Optically secured information retrieval using two authenticated phase-only masks

We propose an algorithm for jointly designing two phase-only masks (POMs) that allow for the encryption and noise-free retrieval of triple images. The images required for optical retrieval are first stored in quick-response (QR) codes for noise-free retrieval and flexible readout. Two sparse POMs are respectively calculated from two different images used as references for authentication based on modified Gerchberg-Saxton algorithm (GSA) and pixel extraction, and are then used as support constraints in a modified double-phase retrieval algorithm (MPRA), together with the above-mentioned QR codes. No visible information about the target images or the reference images can be obtained from each of these authenticated POMs. This approach allows users to authenticate the two POMs used for image reconstruction without visual observation of the reference images. It also allows user to friendly access and readout with mobile devices.

Image hiding algorithm with sparsity constrains. The procedure of designing two POMs P 1 and P 2 that together reconstruct triple images consists of the following steps: 1. Storing three target images (f 1 , f 2 , f 3 ) in QR codes (q 1 , q 2 , q 3 ). 2. Generating two sparse POMs (p 1s , p 2s ) from two images (g 1 , g 2 ), which are used as reference for authentication and differ from the target images. 3. Numerical calculation of the two POMs P 1 and P 2 using the sparse POMs and the QR codes.
To generate two sparse POMs, two secret images g 1 and g 2 need to be respectively encoded in two different POMs by using a modified Fresnel domain GSA 11,12 with the two intensity constraints, i.e., a unit amplitude in the input plane and the image to be encoded in the output plane at a certain distance. Note that the two images used as references are not QR codes used for image encoding. The images g 1 and g 2 are independently encoded into POMs p m and ′ ′ p m in two iterative processes where the propagation distances are d 1 and d 2 . Note that the subscripts m and m′ represent the number of iterations in iterative processes. When p m is illuminated with incident plane wave, an approximation of g 1 can be observed in the object plane at distance d 1 , which can be written by Likewise, we can obtain an approximation of g 2 in the output domain at the distance d 2 , which can be given by Sparse representation of the encrypted data can be successfully used for information authentication in some DRPE-based security systems 14,[21][22][23][24][25] . In this step, two sparse phase functions p 1s and p 2s that used for the calculation of P 1 and P 2 can be randomly extracted from the outcomes of the iterative processes, i.e., p m , ′ ′ p m . Then the following step is numerical calculation of the two authenticated POMs P 1 and P 2 by using the obtained sparse POMs together with the QR codes. A modified double-phase retrieval algorithm (MDPRA) is proposed to achieve this purpose, where the sparse POMs and the QR codes are used as support constraints. In the MDPRA, let functions ( ) P n 1 and ( ) P n 2 respectively denote the two estimates for P 1 and P 2 , where the superscript n represents the nth iteration of the algorithm. In the initial stage, two random POMs can be used as ( ) P 1 1 and ( ) P 2 1 , respectively. The QR codes, i.e., q 1 , q 2 and q 3 , are the three amplitude constraints in the output planes with respect to different distances from the second POM ( ) P n 2 , i.e., z 1 , z 2 and z 3 . Note that the final solutions for P 1 and P 2 are ( + ) P N 1 1 and ( + ) P N 2 1 if the iterative stops after N iterations. The obtained two POMs P 1 and P 2 require authentication before being applied for optical image retrieval. For brevity, only the identifying of P 1 is explained. The reconstructed signal from P 1 given by will be compared with the original image g 1 , by nonlinear correlation in our proposal, where the two parameters d 1 and λ can be used as keys for authentication. The authentication method is described as follows 25,26 : Experimental Simulations and performance analyses. Figure 2(a,b) are two 500 × 500 color images, which can be respectively inserted into the QR codes shown in Fig. 2(d,e) via hyperlinks. When a user scans the QR code containing the hyperlink which automatically redirects the user to the image. Figure 2(c) shows input the input text information (NUS stands for National University of Singapore) and its respective QR code is presented in Fig. 2(f). All of the QR Codes have the size of 500 × 500. Due to its fast readability, great storage capacity and high damage tolerance capability, storing data in QR codes during the processes of optical encryption [27][28][29] and authentication 21-23 holds many practical advantages. Once the reconstructed QR codes are scanned by smartphones or tablets, the target images can be successfully revealed.
In our simulations, the pixel dimensions and the illumination wavelength are set as 8μm × 8μm and λ = 633 nm, respectively. Figure 3(a,c) show two 500 × 500 pixels binary images used as references for authentication. The propagation distances are given by d 1 = 6 cm and d 2 = 8 cm, respectively. Figure 3(b,d) demonstrate two sparse phase functions p 1s and p 2s that are randomly extracted from the outcomes of the iterative processes, p 50 and ′ p 50 , for which both the percentages of the extracted pixels with respect to the pixel size of their originally recovered phase images are 28%.
The two sparse POMs are used as two constraints in the proposed MDPRA, together with the three QR codes. The correlation coefficient (CC) is applied to evaluate the similarity between the recovered images ′ q i and their original images q i , which is defined by where E[] denotes the expected value operator. The CC values get the maximum value of 1 if ′ q i are perfectly correlated with q i . Figure 4 can be generated after the nth iteration, which require authentication before being used for image reconstruction. Evaluation of the correlation outputs can be implemented by using peak-to-correlation (PCE) 23 , which is defined as the ratio between the maximum intensity peak value and the total energy of the output plane, usually indicates the sharpness and height of the output correlation peak. Figure 4 shows the two PCE curves obtained by using all correct authentic keys and the nonlinearity strength ω = 0.4. Different from the PCE curve corresponding to P 2 , the curve with respect to P 1 rises rapidly during the first twenty iterations. After that, the PCE values increased slowly and then reaches a plateau (0.2846) at 145 iteration. However, the PCE values obtained with the second POM P 2 moves up and down in a limited range. It reaches its maximum (0.0390) at 99 iterations. In order to obtain high-quality correlation peak intensity in authentication, the POMs generated after 99 iterations are chosen as the two POMs  used for authentication and image retrieval since the CC values increase very slowly after 100 iterations.
The two POMs P 1 and P 2 designed for triple-image reconstruction and obtained after iteration number of 99 are respectively shown in Fig. 5(a,b). They are required for authentication before being used for image reconstruction. The diffraction patterns of P 1 and P 2 are shown in Fig. 5(c,d), at the distances of d 1 and d 2 respectively, from which no information about the two secret binary images can be observed.
When those two visually unrecognizable images [ Fig. 5(c,d)] are respectively compared with the references, i.e., Fig. 3(a,c), by nonlinear correlation, sharp correlation peaks could be obtained as shown in Fig. 6(a,b), which implies that the two POMs are successfully authenticated with correct parameters.
The security performance of the two POMs is further investigated. When only one of the POMs is placed in the optical scheme shown in Fig. 1, the diffraction patterns in the three object planes are shown in Fig. 7, from which no valuable information about the QR codes could be observed. Note that Fig. 7 only shows the intensity distributions within the area of 4 mm × 4 mm on the output display. The results  shown that neither of the two POMs has the problem of untended information disclosure. The proposed security system can be regard as an information sharing system. Each POM will be sent to different receivers through communication channels. Only when a matched pair of POMs are used for decryption, the primary images can be retrieved by scanning the decrypted QR codes.
Results for the reconstructed images from the two POMs P 1 and P 2 in different object planes using all the correct physical and geometric parameters are shown in Fig. 8(a-c), which have worse quality than the original images due to the effect of energy loss. The energy of the above three recovered images respectively account for about 85%, 84% and 79% of the total energy in their corresponding object planes. Smartphone was used to display the decrypted images by scanning the reconstructed QR codes. As demonstrated in Fig. 8(d-f), the two target color images and the text information can be retrieved without visible loss of information.

Discussions
We developed an algorithm for jointly designing two POMs that allow for the encryption and noise-free retrieval of triple images. Compared with previous works, the proposed algorithm based on sparsity constraints and QR codes has the following features: • This approach allows users to authenticate the two POMs without visual observation of those images used as references for authentication. Since a huge number of differently encoded POMs can be sent out through communication channels 14 , the identifying and matching of the POMs in a simple and direct manner can help increase efficiency. The MPRA with sparsity constraints may expect to be used in the computation of digital holograms and meta-holograms 30-32 for image display and information authentication. • There is no problem of information disclosure in the proposed method. No information can be visually observed from the two POMs and their respective diffraction patterns. Our method can also be used for encryption of three-dimensional objects. • It allows user to friendly access and readout with mobile devices. The target images can be revealed without visible loss of information due to the property of high damage tolerance capability of QR codes. Optically secured information retrieval can be realized when the two designed POMs manufactured by a number of techniques including embossing on plastic films and encoding on photopolymer are placed in the proposed scheme shown in Fig. 1. Methods Double-phase retrieval algorithm with sparsity constraints. Let functions ( ) P n 1 and ( ) P n 2 respectively denote the two solutions for P 1 and P 2 , where the superscript n represents the nth iteration of the algorithm. In the initial stage, two random POMs can be used as ( ) P 1 1 and ( ) P 2 1 , respectively. The QR codes, i.e., q 1 , q 2 and q 3 , are the three amplitude constraints in the output planes with respect to different distances from the second POM ( ) P n 2 , i.e., z 1 , z 2 and z 3 . The process proceeds as follows: (i) The first POM ( ) P n 1 illuminated with incident plane wave is first Fresnel-transformed at the propagation distance z 0 . The resultant wave function can be written as  where the symbol ⨁ denotes a particular way of data embedding. For clarity, the calculation of ( + ) P n 1 1 described by Eq. (9) is explained. We first extract the non-zero pixels of POM p 1s as the data background for POM ( + ) P n 1 1 , and then embed the pixel values of ( ) r n 1 into ( + ) P n 1 1 pixel by pixel but keep the background unchanged.
To generate another POM used for next iteration, we first compute a POM function ( ) r n 2 by using =   where the superscript * denotes conjugation.
(iv) Repeat steps (i)-(iii) until the preset threshold value is satisfied. Suppose the iteration process stop at the Nth iteration. The convergence of the algorithm is demonstrated by the relation between the CC values [between q i and their approximates ′ q i obtained by substituting the two POMs computed with Eqs. (9) and (10) into Eq. (1)] and the iteration numbers.
To sum up, a flowchart of the iterative phase retrieval algorithm is depicted in Fig. 9. So far, we have obtained the two POMs P 1 and P 2 , which require authentication before being applied for optical image retrieval and can be respectively represented by function ( + ) P N  5) and (6), we can readily obtain ′ = ( + ) q U i i N 1 , which implies that the finally recovered images ′ q i could be expected to be closer to their original images q i by increasing the number of iterations before reaching plateau values. It should be pointed out that the MDPRA is designed under the framework of nonlinear double random phase encoding. Different from previously proposed methods of single image encoding 16,17 , here we encode three QR codes into two POMs in Fresnel domain. Two sparse POMs calculated from two reference images for authentication are used as constraints in the iteration process.