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Abstract
An optical security method for multiple-image authentication is proposed based on computational ghost imaging and hybrid non-convex second-order total variation. Firstly, each original image to be authenticated is encoded to the sparse information using computational ghost imaging, where illumination patterns are generated based on Hadamard matrix. In the same time, the cover image is divided into four sub-images with wavelet transform. Secondly, one of sub-images with low-frequency coefficients is decomposed using singular value decomposition (SVD), and all sparse data are embedded into the diagonal matrix with the help of binary masks. To enhance the security, the generalized Arnold transform is used to scramble the modified diagonal matrix. After using SVD again, the marked cover image carrying the information of multiple original images is obtained using the inverse wavelet transform. In the authentication process, the quality of each reconstructed image can be greatly improved based on hybrid non-convex second-order total variation. Even at a very low sampling ratio (i.e., 6%), the existence of original images can be efficiently verified using the nonlinear correlation maps. To our knowledge, it is first to embed sparse data into the high-frequency sub-image using two cascaded SVDs, which can guarantee high robustness against the Gaussian filter and sharpen filter. The optical experiments demonstrate the feasibility of the proposed mechanism, which can provide an effective alternative for the multiple-image authentication.
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1. Introduction
In the past decades, optical information security has received more and more attention in the fields of encryption, authentication, and data hiding. Due to many inherent advantages, such as high degrees of freedom, high speed, and parallel processing ability, numerous methods are developed and demonstrated based on different optical theories and technologies [1–4], since the remarkable double random phase encoding (DRPE) architecture has been proposed in 1995. Using DRPE, although an image can be encrypted into stationary white noise by the aid of two statistically independent random phase masks in the Fourier transform domain, the preliminary information can still be retrieved under plaintext-known and chosen-plaintext attacks. To destroy its intrinsic linearity, the optical security methods have been further extended in other domains obtained with different transforms such as fractional Fourier transform [5–7], Fresnel transform [8,9], fractional Mellin transform domain [10], Gyrator transform domain [11–14], and Radon transform [15]. Moreover, various cryptosystems combined with other technologies including wavelet transform [16], polarized light [17], integral imaging [18], phase retrieval [19], transport of intensity equation [20], correlated imaging [21], compressive ghost imaging [22], elliptic curve [23] and visual secret sharing [24] are developed to ensure the information security. In addition, different multiplexing technologies, such as wavelength [25], azimuth [26], and phase mask [27] multiplexing, also have been suggested to achieve the goal of protecting multiple-image simultaneously.
Recently, due to its significant property of imaging the target object by a bucket detector without spatial resolution, computational ghost imaging has attracted extensive attention, especially in the field of optical information security [28–33]. An image can be easily encrypted to the sparse information using computational ghost imaging, which is constructed with a series of real-valued measurements and considered as the ciphertext output by the cryptosystem. Kang et al. [34] confused the eavesdropper by illuminating a camouflaged image with specific modulated patterns and hiding the secret data into it. Ye et al. [35] stimulated the potential of random lighting patterns in the space-time dimension, and achieved the embedding of a large amounts of information. Wu et al. [36] optimized Hadamard patterns using a 4-connected-region-based method, and imaged the target image with much less measurements. Sui et al. [37] proposed a new mechanism of single-pixel correlated imaging that can ensure less measurements for authenticating multiple images simultaneously. Yuan et al. [38] compressed the information by quantifying measurements and then hiding them into wavelet coefficients of the cover image, which can reduce serious deterioration of the reconstructed image. Ye et al. [39] introduced weighted multiplicative signals in the sender or receiver, and resisted against forgery attacks for computational-ghost-imaging-based encryption and the potential threat of non-invasive imaging. Differing from the traditional computational ghost imaging, Sui et al. [40] reported a new mechanism based on sparse reconstruction with the Barzilai-Borwein gradient projection, where speckle patterns are compressed using compressive sensing to reduce the burden of key management. Zheng et al. [41] enhanced the security of image encryption using an inverse computational ghost imaging, where illumination patterns are determined according to bucket signals. Zhou et al. [42] decomposed the cover image and selected one of components with the low-frequency coefficients. After that, measurements of the watermark encoded by computational ghost imaging are fused with randomly chosen coefficients. Since deep learning can automatically learn features from the training data, it also has attracted more and more researches in the field of optical information security, especially in computational ghost imaging [43,44]. Li et al. [45] solved the narrow working spectrum problem of ptychography by introducing the single-pixel imaging, and obtained the field’s 2D information from the 1D measurements.
As it can be seen from previous works that for a target image encoded using computational ghost imaging, the bucket detector usually needs to record a large number of measurements, which will seriously affect the imaging efficiency because more repetitions of acquisition require more time. To deal with this problem, besides computational ghost imaging an optical security approach for multiple-image authentication is proposed based on generalize Arnold transform and hybrid non-convex second-order total variation. In this method, each original image to be authenticated is encoded into the sparse information using computational ghost imaging, then randomly embedded into a low-frequency sub-image of the cover image with the help of the corresponding binary mask. The sub-image is decomposed using two singular value decompositions, and its decomposed diagonal matrix is scrambled using the generalized Arnold transform. In the authentication process, the quality of reconstructed images can be greatly improved based on hybrid non-convex second-order total variation. Meanwhile, the traditional nonlinear correlation map is modified in order to obtain high discrimination capability, which can easily verify the existence of original images. Also, the corresponding binary mask and iteration number of generalized Arnold transform are used as secret keys, and the security of authentication can be significantly enhanced.
The rest of this paper is organized as follows. In Section 2, along with generalized Arnold transform and hybrid non-convex second-order total variation, the proposed optical multiple-image authentication method based on computational ghost imaging is introduced in detail. In Section 3, a series of optical experiments are carried out to demonstrate its feasibility. Meanwhile, security and robustness are analyzed. Finally, a brief conclusion is described in Section 4.
2. Method description
2.1 Computational ghost imaging
As shown in Fig. 1, an optical schematic of the computational ghost imaging can be used to record one-dimensional spectral data of an object image. In the beginning, a series of illumination patterns are sequentially loaded into the digital projector, and projected on the object image that is located at a certain distance from the projector. The total reflected intensities that are generated by the combination of illumination in the imaging field of view on the object image and speckle patterns are collected by the bucket detector without spatial resolution. Under the illumination of the $i\textrm{ - th}$ illumination pattern, the corresponding measurement denoted as ${B_i}$ acquired by the bucket detector can be mathematically expressed as
where Q is the object image, ${I_i}$ is the $i\textrm{ - th}$ illumination pattern, and $(\mu ,\upsilon )$ is the transverse coordinate of the object image plane. Through calculating the correlation of measured intensities and speckle patterns, the information of the object image can be reconstructed as where $\hat{Q}$ is the reconstructed image, $\left\langle \cdot \right\rangle$ is the ensemble average computation, $\Delta {B_i} = {B_i} - \left\langle {{B_i}} \right\rangle$ and $\Delta {I_i} = {I_i} - \left\langle {{I_i}} \right\rangle$.
Fig. 1. Optical schematic of computational ghost imaging. DLP: digital light projector; BD: bucket detector; IPs: illumination patterns.
To obtain the object image with high visual quality, a large number of measured intensities should be acquired in the imaging process. Moreover, instead of using random speckle patterns in the conventional computational ghost imaging, a series of structured illuminations deserved from the Hadamard matrix are applied to collect measured intensities. For the object image Q with $N \times N$ pixels, the Hadamard matrix with the order ${2^k}$ is constructed with the recursive procedure. Initially, the second order Hadamard matrix is defined as [46]
Then, based on the second order matrix, the Hadamard matrix with the order ${2^k}$ is obtained as
Basically, the Hadamard matrix is both square and symmetric, and composed of +1 and -1 values. Each row of the matrix can be rearranged into a tow-dimensional pattern with $N \times N$ pixels. Thus, a total of ${2^k}$ illumination patterns are generated, which can guarantee that the information of the object image can be efficiently reconstructed even in very low light environment.
2.2 Generalized Arnold transform
As an image scrambling technology, Arnold transform is simple, efficient and easy to operate. Although Arnold transform has the feature of periodicity, i.e., original images can be recovered after a certain number of iterations, it is still widely used in the field of information security, such as stenography, authentication, tamper detection, and data encryption. This is mainly because that this periodicity depends on the size of original images and does not follow any order. Therefore, the number of iterations can be considered as an important secret key in most applications. Moreover, due to its excellent chaotic properties, such as ergodicity and pseudo-randomness, the two-dimensional generalized Arnold transform has been extensively used in image cryptography in recent years. According to an original image with the size of $N \times N$ pixels, the generalized Arnold transform changes the position of a pixel $({x_i},{y_i})$ to a new pixel position, which can be described as [47]
2.3 Hybrid non-convex second-order total variation
The total variation (TV) regularization as an effective regularization method can maintain sharp edges in images and is widely used in the field of image reconstruction. However, the undesired blocky artifacts are usually produced in the reconstructed image. Although higher-order regularization can suppress the block artifacts efficiently, the edges and texture will become blurred. To solve this problem, a combination mode of non-convex higher-order total variation with overlapping group sparse regularizer is used to control the effects of the blocky artifacts, where a re-weighted ${l_1}$ alternating direction method (ADM) is used to deal with the constraints and subproblems [48].
For a blurring model in image processing, a constrained optimization problem can be expressed as
- (1) Compute ${v^{k + 1}} = \mathop {\textrm{argmin} }\limits_{\textrm{v}} \textrm{ }\frac{\rho }{2}\left\|{\textrm{v} - \left( {\nabla {\textrm{u}^{k + 1}} + \frac{{{\mu_1}^k}}{\rho }} \right)} \right\|_2^2 + \phi (\textrm{v} )$, where $\phi (\textrm{v} )= {\sum\limits_i {\left[ {\sum\limits_{j = 0}^K {{{|{\textrm{v}({i + j} )} |}^2}} } \right]} ^{{1 / 2}}}$ and the majorization-minimization (MM) strategy is adopted to yield a convergent solution.
- (2) Compute ${\textrm{w}^{k + 1}} = \max \left\{ {|{{\textrm{x}^{k + 1}}} |- \frac{{{\textrm{t}_i}\omega }}{\rho },0} \right\} \cdot \textrm{sign}({{\textrm{x}^{k + 1}}} )$, where ${\textrm{x}^{k + 1}} = {\nabla ^2}{\textrm{u}^{k + 1}} + \frac{{\mu _2^k}}{\rho }$ and the iterative re-weighted ${l_1}$ (IRL1) algorithm is adopted. In each iteration of the IRL1 algorithm, the weight t is updated by ${\textrm{t}_i} = \frac{{\omega \rho }}{{{{({|{\textrm{w}_i^k} |+ \varepsilon } )}^{1 - p}}}}$ and $\varepsilon$ is a small number.
- (3) Compute ${\textrm{u}^{k + 1}}$ as ${\textrm{u}^{k + 1}} = {({\lambda {\textrm{K}^\textrm{T}}\textrm{K} + \rho {\nabla^\textrm{T}}\nabla + \rho {{({{\nabla^2}} )}^\textrm{T}}{\nabla^2} + \rho \textrm{I}} )^{ - 1}}(\lambda {\textrm{K}^\textrm{T}}\textrm{f} - {\nabla^\textrm{T}}{\mu_1} + \rho {\nabla^\textrm{T}}\textrm{v} - {{({{\nabla^2}} )}^\textrm{T}}$${\mu_2} + \rho {{({{\nabla^2}} )}^\textrm{T}}\textrm{w} - {\mu_3} + \rho \textrm{z} )$, which can be solved in the Fourier domain using the two-dimensional fast Fourier transform due to periodic boundary conditions of the original image.
- (4) Compute ${\textrm{z}^{k + 1}} = \mathop {\textrm{argmin} }\limits_\textrm{z} \textrm{ }\frac{\rho }{2}\left\|{\textrm{z} - \left( {{\textrm{u}^{k + 1}} + \frac{{{\mu_3}^k}}{\rho }} \right)} \right\|_2^2 + \mathcal{I}_\mathcal{C}(\textrm{z} )$, where $\mathcal{I}_\mathcal{C}(\textrm{z} )$ is defined as $\mathcal{I}_\mathcal{C}(\textrm{z} )= 0$ if $\textrm{z} \in \mathcal{C}$ and otherwise $\mathcal{I}_\mathcal{C}(\textrm{z} )= \infty$.
- (5) Update Lagrange multipliers as ${\mu _1}^{k + 1} = {\mu _1}^k + \rho ({{\textrm{v}^{k + 1}} - \nabla {\textrm{u}^{k + 1}}} )$, ${\mu _2}^{k + 1} = {\mu _2}^k + \rho ({{\nabla^2}{\textrm{u}^{k + 1}} - {\textrm{w}^{k + 1}}} )$, and ${\mu _3}^{k + 1} = {\mu _3}^k + \rho ({{\textrm{u}^{k + 1}} - {\textrm{z}^{k + 1}}} )$.
- (6) If the condition $\frac{{{{\left\|{{\textrm{u}^k} - {\textrm{u}^{k + 1}}} \right\|}_2}}}{{{{\left\|{{\textrm{u}^k}} \right\|}_2}}} \le \delta$ is satisfied, where $\delta $ is a predefined threshold, the iteration process will be stopped.
2.4 Generation of the ciphertext and authentication
In this section, the detail of the proposed multiple-image authentication method based on hybrid non-convex second-order total variation is represented. The corresponding diagram of the generation of the ciphertext is plotted in Fig. 2, where the information of multiple original images can be inserted into the cover image with good imperceptibility. Suppose there are L images to be authenticated, and ${Q^{(j )}}\textrm{ }\{ j = 1,2,\ldots ,L\}$ represents the $j\textrm{ - th}$ image and the size of the cover image C is $N \times N$ pixels, the ciphertext generated from multiple original images to be authenticated can be described as follows
- (1) For each image to be verified, a total of K measurements are collected using computational ghost imaging as shown in Fig. 1, where illumination patterns are generated based on the Hadamard matrix. Next, a real-valued sequence denoted as $B_i^{(j )}\textrm{ }\{ j = 1,2,\ldots ,L;\textrm{ }i = 1,2,\ldots ,K\}$ is constructed by the catenation of all measurements. Obviously, there are $K \times L$ real measurements in this sequence.
- (2) Due to its properties of asymmetric, orthogonal and biorthogonal, Daubechies1 wavelet (db1) transform is applied to decompose the cover image C into four sub-images with the size of ${N / 2} \times {N / 2}$ pixels, i.e., $LL,LH,HL$ and $HH$. After that, one of high-frequency sub-images is used to embed the real-valued sequence, which can ensure the visual quality of the marked cover image. Here, the sub-image $HH^{\prime}$ is selected.
- (3) Using SVD, which can efficiently represent the intrinsic algebraic properties of an image, the sub-image $HH$ is decomposed into three part as$$HH = {U_w}SV_w^\textrm{T} = [{{u_{w,1}},{u_{w,2}},\ldots ,{u_{{{w,N} / 2}}}} ]\left( {\begin{array}{cccc} {{\lambda_1}}&0& \cdots &0\\ 0&{{\lambda_2}}& \cdots &0\\ \vdots &0& \ddots &0\\ 0&0& \cdots &{{\lambda_{{N / 2}}}} \end{array}} \right){[{{v_{w,1}},{v_{w,2}},\ldots ,{v_{{{w,N} / 2}}}} ]^\textrm{T}}, $$where S is a diagonal matrix of singular values ${\lambda _i}\textrm{ }\{ i = 1,2,\ldots ,{N / 2}\}$ that arranged in the decreasing order, ${U_w}$ and ${V_w}$ are orthogonal matrices.
- (4) Given L two-dimensional binary masks with the size of ${N / 2} \times {N / 2}$ pixels, which are denoted as $B{M^{(j )}}$ and considered as secret keys, $K \times L$ measurements of all original images can be embedded into the diagonal matrix S as where ${S^{\prime}_i}$ is the modified element that selected from the diagonal matrix S with the help of binary masks, and $\gamma$ is a real-valued attenuation factor to adjust the intensity of measurements. Notably, the values of K pixels are 1 and others are 0 in each binary mask, and these masks cannot be overlapped each other. In the embedding process, scanning each mask, if the value of a pixel is 1, the element in the diagonal matrix S is selected and modified. Thus, the diagonal matrix carrying the information of original images to be verified is obtained.
- (5) To improve the security, the modified diagonal matrix $S^{\prime}$ is further scrambled using generalized Arnold transform. Given the number of iterations as a secret key, denoted as ${K_s}$, the scrambled diagonal matrix $S^{\prime\prime}$ can be obtained using Eq. (5). Using SVD again, $S^{\prime\prime}$ is decomposed into three part as where ${U_v}$ and ${V_v}$ are orthogonal matrices, and $S^{\prime\prime\prime}$ is a diagonal matrix composed of singular values. Notably, the number of iterations of generalized Arnold transform is also used as the secret key of the cryptosystem.
- (6) Along with $U_W$ and $V_W$ obtained using Eq. (9), the diagonal matrix $S^{\prime\prime\prime}$ is used to create the marked sub-image $HH^{\prime}$ as
- (7) Along with sub-images $LL,LH,HL$ and the marked sub-image $HH^{\prime}$, after applying the inverse Daubechies1 wavelet (db1) transform wavelet, the marked cover image $\hat{C}$ containing the information of original images to be verified is reconstructed.
Fig. 2. The generation process of the ciphertext for multiple original images to be verified. Note: two cascaded SVDs are applied in the proposed method
The authentication process of original images is basically in the reversed order as the generation process of the ciphertext. Although this process is relatively simple, some steps should be concerned. In the same time, an enhanced nonlinear correlation algorithm is designed to improve the capability to discriminate the existence of original images. As shown in Fig. 3, main steps to authenticate an original image are sequentially described as follows
- (1) The marked cover image $\hat{C}$ containing the information of all original images is decomposed into four sub-images using the Daubechies1 wavelet (db1) transform, from which the marked high-frequency sub-image $HH^{\prime}$ is obtained.
- (2) Using SVD, the diagonal matrix $S^{\prime\prime\prime}$ can be calculated from the above marked sub-image. Meanwhile, giving ${U_v}$ and ${V_v}$ obtained in the generation process of the ciphertext, the scrambled diagonal matrix $S^{\prime\prime}$ can be recovered using the inverse SVD.
- (3) Using the inverse generalized Arnold transform, the modified diagonal matrix $S^{\prime}$ that measurements of all original images are recorded is recovered. For an original image ${Q^{(j )}}$, along with the corresponding binary mask $B{M^{(j )}}$ and the real-valued attenuation factor $\gamma$, the measurements can be calculated as where ${S^{\prime}_i}$ is the modified element in the generation process of the ciphertext.
- (4) Calculating the correlation of measurements and speckle patterns using Eq. (2), an image ${\hat{Q}^{(j )}}$ is reconstructed, from which the content of the original image can be observed. If the number of measurements is enough high, the reconstructed result is good. With the decrease of measurements, it will become more and more blurry. Therefore, the iterative reconstruction process based on hybrid non-convex second-order total variation is further used to optimize the reconstructed result, where the reconstructed result is considered as a blurring model as described with Eq. (7) and the augmented Lagrangian of solving this problem is described with Eq. (8).
- (5) Calculating the nonlinear correlation map between the reconstructed result and the original image, the authentication process is easily to be implemented, which can be mathematically expressed as [49]$$\textrm{NC(}{\hat{Q}^{(j )}},{Q^{(j )}}\textrm{)} = {\left|{IFT\left\{{{{\left|{FT\left\{{{Q^{(j )}} \times conj\left\{{{{\hat{Q}}^{(j )}}} \right\}} \right\}} \right|}^{p - 1}}FT\left\{{{Q^{(j )}} \times conj\left\{{{{\hat{Q}}^{(j )}}} \right\}} \right\}} \right\}} \right|^2}, $$where $FT\{{\cdot} \} $ and $IFT\{{\cdot} \} $ represent the two-dimensional Fourier transform and inverse Fourier transform, respectively, $conj\{{\cdot} \} $ is used to compute the complex conjugation of the argument, and p is the nonlinearity strength.
- (6) To improve the capability to discriminate the original image, the above nonlinear correlation map is further revised as$$\textrm{NC}({\textrm{NC} < ({({MAX({\textrm{NC}} )- MIN({\textrm{NC}} )} )\ast 0.6} )\textrm{ + }MIN({\textrm{NC}} )} )= 0, $$where $MAX({\cdot} )$ and $MIN({\cdot} )$ represent the maximum and minimum function, respectively, and $\varpi $ is the enhancement coefficient. In this way, the maximum peak in the correlation map can be obviously highlighted, especially when the number of measurements is very low.
3. Experimental results and analysis
In this section, to quantitatively evaluate the feasibility and effectiveness of the proposed multiple-image authentication method, optical experiments have been performed based on the setup as shown in Fig. 1. Due to the limited resource in our laboratory, the bucket detector is replaced with an industrial camera in the following experiments. The model of the projector and the camera is XMING A62S and imaging source DFK 72AUC02, respectively. Meanwhile, besides the reconstruction quality and the security of secret keys, two metrics such as the correlation coefficient (CC) and the peak signal-to-noise ratio (PSNR) are calculated and analyzed. Suppose that Q and $\hat{Q}$ represent an original image and its reconstructed result, respectively, which all have $N \times N$ pixels, and $E({\cdot} )$ denotes the mathematical expectation, the CC between Q and $\hat{Q}$ is calculated as
Suppose that $C$ and $\hat{C}$ represent a cover image and its marked result, respectively, which also have $N \times N$ pixels, the PSNR between $C$ and $\hat{C}$ is utilized to evaluate the transparency degree of the embedded information, which is calculated as
Fig. 3. The authentication process of original images. HNSTV: hybrid non-convex second-order total variation.
In the generation process of the ciphertext, a real-valued attenuation factor $\gamma$ is set to 0.000001, and the number of iterations of generalized Arnold transform is set to 200. In the authentication process of original images, the nonlinearity strength p in Eq. (14) is set to 0.4, and the enhancement coefficient $\varpi$ in Eq. (16) is set to 4.
As shown in Fig. 4(a), the image “Fruit” is used as the cover image with $512 \times 512$ pixels, which is selected from USC-SIPI image database. Four original image to be authenticated are displayed in Fig. 4(b)-(e), which are made by ourself and have $64 \times 64$ pixels. In order to ensure that the corresponding reconstructed results have good visual quality, 4096 structural speckle patterns that generated from the Hadamard matrix are used to collect the reflected intensities in the process of computational ghost imaging, and then these measurements are embedding into the low-frequency sub-image of the cover image. In addition, four corresponding binary masks with $256 \times 256$ pixels are displayed in Fig. 5(a)-(d), which are non-overlapped each other. With the help of these masks, all measurements of original images can be independently embedded into the cover image.
To reduce the influence of environment light and noise, a total of 4096 illumination patterns are generated based on the Hadamard matrix that has the same size as original image to be authenticated. Then, each original image is encoded into 4096 real-valued measurements using computational ghost imaging as shown in Fig. 1. Thus, there are 16384 measurements are recorded for four original images, which will be embedded into the low-frequency sub-image of the cover image with the help of four binary masks. As shown in Fig. 6(a), the marked cover image that carrying the information of four original images is displayed. The PSNR between Fig. 4(a) and Fig. 6(a) is calculated using Eq. (18), and is 54.5651 dB. Obviously, the difference between them cannot be easily observed with naked eyes, and the imperceptibility performance of the proposed method is very good. As shown in Fig. 6(b)-(e), four images reconstructed using Eq. (2) are displayed, where a large number of speckle noise can be distinctly observed. The CC values between them and original images are 0.8450, 0.8410, 0.8984, and 0.8808, respectively. After optimization based on hybrid non-convex second-order total variation, the corresponding CC values can be further improved, and reach 0.8793, 0.8876, 0.9213, and 0.9156, respectively. As shown in Fig. 6(f)-(i), the visual quality of reconstructed images is better than those in Fig. 6(b)-(e).
Fig. 6. (a) The marked cover image, (b)-(e) reconstructed original images using Eq. (2), (f)-(i) reconstructed images using hybrid non-convex second-order total variation.
As discussed in the generation process of the ciphertext, two kinds of secret keys are considered to enhance the security of the proposed method. One is multiple binary masks corresponding to original images, and another is the iteration number used in the generalized Arnold transform. To analyze the impact of these binary masks on the authentication of original images, the affection of data loss rate is investigated at first. Namely, how many pixel values is corrupted in a binary mask, and the reconstructed image will not be authenticated. Through experiments, it is found that the authentication process cannot be implemented if 0.24% of pixels in each mask are randomly selected and corrupted. As shown in Fig. 7(a)-(d), four modified masks are displayed, where the values of 10 pixels with 1 are set to 0 in each mask. Four images are noisily constructed as shown in Fig. 8(a)-(d), from which no any valid information can be observed. The corresponding CC values are -0.0344, 0.0049, 0.0094, and 0.0128, respectively, which are very small. Four nonlinear correlation maps are displayed in Fig. 8(e)-(h), respectively, where no remarkable peak exists in the center position of these distributions. Next, the affection of randomly generated mask is analyzed. As shown in Fig. 9(a), the randomly generated mask is displayed, where the values of 4096 pixels are set to 1 and others are set to 0. Similarly, a noisy image is reconstructed as shown in Fig. 9(b), and no useful information can be obtained. The CC values between it and four original images are 0.0058, -0.0248, 0.0078, and 0.0280, respectively. The correlation maps between it and four original images are displayed in Fig. 9(c)-(f), respectively. Obviously, no significant peak is formed in the center position of these maps, which means that the reconstructed image and all original images are uncorrelated. Therefore, the security of the proposed method can be guaranteed by using binary masks as secret keys of the authentication system.
Fig. 8. (a)-(d) Reconstructed images with corrupted mask, and (e)-(h) corresponding correlation maps.
Fig. 9. (a) Randomly generated mask, (b) reconstructed image, and (c)-(f) four nonlinear correlation maps.
For the iteration number of generalized Arnold transform, the related sensitivity is mainly analyzed, because the high sensitivity as a secret key can prevent any eavesdropper from eavesdropping. As shown in Fig. 10(a)-(b), the reconstructed results of the last original image are displayed when the iteration number is changed with the absolute deviation of 1. The corresponding CC values are 0.0236 and 0.0267, respectively. From these noisy results, no valid information of the original image can be observed. The related CC curve is plotted in Fig. 10(c), from which it can be seen that once the iteration number of generalized Arnold transform is varied, the CC value will quickly drop to 0. For other original images, similar conclusion can be obtained. Therefore, the iteration number can be considered as the secret key due to its high sensitivity. Even without considering the iteration number, because there are ${2^{256 \times 256}}$ situations for each binary mask, the key space to decrypt an original image will be very huge. So, the proposed method has high resistance against the brute-force attack.
Fig. 10. (a)-(b) Reconstructed results when the iteration number is changed with the absolute deviation of 1, and (c) the related CC curve.
To verify the robustness of the proposed method, the resistance against conventional attacks is analyzed. For the occlusion attack, the marked cover image is first occluded with a certain proportion. Then, each original image is recovered and authenticated. In the occlusion process, 25% pixels of the marked cover image are cut from the upper left corner, upper right corner, lower left corner, lower right corner, and central region, respectively. That is, the values of these pixel are set to 0. As shown in Fig. 11(a)-(e), the last original image is recovered with respect to five different situations, from which no distinct information can be observed. Despite this, the original image can be authenticated using the modified nonlinear correlation algorithm. As shown in Fig. 11(f)-(l), the corresponding correlation maps between the last original image and four reconstructed results are displayed, respectively. Obviously, there is a remarkable peak in the center of each map, which indicates the existence of the last original image. For other original images, remarkable peaks also can be obtained.
Fig. 11. (a)-(e) Last original image reconstructed with different occlusions, and (f)-(l) corresponding nonlinear correlation maps.
For the noise attack, two kinds of noise are used to disturb the marked cover image. One is the salt and pepper noise, and another is the Gaussian noise. As shown in Fig. 12(a)-(c), the reconstructed results of the last original image are displayed when the density of salt and pepper noise is set to 0.04, 0.05, and 0.06, respectively. It can be seen intuitively that the visual quality of the recovered image decreases as the noise density increases. Although the content is not easily recognized, the existence of the original image is still implemented with the help of the modified nonlinear correlation maps. As shown in Fig. 12(d)-(f), there is a remarkable peak in each nonlinear correlation distribution. For the Gaussian noise, the marked cover image is disturbed as follows
where $\hat{C}{}^{\prime}$ is the disturbed image, $\eta$ is the noise strength, and G is the Gaussian noise with zero mean and standard deviation 0.001. As shown in Fig. 13(a)-(c), the reconstructed results of the last original image are displayed when the noise strength is set to 0.2, 0.3, and 0.4, respectively. The corresponding nonlinear correlation distributions are displayed in Fig. 13(d)-(f). Similar to the affection of salt and pepper noise, the recovered image becomes increasingly blurred when the noise strength increases. However, the information of the last original image can be confirmed using the nonlinear correlation map. For other original images, the authentication process can be implemented in the same way.Fig. 12. (a)-(c) Last original image reconstructed with different densities of salt and pepper noise, and (d)-(f) corresponding nonlinear correlation maps.
Fig. 13. (a)-(c) Last original image reconstructed with different strengths of Gaussian noise, and (d)-(f) corresponding nonlinear correlation maps.
As a distinctive feature, the information of multiple original images obtained using computational ghost imaging is embedded into the sub-image with high frequency coefficients in the proposed method. In some traditional methods, the related data is usually recorded into components with low or middle frequency, which is helpful to resist against smoothing with the Gaussian filters. Because two cascaded SVDs are applied in the proposed method, which can guarantee high robustness against the Gaussian filter and sharpen filter. As shown in Fig. 14(a)-(c), reconstructed results of the last original image are displayed when the marked cover image is processed using the Gaussian filters with different kernel size. Obviously, the information of original image cannot be clearly distinguished. However, it can be authenticated using the modified nonlinear correlation maps, because there is a remarkable peak in the center of each map as shown in Fig. 14(d)-(f), respectively. In addition, when the marked cover image is processed using the sharpen filters [50], the authentication also can be perfectly implemented. The reconstructed results are displayed in Fig. 15(a)-(b), where the marked cover image is filtered with two kinds of kernel size. The corresponding maps are displayed in Fig. 15(c)-(d), where the remarkable peaks can be obtained in two maps. Similarly, other original images also can be authenticated.
Fig. 14. (a)-(c) Reconstructed image when the marked cover image is processed using Gaussian filters with the kernel of $3 \times 3$, $5 \times 5$, and $7 \times 7$, and (d)-(f) corresponding nonlinear correlation maps.
Fig. 15. (a)-(b) Reconstructed image when the marked cover image is processed using sharpen filters with the kernel of $3 \times 3$ and $5 \times 5$, and (c)-(d) corresponding nonlinear correlation maps.
As an important metric, the number of measurements in the process of computational ghost imaging seriously affects the efficiency of the cryptosystem. To demonstrate the effectiveness of the proposed method, only a small percentage of measurements are embedded into the sub-image with high frequency coefficients that decomposed from the original cover image. Where the sampling ratio equals 0.06, i.e., only 245 measurements are recorded for each original image, the reconstructed results based on hybrid non-convex second-order total variation are displayed as shown in Fig. 16(a)-(d). From these noisy results, it is obvious that no valid information can be obtained. Despite this, the existence of original images still can be verified using the modified nonlinear correlation maps as shown in Fig. 16(e)-(h), where unique remarkable peak can be obtained in the center of each distribution. Therefore, the authentication efficiency of the proposed method is greatly improved.
Fig. 16. (a)-(d) Four reconstructed results when only 245 measurements are used for each original image, and (e)-(h) corresponding nonlinear correlation maps.
4. Conclusion
In summary, an optical multiple-image authentication method is proposed based on computational ghost imaging and Hybrid non-convex second-order total variation. In this method, reconstructed images can be authenticated even when a very low sampling ratio (i.e., 6%) is used in the computational ghost imaging. Therefore, relatively few illumination patterns are used to collect measurements for each original image, and the authentication efficiency can be greatly improved. The nonlinear correlation map is further modified to make the authentication process more recognizable, where a unique remarkable peak is located in the center position and other lower peaks are removed in the distribution. Besides the iterative number of generalized Arnold transform, binary masks are applied as secret keys so that the security level can be efficiently guaranteed. Most importantly, although the sparse information of original images to be authenticated is embedded into the sub-image with high-frequency, high robustness against the Gaussian filter and sharpen filter can be obtained. The feasibility and effectiveness of the proposed method have been demonstrated by a series of optical experiments and analysis. In the future, under the premise of high visual quality of the marked cover image, it is significant to simultaneously embed the information of more images into high-frequency components. It is believed that this research not only enriches functionality of computational ghost imaging but also provides a fascinating boost to the real-time application in the field of optical multiple-image authentication.
Funding
National Natural Science Foundation of China (62031023).
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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