Iterative partial phase encoding based on joint fractional Fourier transform correlator adopting phase-shifting digital holography

Optics Communications 313 (2014) 1–8

Contents lists available at ScienceDirect

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

Iterative partial phase encoding based on joint fractional Fourier transform correlator adopting phase-shifting digital holography Qu Wang a,n, Qing Guo b, Liang Lei a, Jinyun Zhou a a b

School of Physics and Optoelectronic Engineering, Guangdong University of Technology, Guangzhou 510006, China Center for Earth Observation and Digital Earth, Chinese Academy of Sciences, Beijing 100094, China

art ic l e i nf o

a b s t r a c t

Article history: Received 30 August 2013 Received in revised form 21 September 2013 Accepted 24 September 2013 Available online 8 October 2013

In this paper, digital holography based on two-step phase shifting interference (PSI) was applied to realize the iterative partial phase encoding with joint fractional transform correlator (JFTC). By this security system, the primitive image is encoded in two joint fractional power spectra (JFPS) corresponding to different phase-shifting values. The encrypted image can be deduced directly from the JFPSs by digital means, thus eliminating the noise interference of dc and conjugate terms. JFTC not only relaxes the alignment requirement but also avoids the beam splitting required by traditional holography. In the iterative partial phase encoding, the random phase masks (RPMs) are generated by chaotic mapping, and encoding areas are confined by a sequence of random binary masks. To recover the primitive image, decipher must regenerate the partial RPMs with correct chaotic conditions and perform inverse fractional Fourier transforms with correct orders. The decryption process can be realized by JFTC or by totally digital means. Simulation and experimental results have been presented to test security level and feasibility of the scheme. & 2013 Elsevier B.V. All rights reserved.

Keywords: Iterative phase encoding Phase shifting interference Joint fractional transform correlator

1. Introduction Since Réfrégier and Javidi proposed double random phase encoding (DRPE) technique for image encryption in 1990s [1,2], optical encryption and security systems have drawn more and more attentions with their high speed and parallel processing capability. In DRPE, a primitive image can be transformed into a stationary white noise by two statistically independent random phase masks (RPMs), one placed in the input plane and the other in the Fourier plane. Many modified variations of DRPE have also been proposed in different optical transform domains for image encryption [3–6], hiding [7–8] and watermarking [9–11] with larger key space. However, in recent years, the DRPE methods have been proved to be vulnerable against some potential attacks in cryptanalysis [12–14]. As an evolution forms of DRPE, some iterative random phase encryption structures have been reported for enhancing the security strength where more RPMs are introduced as supplement keys [15–17]. Moreover, to reduce the information storage in practical application, these RPMs are generally produced by a two-dimensional chaotic mapping equation with several predefined parameters and a single initial RPM. To remove the need to generate the complex conjugate of the key and accurate alignment required in the conventional DRPE

n

Corresponding author. Tel.: þ 86 2087084387. E-mail address: [email protected] (Q. Wang).

0030-4018/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2013.09.058

based on 4f architectures, several optical encryption systems based on joint transform correlator (JTC) architecture have been reported [18–24]. In Nomura's JTC scheme [18], the primitive image attached with a phase mask is placed side by side with a key mask in the input plane of JTC. The encrypted data are recorded in the joint power spectrum (JPS) as the magnitude squared of the amplitude and phase information. Furthermore, this optical setup is compact because the object and reference beams share a single 2f system. On the other hand, earlier DRPE based methods in general involve the complex encrypted data that need to be recorded holographically. Recent growing interest in transmission of secure information via digital communication channels made that different encryption techniques based on digital holography were developed [24–28]. Recently, Modified JTC systems combined with two-step or three-step phase-shifting interference (PSI) digital holography were also introduced [22,23,27,28]. By twostep PSI technology, the encrypted image can be deduced from only two interference intensity distributions by digital means, thus eliminating the noise disturbance caused by the dc and conjugate terms in the JTC output plane. The fractional Fourier transform (FRT) was introduced to the optical community within the past decade, and has been extensively studied for applications to optical encryption because it can increase the degrees of freedom of the security system design. Recent research has shown that many encryption schemes based on Fourier transform (FT) can be successfully extended into the fractional Fourier domain with key space increased greatly

2

Q. Wang et al. / Optics Communications 313 (2014) 1–8

[4,5,15,29–34]. The joint fractional transform correlator (JFTC) can be considered as an extension of traditional JTC in which FT operations are replaced by FRT operations. Recently Lu and Jin presented a single-channel scheme for color image encryption based on JFTC and phase retrieval algorithm [33]. Simulation results showed that this method provide efficient solution with a strong sense of security. In a previous work, we also constructed a JTC–JFTC hybrid encryption system based on Mach–Zehnder interference architecture [32]. Security level of this system is improved by the fractional order and an additional master key, which has been verified by optical experimental results. Moreover the requirements for stability and alignment in this system are alleviated, since the interferometric operation is reduced to only once. In this paper, we will introduce a new image encryption system by the use of iterative partial random phase encoding and twostep PSI technique in a JFTC setup. To the best of our knowledge, this is the first paper where two-step PSI digital holography technique is applied in the JFTC architecture for iterative phase encoding of primitive image. Application of JFTC can not only relax the strict alignment of phase masks but also avoid the beam splitting for holographic recording of complex information. Moreover, different from conventional iterative phase encoding, a sequence of random binary masks is supplemented as additional keys to limit the encoding area of RPMs. The pixel data at the coordinates where binary masks are equal to be unity are processed by RPMs while the others are retained. On the other hand, the conjugate and zero order terms often encountered in conventional holography can easily removed by the two-step PSI technique in the proposed scheme. During encryption procedure, the complex distribution (object input), containing the secret image and a RPM (reference input), is placed side by side with an inverse FRT of another RPM to constitute the joint input and then fractional Fourier transformed. By adjusting a phase retarder placed in reference beam, two joint fractional power spectra (JFPS) are recorded, from which encoded image in the first step of iteration can be digitally derived. The obtained encoded image is then employed as object input of JFTC in the next step of iteration. Repeating above procedures, eventually, we can obtain the final encrypted image. It is noteworthy that following similar iteration procedures, we can realize decryption via JFTC architecture with conjugate RPMs. Numerical simulations and actual experiments have been performed to verity validity and feasibility of our system. The remaining sections of this paper are organized in the following sequence. In Section 2, the proposed encryption and decryption algorithm are introduced in detail. In Section 3, numerical simulations and experimental results are given to validate the performance of this encryption scheme. Concluding remarks are arranged in the final section.

mathematically defined as ψ k þ 1 ðm; nÞ ¼ F q ½ψ k ðm; nÞ 8 ψ ðm; nÞ=q 0 rψ k ðm; nÞ o q > < k ½ψ ¼ k ðm; nÞ  q=ð0:5  qÞ q r ψ k ðm; nÞ o 0:5 > : F ½1  ψ ðm; nÞ 0:5 r ψ ðm; nÞ o 1 q k k

where ðm; nÞ denote the position index, ψ k ðm; nÞ A ½0; 1Þ and control parameter q A ð0; 0:5Þ. Actually, other chaotic mapping equations, such as logistic map and piecewise nonlinear chaotic map, can also be used as generator of random distribution functions in this paper. An excellent sequence of RDFs ðψ 0 ; ψ 1 ; ψ 2 ⋯Þ can be generated to meet the requirement for phase encoding. Among the function sequence, we arbitrarily select l random functions ψ k0 ; ψ k1 ; …; ψ kl to constitute the required RPMs exp½jφ0 ðm; nÞ; exp½jφ1 ðm; nÞ; …; exp½jφl ðm; nÞ

ð2Þ

where φ0 ðm; nÞ ¼ 2πψ k0 ; φ1 ðm; nÞ ¼ 2πψ k1 ; …; φl ðm; nÞ ¼ 2πψ kl . These RPMs strongly depend on the choice of initial RDF, control parameter and sequence numbers k0 ; k1 ; …; kl , all of which together with the fractional orders of FRT can be considered as keys of iterative phase encoding system. To realize the partial phase encoding, a sequence of random binary masks S1 ; S2 ; …; Sl is supplemented as additional keys to confine the encoding range of RPMs. Firstly, following similar procedures mentioned above, we produced a sequence of random functions from the PWLCM system. Here we can choose initial RDF and control parameters that are different from those used for RPMs. Then an arbitrary threshold value between 0 and 1 was employed to binaries these random functions. The areas where random function values are larger than threshold are set to be unity while other areas are set to be 0. The phase distributions of RPMs used for partial phase encoding can be written as ( Si ðm; nÞ ¼ 1 φi ðm; nÞ φ′i ðm; nÞ ¼ ; 0 r ir l ð3Þ 0 Si ðm; nÞ ¼ 0 which means that only the areas where Si ¼ 1 will be encoded during random phase encoding procedures. The confined RPMs are called partial RPM. The randomness of encoding area will largely improve the security level of the proposed system. In the section 3, numerical simulation and experimental results will be given to verify the influence of these random binary masks on the retrieved images. 2.2. Encryption process and analysis Fig. 1 shows the diagram of optical JFTC architecture we have used to realize the iterative random phase encoding of secret image. Two-step PSI technique will be applied into this setup for digital deduction of encrypted image during each step of iterative encoding. Here the FRT is performed by a transform lens. The distance parameter d of FRT is determined by focal length f and

2. Theoretical analysis and implementing process 2.1. Random partial phase encoding In order to reduce the information load of storage and transition, the RPMs used for iterative partial phase encoding are generated by a chaotic mapping equation [35,36]. A random distribution function (RDF) ψ 0 , which has the same dimensionality with the secret image and distributes uniformly within the range of [0,1), is firstly generated by computer as initial condition of chaotic iterative system. In this paper, the piecewise linear chaotic map (PWLCM) is chosen as generator of RPMs, which can be

ð1Þ

Fig. 1. Optical JFTC architecture for image encryption.

Q. Wang et al. / Optics Communications 313 (2014) 1–8

transform order α in the form: d ¼ f ½1  cos ðπα=2Þ. In the input plane, two spatial light modulators (SLMs) are attached together to display the complex-valued joint input of JFTC, one for amplitude modulation and the other for phase modulation. The CCD placed in the output plane records the real positive JFPS distributions and transits them to a computer for digital processing. Before sending the primitive image into the encryption system, we must firstly generate a series of complex-valued encryption key images ki ðx; yÞ (i ¼ 1; 2; …; l) by performing digital inverse FRTs on the RPMs expðjφ′i Þ. The FRT operations are denoted by ki ¼ F  αi fexpðjφ′Þg, where F  α1 represents the inverse FRT operator with minus transform order  αi . These encryption key images will be used to form the joint input of JFTCs in the subsequent iterative phaseencoding operations. In the initial stage of random iterative phase encoding, the real positive primitive image oðx; yÞ is attached with the input RPM exp½jφ′0 ðx; yÞ to form a complex input image o′ðx; yÞ ¼ oðx; yÞexp½jφ′0 ðx; yÞ. Please note that here encoding area has been confined by binary mask S0 ðx; yÞ according to Eq. (3). The joint input of JFTC is formed by placing the image o′ðx; yÞ and the first key image k1 ðx; yÞ symmetrically in the x-direction with the separation distance 2a. As FRT has shift-variant property, two additional phase only functions, expð 7 j2πpxÞ, where p is a constant value, must be used in the input plane in order to realize the overlapping of FRT spectra. Please note that a phase retarder is placed right before the key image to provide two phase-shifts 0 and arbitrary value δ in ½0; π respectively for subsequent twostep PSI operations. Thus the joint input with phase-shifts 0 and δ can be mathematically expressed as o′ðx  a; yÞexpðj2πpxÞ þ k1 ðx þ a; yÞexpð  j2πpxÞ;

ð4aÞ

o′ðx  a; yÞexpðj2πpxÞ þ k1 ðx þ a; yÞexpð  j2πpxÞexpðjδÞ;

ð4bÞ

where a denotes the shifting coordinates of the joint inputs in the input plane. An expended collimated beam illuminates the input plane perpendicularly, followed by a FRT with order α1 . For the joint input shown in Eq. 4(a), we have   Fα1 k1 ðx þ a; yÞexpð j2πpxÞ n h  πα1  πα1 ¼ exp jπ p 2u1 þp sin cos 2 2  πα1  π io sin α1 þ a 2u1 þ a cos 2 2    πα1 πα1 þ p sin ; v1 Fα1 k1 ðx; yÞ u1 þ a cos 2 2 and   Fα1 o′ðx a; yÞexpðj2πpxÞ

ℑα1 ðu1 ; v1 Þ

  2  

a  p2 πα1 πα1  ¼

exp jπ sin πα1 þ 2u1  p cos þ a sin 2 2 2   α1 F k1 ðx; yÞ ðu1 ; v1 Þ   2   a  p2 πα1 πα1  sin πα1 þ 2u1 p cos  a sin þ exp jπ 2 2 2

2   Fα1 o′ðx; yÞ ðu1 ; v1 Þ



2

2     ¼ Fα1 k1 ðx; yÞ ðu1 ; v1 Þ þ Fα1 o′ðx; yÞ ðu1 ; v1 Þ

h   i   πα1 πα1 þ a sin Fα1 k1 ðx; yÞ þ exp j4πu1  p cos 2 2   ðu1 ; v1 ÞFα1 n o′ðx; yÞ ðu1 ; v1 Þ h  πα1 πα1 i þ a sin þ exp j4πu1  p cos 2 2     ð6Þ Fα1 n k1 ðx; yÞ ðu1 ; v1 ÞFα1 o′ðx; yÞ ðu1 ; v1 Þ: Here * is conjugate operator. For simplicity, the FRT spectra of o′ðx; yÞ and k1 ðx; yÞ are denoted by O′α1 ðu1 ; v1 Þ and K α1 ðu1 ; v1 Þ in the following forms:   ð7aÞ O′α1 ðu1 ; v1 Þ ¼ Fα1 o′ðx; yÞ ¼ A1 ðu1 ; v1 Þexp½jθ1 ðu1 ; v1 Þ   K α1 ðu1 ; v1 Þ ¼ Fα1 k1 ðx; yÞ ¼ exp½jφ′1 ðu1 ; v1 Þ:

ð7bÞ

Please note that the FRT spectrum K α1 ðu1 ; v1 Þ is actually the RPM expðjφ′1 Þ because, as mentioned above, the key image k1 ðx; yÞ is digitally generated by the inverse FRT ( α1 -order) of expðjφ′1 Þ in advance. Thus the JFPS can be rewritten as

2

2 J α1 ðu1 ; v1 Þ ¼ K α1 ðu1 ; v1 Þ þ O′α1 ðu1 ; v1 Þ

h  πα1 πα1 i þ a sin þ exp j4πu1  p cos K α1 ðu1 ; v1 ÞO′n α1 ðu1 ; v1 Þ 2 2 h  πα1 πα1 i n þ a sin K α1 ðu1 ; v1 ÞO′α1 ðu1 ; v1 Þ; þ exp  j4πu1  p cos 2 2 ð8Þ  Let P 1 ðu1 Þ denote the known phase factor 4πu1  p cos πα21 þ a sin πα21 Þ. The obtained JFPS is further simplified as ¼ 1 þ A21 þ 2A1 cos ðθ1  φ′1  P 1 Þ;

ð9Þ

2

ð5aÞ

ð5bÞ

   πα1 πα1  p sin ; v1 ;  Fα1 o′ðx; yÞ u1  a cos 2 2 where Fα1 denotes the FRT operator with order α1 , and ðu1 ; v1 Þ represent the coordinates of α1 -order FRT domain. The FRT will reduce to the conventional FT when the fractional order α1 ¼ 1. To ensure that the FRT spectrum of secret image o′ðx; yÞ overlaps the spectrum of k1 ðx; yÞ in the transform plane, the constant value

M¼

p is taken to be  a cotðπα1 =2Þ according to the shift-variance of FRT. Consequently, the JFPS corresponding to phase shift 0 can be written as below

J α1 ¼ 1 þ A21 þexpðjP 1 ÞK α1 O′n α1 þ expð  jP 1 ÞK nα1 O′α1

n h  πα1  πα1 ¼ exp jπ p 2u1  p sin cos 2 2  πα1  πα1 io sin a 2u1  a cos 2 2

 2 2m1 J αδ 1 þJ α1 þ 4 sin ðP 1 þ δÞ 

3

where jK α1 ðu; vÞj ¼ 1, because K α1 ðu; vÞ is a pure phase function (Eq. (7b)). Using similar method, for the joint input with phase shift δ (Eq. (4b)), we can get JFPS as below J αδ 1 ¼ 1 þ A21 þ expðjP 1 ÞK α1 expðjδÞO′n α1 þ expð  jP 1 ÞK nα1 expð  jδÞO′α1 ¼ 1 þ A21 þ 2A1 cos ðθ1  φ′1  P 1  δÞ:

ð10Þ

For simplicity, here, we omit the deriving procedures of J αδ 1 . Applying the two-step PSI technique [27], we can obtain the complex-valued encrypted image E1 of the first iteration by digital means from the JFPSs J α1 and J αδ 1 , the phase shift value δ and the known phase distribution P 1 . The derivation procedures can be mathematically given by E1 ðu1 ; v1 Þ ¼ A1 ðu1 ; v1 Þexp½jðθ1  φ′1 Þ   J α1  M cos ðP 1 þ δÞ  1 J α1  J α1 cos ðP 1 þ δÞ þj Mþ δ ; ¼ 2 2 sin ðP 1 þ δÞ 2 sin ðP 1 þ δÞ ð11Þ where

rhffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i2  2 2m1 Jαδ 1 þJ α1 þ 4 sin ðP 1 þ δÞ  8m1 m2 ; 4m1

4

Q. Wang et al. / Optics Communications 313 (2014) 1–8

m1 ¼ 1  cos ðP 1 þ δÞ  2 2 and m2 ¼ J αδ 1 þ ðJ α1 Þ2 þ 4 sin ðP 1 þ δÞ 2J αδ 1 J α1 cos ðP 1 þ δÞ: By Eq. (11), the encrypted image of the first iteration E1 ðu1 ; v1 Þ is exactly derived from two JFPSs. The influences of zero order terms (the first two terms of Eqs. (9) and (10)) and the conjugate terms (contained in cosine terms) will not appear in the encrypted results. The beam splitting often used in conventional holographic recording has also been circumvented. Moreover the JFTC architecture avoids the precise alignment of RPM. The encrypted image E1 is then sent to the input plane of the next iteration, constituting the joint input of the next iteration with the second key image   k2 ðu1 ; v1 Þ ¼ F  α2 exp½jφ′2 ðu2 ; v2 Þ , where ðu2 ; v2 Þ denote the coordinates of α2 -order FRT domain. By using two-step PSI method again, we can digitally extract the encrypted image E2 ðu2 ; v2 Þ of this stage of phase encoding. After repeating above-mentioned procedures l times, the final encrypted image is achieved as      El ðul ; vl Þ ¼ Fαl ⋯ Fα2 Fα1 o′ðx; yÞ     expð  jφ′1 Þ expð  jφ′2 Þ ⋯ expð  jφ′l Þ; ð12Þ where ðul ; vl Þ denote the coordinates of αl -order FRT domain. It is impossible to recover the primitive image by inverse FRTs directly on the encrypted image El without knowing the encrypted RPMs. During the whole encryption procedures, the fractional orders of

FRTs, the chaotic control parameters and the initial conditions used to generate the complete RPMs exp jφi , and the random binary masks Si (i ¼ 1; 2; …; l), are all security keys. Sensitivity of chaotic mapping equation provides a considerably large key space for our security system to resist against potential brute force attacks.

2.3. Decryption process and analysis The decryption procedures of our system can be considered as the inverse procedures of Eq. (12), which can be realized either by totally digital means or by optoelectronic JFTC architecture based on two-step PSI digital holography. Firstly, decipher must use correct chaotic initial conditions and control parameters to generate the conjugates of partial RPMs, expð  jφ′i Þ (i ¼ 1; 2; …; l 1). Performing digital FRTs on these RPMs,  a seriesof decryption key images are obtained as dki ¼ Fαi þ 1 expð  jφ′i Þ (i ¼ 1; 2; …; l  1) beforehand. In the optoelectronic decryption, the joint input distribution, containing the final encrypted image multiplied by RPM El expðjφ′l Þ and decryption key image dkl  1 , is displayed in the input plane. As done in the encryption procedures, to realize the spatial overlapping of inverse FRT spectra, the joint input is modulated by two additional phase only functions in the following

 Fig. 2. (a) Primitive image Lena; (b) phase distribution of RPM exp jφ3 ; (c) binary mask S3 ; (d) JFPS distribution for phase shifting δ ¼ π=3 in the third iteration; (f) The real part and (g) imaginary part of final encrypted image.

Q. Wang et al. / Optics Communications 313 (2014) 1–8

form:

3. Numerical simulation and experimental results

El ðul  a; yÞexp½jφ′l ðul  a; yÞexpðj2πpul Þ

3.1. Numerical simulations

þ dkl  1 ðul þ a; vl Þexpð  j2πpul Þ;

ð13Þ

where p ¼  a cotð παl =2Þ. Phase retarder placed before the decryption key image can introduce a constant phase shift δ within ½0; π. Then an inverse FRT with order  αl is then performed by lens and recorded by CCD. Subsequent procedures are very similar to the encryption. Finally, from the JFPSs corresponding to two phase shifts 0 and δ, the following decrypted image is obtained by digital means as below El  1 expðjφ′l  1 Þ ¼ F  αl fEl expðjφ′l Þgexpðjφ′l  1 Þ;

ð14Þ

which will be used as object input of the next iteration. Repeating above procedures l times with correct FRT orders and RPMs, we can eventually retrieve the primitive image without error. In the next section, we will give some numerical simulations and experimental results to demonstrate performance of the proposed scheme.

5

Firstly, we verify the security level of the encryption system by numerical simulations at the Matlab platform. Fig. 2(a) shows the primitive gray-level image, a normalized Lena with the size of 256  256 pixels. In our simulations, The iteration number was l ¼ 5. The chaotic parameter of PWLCM used for generating the RPMs and the corresponding binary masks were chosen to be q1 ¼ 0:27 and q2 ¼ 0:34 respectively. Two statistically independent RDFs, which distributes in the range of [0,1) and have the same dimensional size as the primitive image, were produced by computer as initial conditions of PWLCM for RPMs and random binary masks respectively. Based on the iterative laws given in Eq. (1), two sequences of RDFs can be generated. In the first sequence, RDFs numbered by 26, 39, 73, 45, 81 and 69 were selected to form the random phases φ0 ; φ1 ; …; φ5 respectively while in the other sequence, RDFs numbered by 35, 42, 59, 84, 21 and 66 were stochastically selected to generate the corresponding binary masks S0 ; S1 ; …; S5 . Here the binary operations were performed with the threshold value 0.5. In the iterative phase

Fig. 3. (a) decrypted image obtained with correct keys; (b) decrypted image obtained with wrong order of inverse FRT α2 ¼  0:59; (c) decrypted image when incorrect  initial RDF ψ 0 or (d) incorrect control parameter q1 ¼ 0:27þ 10  16 is used to generate RPMs; (e) decrypted image obtained directly from the RPMs exp jφi (i ¼ 0; 1; …; 5); (f) recovered image when two random binary masks, S2 and S4 , were mistakenly exchanged for decryption; (g) recovered image when incorrect S3 was applied for regenerating RPMexpðjφ′3 Þ.

6

Q. Wang et al. / Optics Communications 313 (2014) 1–8

encoding procedures, fractional orders of FRTs are taken to be α1 ¼ 0:24, α2 ¼ 0:58, α3 ¼ 0:36, α4 ¼ 0:72 and α5 ¼ 0:65. 2  Fig. (b) and (c) illustrate the phase distribution of RPM exp jφ3 and binary mask S3 respectively. The RPM expðjφ′3 Þ directly used for partial phase encoding is displayed in Fig. 2(d). Phase shifting values for two-step PSI technique are 0 and π=3. In Fig. 2(e), we present one of the both JFPSs (δ ¼ π=3) in the third iteration of partial phase encoding. The real part and imaginary part of final encrypted image are shown in Fig. 2(f) and (g) respectively, from

5000 4000

MSE

3000

which one cannot any silhouette related to the primitive Lena image. During decryption procedures, the encrypted image, along with the initial conditions, control parameters of PWLCM, sequences numbers of RDFs and fractional orders of FRT, are provided to the legal decipher. For correct decryption, he must firstly regenerate the decryption RPMs expðjφ′i Þ(i ¼ 0; 1; ⋯; 5) for partial phase encoding from these known data through chaotic iteration. The correct decrypted image, as shown in Fig. 3(a), can be successfully obtained with all keys correctly applied. The undistorted Lena image demonstrates that interference cause by zero-order terms and conjugate terms in conventional JTC-based schemes can be effectively avoided by applying the two-step PSI digital holography technique. In this paper, we use mean square error (MSE) to measure the variation between primitive image and decrypted image. In discrete form, the MSE between two images is defined as MSE ¼

2000 1000 0 −20

−10

0

10

q1 Fig. 4. MSE curve corresponding to the change of q1 .

20 10−17

M N 1 ∑ ∑ ½I p ði; jÞ  I d ði; jÞ2 ; MN i j

ð15Þ

where Mand N are image size. I p and I d represent the primitive image and the retrieved image respectively. The MSE value of Fig. 3(a) is 8:42  10  23 . Fig. 3(b) display the noisy recovered image (MSE ¼ 3647) when only one fractional order of FRT is wrongly used (  α2 ¼  0:59) for decryption. If incorrect initial RDF ψ 0 or incorrect control parameter q1 ¼ 0:27 þ 10  16 is utilized to generate the RPM sequences, one can only retrieve the wrong noisy results shown in Fig. 3(c) (MSE ¼ 3620) and (d) (MSE ¼ 4120) respectively. Please note that when the wrong key was used, the other keys remained to be true. Fig. 3(d) demonstrates the high

Fig. 5. (a) Amplitude of encrypted image corrupted by white additive Gaussian noise (s ¼ 0:5); (b) recovered image corresponding to (a); (c) amplitude of encrypted image cut by25%; (d) recovered image corresponding to (c).

Q. Wang et al. / Optics Communications 313 (2014) 1–8

7

Fig. 6. (a) Primitive image for actual experiment; (b) amplitude of final encrypted image; (c) decrypted image obtained with correct keys; (d) decrypted image when iterative number is l ¼ 4; (e) recovered image obtained from incorrect RPMs or (f) incorrect order of inverse FRT α3 ¼  0:56; (g) recovered image when RDFs numbered by  26, 39, 72 and 45 were chosen to regenerate the RPMs. (h) decrypted image obtained directly from the RPMs exp jφi (i ¼ 1; 2; 3).

sensitivity of the decrypted results to the control parameter of chaotic mapping. To further prove the sensitivity in detail, in Fig. 4, we give the MSE values corresponding to different changes of q1 around the correct value. From Fig. 4, we can find the MSEs increase greatly as the q1 deviates from the required value with a very small error. Application of partial phase encoding can further enhance the security level of the proposed scheme, which can be verified by the recovered image presented in Fig. 3(e) (MSE ¼ 3874). Fig. 3(e) is  obtained directly from the complete RPMs exp jφi (i ¼ 0; 1; …; 5) without any binary masks involved (MSE ¼ 3936). Fig. 3(f) presents the recovered result (MSE ¼ 3862) when two random binary masks, S2 and S4 , were mistakenly exchanged during decryption procedures. If the illegal decipher does not know only one random binary mask, he can recover some silhouette of the primitive image, as shown in Fig. 3(g) (MSE ¼ 2317), which was produced when incorrect S3 was applied for regenerating the partial RPM expðjφ′3 Þ. In spite of this, the quality of recovered image deteriorates seriously. From above results, one can conclude that the random binary masks used for partial phase encoding can be regarded as additional protection of secret information in the proposed system. An eligible encryption system must have certain robustness against noise and occlusion attacks. Fig. 5(a) shows the amplitude of encrypted image corrupted by white additive Gaussian noise

with standard deviation s ¼ 0:5. The corresponding decryption result shown in Fig. 5(b) was gotten with correct decryption keys (MSE ¼ 1362). Fig. 5(c) shows the amplitude distribution of encrypted image cut by25%, where the black area is filled with zeros, and corresponding result is given in Fig. 5(d) (MSE ¼ 1287). Although both results are affected by noise interference, one can still discern the remaining visual information of original Lena image apparently. 3.2. Experimental results Next we give some experimental results to support the feasibility of the scheme in practical application. Owing to the current resource limitation in our laboratory, the encryption of primitive image was performed with PSI-based optoelectronic JFTC setup but the decryption was done by totally digital means. In the actual experimental setup, two translucent Holoeye LC2002 SLMs with a pixel size of 32 μm were employed to display the complex information of joint input of JFTC. In the joint input, the area of both input windows was 2.0  2.0 mm2, and the distance between windows was 2.8 mm. The incident collimated beam was provided by a laser with wavelength 632 nm. The focal length of transform lens for FRT was 200 mm. For simplicity of implementation setup,

8

Q. Wang et al. / Optics Communications 313 (2014) 1–8

the transform orders of FRTs were fixed to be 0.6 in the iterative encoding. Thedistance   parameter can be calculated by the relation d ¼ f 1  cos πα=2 . The intensity distributions (JFPS) of digital holography were recorded by a CCD (Sony, Japan: 752  582 pixels, pixel size 6.5  6.25 μm). An Arcoptix phase retarder with aperture of 10 mm was placed in the optical path of reference beam to introduce phase-shifting (δ ¼ 0; π=3) for digital holography. In our experiments, the iterative number was l ¼ 3 and control parameters of PWLCMs were chosen to be the same as those used in numerical simulations. Two sequences of RDFs were generated from two computer-generated initial values respectively via chaotic iteration. The RPMs were produced from the RDFs numbered by 26, 39, 73 and 45 in the first sequence while the corresponding random binary masks were produced from the RDFs numbered by 35, 42, 59 and 84 in the second sequence (threshold value 0.5). Fig. 6(a) and (b) present the primitive image, binary “E” letter, and the amplitude distribution of final encrypted image. By using correct decryption keys and fractional orders, one can obtain the recovered result shown in Fig. 6(c). The degradation of the image quality was mainly due to the inherent speckle noise and the limited resolution of experimental instruments. During iterative phase encoding, frequent optoelectronic transformations will lead to accumulation of noise, which has gotten to be a bottleneck problem for iterative phase encoding in practical application. In our case, iterative number l ¼ 3 is a satisfying trade-off between the security strength and decryption quality. In Fig. 6(d), we exhibit a decrypted image when iterative number is l ¼ 4. The quality degrades so seriously that one almost cannot discern the existence of letter “E”. Fig. 6(e) and (f) show the recovered results obtained from incorrect RPMs or incorrect fractional order of inverse FRT α3 ¼  0:56. The incorrect RPMs were generated from PWLCM with wrong control parameter q1 ¼ 0:27 þ 10  12 . Besides the control parameters, initial RDFs of PWLCM and that fractional orders of FRTs, decipher also must know the sequence numbers of RDFs that are selected for synthesizing the RPMs. The noisy results shown in Fig. 6(g) was obtained when RDFs numbered by 26, 39, 72 (wrong, correct number 73) and 45 were chosen to regenerate the RPMs. Finally, one cannot  get the correct primitive image by using the complete RPMs exp jφi (i ¼ 1; 2and 3) directly for decryption. Corresponding result is given in Fig. 6(h).

the proposed system more stable and more compact in practical use. In partial phase encoding, the RPMs are generated by PWLCM system and confined by a corresponding sequence of random binary masks. Compared with conventional iterative phase encoding schemes, the additional random binary masks provide new protection for the secret information. Illegal attacker cannot retrieve the primitive image only using the complete RPMs in which the encoding area is not confined by the random binary masks. Initial conditions, chaotic control parameters and sequence numbers of RPMs are the main keys of our system while the fractional orders of FRTs can be regarded as the additional keys. Some numerical simulations and actual experimental results have been given to verify the effectiveness and feasibility of the proposed scheme.

Acknowledgments This research is financially supported by the program of the National Natural Science Foundation of China (61101204, 61107029) and the Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (LYM10070). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]

4. Conclusions In this paper, we build up an optical encryption system based on JFTC architecture where iterative partial random phase encoding and two-step PSI holography technique are combined together. JFTC architecture helps relax the precise alignment of RPMs often required in previous iterative phase encoding schemes. Application of digital holography technique based on two-step PSI eliminates the noisy disturbance originating from dc and conjugation terms on the decryption results. Deriving encrypted image only need to record two interference intensity distributions (JFPSs) in each step of iterative encoding. Moreover beam splitting often used in the conventional holographic recording is also avoided, which makes

[22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]

P. Refregier, B. Javidi, Optics Letters 20 (7) (1995) 767. B. Javidi, Physics Today. 50 (1997) 27. O. Matoba, B. Javidi, Optics Letters 24 (1999) 762. G. Unnikrishnan, J. Joseph, K. Singh, Optics Letters 25 (12) (2000) 887. G. Unnikrishnan, K. Singh, Optics Engineering 39 (11) (2000) 2853. G. Situ, J. Zhang, Optics Letters 29 (14) (2004) 1584. Y. Shi, G. Situ, J. Zhang, Optics Letters 32 (13) (2007) 1914. S. Kishk, B. Javidi, Applied Optics 41 (2002) 5462. N. Takai, Y. Mifune, Applied Optics 41 (2002) 865. S. Kishk, B. Javidi, Optics Letters 28 (2003) 167. H. Chang, C. Tsan, Applied Optics 44 (2005) 6211. A. Carnicer, M. Montes-Usategui, S. Arcos, I. Juvells, Optics Letters 30 (2005) 1644. X. Peng, P. Zhang, H. Wei, B. Yu, Optics Letters 31 (2006) 1044. X. Peng, H. Wei, P. Zhang, Optics Letters 31 (2006) 3261. Y Zhang, C. Zheng, N. Tanno, Optics Communications 202 (2002) 277. Z. Liu, L. Xu, C. Lin, J. Dai, S. Liu, Optics and Lasers in Engineering 49 (2011) 542. Z. Liu, M. Yang, W. Liu, S. Li, M. Gong, W. Liu, S. Liu, Optics Communications 285 (2012) 3921. T. Nomura, B. Javidi, Optics Engineering 39 (9) (2000) 2031. E. Rueda, J.F. Barrera R., R. Henao, R. Torroba, Optics Communications 282 (16) (2009) 3243. E. Rueda, C. Ríos, J.F. Barrera, R. Torroba, Applied Optics 51 (11) (2012) 1822. E. Rueda, J.F. Barrera, R. Henao, R. Torroba, Optics Engineering 48 (2009) 027006. C. Mela, C. Iemmi, Optics Letters 31 (17) (2006) 2562. X. Shi, D. Zhao, Y. Huang, Optics Communications 297 (2013) 32. C.L. Chen, L.C. Lin, C.J. Cheng, Opt. Eng. 47 (2008) 068201. A. Nelleri, J. Joseph, K. Singh, Optics Engineering 47 (2008) 115801. R. Henao, E. Rueda, J.F. Barrera, R. Torroba, Optics Letters 35 (2010) 333. X. Meng, L. Cai, X. Xu, X. Yang, X. Shen, G. Dong, Y. Wang, Optics Letters 31 (2006) 1414. E. Tajahuerce, O. Matoba, S. Verrall, B. Javidi, Applied Optics 39 (2000) 2313. S.K. Rajput, N.K. Nishchal, Applied Optics 51 (10) (2012) 1446. Z. Zhong, et al., Optics Communications 285 (1) (2012) 18. Q. Wang, Q. Guo, J. Zhou, Optics Communications 285 (21–22) (2012) 4317. Q. Wang, Q. Guo, L. Lei, J. Zhou, Optics Engineering 52 (4) (2013) 048201. D. Lu, W. Jin, Chinese Optics Letters 9 (2) (2011) 021002. Z. Liu, S. Liu, Optics Communications 275 (2007) 324. Z. Liu, Q. Guo, L. Xu, M.A. Ahmad, S. Liu, Optics Express 18 (2010) 12033. C. Jeffries, J. Perez, Physical Review A 26 (4) (1982) 2117.