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Abstract
The principle of computational ghost imaging (GI) offers a potential application in optical encryption. Nevertheless, large numbers of keys composed of random or specific patterns set an obstacle to its application. Here, we propose a series of pattern compression methods based on computational GI, in which thousands of patterns are replaced by a single standard image (i.e., two-dimensional data), a sequence of numbers (i.e., one-dimensional data) or the fractional part of an irrational number (i.e., zero-dimensional data). Different pattern compression methods are tested in both simulations and experiments, and their error tolerances in encryption are further discussed. Our proposed methods can greatly reduce the pattern amount and enhance encryption security, which pushes forward the application of computational GI, especially in optical encryption.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Instead of capturing the object image directly, ghost imaging (GI) applies the correlation calculations of intensity fluctuations to retrieve the object information. Since the first GI experiment was reported in 1995 [1], GI has attracted considerable research attention in the last two decades [2–29]. Due to the indirect imaging property, GI has shown great potential in different applications, such as turbulence-free remote sensing and detection [8–11], super-resolution imaging [13], and so on.
To simplify the GI setup, computational GI was proposed [7], where only a pattern generator and a single-pixel camera were required in experimental configuration. In computational GI, the object information can be reconstructed by calculating the correlation between a series of computer-generated patterns and their object-interacting intensity signals detected by a single-pixel camera. This special imaging and reconstruction processes, therefore, offers an optical encryption scheme, where patterns and intensity signals act as the keys and cipher text, respectively [18]. Based on the principle of computational GI, different interesting image encryption methods were proposed, such as grayscale and color image encryption [19], multiple images encryption [16,17,20,21], encryption together with metasurfaces [22,23], ghost identification in big data environment [24], secured broadcast imaging [30], etc. Although the random (or specific) patterns in computational GI can act as the encryption and decryption keys, their great amount always puts a burden on the data transmission, which is an intrinsic shortcoming of computational GI in secure communication applications.
In order to reduce the amount of keys in computational GI encryption, we here propose a series of pattern compression methods and test them in both simulations and experiments. We first employ a single grayscale or binary base image together with specific rules to reduce conventional 3D pattern data into 2D. Using the Kronecker product of 1D sequence consists of 1 and −1 and Hadamard matrix, we then compress the pattern data into 1D. Furthermore, we apply the fractional part of an irrational number to generate the patterns in computational GI, which compresses the information of patterns into a single irrational number, i.e. 0D. Without a heavy transmission of keys like thousands of random patterns during the encryption process, our methods reduce the data amount of pattern from 3D to 2D, and further into 1D and 0D, which push forward the application of computational GI in optical encryption.
2. Model and method
In computational GI, different computer-generated patterns $\{X^{m\times n}\}$ are used to replace the spatially resolved measurements of the reference beam in traditional two-detector GI [7]. As shown in Fig. 1(a), the controllable light speckles are generated by a stable source together with a spatial light modulator or digital micromirror device which can offer active illumination patterns [7,18,22,31,32]. A single-pixel detector is then employed to record the bucket signals $\{Io\}$ of the transmissive or reflective signals after the light speckles interacting with the target. Usually, $N$ $(N\gg m\times n)$ measurements together with $N$ random patterns are needed to reconstruct the ghost image based on the second-order correlation function defined as follows:
Fig. 1. (a) Schematic setup of computational GI. (b) Optical encryption and decryption with random patterns based on the principle of computational GI. (c) Pattern compression methods with a grayscale or binary base image, a 1D sequence, and an irrational number.
Compared with GI algorithm, compressive ghost imaging (CGI), which applies compressed sensing (CS) algorithm, can overcome the Nyquist limit and needs much less measurements $(N\ll m\times n)$ to recover the image. In CGI, the random matrix $X^{m\times n}$ is resized into a row vector $(1\times A, A=m\times n)$, and the set $\{X^{m\times n}\}$ of $N$ measurements is rewritten into a two-dimensional matrix $B$ $(N\times A)$. Meanwhile, the bucket signal $\{Io\}$ is expressed as a column vector $I_{O}^{CGI}(N\times 1)$ [10,22,33,34]. Different CGI algorithms were developed to reconstruct high-quality ghost images efficiently [35]. We here choose CS-based total variation (TV) regularization method [35] in simulations below. The TV minimization model has a huge capability in preserving the edges of image because the gradient’s integral of a natural image is statistically low. Augmented Lagrangian method(ALM) is applied to restore the image efficiently. The target image’s total variation is calculated using $L_{2}$ -norm with the gradient calculation matrix $H$. The model becomes
where $T$ is the original target. By introducing a Lagrange multiplier $y$ to incorporate the equality constraints into the objective function, the Lagrange function is minimized asBased on the principle of computational GI, an optical encryption and decryption scheme was recognized [18], as shown in Fig. 1(b). By using $N$ random patterns, Alice can encode a target image Baboon into $N$ intensity signals, i.e., ciphertext $\{I_{O}\}$. To decode the information, Bob needs to receive the ciphertext $\{I_{O}\}$ and the keys (i.e., a series of random patterns) and applies one of GI algorithms (e.g., GI or TV) to recover the image. As shown in the red box, the quality of recovered image is better with TV algorithm, so in the following discussion, we use TV algorithm to recover images. From the perspective of cryptography, the random patterns here play the role of keys. Whereas in view of Fourier expansion, the random patterns play the role of bases to represent the target image, with $\{I_{O}\}$ as coefficients. These random and nonorthogonal bases inspire us to propose a series of pattern compression methods for the key preparation. As shown in Fig. 1(c), to encode a target image with $n\times n$ pixels in size, $n^{2}$ random patterns $\{X^{n\times n}\}$ are usually used before, and the total amount of data is $n^{4}$. Here, they can be replaced by a grayscale or binary base image in ${n\times n}$ pixels, a 1D sequence with ${1\times n}$ data or even an irrational number (only 1 data). Therefore, the amount of pattern data can be reduced from $n^{4}$ (3D) to $n^{2}$ (2D), $n$ (1D) and $n^{0}$ (0D).
The detailed pattern compression methods are shown in Fig. 2. In Fig. 2(a), a grayscale base image (Peppers) or a random binary base image together with a rule of pixel replacement (e.g., Left-shift) are used to generate a series of patterns. For Left-shift rule, all pixels of the base image will move $k$ positions to the left in the $k$-th transformation, which generates the $k$-th pattern. After $n\times n$ transformations, we can obtain $n^{2}$ different patterns as the encryption keys as shown in Fig. 2(a). In this method, the base image can be regarded as the "start value" of the linear congruential generator, and the rule of pixel replacement is the recurrence relation to generate a series of patterns.
Fig. 2. 2D, 1D and 0D pattern compression methods. (a) 2D: Grayscale and binary base images with Left-shift rule. (b) 1D: The Kronecker product of 1D sequence and Hadamard matrix. (c) 0D: The fractional part of irrational number $\sqrt {2}$ with Odd-Even rule.
To further decrease the data amount of pattern, we propose the Kronecker product of a 1D sequence and Hadamard matrix to generate patterns as shown in Fig. 2(b). Firstly, a 1D sequence of 1 and −1 $(1\times 64)$ is transformed into a square matrix (8$\times$8), called "S". A pink square is used to represent S and a yellow square represents -S. To generate 4096 patterns, Hadamard$_{512\times 512}$ matrix here is used for Kronecker product, in which process S keeps the prototype or multiplies each pixel by −1 to obtain -S according to the element of Hadamard matrix. The resulted matrix ($4096\times 4096$) can be regarded as a combination of S and -S (pink and yellow squares). Specifically, when the element $h_{(i,j)}$ in row $i$ and column $j$ of Hadamard$_{512\times 512}$ is 1, corresponding elements in row $8(i-1)$ to $8i$ and column $8(j-1)$ to $8j$ of the resulted matrix is S. When $h_{(i,j)}$ is −1, then corresponding elements of the resulted matrix is -S. Then, each row $(1\times 4096)$ of the resulted matrix represents a pattern we needed. Extracting each row of the resulted matrix and resized it into $64\times 64$, we therefore obtain 4096 patterns. In this method, we generate 4096 patterns ($64\times 64$ pixels) with only 64 binary numbers, indicating a pattern data reduction into 1D.
Furthermore, an irrational number which has an infinite number of fractional part inspire us to reduce the pattern data into 0D. As shown in Fig. 2(c), irrational number $\sqrt {2}$ together with an operation (e.g., Odd-even) on each element can generate a series of patterns. For Odd-even operation, the element value takes 0 (or 1) when the fractional part of $\sqrt {2}$ is odd (or even), as shown in Fig. 2(c). After Odd-even operation, a sequence of 0-1 numbers are obtained and is used to generate patterns by arranging the sequence in order. Besides, the fractional part of irrational numbers can also employ Left-shift rule to generate a series of patterns. For example, choose $n\times n$ elements from $\sqrt {2}$ to construct a base image. Then, together with different rules (i.e., each element undergoes a transformation according to Left-shift), the base image can generate $n^{2}$ patterns. From the comparison of Figs. 1(b),2(a),2(b) and 2(c), thousands of random patterns can be replaced by a single intriguing base image, a 1D sequence or an irrational number together with a specific rule or operation, which largely reduce the amount of pattern data for transmission from $n^{4}$ to $n^{2}$, and further into $n$ and even $n^{0}$.
3. Results and analysis
Compressive sensing relies on the sparsity of target image and incoherence of patterns to achieve high-quality recovered images [36]. Here, we apply the mutual coherence to analyze the quality of patterns from three methods: the smaller the coherence, the fewer patterns are needed [37]. Using the equation
where $d_{i}$ and $d_{j}$ represent different columns in the projection matrix, $\left \langle \cdots \right \rangle$ represents the inner product operation and $\left | \cdots \right |$ represents to obtain the absolute value, we obtain the mutual coherence 878, 1122, 3072 and 1166 for the "Peppers" and binary base image, 1D sequence and $\sqrt {2}$, respectively. For a series of random patterns, the mutual coherence is about 1000. Thus, patterns generated using the base image and fractional part of $\sqrt {2}$ is similar to random patterns. But for the 1D sequence method, the mutual coherence is larger, which means that patterns have higher correlation.In the pattern compression method in 2D and 1D, the error can originate from pixel-errors in both base images and 1D sequence. Peak signal-to-noise ratio (PSNR) is employed here to evaluate the image quality, which is defined as
where MAX=255 denotes the maximum possible pixel value of the 8-bit images, and MSE denotes the mean square error given by $\frac {1}{m\times n} {\textstyle \sum _{i,j}^{}} [T_{re}\left ( x_{i},y_{i} \right ) -T\left ( x_{i},y_{i} \right ) ]^{2}$, where $T_{re}\left ( x_{i},y_{i} \right )$ and $T\left ( x_{i},y_{i} \right )$ denote the pixel values of the recovered image and the object, respectively. To test the universality of the error tolerance, the error pixels are chosen randomly. As shown in Figs. 3(a) and 3(b), we choose "Peppers" and a binary base image to discuss the pixel-error tolerance, where red dots represent pixel errors. Without error, our proposed 2D pattern compression method can recover the "Baboon" target perfectly using TV algorithms. When the error ratio increases, the quality of recovered images decreases. In comparison, "Peppers" base image is more robust against the error and target image can be identified even with around 30% error ratio. For 1D sequence method as shown in Fig. 3(c), although a perfect recovered image can be obtained without error, a strong strip-like background will appear when the error ratio increases. From the error tolerance study, one can conclude that with a heavier pattern compression rate, the imaging reconstruction will become more sensitive to the pattern error.Fig. 3. Error tolerance of the pattern compression methods in 2D and 1D. (a) "Peppers" base image with error. (b) Binary base image with error. (c) 1D sequence of −1 and 1 with error.
Besides, the error can also originate from bucket intensity values. As shown in Fig. 4, we add different ratios of Gauss-like noise to the bucket intensity values to imitate the experimental situation. PSNRs of four cases generally show a downward trend as the noise ratio increases, and patterns generated from binary base image have a better noise tolerance under the same noise ratio in comparison with other cases. From the simulation, patterns generated by "Peppers" base image leads to a reconstruction of nothing when the noise is greater than 10%. In particular, PSNR value for the case of 1D sequence shows a fluctuation as the noise ratio increases since the bucket intensity values are not equivalent in this case. Many coefficients take on negligibly small values in this case. Therefore, the error will give a greater impact when it is added into a larger intensity value, which is similar to the case of Hadamard patterns [38]. From Fig. 4, we can see that our patterns compression methods are not so robust against the error from the intensity values, i.e. a high-noise-level environment in experiment.
Fig. 4. Noise tolerance study of computational GI with pattern compression methods. PSNR results and four recovered images in "Peppers" base image, random binary base image, 1D sequence and $\sqrt {2}$ are marked, respectively.
Except for pattern data, one can further reduce the amount of ciphertext $\{I_{O}\}$ during the encryption process, which is equivalent to an undersampling problem. The sampling rate effect of our methods is plotted in Fig. 5. Again, "Baboon" $(64\times 64$ pixels) acts as the target, and a grayscale image "Peppers" and a random binary image combined with Left-shift rule are chosen to generate two sets of patterns. The Kronecker product of 1D sequence and Hadamard matrix generates another set of patterns. The fractional part of $\sqrt {2}$ together with Odd-Even operation generates the last set of patterns.
Fig. 5. PSNR curves of the four cases as the sampling rate changes with "Peppers" base image, random binary base image, 1D sequence and $\sqrt {2}$.
We can see that four cases show a similar trend as the sampling rate changes in Fig. 5. Nonetheless, 1D sequence always gives a better recovered image quality when the sampling rate is larger than 95%. In comparison, the quality of recovered image with $\sqrt {2}$ is better than other cases when the sampling rate is less than 90%. Both the curves and the recovered images show that TV algorithm can recover a satisfactory image at a rate > 50% for "Peppers" base image, random binary base image and 1D sequence, while $\sqrt {2}$ case can recover a satisfactory image at a lower rate around 30%.
Although our pattern compression methods are designed for computational GI-based encryption instead of imaging detection, we also try to test their feasibility in imaging experiments. Figure 6(a) shows the experimental setup of computational GI. A projector (Epson, EP-970) is employed to project patterns generated with our pattern compression methods. The light intensity reflected from the target image is detected by a single-pixel detector (Thorlabs, PDA100A2) and recorded by a computer-hosted data acquisition card (NI, PCIe-6251). Two base images ("Peppers" and random binary base image, $64\times 64$ pixels) combined with Left-Shift rule, the Kronecker product of 1D sequence ($1\times 64$) and Hadamard$_{512\times 512}$ matrix, and the fractional part of $\sqrt {2}$ combined with Odd-Even operation are employed to generate 4096 patterns and tested in experiments. Recovered images with simulation and experimental results using GI algorithm are shown in Fig. 6(b). All four cases can recover the target image but with a low imaging quality, which might be due to the environmental noise in experiment.
Fig. 6. Experimental test of computational GI with pattern compression methods. (a) Schematic of the experimental setup. (b) The target image and simulation and experimentally recovered images by using "Peppers" base image, binary base image, 1D sequence of 1 and −1, and the fractional part of $\sqrt {2}$, respectively.
Nevertheless, it should be mentioned that the mission of our pattern compression methods is not for experimental detection but for image encryption. Computational ghost imaging can be applied in optical encryption by using random patterns as the key. But their large amount places a heavy burden on data transmission and can easily attract the eavesdropper’s attention and have low security. With the proposed pattern compression methods, keys are hidden into a base image, a 1D sequence, or an irrational number. Even if the eavesdropper obtains the limited information, it is difficult to generate suitable patterns for decryption. Besides, specific rules or operations are additional keys which can be pre-agreed or steganographically written for transmission to improve the security.
4. Conclusions
In conclusion, we have proposed a series of pattern compression methods for image encryption, based on the principle of computational GI. It has shown that pattern compression method can effectively reduce data transmission and increase the encryption security simultaneously. By disguising the encryption keys into a base image, a sequence of numbers, or an irrational number, together with corresponding rules as an additional key, our pattern compression methods steer the computational GI towards the image encryption in a more practical way.
Funding
Multi-Year Research Grant of University of Macau (MYRG2020-00082-IAPME); Science and Technology Development Fund from Macau SAR (FDCT) (0062/2020/AMJ); National Natural Science Foundation of China (11735005, 11474027); Interdiscipline Research Funds of Beijing Normal University.
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.
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