Open Access
Add to BibSonomy
Abstract
We propose a speckle-based optical encryption scheme by using complex-amplitude coding and deep learning, which enables the encryption and decryption of complex-amplitude plaintext containing both amplitude and phase images. During encryption, the amplitude and phase images are modulated using a superpixel-based coding technique and feded into a digital micromirror device. After passing through a 4f system, the information undergoes disturbance modulation by a scattering medium, resulting in a diffracted speckle pattern serving as the ciphertext. A Y-shaped convolutional network (Y-Net) model is constructed to establish the mapping relationship between the complex-amplitude plaintext and ciphertext through training. During decryption, the Y-Net model is utilized to quickly extract high-quality amplitude and phase images from the ciphertext. Experimental results verify the feasibility and effectiveness of our proposed method, demonstrating that the potential of integrating speckle encryption and deep learning for optical complex-amplitude encryption.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical security technology has emerged as one of the critical areas of research in information security due to its efficient processing of large-capacity and multi-dimensional data through high-speed parallel operations. Since the introduction of the double random phase encoding (DRPE) technique by Refregier and Javidi [1], the field of optical encryption has experienced significant advancements. Various new methods have been proposed with the aim of enhancing the performance of encryption systems [2–13], including those utilizing DRPE [2,3], interference principle [4], cascaded diffraction [5], joint transform correlation [6], ghost imaging [7,8], integral imaging [9,10], and other innovative approaches [11–15]. However, many of these methods require obtaining the phase information of the encrypted result, which necessitates complex interference optical paths for digital holography and can potentially degrade decryption quality. Additionally, there has been limited research on encrypting complex amplitude information. Recently, superpixel-based phase and amplitude modulation methods have been employed for preprocessing complex amplitude information, specifically designed to be compatible with digital micromirror (DMD) devices [16]. Security schemes employing holographic projection for speckle encryption have also been proposed [17,18], simplifying the optical setup. By integrating complex amplitude processing methods with holographic encryption, it becomes possible to achieve optical speckle encryption of complex amplitude information without the need for complex optical configurations for phase recording.
In recent years, deep learning techniques, which provide high-quality solutions in several computational imaging during the past few years [19–22], have also been applied in the field of optical image encryption. Initially, the focus was primarily on analyzing the security of existing optical image encryption systems [23–29]. However, different attack methods not only test the security of encryption systems but also provide new insights for designing novel optical image encryption systems. More recently, researchers have started to explore the application of deep learning techniques in developing new encryption systems [18,30–36]. For instance, Chen et al. encoded plaintext images into speckle patterns using complex scattering media and decoded them using a Convolutional Neural Network (CNN) model [30]. Wang et al. presented an experimental scheme for optical encryption and decryption using random binary mask and deep learning [32]. Zhao et al. proposed a speckle-based optical cryptosystem where face images can be decrypted from random speckles using a highly trained neural network [34]. These recent studies indicate the significant potential of deep learning in enhancing both the security and efficiency of optical encryption systems.
In this paper, we present a speckle-based optical encryption method based on complex amplitude coding and deep learning. During the encryption process, a complex amplitude object, comprising amplitude and phase images, is firstly encoded into a binary pattern using a superpixel-based coding technique. This binary pattern is subsequently fed into a DMD. The projection beam modulated by the DMD passes through a 4-f optical system and a scattering element to generate speckle patterns on a CCD, which are stored as ciphertext. We design a Y-shaped convolutional neural network model called Y-net to directly recover the amplitude and phase images from the ciphertext. This model is trained using paired amplitude-phase images and their corresponding speckle patterns. Unlike recently proposed speckle-based encryption methods, our approach successfully encrypts complex amplitude targets that contain both amplitude and phase images, and utilizes deep learning technique for high-quality and efficient complex amplitude decryption.
2. Principles
Our proposed encryption and decryption method is illustrated in Fig. 1. As show in Fig. 1(a), the encryption process can be summarized as follows:
- (1) The complex amplitude plaintext, which includes amplitude and phase images, is encoded into a binary pattern and input to the digital DMD.
- (2) The modulated beam from the DMD passes through a 4-f system, which includes two lenses and a circular aperture spatial filter, and before reaching the CCD, the optical field undergoes modulation by a scattering element. The CCD records the resulting noisy pattern, which is saved as the ciphertext.
- (3) A Y-shaped neural network is trained using multiple pairs of “complex amplitude plaintext-ciphertext” to learn the mapping between ciphertext and plaintext, enabling decryption of the ciphertext to obtain the original complex amplitude plaintext.
In our method, optical parameters such as diffraction distance, wavelength, and scattering media are not employed as decryption keys nor stored alongside the ciphertext for that purpose. This ensures protection against potential attacks wherein these parameters could be exploited to create ciphertexts or acquire plaintext-ciphertext pairs directly. We will provide a detailed explanation of this in Section 3 later. Conversely, network parameters should be treated as confidential encryption keys and securely stored.
Figure 2 demonstrates the complex amplitude encoding principle based on the superpixel approach. In Fig. 2(a), patterns input into the DMD are designed to modulate the incident light field. After passing through a 4-f double-lens configuration, the desired complex amplitude image is generated on the target plane. Both the DMD plane and the target plane are divided into superpixels, with each superpixel comprised of 4 × 4 pixels. The phase response on the target plane depends on the pixel position on the DMD plane, which is achieved by introducing additional phase factors through two slightly off-axis lenses. To suppress high spatial frequency information, a circular aperture spatial filter is placed between the lenses. This filter blurs and averages the amplitudes and phases of neighboring pixels on the target plane, effectively rendering individual DMD pixels inside a superpixel indistinguishable. The position and size of the filter aperture are carefully designed to achieve an equivalent result on the target plane, where the incident light field is multiplied by a virtual phase mask. The number of pixels in this virtual phase mask is the same as that of the DMD plane and consists of local blocks of 4 × 4 pixels. For example, if each of the entire DMD plane and phase mask comprises 256 × 256 pixels, they would be composed of 64 × 64 local blocks, with each block corresponding to a superpixel. The location of the spatial filter aperture determines a uniform distribution of phase within each local block of the phase mask, ranging from 0 to 2π. By utilizing the filter design and phase distribution from a previous study [15], as depicted in Fig. 2(b), the pixel positions within a superpixel can encode 16 distinct phase values. In Fig. 2(c), a DMD projection pattern is illustrated, where pixel states can be flexibly controlled between “on” and “off” states to achieve desired encoding. Each superpixel can encode a complex amplitude value by calculating the inner product between its 4 × 4 binary-encoded DMD pixels and their corresponding 4 × 4 pixels in the virtual phase mask. Assuming the DMD pixels are encoded as shown in Fig. 2(c), where only the three indicated binary pixels are illuminated, the resulting superpixel in the target plane can approximate a complex amplitude that is equal to the sum of the three dots depicted in Fig. 2(d). The three dots depicted in Fig. 2(d) represent three complex amplitude values in the target plane encoded by the proposed method. Esuperpixel represents expected value of the complex amplitude of a surperpixel in the target plane. Esuperpixel can be approximately equal to the sum of three complex amplitude values encoded by the proposed method.
Fig. 2. Principle of superpixel-based encoding. (a) Optical setup of complex amplitude modulation system. (b) Periodic phase mask for a single superpixel. (c) A 4 × 4 binary pattern on the DMD, where green indicates the “on” state of the DMD. (d) Combination of phase values for binary coding, forming a complex light field.
The encoding and modulation of each superpixel are accomplished independently. The 4 × 4 binary block patterns may be encoded with a total of 216 = 65536 potential permutations for each superpixel. However, the different encoded patterns of the DMD may produce the same superpixel value. As a result, the greatest number of complex amplitude values that the superpixel may represent is 6561. The 6561 values can be kept in a lookup table and correspond to a 4 × 4 DMD pixel block. An arbitrary complex amplitude value after normalizing can be mapped to one of the adjacent values that are among 6531. The complex amplitude light field made up of multiple superpixels is encoded into a DMD array using the superpixel-based technique. For example, when encoding a complex amplitude value Esuperpixel, we start by finding the adjacent complex amplitude values of Esuperpixel through a lookup table. The corresponding array from the lookup table, as shown in Fig. 2(c), is then loaded onto the DMD. Through the optical path depicted in Fig. 2(a), the target plane obtains the adjacent value, which is represented by the sum of the three green dots in Fig. 2(d) and denoted as $\left( {\frac{{2\pi }}{8} + \frac{{4\pi }}{8} + \frac{{16\pi }}{8}} \right)$. This sum is approximately equal to the original complex amplitude value Esuperpixel.
During the decryption process, the user needs to employ the correct network parameters and input the ciphertext into the network. Subsequently, the network will generate the amplitude and phase images, as illustrated in Fig. 1. Recovering complex amplitude information from speckles has always been a challenging undertaking due to potential influences like optical noise and other factors. However, with the development of computer technology and algorithms, neural networks have achieved significant success in dealing with various problems, including imaging through scattering media [18–22]. In this study, we propose a one-to-two CNN based on Y-net architecture to recover amplitude and phase images from ciphertexts.
Figure 3 depicts the structure diagram of our proposed network. It consists of one down-sampling path and two up-sampling paths. The down-sampling path contains five down-sampling layers linked by four max pooling operations with a kernel size of 2 × 2 and a stride size of 2. Each down-sampling layer consists of two sub-layers, which are both made up of a convolutional layer (Conv), a batch normalization (BN), and a rectified linear unit (ReLU). The convolutional layers have a kernel size of 3 × 3, a stride of 1, and a padding of 1. Similarly, the up-sampling path contains five up-sampling layers with the same structure as the down-sampling layers. It is connected by four transposed convolutional operations with a kernel size of 2 × 2 and a stride of 2. An additional convolution layer at the output is introduced to further reduce the number of channels in the network. During the network training process, the feature maps of the input image are successively extracted via the down-sampling layer and transmitted. At the last layer of down-sampling, the extracted feature map is divided into two channels and input up-sampling layer after carrying out the transposed operation respectively. Besides, each successively decimated signal from the down-sampling path is then inputted into correspond to the up-sampling layers in the up-sampling path through skip connections, resulting in increasing the dimension of the input up-sampling layer by a factor of 2. Finally, network output is produced by the convolutional layer after passing the up-sampling path.
This network employs a mixed loss function to access the similarity between the output and the ground truth. It utilizes metrics such as normalized partial cross correlation (NPCC), structural similarity index (SSIM), peak signal-to-noise ratio (PSNR), and mean squared error (MSE) for evaluation purposes. The loss functions for amplitude and phase are referred to as $los{s_1}$ and $los{s_2}$ with their mathematical expression given by
where y and $\hat{y}$ represent the original and decrypted images, respectively, and the metrics involved are defined asIn the equations above, mean(y) and mean($\hat{y}$) represent the mean values of y and $\hat{y}$, respectively. σ(y) and σ($\hat{y}$) are the standard deviations of y and $\hat{y}$, respectively. cov is the covariance of y and $\hat{y}$. ${c_1}$, ${c_2}$ and ${c_3}$ are three very small constants to prevent division by 0 in SSIM. I is the luminance similarity, C is the contrast similarity and S is the structure similarity.
The overall loss function of the network, $Loss$, is determined by
3. Results and discussion
In the optical experiments, a collimated and extended beam was generated using a semiconductor-pumped laser (DH-SL532-50A) with a wavelength of 532 nm. The beam illuminated a DMD (ViALUX-7001VIS) consisting of 1024 × 768 mirrors, each with a size of 13.6 μm. Both lenses in the 4-f optical system have a focal length of 10 cm. The distance between the DMD and the diffuser was set to d1 = 40 cm, while the distance between the diffuser and the CCD detector was d2 = 8 cm. For the experimental validation, images with dimensions of 64 × 64 pixels were selected from the Fashion-MNIST [37] and MNIST databases [38] as amplitude and phase object representations, respectively. These databases were chosen due to their accessibility and widespread use in deep learning research. Prior to being fed into the DMD, these images were encoded using the superpixel method to generate the projection pattern with a size of 256 × 256 pixels. During the training process, we applied random sparsity treatment to the original data. Firstly, we randomly generated a binary matrix where the ratio of zeros to the total number of pixels determines the sparsity level. Next, we multiplied the ciphertext by the random sparse matrix, randomly assigning zero values to certain pixels. This process represents the random sparsity treatment applied to the ciphertext. For example, we can perform two rounds of 50-100% random sparsification on the initial 10,000 ciphertexts, resulting in a sparse dataset containing 20,000 instances. This methodology enriches the dataset, strengthens the model’s learning capabilities, and enhances its resistance to ciphertext loss.
We randomly selected 10,000 amplitude and phase objects from Fashion-mnist database and MNIST handwritten digit database. An additional set of 100 objects were designated as the test dataset. These objects were resized to dimensions of 64 × 64 pixels and encoded into DMD projection patterns with a pixel size of 256 × 256. The optical encryption results (speckle patterns) and corresponding label images were further resized to 128 × 128 pixels for training. During training, random sparsity ranging from 50% to 100% was applied to the obtained ciphertext, expanding the dataset and enhancing the network’s resistance to interference. A total of 20,000 object-speckle pattern pairs were employed for training. The adaptive moment estimation (Adam) optimization algorithm was employed with a learning rate set to 0.0001, gradually decreasing it as the training epochs progressed. The neural network was implemented using the Pytorch framework. The training process consisted of 50 epochs and was executed on an NVIDIA RTX A4000 GPU. It took approximately 15.0 hours to complete the training of the CNN model.
Figure 4 presents the results of our predictions by evaluating each pre-trained model using test sets. The first and second lines in Fig. 4 depict the ground truth images of amplitude and phase, while the third line shows the encrypted images obtained from the optical scheme. The reconstructed amplitude and phase images are displayed in the fourth and fifth lines of Fig. 4, respectively. The results demonstrate that our proposed Y-net model can effectively reconstruct high-quality images, as evidenced by the resemblance between the reconstructed and ground truth images. The image resemblance results strongly indicate the good performance of our Y-net model in reconstructing high-quality images. To assess the performance of the reconstructed images quantitatively, we calculated the SSIM and PSNR for the test images. SSIM measures the structural similarity between the ground truth and the predicted image, while PSNR evaluates the average difference between them. For the decrypted amplitude images shown in Fig. 4, the average PSNR and SSIM values were 19.2 dB and 0.69, respectively. For the decrypted phase images, the average PSNR and SSIM values were found to be 19.5 dB and 0.84, respectively.
Fig. 4. Experiment results. (a) Original amplitude images. (b) Original phase images. (c) Ciphertext. (d) Reconstructed amplitude images. (e) Reconstructed phase images.
In our approach, the pre-trained learning model and its parameters are utilized as secure keys. We initially decrypt the data using a different model trained on fabricated complex amplitude plaintext-speckle ciphertext pairs to verify the model key. The decryption results obtained with this network model are shown in Fig. 5. When errors occur in this model, the decrypted images fail to provide meaningful information about the original images. This result emphasizes the significant influence of network model on image reconstruction using the CNN model, reflecting their dual potential as both crucial elements for accurate reconstruction and as secure keys.
Fig. 5. Decrypted results using incorrect network models. (a) Plaintext amplitude images (b) Plaintext phase images. (c) Decrypted amplitude and (d) phase images using incorrect CNN models.
We conducted additional tests to evaluate the impact of unauthorized users eavesdropping on network parameters and examine the significance of network parameter keys in decryption. In these tests, we simulated the scenario of secret key eavesdropping by randomly setting specific network parameters (w and b) to 0. For instance, if 10% of the network parameters are set to 0, it indicates that 90% of the key has been compromised. Figure 6 shows the results of the decrypted image quality under different eavesdropping rates for parameters w and b. These results were obtained through decryption of a single ciphertext. It indicates that the eavesdropping rate significantly affects the decrypted image quality when using the proposed scheme for image decryption. As the eavesdropping rate increases, the quality of the decrypted images gradually improves. When the eavesdropping rate is below 95%, we observed no leakage of any plaintext information in the decrypted image. Hence, even if certain network parameters are intercepted by unauthorized users, the decrypted image remains highly secure. However, when the eavesdropping ratio exceeds 95% but remains below 99%, we noticed that some outline information in clear text is obtained during the image decryption process, resulting in a loss of details. Only when the eavesdropper intercepted more than 99% of the key could a high-quality decrypted image be recovered. In Fig. 6, it can be observed that there are noticeable sudden drops in the phase image quality. These drops may be attributed to the loss of crucial parameters in the network model. When some of network’s parameters are randomly lost, these vital parameters, which play a key role in the output process of the phase image, are either missing or have incorrect values. As a result, the network produces unreliable or distorted results. These results demonstrate that even if a significant portion of the network parameters are eavesdropped, meaningful information remains concealed until an extremely high threshold eavesdropping rate is reached. Therefore, our scheme proves to be well-suited for secure image transmission scenarios.
Fig. 6. Analysis of reconstruction quality under different eavesdropping rates for parameter security keys (w and b). The left side presents the PSNR of the individual image at varying eavesdropping rates, while the right side exhibits the SSIM the individual image for each eavesdropping rate.
We also conducted several experiments using varying numbers of plaintext-ciphertext pairs for network training to assess their impact on decryption quality, as depicted in Fig. 7. The results indicate that reducing the number of pairs during network training leads to a decline in decryption quality and reliability. When 10,000 pairs are used for training, the decrypted results retain sufficient information from their original images. However, using only 5,000 pairs results in deteriorated quality of amplitude images obtained through network decryption, along with inaccurate predictions of phase images. With just 1,000 pairs, the amplitude image contours in the decryption become distorted, and all predictions of the phase images fail. These findings emphasize the impracticality of attackers decrypting the proposed encryption system with only a few intercepted plaintext-ciphertext pairs. The experimental results suggest that a training set containing at least 10,000 pairs is necessary to achieve accurate and clear decryption results. Although approximately 500 pairs may yield seriously blurry amplitude decryption outcomes, further analysis will demonstrate that eavesdroppers would encounter significant difficulties in generating such a large number of plaintext-ciphertext pairs due to our ciphertext generation process and the composition of the decryption keys, thereby ensuring the security of the system.
Fig. 7. Reconstructed images obtained using a network trained with varying numbers of plaintext-ciphertext pairs: (a) 20,000 (b) 10,000 (c) 5,000 (d) 2,500 (e) 1,000 (f) 500.
In our approach, we avoid using optical parameters such as diffraction distance, wavelength, and scattering medium as decryption keys or storing them with the ciphertext. This prevents potential attacks where adversaries could exploit these parameters to generate ciphertext or directly obtain plaintext-ciphertext pairs. Figure 8 shows that even small deviations in diffraction distance can result in incorrect decryption, causing severely blurred amplitude image contours and inaccurate phase images. The high sensitivity of our network to diffraction distance makes it challenging for eavesdroppers to obtain precise plaintext-ciphertext pairs, significantly weakening their ability to break the encryption scheme based on optical parameters. Therefore, by abstaining from providing these optical parameters, we greatly enhance the security of our encryption scheme.
Fig. 8. Reconstructed amplitude and phase images with different diffraction distances deviating from the original distance: (a) -2 mm, (b) -1 mm, (c) 0 mm, (d) 1 mm, and (e) 2 mm.
Next, we investigated the impact of data loss on decryption. To simulate data loss, we randomly set certain pixel values in the ciphertext to zero. This randomization process compromises the integrity of the encrypted image and may cause potential issues during decryption. Figure 9 present a series of decryption results for ciphertexts that experienced varying percentages of data loss: 10%, 20%, 30%, 40%, and 50%, respectively. As shown in the Fig. 9, with the loss of ciphertext information, image details gradually diminish. Nevertheless, the decryption results using the pre-trained model still deliver satisfactory visual effects despite the data loss.
Fig. 9. Robustness test against data loss. Rows one to five depict the decrypted images from the ciphertext with data loss percentages of 10%, 20%, 30%, 40%, and 50%, respectively.
Furthermore, to quantitatively evaluate the quality of the reconstructed images under different data loss ratios, we calculated their PSNR and SSIM. Table 1 presents the computed PSNR and SSIM values, comparing the reconstructed images with the ground truth. Importantly, our evaluation demonstrated that the reconstructed images achieved high quality without requiring retraining of the CNN model. This highlights the robustness of our proposed method in effectively handling data loss scenarios.
Table 1. The PSNR and SSIM of decrypted result from the cyphertext with percent of data loss
Finally, we conducted a comparative analysis of the proposed method with several recently proposed methods related to optical imaging using DMD or through complex scattering media. For this purpose, we compared our method with those by Goorden et al. [15], Ruan et al. [39], and Zhou et al. [30]. The comparison revealed differences in functionality and research objectives among these methods. Our method focuses on encryption, aligning with Zhou et al.’s approach. In contrast, Goorden et al. primarily concentrate on imaging, while Ruan et al. emphasize information transmission. Goorden et al.’s method does not involve keys and encryption performance. On the other hand, Ruan et al. utilize scattering media and its position as keys. Both our method and Zhou et al.’s work employ network parameters as the decryption keys. When it comes to coding capability, Ruan et al. and Zhou et al. solely address amplitude images, whereas our method allows for the encoding of complex amplitude images. Regarding reconstruction methods, Goorden et al. use the 4f system reconstruction method, Ruan et al. adopt the optical-channel-based intensity streaming method, and Zhou et al. employ CNN. In contrast, our proposed method utilizes the Y-net reconstruction method designed specifically for recovering complex amplitude images. The summary of the comparative analysis, including the encryption capability of our proposed method, is provided in Table 2 for clarity.
Table 2. Comparative analysis
4. Conclusion
In conclusion, our work has successfully demonstrated the encryption and decryption of complex amplitude images without the need for complex optical configurations to record phase. This achievement is made possible by integrating complex amplitude preprocessing with speckle-based optical encryption. The decryption process effectively utilizes an improved Y-Net model as an encrypted image processor. Experimental results have validated the scheme’s robust security, resilience, and data loss protection capabilities. Our proposed approach encompasses the encryption of complex amplitude targets that include both amplitude and phase images, and leverages deep learning techniques for high-quality and efficient complex amplitude decryption. This advancement enhances both the encryption capacity and capability of speckle-based optical encryption systems, offering an effective alternative solution for data protection.
Funding
National Natural Science Foundation of China (62375245, 61975185, 61575178); Scientific Research and Developed Fund of Zhejiang University of Science and Technology (F701108L03).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. P. Refrégier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995). [CrossRef]
2. B. Javidi, G. Zhang, and J. Li, “Encrypted optical memory using double-random phase encoding,” Appl. Opt. 36(5), 1054–1058 (1997). [CrossRef]
3. G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29(14), 1584–1586 (2004). [CrossRef]
4. Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008). [CrossRef]
5. W. Chen, X. Chen, and C. J. R. Sheppard, “Optical image encryption based on diffractive imaging,” Opt. Lett. 35(22), 3817–3819 (2010). [CrossRef]
6. T. Nomura and B. Javidi, “Optical encryption using a joint transform correlator architecture,” Opt. Eng. 39(8), 2031–2035 (2000). [CrossRef]
7. P. Clemente, V. Durán, V. Torres-Company, et al., “Optical encryption based on computational ghost imaging,” Opt. Lett. 35(14), 2391–2393 (2010). [CrossRef]
8. Y. Kang, L. Zhang, H. Ye, et al., “One-to-many optical information encryption transmission method based on temporal ghost imaging and code division multiple access,” Photonics Res. 7(12), 1370–1380 (2019). [CrossRef]
9. X. Li, Y. Wang, Q. Li, et al., “Optical 3D object security and reconstruction using pixel evaluated integral imaging algorithm,” Opt. Express 27(15), 20720–20733 (2019). [CrossRef]
10. W. Song, Q. Cheng, Y. Liu, et al., “Three-dimensional image authentication using binarized images in double random phase integral imaging,” Chin. Opt. Lett. 17(5), 051002 (2019). [CrossRef]
11. Z. Ren, J. Guo, H. Liu, et al., “Light-field 3D encryption based on the monocular depth rendering,” Opt. Lett. 47(18), 4762–4765 (2022). [CrossRef]
12. S. Jiao, D. Zhang, C. Zhang, et al., “Data hiding in complex-amplitude modulation using a digital micromirror device,” Opt. Lasers Eng. 138, 106455 (2021). [CrossRef]
13. N. Zhou, L. Tong, and W. Zou, “Multi-image encryption scheme with quaternion discrete fractional Tchebyshev moment transform and cross-coupling operation,” Signal Process. 211, 109107 (2023). [CrossRef]
14. H. Zhou, X. Li, Z. Xu, et al., “Correlated triple hybrid amplitude and phase holographic encryption based on a metasurface,” Photonics Res. 10(3), 678–686 (2022). [CrossRef]
15. J. Wang, G. Zhu, W. Zhang, et al., “Cascaded metasurface for separated information encryption [Invited],” Chin. Opt. Lett. 21(2), 020003 (2023). [CrossRef]
16. S. A. Goorden, J. Bertolotti, and A. P. Mosk, “Superpixel-based spatial amplitude and phase modulation using a digital micromirror device,” Opt. Express 22(15), 17999–18009 (2014). [CrossRef]
17. X. Wang and S. Mei, “Information authentication using an optical dielectric metasurface,” J. Phys. D: Appl. Phys. 50(36), 36LT02 (2017). [CrossRef]
18. X. Wang, W. Wang, H. Wei, et al., “Holographic and speckle encryption using deep learning,” Opt. Lett. 46(23), 5794–5797 (2021). [CrossRef]
19. Y. Li, Y. Xue, and L. Tian, “Deep speckle correlation: a deep learning approach toward scalable imaging through scattering media,” Optica 5(10), 1181–1190 (2018). [CrossRef]
20. M. Lyu, H. Wang, G. Li, et al., “Learning-based lensless imaging through optically thick scattering media,” Adv. Photonics 1(3), 1 (2019). [CrossRef]
21. S. Zhu, E. Guo, J. Gu, et al., “Imaging through unknown scattering media based on physics-informed learning,” Photonics Res. 9(5), B210–B219 (2021). [CrossRef]
22. X. Zhao, X. Nie, Z. Yi, et al., “Imaging through scattering media via spatial-temporal encoded pattern illumination,” Photonics Res. 10(7), 1689–1694 (2022). [CrossRef]
23. L. Zhou, Y. Xiao, and W. Chen, “Learning-based attacks for detecting the vulnerability of computer-generated hologram-based optical encryption,” Opt. Express 28(2), 2499–2510 (2020). [CrossRef]
24. L. Chen, B. Peng, W. Gan, et al., “Plaintext attack on joint transform correlation encryption system by convolutional neural network,” Opt. Express 28(19), 28154–28163 (2020). [CrossRef]
25. S. Jiao, Y. Gao, T. Lei, et al., “Known-plaintext attack to optical encryption systems with space and polarization encoding,” Opt. Express 28(6), 8085–8097 (2020). [CrossRef]
26. S. Yuan, L. Wang, X. Liu, et al., “Forgery attack on optical encryption based on computational ghost imaging,” Opt. Lett. 45(14), 3917–3920 (2020). [CrossRef]
27. X. Liu, X. Meng, Y. Wang, et al., “Known-plaintext cryptanalysis for a computational-ghost-imaging cryptosystem via the Pix2Pix generative adversarial network,” Opt. Express 29(26), 43860–43874 (2021). [CrossRef]
28. W. He, S. Pan, M. Liao, et al., “A learning-based method of attack on optical asymmetric cryptosystems,” Opt. Lasers Eng. 138, 106415 (2021). [CrossRef]
29. M. Liao, S. Zheng, S. Pan, et al., “Deep-learning-based ciphertext-only attack on optical double random phase encryption,” Opto-Electron. Adv. 4(5), 200016 (2021). [CrossRef]
30. L. Zhou, Y. Xiao, and W. Chen, “Learning complex scattering media for optical encryption,” Opt. Lett. 45(18), 5279–5282 (2020). [CrossRef]
31. K. Inoue and M. Cho, “Amplitude based keyless optical encryption system using deep neural network,” J. Vis. Commun. Image Represent. 79, 103251 (2021). [CrossRef]
32. X. Wang, H. Wei, M. Jin, et al., “Experimental optical encryption based on random mask encoding and deep learning,” Opt. Express 30(7), 11165–11173 (2022). [CrossRef]
33. F. Wang, R. Ni, J. Wang, et al., “Invertible encryption network for optical image cryptosystem,” Opt. Lasers Eng. 149, 106784 (2022). [CrossRef]
34. Q. Zhao, H. Li, Z. Yu, et al., “Speckle-Based Optical Cryptosystem and its Application for Human Face Recognition via Deep Learning,” Adv. Sci. 9(25), 2202407 (2022). [CrossRef]
35. Q. Zhou, X. Wang, M. Jin, et al., “Optical image encryption based on two-channel detection and deep learning,” Opt. Lasers Eng. 162, 107415 (2023). [CrossRef]
36. Y. Qin, Y. Wan, Q. Gong, et al., “Deep-learning-based cross-talk free and high-security compressive encryption with spatially incoherent illumination,” Opt. Express 31(6), 9800–9816 (2023). [CrossRef]
37. H. Rasul, K. Xiao, and R. Vollgraf, “Fashion-mnist: a novel image dataset for bench-marking machine learning algorithms,” arXiv, arXiv:1708.07747 (2017). [CrossRef]
38. Y. LeCun, C. Cortes, and C. Burges, “MNIST handwritten digit database,” AT&T Labs (2010).
39. H. Ruan, J. Xu, and C. Yang, “Optical information transmission through complex scattering media with optical-channel-based intensity streaming,” Nat. Commun. 12(1), 2411 (2021). [CrossRef]
