Two-Step Integral Imaging Coding Based Three-Dimensional Information Encryption Approach

We propose a two-step integral imaging coding based three-dimensional (3D) information encryption approach in this paper. In this approach, a synthetic aperture integral imaging system is applied to acquire a set of parallax images including spatial and angular information of 3D scene. In the encryption process, two-step coding is performed. In the first step, the acquired parallax images are encrypted firstly by double random-phase coding in the Fresnel domain. In the second step, these encrypted parallax images are encoded into a cipher image by mapping algorithm which is used to generate elemental image array of integral imaging. In the decryption process, an inverse operation is performed. The experimental results demonstrate the feasibility, security, and robustness of the proposed approach.


Introduction
For the past decade, the autostereoscopic techniques without any auxiliary devices have gained more attention.Integral imaging (II) is considered as one of the attractive threedimensional (3D) technologies since it can provide both horizontal and vertical parallaxes and quasi-continuous viewing angles [1][2][3][4].II consists of pickup process and reconstruction process.In the pickup process, the conventional II system inserts a microlens array in front of the image sensor to capture multiple images of a 3D scene from different perspectives [3,5].In this way, 3D information is limited because the resolution of elemental image is restricted by image sensor resolution and microlens size, and the aberrations and diffraction also degrade the image quality [6,7].To overcome these disadvantages, II pickup process can be implemented by using a camera array which can capture a larger field of view and high-resolution parallax images [8].A synthetic aperture integral imaging (SAII) system in which a single camera shifts on a 2D plane to capture different perspective information can take the place of the camera array to reduce cost and the complexity of devices [9].In the reconstruction process, optical integral imaging reconstruction (OIIR) and computational integral imaging reconstruction (CIIR) can be implemented to reconstruct the 3D information [10][11][12].With the rapid development of 3D technologies, the 3D information encryption has become an urgent problem due to 3D information transmission security.
Optical encryption methods have attached much attention because they have various merits [13][14][15][16][17][18][19][20][21][22][23][24][25].Double random-phase encoding in the Fresnel domain was proposed in 2004, and the encryption method is more flexible and compact because it is lensless and has higher security [26].In recent years, some image encryption methods have also been proposed based on II [27][28][29].An image encryption method using II and pixel scrambling technique was proposed [30].In this method, the image is scrambled firstly by pixel scrambling technique; then the scrambled image is picked up through a microlens array.Finally, the acquired elemental image array (EIA) is scrambled again to obtain encrypted image.However, the method is only applied to encrypt a 2D image.3D image encryption techniques based on computational integral imaging were proposed [31].In the method, computational integral imaging is applied to pick up double images to generate EIA; then the EIA is encrypted by a cryptographic technology.However, the technique cannot be applied to the real 3D scene.A 3D information encryption system based on II and multiple            chaotic maps was proposed [32].The system uses three chaotic maps to encrypt, respectively, three channels of EIA acquired by II.The encryption system is applied to real 3D scene.
In this paper, we propose a two-step coding 3D information encryption approach which is applied to real 3D scene based on an SAII system and double random-phase encoding (DRPE) in Fresnel domain, which improves further security and robustness.Meanwhile, the approach can reconstruct the flexible 3D information according to the II system configuration.In the process of two-step coding, parallax images encrypted by DRPE in Fresnel domain are interweaved to generate the cipher image using the mapping algorithm.In addition, the robustness of the proposed method is very high due to the data redundancy property of EIA.The experimental results verify the security and robustness of the proposed encryption approach, and the 3D information can be reconstructed flexibly by OIIR and CIIR according to II system configuration.

Theoretical Analysis
The proposed 3D information encryption approach is illustrated in Figure 1, and it consists of three parts: pickup, twostep coding, and reconstruction.In the pickup part, the 3D information is acquired firstly by using a SAII system, and  ×  parallax images which record different perspectives of 3D scene can be obtained.In the two-step coding part, the DRPE in Fresnel domain is applied to encrypt parallax images with different secret keys.Then a mapping algorithm

Parallax images
Cipher image is applied to interweave the encrypted parallax images to generate the cipher image.In the 3D information reconstruction part, the cipher image is divided into  ×  images by using the inverse process of the EIA mapping algorithm.Then the decryption algorithm is applied to revive parallax images, and the 3D information is reconstructed flexibly by OIIR or CIIR.

Acquisition of 3D Information
Using SAII System.In this part, the 3D information is acquired by an SAII system, in which a single camera is shifted on a 2D plane to capture different perspectives of the 3D scene.As shown in Figure 2, the SAII system can be considered as a camera array which is composed of × cameras.These cameras are located at the same plane, and the distance between adjacent cameras is .The optical axes of the cameras converge to point  which is located at the center of the central depth plane (CDP).Based on this SAII system, a set of parallax images recording 3D information of the scene are acquired.

Analysis of the Two-Step
Coding.The proposed encryption approach performs two-step coding to improve the security and robustness.The schematic of the encryption process is shown in Figure 3.In the first step, we use DRPE in Fresnel domain to encrypt recorded parallax images.The locations of the first random-phase mask RPM 1 , , the second random-phase mask RPM 2 , , and the encrypted image are defined as the input plane, the relay plane, and the output plane, respectively.For convenience, we define the (, )th parallax image and its corresponding encrypted image to be  , (, ) and  , (  ,   ), respectively.exp[(, )] and exp[(  ,   )] are used to represent RPM 1  , and RPM 2 , .The parallax image is located on the position which is attached to the input plane.A plane wave with  wavelength is used to illuminate the input image perpendicularly.The parallax image  , (, ) is modified by RPM 1 , , then travelling the distance  1  , to relay plane.In the case of satisfying Fresnel approximation, complex amplitude  , (  ,   ) in the relay plane can be expressed as where x and ŷ represent the spatial frequencies and FT represents Fourier transform.For convenience, we assume Then  , (  ,   ) is modified by RPM 2 , , then travelling the distance  2  , to the output plane to obtain the encrypted parallax image, and the complex amplitude  , (  ,   ) in the output plane can be expressed as follows: where Δ is the pixel size of the encrypted parallax images, p represents the pitch of the virtual microlens, and × is the resolution of the parallax images.The proposed encryption approach has multiple group secret keys, and we can adopt different secret keys or the same secret keys to encrypt each parallax image.And, the parameters of the mapping algorithm can be considered as a part of secret keys.Therefore, the approach has great security.

Analysis of 3D Information Reconstruction.
In decryption process, at first, the cipher image is divided into × images by using inverse EIA mapping algorithm.Then, the encrypted parallax images are decoded through the following operation: The  ×  parallax images are decrypted by corresponding secret keys, respectively.The 3D information can be recovered by II reconstruction which is performed by two methods: OIIR and CIIR.In the OIIR, 3D information is recovered according to the principle of reversibility of the light rays, which is the inverse of the pickup stage.In the CIIR technique, according to the geometric optics, 3D information can be constructed through mapping inversely parallax images into object space, as shown in Figure 4.The reconstructed image located the depth of  can be expressed as follows: where  and  denote the pixel coordinates of the reconstructed image,  , is the parallax image of the location of (, ) in the camera array,  is the gap between reconstructed image and virtual microlens array,   ×   represents the size of elemental image, and (, ) represents the overlapping number matrix.

Experiment Results and Discussion
3.1.Experiment Results.In order to demonstrate the effectiveness of the proposed approach, we implement an experiment using two 3D objects including a Rubik's cube and a model car.The SAII pickup system consists of a camera and a motorized translation stage.The experiment setup is illustrated in Figure 5. Table 1 shows the experimental parameters.
In the experiment, we use the SAII pickup system to acquire 11 × 11 parallax images which record spatial and angular information of the 3D scene, and a part of parallax images are illustrated in Figure 6.For the sake of simplicity, these acquired parallax images are encrypted with the same secret keys which are generated by random number generator under the Fresnel approximation.The mapping algorithm is applied to generate the cipher image.
The decrypted parallax images are acquired by the decryption process which is a reverse operation of the encryption process.In the proposed approach, 3D information can be reconstructed flexibly according to II system configuration.The result of OIIR shows different perspectives of 3D scene, as shown in Figure 7.The results of the  reconstructed 3D images by CIIR which are focused on headstock, tailstock, and Rubik's cube are shown in Figure 8.It is evident that the 3D information can be reconstructed successfully.

Performance Analysis
and MSE can be calculated by where

Security Analysis.
The sensitivity to the secret keys is an important indicator to certify the security of the proposed approach.The proposed encryption approach tremendously improves the security because of the two-step coding operation.Based on the two-step coding process, the cipher image cannot be decrypted directly by the secret keys of DRPE in Fresnel domain.To demonstrate the security of the proposed method, we simulate the case in which attackers obtain a part of the secret keys.Firstly, the first decoding is not performed, the secret keys of DRPE in Fresnel domain are used to decrypt the cipher image, and the decrypted image is shown in From the results, it is very difficult for attackers to obtain 3D information.
Statistical information can be used to speculate the image information, and autocorrelation between the adjacent pixels can reflect the statistical information of image.We calculate the values of autocorrelation and the results are shown in Figure 10.Figures 10(a) and 10(b) show the autocorrelation values of the EIA and the cipher image, respectively.From the results, we can see that the autocorrelation of the cipher image is much weaker than EIA, which confirms that the proposed approach has high robustness against statistical attack.

Robustness Analysis.
To certify the robustness of the proposed encryption approach against different attacks, the Gaussian noise and occlusion attacks are adopted.We add the zero-mean Gaussian noise with different variances to the cipher image and reconstruct the corresponding 3D images by using CIIR.2. From the results, we can know that the 3D information can be clearly recognized even if the cipher image suffers from the zeromean Gaussian noise with the variance of 0.2.Also, the cipher image which is occluded 50% is used to test the robustness of the encryption approach against occlusion attacks.Figure 12 3.From the reconstructed images and the calculated PSNR values, we can see that the 3D information can be reconstructed completely when the cipher image is occluded 50%.

Plaintext Attack Analysis.
Plaintext attacks can be implemented by attackers to obtain information, for instance, the known-plaintext attack and the chosen-plaintext attack.The known-plaintext attack is an attack method, in which the attackers can acquire original images and corresponding encrypted images.The information can be used to deduce   secret keys.In the chosen-plaintext attack, it is presumed that the attackers can gain the encrypted image for each original image; therefore, the secret keys can be deduced by these pairs of original images and encrypted images.
In our two-step integral imaging coding approach, the DRPE in Fresnel domain and a mapping algorithm which is used to generate elemental image array of integral imaging are used.Each pixel of the cipher image derives from encrypted parallax images.The location of the pixel is decided by the parameters of mapping algorithm, and the value of the pixel is decided by the parameters of DRPE in Fresnel domain.The parameters of DRPE in Fresnel domain and the mapping algorithm can be both considered as secret keys.The security analysis confirms that our approach provides a high security due to big key space and two-step coding.It is very difficult for attackers that information is acquired by plaintext attacks.

Conclusions
We have presented a two-step integral imaging coding based 3D information encryption approach.In this approach, an SAII pickup system is used to transform 3D information to a set of parallax images which record different perspectives of 3D scene.DRPE in Fresnel domain is used to encrypt parallax images.The mapping algorithm is introduced to translate encrypted parallax images to a cipher image, which improves greatly the security because the cipher image cannot be decrypted straightway by decryption algorithm unless it is remapped inversely, and the parameters of the SAII pickup system can be used as the secret keys.Meanwhile, due to the property of DRPE in Fresnel domain, autocorrelation between the adjacent pixels of the cipher is much weaker, and the robustness against different attacks is higher.The

Figure 1 :
Figure 1: Schematic of the proposed encryption approach.

Figure 3 :
Figure 3: Schematic of the 3D information encryption.

Figure 5 :
Figure 5: Experimental setup of the SAII pickup system.
Figure 9(a).Secondly, the cipher image is decoded by the inverse mapping algorithm to generate a set of encrypted parallax images.Then secret keys with a wrong parameter of DRPE in Fresnel domain are used to decode one of the images; the resulting image is shown in Figure 9(b).

Figure 7 :
Figure 7: Reconstructed 3D images by OIIR with different perspectives: (a) left view, (b) front view, (c) right view, (d) top view, and (e) bottom view.
Figures 11(a)-11(c) show the reconstructed 3D images under the variance of 0.1; Figures 11(d)-11(f) show the reconstructed 3D images under the variance of 0.2.The values of PSNR are calculated and shown in Table (a) shows the vertically occluded cipher image.Correspondingly, the reconstructed 3D images focusing on different objects are obtained and shown in Figures 12(b)-12(d), respectively.The horizontally occluded cipher image is shown in Figure 12(e).The reconstructed 3D images in the different depth planes are shown in Figures 12(f)-12(h), respectively.The values of PSNR are calculated and shown in Table

Table 1 :
Experimental parameters of the SAII system.
and  represent the original image and reconstructed image, respectively, and  and  represent the pixel coordinates of the processed image.We calculate the average PSNR values of three channels of the reconstructed 3D images which focus on different depths, and the values are 38.25 dB, 37.58 dB, and 38.76 dB, respectively.The high PSNR values demonstrate the excellent quality of the proposed approach.

Table 2 :
PSNR values of CIIR from the cipher image that suffered from Gaussian noise.

Table 3 :
PSNR values of reconstructed 3D images by CIIR from the cipher image occluded 50%.