1. Introduction
Optical encryption, such as double random phase encryption (DRPE) [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13], has been an important technique for protecting the private data recently. It can encrypt the primary data by using a 4
f imaging system with two different random phase masks. For encryption in DRPE, the primary data are multiplied by the first random phase mask, which follows uniform distribution with range [0, 2
]. Then, it passes through the first imaging lens in the 4
f imaging system, which means Fourier transform. Now, it is in the spatial frequency domain. After multiplying it by the second random phase mask, which also follows uniform distribution with range [0, 2
], it passes through the second imaging lens in the 4
f imaging system, which means the inverse Fourier transform. Finally, the encrypted data can be generated. For decryption in DRPE, the encrypted data passes through the first imaging lens in the 4
f imaging system, and it is multiplied by the complex conjugate of the second random phase mask (i.e., key information) used in the encryption of DRPE. Then, it passes through the second imaging lens in the 4
f imaging system and is recorded by an image sensor such as a charge-coupled device (CCD). Finally, the decrypted data can be obtained. Thus, the processing speed of DRPE is the same as the speed of light. However, DRPE has two main drawbacks. First, for encryption, it needs to record the complex-valued data because it uses Fourier transform. It is difficult to obtain this complex-valued data by the conventional image sensor. To solve this problem, the holographic recording technique may be utilized since it can record both the amplitude and phase data by the conventional image sensor. Second, when attackers know the key information (i.e., the complex conjugate of the second random phase mask), the primary data may be revealed easily. To overcome this problem, photon-counting DRPE [
14,
15,
16], three-dimensional (3D) photon-counting DRPE [
17], and artificially intelligence (AI) approach [
18] were proposed.
Photon-counting DRPE uses computational photon-counting imaging for recording the amplitude of the encrypted data in the encryption of DRPE. This means that encrypted data by photon-counting DRPE has more sparse amplitudes than the conventional DRPE. Thus, even though the key data are attacked, the primary data may not be recognized by human eyes. It may be recognized by matched filters such as a correlation filter, where receivers must know the primary data. Thus, it is not practical. To overcome this problem, a 3D photon-counting DRPE was proposed. It utilizes integral imaging to record the encrypted data and reconstruct the decrypted data. Since integral imaging can record multiple 2D images with different perspectives from 3D objects at a certain depth, using volumetric computational reconstruction (VCR) [
19,
20] and the statistical estimation methods [
17] such as maximum likelihood estimation (MLE) or Bayesian approaches, the decrypted data can be reconstructed. In addition, the reconstruction depth can be another key data in 3D photon-counting DRPE. It is apparent that the primary data can be decrypted when both the phase information and the reconstruction depth are known. Therefore, it can enhance the security level of DRPE.
However, DRPE still requires two different random phase masks for encryption. In addition, for decryption, it needs taking the absolute value of the decrypted data because the decrypted data is complex-valued. To solve these problems, in this paper, we propose a new optical encryption method which uses the additive white Gaussian noise (AWGN) with zero mean and unit variance, and the single random phase mask after the first imaging lens (i.e., after Fourier transform). It can decrypt the primary data without taking the absolute value of the decrypted data, and its key information is still the same as that of DRPE. Moreover, it can enhance the security level because its 2D decrypted data may seem to be noisy data caused by AWGN. However, the 3D decrypted data can be reconstructed by VCR at the reconstruction depth since VCR has the average effect. This means that the 2D decrypted data are overlapping each other so that the average value of the overlapped noise data goes to zero by AWGN. To verify the validity of our proposed method, we implement the simulation and calculate peak sidelobe ratio (PSR) and structural similarity (SSIM) as the performance metric.
This paper consists of the following sections. We present the basic concept of DRPE and VCR integral imaging in
Section 2. Then, we describe our proposed method, which is called single random phase encryption (SRPE) in
Section 3. Simulation results are shown in
Section 4. Finally, we conclude with future work in
Section 5.
4. Simulation Results
To show the ability of 3D SRPE, we recorded multiple images with different perspectives (i.e., elemental images) by SAII. A 10 (H) × 10 (V) camera array is used, where the focal length
f is 50 mm, the pitch between elemental images
p is 10 mm in both
x and
y directions, the sensor size is 36 (H) mm × 36 (V) mm, and the resolution of each elemental image is 1000 (H) × 1000 (V) pixels. The 3D object is a car with the licence plate “SMC 5475” which is located 323 mm from the camera array. AWGN follows normal distribution with zero mean and unit variance
. We generated 1000 encrypted data randomly by Equation (
5) for each elemental image. Then, the decrypted data were obtained by Equation (
6). To improve the visual quality of the decrypted data, we used Equation (
7) with 1000 decrypted data for each elemental image. Finally, using Equations (
3) and (
8) with 10 (H) × 10 (V) elemental images, the reconstructed 3D images at 323 mm was obtained.
Figure 13 shows 2D results by SRPE. To compare the visual quality of the decrypted data, the license plate is enlarged for each decrypted result.
Figure 13a is the primary data, which are used as the reference.
Figure 13b is the encrypted data by SRPE.
Figure 13c–f shows the decryption results via various generations (i.e.,
,
,
, and
) of SRPE. As shown in
Figure 13, it is apparent that the visual quality of the decrypted data depends on the number of generations of SRPE. Thus, the result shown in
Figure 13f has the best visual quality compared to the others. For numerical comparison, we calculated the structural similarity (SSIM) as shown in
Table 1. It is noticed that the similarity can be improved by increasing the number of generations of SRPE. However, when the number of generations of SRPE increases, the processing speed of SRPE is slow. Therefore, to obtain the reasonable visual quality and processing speed, 3D SRPE was implemented.
Figure 14 shows 2D and 3D results by SRPE via various generations of SRPE. It is noticed that 3D results have better visual quality than 2D results. In addition, as we increase the number of generations of SRPE, the visual quality of the decrypted data is improved. However, even though the number of generations of SRPE increases, the visual quality of the decrypted data is limited (i.e., saturation). For numerical comparison, we calculated the SSIM and SSIM ratio between the 2D and 3D results as shown in
Table 2. When
, the 3D results have 4.68 times the SSIM than the 2D results. It is remarkable that the visual quality of 3D results with
is dramatically improved compared to the others. This means that a lot of generations of SRPE are not required. In addition, when
, SSIM is 1, which means that the primary data and the decrypted data are the same as each other (i.e., perfect decryption).
In 3D SRPE, since the reconstruction depth is another key information, we need to show that the primary data are revealed at the only correct reconstruction depth. Thus, we found the correlation between the primary data and the decrypted data via different reconstruction depths by using
kth low nonlinear correlation filter [
22]. It has non-linearity factor
k, which is
real number. The filter is defined as the following [
22]:
where
are the amplitudes of Fourier transformed reference and target images,
are the phases of Fourier transformed reference and target images, and
is the correlation result between the reference and target images, respectively.
In addition, for numerical analysis, peak sidelobe ratio (PSR) was calculated by the following [
22]
where
is the maximum value of the correlation result by Equation (
9),
is the mean value of the correlation result, and
is the standard deviation of the correlation result. When PSR value is high, the correlation is strong.
Figure 15 shows PSR results for the decrypted data with various generations of SRPE via different reconstruction depths. Here, the reference image is the 3D reconstructed image at 323 mm obtained by using VCR and elemental images, as shown in
Figure 5. All decrypted data have the highest PSR at 323 mm. When
, PSR value at 323 mm is 1302.358209. On the other hand, when
, PSR value at 323 mm is 2255.662650. This means that the reconstruction depth can be another key information in SRPE.
Moreover, for the encryption efficiency comparison, we measured the processing time between DRPE and SRPE. System specification used for comparison is shown in
Table 3. The 10 (H) × 10 (V) elemental images, as shown in
Figure 5, which has 1000 (H) × 1000 (V) color pixels, are encrypted and decrypted. For SRPE, the number of observations is set as 1 and 10. The processing time is shown in
Table 4.
As shown in
Table 4, for single observation, the encryption and decryption processing time of SRPE is slightly less than that of DRPE. However, since SRPE requires multiple observations to improve the visual quality of the decrypted data, the processing time of SRPE is much more than DRPE for multiple observations. It is limitation of SRPE.
5. Conclusions
In this paper, we have proposed a new optical 3D encryption method which uses the single random phase mask and AWGN. Our method, SRPE, can encrypt the primary data through the same 4f imaging system as DRPE by using the single random phase mask and AWGN. The decryption process of SRPE is almost the same as DRPE, but it does not need the absolute value operator. When attackers know the key information, the primary data can be revealed easily in DRPE. On the other hand, in SRPE, it is difficult to observe the primary data because the decrypted data still have noise caused by AWGN. To improve the visual quality of the decrypted data in SRPE, expectation operator can be utilized. In addition, to enhance the security level in SRPE, integral imaging can be applied. Thus, the reconstruction depth can be another key information. In our proposed method, the primary data can be revealed when attackers knows the key information, the reconstruction depth, and multiple decrypted data. Thus, it may be impossible to observe the primary data in SRPE. Therefore, we believe that our method can be used for various applications that consider the private information. However, our method has several drawbacks. To encrypt the primary data, SRPE needs a lot of encrypted data for decryption. Thus, its processing speed is a critical problem. In addition, it needs a method to record the complex-valued data because the encrypted data are complex-valued. In our opinion, this may be solved by introducing the holography technique. We will investigate these issues in future work.