Image multiplexing and authentication based on double phase retrieval in fresnel transform domain

Optics Communications 389 (2017) 150–158

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

Image multiplexing and authentication based on double phase retrieval in fresnel transform domain

MARK

⁎

Hsuan-Ting Changa, Che-Hsian Lina, Chien-Yue Chenb, a b

Photonics and Information Laboratory, Department of Electrical Engineering, National Yunlin University of Science and Technology, Taiwan Graduate Institute of Color and Illumination Technology, National Taiwan University of Science and Technology, Taiwan

A R T I C L E I N F O

A BS T RAC T

Keywords: Image multiplexing Manipulations of wavelength and position Double-phase retrieval algorithm Phase-locked system Authentication system Computer simulation

An image multiplexing and authentication method based on the double-phase retrieval algorithm (DPRA) with the manipulations of wavelength and position in the Fresnel transform (FrT) domain is proposed in this study. The DPRA generates two matched phase-only functions (POFs) in the different planes so that the corresponding image can be reconstructed at the output plane. Given a number of target images, all the sets of matched POFs are used to generate the phase-locked system through the phase modulation and synthesis to achieve the multiplexing purpose. To reconstruct a target image, the corresponding phase key and all the correct parameters in the FrT are required. Therefore, the authentication system with high-level security can be achieved. The computer simulation verifies the validity of the proposed method and also shows good resistance to the crosstalk among the reconstructed images.

1. Introduction The developments of optical technology on image encryption and authentication have received great attention since the last two decades. Many optics-based image encryption/authentication algorithms with distinct optical structures were proposed for the diverse applications [1–20]. First, Refregier et al. proposed the double random phase encryption (DRPE) method based on the 4-ƒ optical structure using the optical Fourier transform (FT) [1]. Then, various optical encryption/ authentication methods on the basis of FT [2–4,6,7], Fractional Fourier transform (FrFT) [5,8,9], and Fresnel transform (FrT) [10– 12] are developed. The DRPE algorithm utilizes two random phase functions to encrypt the plain image as the encrypted data, which consist of both the amplitude and phase information. In decryption, the conjugate phase information is required for restoring the plain image at the output plane. Thereafter, successive studies on optical encryption [1–5] have been developed. As to the image authentication problems, the phase functions are retrieved to reconstruct the target images. To solve such a problem, a lot of studies introduce the iterative processes based on the DRPE [6–12] to retrieve the phase functions which can be used to reconstruct the target images at the output plane. Wang et al. proposed an alternative [2], which, with iterations, encoded the original image to the phase-only function (POF) on the Fourier plane in a 4-f structure. Li et al. modified such a method [3] by encrypting the original image as the POF at the input plane and

⁎

Corresponding author. E-mail address: [email protected] (C.-Y. Chen).

http://dx.doi.org/10.1016/j.optcom.2016.11.061 Received 1 November 2016; Accepted 23 November 2016 0030-4018/ © 2016 Published by Elsevier B.V.

achieved more convenient applications. Chang et al. further modified the method to retrieve double POFs at both the input and Fourier planes to achieve higher security and better image reconstruction quality [6]. In addition to the FT, Situ and Zhang proposed an iterative algorithm [12] to separately retrieve two POFs at the input and FrT plane without using any lens to make the implementation easier. Compared with the previous methods, encoding an original image to the POFs on the Fresnel plane through the iterative algorithm provides the following advantages: (1) The fewer optical components are used since there is no lens required. (2) In addition to the POFs as the keys, the wavelength and position parameters in the FrT can also be the keys to obtain higher security. (3) The encrypted data could be directly delivered through the information channel as the public key. Then legal users can complete the decryption with correct FrT parameters and the secret phase key. In addition to the single image encryption, a lot of methods especially for multiple-image encryption have been proposed. Situ and Zhang proposed the wavelength- and position-based optical image multiplexing methods in the FrT domain [14,15]. The double-phase encryption is utilized for encoding each original image. Then, the final encrypted image is simply generated with superposition. In decryption, however, the crosstalk among the images limits the number of images for multiplexing. This method is especially not suitable for multiple grayscale images because the decrypted image quality would be significantly worse than the binary ones. Hwang et al. further utilized

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structed images can be observed. The rest of this paper is organized as following: The DPRA in the FrT domain is described in Section 2. Section 3 deals with the proposed image encryption and multiplexing method. The numerical simulation results are provided in Section 4. Finally, the conclusions are drawn in Section 5.

the modified Gerchberg-Saxton algorithm (MGSA) [11] for the multiple image encryption based on wavelength and position multiplexing [16–18] to improve the cryptosystem performances on crosstalk resistance and image quality. Compared to the single POF method [16], the methods using two POFs [17,18] in the encryption process can enhance the security of the cryptosystem. Moreover, more information, such as the distance between two POFs in the optical architecture and the wavelength of incident light, are required for accurately recovering the target images in decryption and this does enhance the system security. However, there is a disadvantage in our previous methods [17,18]. If one of the double POFs is known and the other is not, the silhouette of the target image is still visible. Therefore, the security of such an authentication system is not high enough. On the other hand, according to the analyses on the system parameters in past methods, there are certain restrictions on the optical architecture in order to acquire better quality on the reconstruction images [21]. In addition to the MGSA for generating the double POFs in the optical image multiplexing systems, Chang et al. proposed the multiplephase retrieval algorithm (MPRA) [6] based on the DRPE and projection onto constraint sets (POCS) algorithms [23]. Through the threshold criteria for the iterated images at the reconstruction plane and the phase only constraint, more than two POFs located on the FT and inverse Fourier transform (IFT) planes are determined when the convergence condition has been achieved. The MPRA can arbitrarily select the number (at least two) of retrieved POFs and can also be easily extended to retrieve the POFs at the planes in the FrFT and FrT domains. For the case of retrieving two POFs in the MPRA, which can be named as the double-phase retrieval algorithm (DPRA), both the convergence speed of the iteration process and the reconstructed image quality are better than that in the previous methods [14,15]. Therefore, in this paper, a different optical image encryption and multiplexing method based on the DPRA in the FrT domain is proposed. By using the similar phase modulation schemes in our previous wavelength and position multiplexing methods for multiple images [17,18], each target image can be encrypted into a secret phase key with a common phase lock, which is the summarization of all the modulated POFs. The proposed method can achieve higher system security than that in our previous methods [17,18] because no visible silhouettes in the recon-

2. Double-phase retrieval algorithm (DPRA) The random phase encoding in the MPRA [6] can be used to generate two or more POFs for a given target image. During each iteration process in the MPRA, only a POF would be modified according to the phase constraint, while all the other POFs are fixed. The POFs located in the different positions are determined by using the FT and inverse FT in the frequency and spatial domains, respectively, according to the constraints in the iteration process in the MPRA. When only two POFs are considered in this method, a phase function can be used as the key and the other is used as the lock for the security system so that the target images can only be recovered by using the matched pairs of keys and locks. The original MPRA retrieves the POFs in both the spatial and FT domains. In the proposed DPRA, alternatively, two POFs are separately retrieved from the spatial and FrT domains according to a given target image. Compared with the FT, the higher flexibility of FrT parameters can result in the more secured POFs. In addition, the lens required in the optical FT can be discarded in the optical architecture for the FrT. Let h(x0, y0) be the object (i.e. the POF in this study) at the original plane and H(x, y) be the designated target image at the output FrT plane. The general FrT equation can be expressed as [22]

exp ( H (x, y)=FrT{h (x 0 , y0)}= ( y − y0)2 ]}dx 0 dy0 ,

i 2πz ) λ

iλz

∬ h (x0, y0)exp { iπλz [(x − x0)2 + (1)

where λ is the wavelength and z is the diffraction distance between the input and output planes. Fig. 1 shows the block diagram of the proposed FrT-based DPRA, which can be divided into the upper-right and the bottom-left parts by

Fig. 1. Block diagram of the proposed FrT-based DPRA.

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Fig. 2. Optical structures of the proposed DPRA while determining the POFs during the (a) odd-number and (b) even-number iterations.

preset threshold value τ th. Otherwise, the pixel values in the iterated image g k (x 0 , y0) are modified to satisfy the expected image constraint shown below:

using a dashed line, and the corresponding optical structures are shown in Fig. 2(a) and (b), respectively. In the upper-right part, two initial random POFs φ0 (x 0 , y0)and Ψ0 (x, y) distributed between 0 and 2π are first generated and the initial image is determined by using Eq. (2):

⎧ g k (x 0 , y ),if |g (x 0 , y )−g k (x 0 , y ) | ≤ γ 0 0 0 g k (x 0 , y0)=⎨ , k ⎩ g (x 0 , y0),if |g (x 0 , y0)−g (x 0 , y0) | > γ ⎪

g 0 (x 0 , y0)=IFrT{FrT{exp[iφ0 (x 0 , y0]}exp[i Ѱ

0 (x , y)]}

(2)

⎪

where FrT{} and IFrT{} represent the FrT shown in Eq. (1) and the inverse FrT, respectively. Let the POF φk (x 0 , y0) in the spatial domain be fixed and the POF k (x, y) in the FrT domain can be modified in the kth iteration in this part. In the lower-left part, on the other hand, the POF k (x, y) is fixed, while the other POF φk (x 0 , y0) can be modified. During the kth iteration, where k is an the odd number, the POF k (x, y) is determined with the POF φk−1 (x 0 , y0) acquired from the previous iteration. When k is an even number, the POF φk (x 0 , y0) is determined with the POF k (x, y). Eqs. (3) and (4) show the determination of the POF k (x, y) with the procedure described in Fig. 1. A complex function is first calculated

P 2k (x, y)=

FrT{g k −1 (x 0 , y0)} FrT{exp[iφk −1 (x 0 , y0)]}

(3)

P 2k (x,

y) can be calculated by using Eq. (4),

Then the phase part of k (x ,

y) = arg {P 2k (x, y)},

where γ denotes the threshold value for determining whether the pixel values in the iterated image will be modified or not during the iteration process. When the kth iterated image g k (x 0 , y0) is generated, the second phase key Ψk (x, y) acquired in the upper-right part in Fig. 1 is fixed while dealing with the lower-left part and is used for determining the first POF φk+1 (x 0 , y0). Eq. (9) is used when determining φk+1 (x 0 , y0) in Fig. 1:

⎧ FrT{g k (x 0 , y0)} ⎫ ⎬. P1k +1 (x 0 , y0)=IFrT ⎨ ⎩ exp[i Ѱ k (x, y)] ⎭

0,

φk +1 (x 0 , y0)=arg {P1k +1 (x 0 , y0)},

}exp[i Ѱ

k (x ,

to obtain the POF in the FrT domain. To generate the (k+1) iterated image g k+1 (x 0 , y0)at the output plane, the first phase term exp[iφk+1 (x 0 , y0)] is Fresnel transformed and then multiplied by the second phase term exp[i Ѱ k (x, y)]. The IFrT is then utilized for generating the iterated image as shown in Eq. (11).

g k +1 (x 0 , y0)=IFrT{FrT{ exp[iφk +1 (x 0 , y0)]}exp[i Ѱ }exp[i Ѱ

y)]}=IFrT{ϕk −1 (x, y)

where ϕk −1 (x, y)=FrT{exp[iφk −1 (x 0 , y0)] denotes the complex function in the FrT domain. Fig. 2(a) shows the corresponding optical architecture. To check whether the iterated image g k (x 0 , y0) has converged to the target image, the mean-squared error (MSE) between the iterated and target images is calculated for the similarity measurement.

MSE[g (x 0 , y0), g k (x 0 , y0)] =

1 M×N

M

⎪

x 0 =1 y0 =1

(12)

The upper-right and lower-left parts are repeatedly performed until the correlation coefficient ρ converges or the MSE is smaller than a preset threshold τ th. Then the two final phase keys φˆ (x 0 , y0) and ˆ (x, y) at the input and FrT planes, respectively, can be determined. To reconstruct the target image, the first phase term at the input plane exp[iφˆ (x 0 , y0)] is Fresnel transformed and then is multiplied by the y)]. The IFrT is then performed to second phase term exp[i Ѱ ˆ (x, obtain the approximation image gˆ (x 0 , y0) of the target image as shown in Eq. (13).

where M × N denotes the image size. Another comparison can be made by calculating the correlation coefficient (CC) ρ between the two images:

COV (g , g k ) , σg σ g k

(11)

⎪

[g (x 0 , y0) − g k (x 0 , y0)]2 , (6)

ρ=

y)]}=IFrT{ɸk +1 (x, y)

⎧ g k +1 (x 0 , y ),if |g (x 0 , y )−g k +1 (x 0 , y ) | ≤ γ 0 0 0 . g k +1 (x 0 , y0)=⎨ k +1 ⎩ g (x 0 , y0),if |g (x 0 , y0)−g (x 0 , y0) | > γ

N

∑ ∑

k (x ,

k (x , y)]}.

Fig. 2(b) shows the corresponding optical architecture. By calculating the MSE or the correlation coefficient ρ , the difference between the iterated image g k (x 0 , y0) and the target image g(x 0 , y0) can be observed. The iteration process stops when the correlation coefficient ρ converges or the MSE is smaller than a preset threshold τ th. Otherwise, the pixel values in the iterated image g k+1 (x 0 , y0) are modified to satisfy the image constraint shown in Eq. (12).

(5)

k (x , y)]},

(10) th

(4)

y0)=IFrT{FrT{ exp[iφk −1 (x 0 , y0)]}exp[i Ѱ

(9)

The phase term of P1k +1 (x 0 , y0) could be calculated by using the equation:

to acquire the POF in the FrT domain. To generate the kth iterated image g k (x 0 , y0) at the output plane, the phase term exp[iφk−1 (x 0 , y0)] is Fresnel transformed and then is multiplied by the phase term exp[i Ѱ k (x, y)]. The IFrT is then utilized to determine the iterated image as shown in Eq. (5):

g k (x

(8)

(7)

where COV (g ,g k ) represents the covariance between the two images g and g k , and σg andσ g k are the standard deviations of the two images. The ρ value closer to 1 shows that the iterative image g k (x 0 , y0)is closer to the target image g. The iteration process stops when the correlation coefficient reaches to a threshold value ρth or the MSE is smaller than a

gˆ (x 0 , y0)=IFrT{FrT{ exp[iφˆ (x 0 , y0)]}exp[i Ѱ ˆ (x, y)}exp[i Ѱ ˆ (x, y)]}, 152

ˆ (x , y)]}=IFrT{ɸ (13)

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ˆ (x, y) is generated on the FrT plane after applying the FrT to where ɸ the phase term exp[iφˆ (x 0 , y0)]. 3. DPRA-based wavelength and position multiplexing and authentication In Section 2, each target image can be encrypted by using the FrT with the various wavelength and position parameters to obtain the two POFs, in which one is placed at the input plane with the coordinate (x0, y0) and the other is at the FrT plane with the coordinate (x, y). Fig. 3(a) and (b) show the optical architectures of the proposed methods for wavelength multiplexing and position multiplexing, respectively. To achieve the image multiplexing, the POFs corresponding to different images at the FrT plane are synthesized to as a single phase-only mask (POM). When the nth image gˆnz (x 0 , y0) needs to be decrypted, the corresponding POFnφn (x 0 , y0) is placed at the input plane. The distance between the synthesized POM and the output plane corresponds to the diffraction distance zn of the nth image. In order to avoid the crosstalk among the overlapping reconstructed images, the POFs at the FrT plane retrieved from each image are processed with the phase modulation scheme [16] before the synthesis. Therefore, the reconstructed target images can be distributed at different locations at the output plane. Fig. 4 shows the system diagram of the proposed method for the wavelength or position multiplexing schemes of multiple images based on the DPRA. First, the FTs in the MPRA [6] is modified as the FrT in

Fig. 4. System diagram of the proposed method for the wavelength or position multiplexing schemes of multiple images based on DPRA.

the proposed DPRA without using any lens to encrypt the N target images gn (x 0 , y0), n=1~N. The target images corresponding to the distinct wavelength λ n and position distance zn parameters in the FrT, can be decomposed into two POFs, {φˆ λn (x 0 , y0), ˆ λn (x, y)} and {φˆ (x 0 , y ), ˆz (x, y)}, respectively, according to the block diagram zn

0

n

shown in Fig. 1. Each pair of the POFs would satisfy the following equation:

IFrT{FrT{exp[iφˆ λn (x 0 , y0)]; λ n; zs}×exp[i Ѱ ˆ λn (x, y)]; λ n; z}=gˆnλ (x 0 , y0)exp [iε (x 0 , y0)],

(14)

or

IFrT{FrT{exp[iφˆzn (x 0 , y0)]; λ; zs}×exp[i Ѱ ˆzn (x, y)]; λ; zn}=gˆnz (x 0 , y0)exp [iε (x 0 , y0)],

(15)

where zs denotes the separation distance between the two POFs, gˆnλ (x 0 , y0) or gˆnz (x 0 , y0), together with the phase term ε (x 0 , y0), is the reconstructed target image at the output plane. Phase modulation is then utilized for the total N POFs, ˆ λn (x, y) or ˆzn (x, y), at the FrT plane.

Ψ'λn (x, y)=Ѱ ˆ λn (x, y)+

Fig. 3. The optical architectures for the (a) wavelength multiplexing and (b) position multiplexing schemes based on the proposed DPRA.

2π (μn x 0 + νn y0) , λ nz

Fig. 5. The nine target images used in the computer simulation.

153

(16)

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or

ˆ z (x, y)+ 2π (μn x 0 + νn y0) , Ψ'zn (x, y)=Ψ n λz n

(17)

where μn and νn are the preset deviations of reconstructed images at the output plane along x 0 andy0 directions, respectively. Finally, among the N phase signals, the phase term Ψ'λn (x, y) or ' Ψ zn (x, y) could be synthesized as a POMλ or POMz by using the modulation scheme shown in Eqs. (16) or (17) in order to achieve the wavelength or position multiplexing. N

Fig. 6. Comparison of the convergence speed and image quality for the methods based on the proposed DPRA and our previous MGSA.

POMλ=exp{ ∑ [i Ψ'λn (x , y)]}, n =1

(18)

or N

POMz=exp{ ∑ [i Ψ'zn (x , y)]}, n =1

(19)

Several target images would be encoded to individual POFs, φˆ λn (x 0 , y0) or φˆzn (x 0 , y0), and a POMλ or POMz as the phase-locked system, after being encrypted with the proposed wavelength or position multiplexing. In the decryption, each target image gn (x 0 , y0) could be reconstructed as the approximated target image gˆnλ (x 0 , y0) or gˆnz (x 0 , y0) with the corresponding phase key and the lock, as shown below:

Fig. 7. Examples of the synthesized phase information of a given target image for: (a) wavelength multiplexing: POMλ and (b) position multiplexing: POMz.

Table 1 The target images can only be correctly reconstructed when the matched POFs are used in the proposed method. Otherwise, there are noise-like images obtained.

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⎧ IFrT ⎨FrT{exp[iφˆ λn (x 0 , y0)]; λ n; zs}× ⎩ N

⎡

∑n =1 exp ⎢i Ψˆλn (x, y)+ ⎣

which is generated because of crosstalk or the wrong phase keys used in decryption. Note that only using the matched phase key and lock with the corresponding encryption parameters can reconstruct the approximated target image. Otherwise, a noise-like image is obtained. The noise caused from crosstalk increases as the number of synthesized POFs increases because that the POFs are summarized to achieve the multiplexing purpose at the FrT plane.

⎫ 2π (μn x 0 + νn y0) ⎤ ⎥; λ n; z⎬ ⎦ λ nz ⎭

⎫ ⎧ N =IFrT ⎨FrT{exp[iφˆ λn (x 0 , y0)]; λ n; zs}× ∑n =1 exp[i Ψ'λn (x, y)]; λ n; z⎬ ⎭ ⎩ =IFrT{FrT{exp[iφˆ λn (x 0 , y0)]; λ n; zs}×POMλ; λ n; z}=gˆnλ (x 0 −μn , y0 −νn ) exp [iε (x 0 , y0)]

+ ℵ(x 0 , y0)≈gˆ nλ (x 0 −μn , y0 −νn )+ℵ(x 0 , y0)=gˆnλ

(xn , yn)+ℵ(x 0 , y0),

4. Computer simulation (20)

The computer simulation is performed to verify the proposed image wavelength- and position-multiplexing schemes based on the DPRA. The computer is equipped with Intel Core(TM) i5-4750 CPU@ 3.30 GHz and 4 G DRAM, and Matlab R2011b is utilized for the simulation. Fig. 5 shows the nine input grayscale target images, which are separately corresponding to gn(x 0 , y0), n=1–9, of the size 64 ×64 . The POF of each target image sent to the DPRA is of size 800×600 . Fig. 6 shows the comparison result of the convergence speed and image quality for the methods based on the proposed DPRA and our previous MGSA, in which the reconstructed images converge at the 10th and 40th iterations, respectively. The corresponding computation periods are 4.6 and 5.8 s, which represents 20.7% improvement on the speed. In addition, the image quality in the proposed DPRA is also better than that in the MGSA. Regard to the wavelength multiplexing, the encryption parameters are set as follows: the fixed position distance z=1.3 m, and the

or

⎧ ⎤ ⎡ N ˆ z (x, y)+ 2π (μn x 0 + νn y0) ⎥ IFrT ⎨FrT{exp[iφˆzn (x 0 , y0)]; λ; zs}× ∑n =1 exp ⎢i Ψ ⎦ ⎣ n λz n ⎩ ⎫ ⎧ ; λ; zn⎬=IFrT ⎨FrT{exp[iφˆzn (x 0 , y0)]; λ; zs}× ⎩ ⎭ N

⎫

∑n =1 exp[i Ψ'zn (x, y)]; λ; zn⎬ ⎭

=IFrT{FrT{exp[iφˆzn (x 0 , y0)]; λ; zs}×POMz; λ; zn}=gˆnz (x 0 −μn , y0 −νn )exp [iε (x 0 , y0)]

+ ℵ(x 0 , y0)≈gˆ nz

(x 0 −μn , y0 −νn )+ℵ(x 0 , y0)=gˆnz (xn , yn)+ℵ(x 0 , y0),

(21)

where ℵ(x 0 , y0) denotes the noise or interference at the output plane,

Table 2 The reconstruction results of our previous wavelength multiplexing method (z=1.3 m, λ n=390 + 10n nm,n=1–9). In addition to the correctly reconstructed images in the diagonal, all the other images reconstructed by using the mismatched phase keys are still recognizable.

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one of the matched POFs is used. Tables 2 and 3 show the reconstruction results of our previous position and wavelength multiplexing methods, respectively, based on the double phase retrieval method in Ref. [11]. As shown in both tables, the images in the diagonals are correctly reconstructed by using the matched POFs φ and ψ. However, the other images are still visible and can be easily correlated to the original images. Such systems are not secure enough because the POF ψ has the much higher dominant effect than the POF φ during the reconstruction process. Similar results are obtained for both the proposed wavelength and position multiplexing schemes. Here only the results of position multiplexing are discussed for simplicity. First, the phase key φˆz3 (x 0 , y0) and the position z3=1.03 m are used in decryption. Fig. 8(a) shows the decrypted image at the reconstruction plane. Fig. 8(b) shows the enlarged reconstruction image gˆ3z (x 0 , y0). In another case, the phase key φˆz8 (x 0 , y0) and the position z8=1.08 m are used. Fig. 8(c) and (d) show the entire decryption image and the enlarged reconstruction image gˆ8z (x 0 , y0) at the reconstruction plane, respectively. The correlation coefficients ρ between the reconstructed images shown in Fig. 8(b) and (d) and the original target image are ρ=0.903 and ρ=0.943, respectively, and both present high similarity. The deviation for phase modulation is set (μn , νn )=(αDw, βDh), where α and β are within the range [−3, 3], and Dw and Dh are the width and height of the target image, respectively. Nevertheless, when the positions z3=1.03 m and z8=1.08 m are used, but with wrong phase key, only noise images are reconstructed. Fig. 9(a) and (b) show the

wavelengths λ n =390+10nnm , corresponding to the target images gn(x 0 , y0), n=1–9. Regard to the position multiplexing, the encryption parameters are set as follows: the fixed wavelength λ=632.8nm , and the positions zn =0.99+0.01nm , corresponding to the target images. The initial (k=0) phase keys φλ0n (x 0 , y0) or φz0n (x 0 , y0), n=1–9, are randomly generated. The encryption is then proceeded as shown in Figs. 1 and 4 to generate the nine phase keys φˆ λn (x 0 , y0) or φˆzn (x 0 , y0) for recording the image information and the corresponding synthesized phase functions POMλ or POMz. Fig. 7(a) and (b) show the examples of synthesized POMλ and POMz for wavelength and position multiplexing, respectively. After applying the DPRA for different target images, the distinct pairs of phase keys are determined and only the matched POFs can be used to successfully reconstruct the target image in decryption. In our simulation, both the correct and wrong phase keys and the FrT parameters are tested. When the correct the phase key φˆ λn (x 0 , y0) or φˆzn (x 0 , y0) and the corresponding synthesized phase functions POMλ or POMz through the FrT are multiplied and then using the IFrT with the correct FrT parameters, the target image can be successfully reconstructed at the designated location. Table 1 shows an example of the reconstruction results of the different combinations of the double POFs retrieved for the nine target images. In this table, the target images shown in the diagonal direction are successfully reconstructed because the two matched POFs are used. On the other hand, the other images which are reconstructed with the mismatched POFs are noise-like images and none of the visible silhouette can be observed even though

Table 3 The reconstruction results of our previous position multiplexing method ( λ=632. 8nm, zn=0.99 + 0.01n m, all the other images reconstructed by using the mismatched phase keys are still recognizable.

156

n=1–9). In addition to the correctly reconstructed images in the diagonal,

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Fig. 8. (a) The entire image reconstruction plane decrypted with the phase key φˆz3 (x 0 , y0) and the corresponding position parameter z3=1.03 m; (b) The enlarged approximated image gˆ3z (x 0 , y0) retrieved at the reconstruction plane; (c) The entire reconstruction image plane decrypted with the phase key φˆz8 (x 0 , y0) and the corresponding position parameter z8=1.08 m; (d) The enlarged approximated image gˆ8z (x 0 , y0) retrieved at the reconstruction plane.

Fig. 10. (a) Comparison of the correlation coefficients under the different numbers of encrypted images among the proposed and the previous wavelength multiplexing methods in Refs. [14] and [17]; (b) Comparison of the correlation coefficients under the different numbers of encrypted images among the proposed and the previous position multiplexing methods in Refs. [15] and [18].

wavelength multiplexing scheme, when the incident light wavelength λ1′ is different from the wavelength λ1 used in encryption, the decrypted image quality obviously decreases. Fig. 11(a) shows the effects caused by the deviation of the wavelength λ1′, whose range is from -30 nm to +30 nm , and the gap between two consecutive wavelengths is 1.875 nm. Regard to the position multiplexing scheme, when the distance z1′ between the input and FrT planes or between the FrT and the output planes is different from the distance z1 set in encryption, the decrypted image quality decreases. Fig. 11(b) shows the effects of the deviation on the distance z1′ in decryption. The deviation is within the range from −0.006 to 0.006 m, and the gap between the two consecutive positions is 4 mm. As shown in Fig. 11(a) and (b), when the parameter deviation increases, in either the wavelength or position multiplexing schemes, the correlation coefficients between the reconstructed and target images decrease. Thus the proposed image multiplexing system presents high sensitivity to both the wavelength and position. The correlation coefficient ρ of the reconstructed image will be lower than 0.5 when the wavelength deviation is larger than 1.875 nm or the position deviation Δz is larger than 4 mm.

Fig. 9. The enlarged noise images reconstructed with (a) the correct position parameter z3=1.03 m but a wrong phase key at the same position in Fig. 8(a); (b) the correct position parameter z8=1.08 m but a wrong phase key at the same position in Fig. 8(c).

enlarged noise images retrieved from the reconstruction plane. The correlation coefficients ρ between the two decrypted noise images shown in Fig. 9(b) and (d) and the original target image are ρ=0.0023 and ρ=0.0024, respectively. Fig. 10(a) and (b) show the comparison results of the correlation coefficients among the decrypted target images in the proposed method, the wavelength multiplexing schemes in Refs. [14] and [17], and the position multiplexing schemes in Refs. [15] and [18]. As shown in the both figures, the proposed method achieves the highest correlation coefficients in either the wavelength or position multiplexing schemes when the number of encrypted images is less than four. The correlation coefficients are still higher than 0.85 when the number of encrypted images is nine. In addition to the number of encrypted images, the parameter errors in the decryption process also degrade the image quality. Both the wavelength and position multiplexing schemes are tested to evaluate the error effects on the image quality. Regard to the 157

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security level will be investigated. If the alignment issue of the two POMs and other optical devices can be solved, the optical experiments will be conducted as well. The number of POMs can be more than two in the proposed system if the higher security is desired. Acknowledgment This research is partially supported by Ministry of Science and Technology (MOST) Taiwan under contract numbers MOST 103-2221E-224-035-MY2 and MOST 105-2221-E-224-021-MY2. References [1] P. Refregier, B. Javidi, Optical image encryption based on input plane and Fourier plane random encoding, Opt. Lett. 20 (1995) 767–769. [2] R.K. Li, I.A. Watson, C. Chatwin, Random phase encoding for optical security, Opt. Eng. 35 (1996) 2464–2469. [3] Y. Li, K. Kreske, J. Rosen, Security and encryption optical systems based on a correlator with significant output images, Appl. Opt. 39 (2000) 5295–5301. [4] C.H. Yeh, H.T. Chang, H.C. Chien, C.J. Kuo, Design of cascaded phase keys for a hierarchical security system, Appl. Opt. 41 (2002) 6128–6134. [5] G. Unnikrishnan, J. Joseph, K. Singh, Optical encryption by double-random phase encoding in the fractional Fourier domain, Opt. Lett. 25 (2000) 887–889. [6] H.T. Chang, W.C. Lu, C.J. Kuo, Multiple-phase retrieval for optical security systems using random phase encoding, Appl. Opt. 41 (2002) 4825–4834. [7] G. Situ, J. Zhang, A cascaded iterative Fourier transform algorithm for optical security applications, Optik 114 (2003) 473–477. [8] Y. Zhang, C.H. Zheng, N. Tanno, Optical encryption based on iterative fractional Fourier transform, Opt. Commun. 202 (2002) 277–285. [9] Z. Liu, S. Liu, Double image encryption based on iterative fractional Fourier transform, Opt. Commun. 275 (2007) 324–329. [10] G. Situ, J. Zhang, Double random-phase encoding in the Fresnel domain, Opt. Lett. 29 (2004) 1584–1586. [11] H.-E. Hwang, H.T. Chang, W.-N. Lai, Fast double-phase retrieval in Fresnel domain using modified Gerchberg-Saxton algorithm for lensless optical security systems, Opt. Expr. 17 (2009) 13700–13710. [12] G. Situ, J. Zhang, A lensless optical security system based on computer-generated phase only masks, Opt. Commun. 232 (2004) 115–122. [13] S. Liu, Q. Mi, B. Zhu, Optical image encryption with multistage and multichannel fractional Fourier-domain filtering, Opt. Lett. 26 (2001) 1242–1244. [14] G. Situ, J. Zhang, Multiple-image encryption by wavelength multiplexing, Opt. Lett. 30 (2005) 1306–1308. [15] G. Situ, J. Zhang, Position multiplexing for multiple-image encryption, J. Opt. A: Pure Appl. Opt. 8 (2006) 391–397. [16] H.-E. Hwang, H.T. Chang, W.-N. Lai, Multiple-image encryption and multiplexing using modified Gerchberg-Saxton algorithm and phase modulation in Fresnel transform domain, Opt. Lett. 34 (2009) 3917–3919. [17] H.T. Chang, H.-E. Hwang, C.L. Lee, M.-T. Lee, Wavelength multiplexing multipleimage encryption using cascaded phase-only masks in Fresnel transform domain, Appl. Opt. 50 (2011) 710–716. [18] H.T. Chang, H.-E. Hwang, C.L. Lee, Position multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domain, Opt. Comm. 284 (2011) 4146–4151. [19] M. Paturzo, P. Memmolo, L. Miccio, A. Finizio, P. Ferrarro, A. Tulino, B. Javidi, Numerical multiplexing and demultiplexing of digital holographic information for remote reconstruction in amplitude and phase, Opt. Lett. 33 (2008) 2629–2631. [20] X.F. Meng, L.Z. Cai, Y.R. Wang, X.L. Yang, X.F. Xu, G.Y. Dong, X.X. Shen, H. Zhang, X.C. Cheng, Hierarchical image encryption based on cascaded iterative phase retrieval algorithm in the Fresnel domain, J. Opt. A: Pure Appl. Opt. 9 (2007) 1070–1075. [21] H.T.Chang, H.-E.Hwang, M.-D.Wu, Analysis of system parameters in double-phase wavelength and position multiplexing for multiple image encryption, in: Proceedings of the 2012 National Symposium on Telecommunications, Changhua Taiwan, Nov., 2012, pp.16–17. [22] J.W. Goodman, Introduction to Fourier Optics, 3rd ed., Roberts and Company Publishers, Greenwood Village, Colorado, USA, 2004 Ch. 4. [23] J. Rosen, Learning in correlators based on projection onto constraint sets, Opt. Lett. 18 (1993) 1183–1185.

Fig. 11. Sensitivity analyses of the correlation coefficients on the reconstructed images when there are parameter deviations on (a) wavelength and (b) position in decryption.

5. Conclusions The optical image multiplexing and authentication method based on the DPRA is proposed in this study. The parametric characteristics of wavelength and position in the FrT are utilized to achieve the image multiplexing and authentication. Compared with the previous studies which also based on the double-POF architecture, the number of required parameters in the FrT and the reconstructed image quality are similar. However, the proposed method provides higher security because only the noise-like image can be decrypted when any of the keys is wrong during the reconstruction process. In our future work, more analysis of the parameter sensitivity on system performance and

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