlinear Transform In the nonlinear transform operation, we use to two common schemes: (1) Power-law transform: (2) Log-sigmoid transform: In convert part, we using the random phase key h to convert by FrT is shown in Eq. (3):
H(x2,y2) exp[ jψH(x2,y2)] = FrT{exp[ jh(x1,y1)];z1}
(3)
84
H.-T. Chang and Z.-Y. Wu
Then we use the part of amplitude H to calculate the signal intensity O, which is shown in Eq. (4): O(x2, y2) = | H(x2,y2) |2
(4)
Then we normalize O to [0, 1] to obtain O’. Finally, let O’ convert to Ψ by using the nonlinear transform and normalize Ψ to [-π, π]. Then we store the result to P. 2.3 Combination Calculation After the previous of pure phase s and the target image g’, in order to combine the results, we divided s to P to obtain the phase mask φ is shown in Eq. (5): exp[ jφ ( x 2 , y 2 )] =
exp[ js ( x 2 , y 2 )] exp[ jP ( x 2 , y 2 )]
(5)
After the phase ϕ is obtained, that can make our operation s to achieve a coherent. Entering a Correct phase key get through calculus will obtain the target image g’ in this system, we get keys of distance parameters z1, z2 and the modulation parameters used in the nonlinear transform.
Fig. 3. Optical Fresnel transform between two planes
3 Simulation Results The computer simulation is done by using MATLAB. The size of target image is 256 × 256, such as Fig. 4(a). The distance of FrT, z2 = 20cm, wavelength λ = 632.8 nm. The iteration number is set to 100, the threshold value of MSE is set as five. After the above calculation obtain phase mask s and approximated image g’ is shown in Fig. 4(b) and 4(c), respectively. The CC values between g’ and g is 0.9928. We use a random phase to be the input phase key, such as Fig. 5(a). Set the convert distance z1 to 12 cm and the wavelength λ to 632.8 nm for FrT. After the signal intensity is obtained,
Optical Image Encryption Based on Joint Fresnel Transform Correlator
(a)
(b)
85
(c)
Fig. 4. (a) The target Len image; (b) the reconstructed image g’; (c) Phase key s
we use the Log-sigmoid transform as the nonlinear transformation, in which the parameters are a = -5 c = 10. After the processing of signal conversion, the pure phase P can be obtained, such as that shown in Fig. 5(b). Then following Eq. (5), we can obtain the other phase φ, which is shown in Fig. 5(c). After an overall operation, the obtained image is shown in Fig. 5(d) which is similar to that in Fig. 4(b). Consider the sensitivity of different system parameters. Following the previous setting, if we enter the distance parameter z1 = 19 cm, and 15 cm, the reconstructed images are shown in Figs. 6(a) and 6(b), respectively. Figure 7 shows the variation of
，
(a)
(b)
(c)
(d)
Fig. 5. (a) The random phase key; (b) Phase key P; (c) Phase key ϕ; (d) The verified image g’
(a)
(b)
Fig. 6. The reconstructed images under incorrect distance parameters; (a) z1 = 13cm; (b) z1 = 17cm
86
H.-T. Chang and Z.-Y. Wu
CC values between the original image and reconstructed image for z1 in the range from 1cm to 50 cm (Take a sample of each cm). If we enter the distance parameter z2 = 19 cm and 15 cm, the reconstructed images are shown in Figs. 8(a) and 8(b), respectively. We plot the variation of the CC values between the original image and reconstructed image for the z2 parameter from 1 cm to 50 cm (Take a sample in each cm), which is shown in Fig. 9. 1 0.9 0.8
Correlation coefiicient
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
10
20
30
40
50
၏ᠦᑇ z1(cm)
Fig. 7. Variation of the CC values between original image and reconstructed image for the z1 parameter
(a)
(b)
Fig. 8. The reconstructed images for the incorrect distance parameters: (a) z2 = 19cm; (b) z2 = 15cm
Considering the wavelength λ = 682.8nm, the reconstructed image is shown in Fig. 10(a). When the wavelength λ = 832.8nm, the reconstructed image is shown in Fig. 10(b). We plot the variation of the CC values between the original image and reconstructed image for the wavelength λ ranging from 332.8 nm to 832.8 nm (Take a sample in each 10nm), which is shown in Fig.11.
Optical Image Encryption Based on Joint Fresnel Transform Correlator
87
1.2 1
Correlation coefficient
0.8 0.6 0.4 0.2 0 -0.2
0
5
10
15
20
25
30
35
40
45
50
၏ᠦᑇ z2(cm)
Fig. 9. Variation of the CC values between the original image and reconstructed image for the z2 parameter
(a)
(b)
Fig. 10. The reconstructed images of two wavelengths: (a) λ = 682.8nm; (b) λ = 832.8nm
0.9 0.8
Correlation coefficient
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1
350
400
450
500
550
600
650
700
750
800
٠ं९ λ(nm)
Fig. 11. Variation of the CC values between the original image and reconstructed image for the λ parameter
88
H.-T. Chang and Z.-Y. Wu
However, we found that the parameter can’t use for all real numbers in the nonlinear transform. Inappropriate parameters will make the system wrong and operational failure, finally decrease the safety. Trough the computer simulation, we try to found the appropriate interval of parameters. We use the incorrect phase key to replace the phase key h1 to test for Log-sigmoid transform. In processing of parameters, if a = 0, converted values will make no effects in system. We plot the varies of MSE and CC between original image and reconstructed image for a in -10 to 10 and c in -10 to 10, which are shown in Figs. 12 and 13, respectively.
Mean square error
8000
6000
4000
2000
0 10 5
10 5
0
0
-5
-5 -10
a
-10
c
Fig. 12. Variation of the MSE values between original image and reconstructed image under the various nonlinear parameters a and c
Correlation coefficient
1
0.5
0
-0.5 10 5
10 5
0
0
-5 a
-5 -10
-10
c
Fig. 13. Variation of the CC values between the original image and reconstructed image under the various nonlinear parameters a and c
4
Conclusion
In this paper, we proposed an optical encrypt system based on the joint Fresnel transform. The frequency part of an image is converted to a pure phase mask. Then, with
Optical Image Encryption Based on Joint Fresnel Transform Correlator
89
the nonlinear transform methods, the target can be reconstructed by using two phase keys in the overall optical system. The proposed structure is simpler than that in conventional joint transform correlator architecture because of using no lenses in the proposed system. Moreover, the wavelength and the distances between two phases are two additional encryption parameters which can further enhance the system security level. The quality of system is estimated using the values of MSE and CC between the decrypted and original images.
Acknowledgment This work is partially supported by National Science Council, Taiwan under the contract number NSC 99-2221-E-224-004.
References [1] Rosen, J., Javidi, B.: Security optical systems based on a joint transform correlator with significant output images. Opt. Eng. 40(8), 1584–1589 (2001) [2] Joseph, R.: Learning in correlators based on projections onto constraint sets. Opt. Lett 18(14), 2165–2171 (1993) [3] Chang, H.T., Lu, W.C., Kuo, C.J.: Multiple-phase retrieval for optical security systems by use of random-phase encoding. Appl. Opt. 41(23), 4825–4834 (2002) [4] Chang, H.T.: Image encryption using separable amplitude-based virtual image and iteratively retrieved phase information. Opt. Eng 40(10), 2165–2171 (2001) [5] Refregier, P., Javidi, B.: Optical image encryption based on input plane and Fourier plane random encoding. Opt. Lett 20(7), 767–769 (1995) [6] Towghi, N., Javidi, B., Luo, Z.: Fully phase encrypted image processor. J. Opt. Soc. Am. A 16(8), 1915–1927 (1999) [7] Fienup, J.R.: Phase retrieval algorithm: a comparison. Appl. Opt. 22(15), 2758–2769 (1982) [8] Sun, H., Kwok, W.: Concealment of damaged block transform coded images using projection onto convex sets. IEEE Transactions on Image Processing 4(4), 470–477 (1995) [9] Chang, Y.C., Chang, H.T., Kuo, C.J.: Hybrid image cryptosystem based on dyadic phase displacement in the frequency domain. Optics Communications 236(4-6), 245–257 (2004) [10] Yamamoto, H., Hayasaki, Y., Nichida, N.: Securing information display by use of visual cryptography. Optics Letters 28(17), 1564–1566 (2003) [11] Chuang, C.H., Lin, G.S.: Optical image cryptosystem based on adaptive steganography. Optical Engineering 47(4), 470021–470029 (2008) [12] Chang, H.T., Chen, C.C.: Fully phase asymmetric image verification system based on joint transform correlator. Optics Express 14(4), 1458–1467 (2006) [13] Nomura, T., Javidi, B.: Optical encryption using a joint transform correlator architecture. Optical Engineering 39(8), 2031–2035 (2000) [14] Chen, L., Zhao, D.: Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms. Optics Express 14(19), 8552–8560 (2006)
Abstract. Optical verification systems can make image hide in high-security phase key through the phase retrieval operation Conventionally, optical verification systems based on the joint transform correlator usually use Fourier transform. In this paper we propose the joint Fresnel transform architecture and use the methods of the projection onto constraint set and nonlinear transform to determine the pure phase key. As shown in our simulation results, the advantages of the proposed method include that our system is lensless and the additional wavelength and distance parameters enhance the system security. Keywords: Image encryption, Fresnel transform, phase retrieval, nonlinear transform.
1 Introduction In current communication systems, cryptography [9-11] is an important issue because the information is easily attacked, stolen, or forgery. Encryption for information is indispensable. The developing of systems and technologies in optic can create more developmental and application. Compared to the previous digital encryption, optical encryption is faster, the computing speed equivalent to the light, and having characteristics of parallel processing in image signal. The other, the optical encryption can store in the form of phase and amplitude. In contrast, the optical encryption has a large space for development. The demand on the device, optical system needing more precise of instruments in encrypting and decrypt, so the invasion risk is reduced. Previous joint transform correlators (JTCs) calculate far-field projection by using the optical Fourier transform (FT), then through CCD(Charge-Coupled Device) to receive image intensity, and implement IFT (Inverse Fourier Transform) to get target image. Reference to this framework, we propose FrT (Fresnel Transform) [14] to calculus near-field projection, thus can making lensless and increase distance parameters. In the latter segment of this system, we using POCS (Projection Onto Constraint Set) [2], [8], [12], [13] to iterate the reconstruct information to pure phase mask [3~7]. To make all of the operations based on pure phase, we using nonlinear transform and normalize to convert amplitude to phase. We reference to the iterative encrypt system *
Corresponding author. [email protected]
J.-S. Pan, S.-M. Chen, and N.T. Nguyen (Eds.): ICCCI 2010, Part I, LNAI 6421, pp. 81–89, 2010. © Springer-Verlag Berlin Heidelberg 2010
82
H.-T. Chang and Z.-Y. Wu
for one image and one random phase mask based on FrT, and two types of nonlinear conversion. According to the idea of using near-field projection and the algorithm of pure phase mask, we proposed the joint Fresnel transform system as shown in Fig. 1. Section 2 is divided into optical iterative encryption and nonlinear transform, we will introduce the calculus process of POCS algorithm for enhance the quality of reconstruction, and the use of nonlinear transformation. The experiment of Section 3, we using image example to test this algorithm by MATLAB, and using the results of MSE (Mean square error) and CC (Correlation coefficient) to verify the quality of this algorithm. The final section, we referred to conclusions of this algorithm and the development of future applications.
Fig. 1. The proposed joint Fresnel transform system for optical image encryption
2 Encryption algorithm 2.1 POCS The flow chart of our system divided into the iteration part (red block) and the transform part (green block), which are shown in Fig. 2. In conversion operation of signals, we use the near-field projection for FrT. FrT{Η (x,y);z} =
h
,
, where k
.
Optical Image Encryption Based on Joint Fresnel Transform Correlator
83
In the iteration part, we give an original image g and it is transformed with the FrT, the transform distance is z2 (the block diagram is shown in Fig. 3), such as Eq. (1): t(x2,y2)exp[js(x2,y2)] = IFrT{g(x3,y3);z2}
(1)
t is the part of amplitude and s is the part of phase. We extract s to convert by FrT, such as Eq. (2): g’(x3,y3)exp[ jψ(x3,y3)] = FrT{exp[ js(x2,y2)];z2}
(2)
We make the amplitude g’ to normalize, judge with g for pixel error. If some pixel error is greater than threshold that we enter, the pixel of g’ can replace by the same position pixel of g. After the judge, we using the new image to multiplied by ψ , and continue to execute the next loop by back to Eq. (1), and so on.
Fig. 2. The systematic block diagram of the proposed joint Fresnel transform system
2.2 Nonlinear Transform In the nonlinear transform operation, we use to two common schemes: (1) Power-law transform: (2) Log-sigmoid transform: In convert part, we using the random phase key h to convert by FrT is shown in Eq. (3):
H(x2,y2) exp[ jψH(x2,y2)] = FrT{exp[ jh(x1,y1)];z1}
(3)
84
H.-T. Chang and Z.-Y. Wu
Then we use the part of amplitude H to calculate the signal intensity O, which is shown in Eq. (4): O(x2, y2) = | H(x2,y2) |2
(4)
Then we normalize O to [0, 1] to obtain O’. Finally, let O’ convert to Ψ by using the nonlinear transform and normalize Ψ to [-π, π]. Then we store the result to P. 2.3 Combination Calculation After the previous of pure phase s and the target image g’, in order to combine the results, we divided s to P to obtain the phase mask φ is shown in Eq. (5): exp[ jφ ( x 2 , y 2 )] =
exp[ js ( x 2 , y 2 )] exp[ jP ( x 2 , y 2 )]
(5)
After the phase ϕ is obtained, that can make our operation s to achieve a coherent. Entering a Correct phase key get through calculus will obtain the target image g’ in this system, we get keys of distance parameters z1, z2 and the modulation parameters used in the nonlinear transform.
Fig. 3. Optical Fresnel transform between two planes
3 Simulation Results The computer simulation is done by using MATLAB. The size of target image is 256 × 256, such as Fig. 4(a). The distance of FrT, z2 = 20cm, wavelength λ = 632.8 nm. The iteration number is set to 100, the threshold value of MSE is set as five. After the above calculation obtain phase mask s and approximated image g’ is shown in Fig. 4(b) and 4(c), respectively. The CC values between g’ and g is 0.9928. We use a random phase to be the input phase key, such as Fig. 5(a). Set the convert distance z1 to 12 cm and the wavelength λ to 632.8 nm for FrT. After the signal intensity is obtained,
Optical Image Encryption Based on Joint Fresnel Transform Correlator
(a)
(b)
85
(c)
Fig. 4. (a) The target Len image; (b) the reconstructed image g’; (c) Phase key s
we use the Log-sigmoid transform as the nonlinear transformation, in which the parameters are a = -5 c = 10. After the processing of signal conversion, the pure phase P can be obtained, such as that shown in Fig. 5(b). Then following Eq. (5), we can obtain the other phase φ, which is shown in Fig. 5(c). After an overall operation, the obtained image is shown in Fig. 5(d) which is similar to that in Fig. 4(b). Consider the sensitivity of different system parameters. Following the previous setting, if we enter the distance parameter z1 = 19 cm, and 15 cm, the reconstructed images are shown in Figs. 6(a) and 6(b), respectively. Figure 7 shows the variation of
，
(a)
(b)
(c)
(d)
Fig. 5. (a) The random phase key; (b) Phase key P; (c) Phase key ϕ; (d) The verified image g’
(a)
(b)
Fig. 6. The reconstructed images under incorrect distance parameters; (a) z1 = 13cm; (b) z1 = 17cm
86
H.-T. Chang and Z.-Y. Wu
CC values between the original image and reconstructed image for z1 in the range from 1cm to 50 cm (Take a sample of each cm). If we enter the distance parameter z2 = 19 cm and 15 cm, the reconstructed images are shown in Figs. 8(a) and 8(b), respectively. We plot the variation of the CC values between the original image and reconstructed image for the z2 parameter from 1 cm to 50 cm (Take a sample in each cm), which is shown in Fig. 9. 1 0.9 0.8
Correlation coefiicient
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0
10
20
30
40
50
၏ᠦᑇ z1(cm)
Fig. 7. Variation of the CC values between original image and reconstructed image for the z1 parameter
(a)
(b)
Fig. 8. The reconstructed images for the incorrect distance parameters: (a) z2 = 19cm; (b) z2 = 15cm
Considering the wavelength λ = 682.8nm, the reconstructed image is shown in Fig. 10(a). When the wavelength λ = 832.8nm, the reconstructed image is shown in Fig. 10(b). We plot the variation of the CC values between the original image and reconstructed image for the wavelength λ ranging from 332.8 nm to 832.8 nm (Take a sample in each 10nm), which is shown in Fig.11.
Optical Image Encryption Based on Joint Fresnel Transform Correlator
87
1.2 1
Correlation coefficient
0.8 0.6 0.4 0.2 0 -0.2
0
5
10
15
20
25
30
35
40
45
50
၏ᠦᑇ z2(cm)
Fig. 9. Variation of the CC values between the original image and reconstructed image for the z2 parameter
(a)
(b)
Fig. 10. The reconstructed images of two wavelengths: (a) λ = 682.8nm; (b) λ = 832.8nm
0.9 0.8
Correlation coefficient
0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1
350
400
450
500
550
600
650
700
750
800
٠ं९ λ(nm)
Fig. 11. Variation of the CC values between the original image and reconstructed image for the λ parameter
88
H.-T. Chang and Z.-Y. Wu
However, we found that the parameter can’t use for all real numbers in the nonlinear transform. Inappropriate parameters will make the system wrong and operational failure, finally decrease the safety. Trough the computer simulation, we try to found the appropriate interval of parameters. We use the incorrect phase key to replace the phase key h1 to test for Log-sigmoid transform. In processing of parameters, if a = 0, converted values will make no effects in system. We plot the varies of MSE and CC between original image and reconstructed image for a in -10 to 10 and c in -10 to 10, which are shown in Figs. 12 and 13, respectively.
Mean square error
8000
6000
4000
2000
0 10 5
10 5
0
0
-5
-5 -10
a
-10
c
Fig. 12. Variation of the MSE values between original image and reconstructed image under the various nonlinear parameters a and c
Correlation coefficient
1
0.5
0
-0.5 10 5
10 5
0
0
-5 a
-5 -10
-10
c
Fig. 13. Variation of the CC values between the original image and reconstructed image under the various nonlinear parameters a and c
4
Conclusion
In this paper, we proposed an optical encrypt system based on the joint Fresnel transform. The frequency part of an image is converted to a pure phase mask. Then, with
Optical Image Encryption Based on Joint Fresnel Transform Correlator
89
the nonlinear transform methods, the target can be reconstructed by using two phase keys in the overall optical system. The proposed structure is simpler than that in conventional joint transform correlator architecture because of using no lenses in the proposed system. Moreover, the wavelength and the distances between two phases are two additional encryption parameters which can further enhance the system security level. The quality of system is estimated using the values of MSE and CC between the decrypted and original images.
Acknowledgment This work is partially supported by National Science Council, Taiwan under the contract number NSC 99-2221-E-224-004.
References [1] Rosen, J., Javidi, B.: Security optical systems based on a joint transform correlator with significant output images. Opt. Eng. 40(8), 1584–1589 (2001) [2] Joseph, R.: Learning in correlators based on projections onto constraint sets. Opt. Lett 18(14), 2165–2171 (1993) [3] Chang, H.T., Lu, W.C., Kuo, C.J.: Multiple-phase retrieval for optical security systems by use of random-phase encoding. Appl. Opt. 41(23), 4825–4834 (2002) [4] Chang, H.T.: Image encryption using separable amplitude-based virtual image and iteratively retrieved phase information. Opt. Eng 40(10), 2165–2171 (2001) [5] Refregier, P., Javidi, B.: Optical image encryption based on input plane and Fourier plane random encoding. Opt. Lett 20(7), 767–769 (1995) [6] Towghi, N., Javidi, B., Luo, Z.: Fully phase encrypted image processor. J. Opt. Soc. Am. A 16(8), 1915–1927 (1999) [7] Fienup, J.R.: Phase retrieval algorithm: a comparison. Appl. Opt. 22(15), 2758–2769 (1982) [8] Sun, H., Kwok, W.: Concealment of damaged block transform coded images using projection onto convex sets. IEEE Transactions on Image Processing 4(4), 470–477 (1995) [9] Chang, Y.C., Chang, H.T., Kuo, C.J.: Hybrid image cryptosystem based on dyadic phase displacement in the frequency domain. Optics Communications 236(4-6), 245–257 (2004) [10] Yamamoto, H., Hayasaki, Y., Nichida, N.: Securing information display by use of visual cryptography. Optics Letters 28(17), 1564–1566 (2003) [11] Chuang, C.H., Lin, G.S.: Optical image cryptosystem based on adaptive steganography. Optical Engineering 47(4), 470021–470029 (2008) [12] Chang, H.T., Chen, C.C.: Fully phase asymmetric image verification system based on joint transform correlator. Optics Express 14(4), 1458–1467 (2006) [13] Nomura, T., Javidi, B.: Optical encryption using a joint transform correlator architecture. Optical Engineering 39(8), 2031–2035 (2000) [14] Chen, L., Zhao, D.: Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms. Optics Express 14(19), 8552–8560 (2006)
