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JPEG image steganography payload location based on optimal estimation of cover cofrequency subimage
EURASIP Journal on Image and Video Processing volume 2021, Article number: 1 (2021)
Abstract
The excellent cover estimation is very important to the payload location of JPEG image steganography. But it is still hard to exactly estimate the quantized DCT coefficients in cover JPEG image. Therefore, this paper proposes a JPEG image steganography payload location method based on optimal estimation of cover cofrequency subimage, which estimates the cover JPEG image based on the Markov model of cofrequency subimage. The proposed method combines the coefficients of the same position in each 8 × 8 block in the JPEG image to obtain 64 cofrequency subimages and then uses the maximum a posterior (MAP) probability algorithm to find the optimal estimations of cover cofrequency subimages by the Markov model. Then, the residual of each DCT coefficient is obtained by computing the absolute difference between it and the estimated cover version of it, and the average residual over coefficients in the same position of multiple stego images embedded along the same path is used to estimate the stego position. The experimental results show that the proposed payload location method can significantly improve the locating accuracy of the stego positions in low frequencies.
1 Introduction
Digital steganography is the technique that embeds information, known as the payload, into the redundant parts of multimedia data such as digital images, video, audio, and text, termed the cover, to conceal secret communications. In the past decades, a series of steganographic algorithms have been proposed with image, text, audio, or video as cover [1,2,3,4,5,6,7,8]. Correspondingly, many steganalysis algorithms also have been proposed to detect the stego object [9,10,11,12,13,14]. However, in real life, the investigators often not only satisfy with distinguishing the cover objects and the stego objects, but also are eager to extract the hidden information. Compared with the detection of the stego objects, the extraction of hidden information is much more difficult and requires more clues, such as the stego key space, the stego positions, and the selection scheme of stego positions. The technique to identify the stego positions is referred as steganography payload location. In [15, 16], Yang et al. and Liu et al. have reported that when the selection scheme of stego positions is known, if the investigator can locate the steganography payload with the accuracy higher than randomly guessing, he (or she) can extract the hidden information by a collision attack.
Although Quach [17] has proved the locatability of modified pixels in a single stego image, the actual steganography payload algorithms designed for a single stego image can only locate the steganography payload with low accuracy because it is very difficult to precisely estimate the cover of the given stego image and about half of the stego elements are still unchanged [18]. However, for the convenience of communication, many communication participants use the same key in a certain period of time and limit the embedding ratio. At this point, if they use multiple images with the same size to embed a large amount of data, the investigator may possess a number of stego images each containing payload at the same locations. Under such a scenario, in 2008, Ker [19] firstly proposed a payload location algorithm based on weighted stegoimage (WS) residuals for least significant bit (LSB) replacement. After that, many payload location algorithms have been proposed for spatial image steganography under this condition. Chiew and Pieprzyk [20] modified Ker’s algorithm to locate the payload of binary image replacement steganography under the same condition. Ker and Lubenko [21] proposed a payload location algorithm for LSB matching, which filters the horizontal, vertical, and diagonal wavelet subbands of stego images by Wiener filter, and locates the stego pixel positions according to the absolute sum of the wavelet residuals in the same positions of multiple images embedded messages into the same positions. Quach [22, 23] proposed several payload location algorithms for LSB replacement and LSB matching, which employ the Viterbi decoding algorithm or Quadratic PseudoBinary Optimization (QPBO) algorithm to find the optimal estimate of the cover image, and compute the residuals between the estimated cover images and the stego images to locate the payload. Gui et al. [24] proposed a payload location algorithm for LSB matching steganography by fusing the mean of 4 neighborhood pixels and 8 residuals computed along 8 different directions by the algorithm proposed by Quach [22]. Liu et al. [25] proposed a payload location algorithm for embedding messages into the spatial images subjected to JPEG compression by LSB replacement or LSB matching, which estimates the cover images by JPEG recompressing the stego images and decompressing the recompressed versions. Yang et al. [15] proved the properties of the optimal stego subset of the multiple least significant bits (MLSB) steganography, then proposed a payload location algorithm and a stego key recovery algorithm based on the optimal stego subset. Sun et al. [26] proposed a payload location algorithm base on a tailored deep neural network (DNN) equipped with the improved feature named the “mean square of adjacency pixel difference.”
The above algorithms can locate the payload of LSB replacement, LSB matching, and MLSB replacement steganography with high accuracy and even can be used to estimate groups in group parity steganography or extract the hidden message for some special cases. However, they cannot work for the steganography algorithms with JPEG image as cover.
When the messages are embedded into the JPEG images, recently, the authors [27] proposed a payload location method based on cofrequency subimage filtering for a category of pseudorandom scrambled JPEG image steganography. The accuracy of this payload location method is influenced by the fidelity of the estimated cover images and can be improved if a more precise estimator can be designed.
Activated by the optimal cover estimation method proposed by Quach in [22] for spatial image steganography, this paper proposes a payload location method for JPEG image steganography based on the optimal estimation of cover cofrequency subimage. Instead of directly applying the maximum a posterior (MAP) probability algorithm to the given stego spatial image to estimate the cover spatial image by the method in [22], the proposed method divides the stego JPEG image into 64 cofrequency subimages, then applies the MAP algorithm to estimate the optimal cover cofrequency subimages, and combines them to obtain the optimal cover JPEG image. This makes use of the correlation between the coefficients in the same position of adjacent blocks with a size of 8 × 8.
The structure of this paper is as follows: Section 2 briefly introduces the random JPEG image steganography targeted in this paper. Section 3 proposes the payload location method based on the optimal estimation of cover cofrequency subimage. Section 4 gives a specific payload location algorithm for F5 steganography. Section 5 presents the experimental results and the discussions. Finally, the paper is summarized in Section 6.
2 Related work—Pseudorandom JPEG image steganography
In order to improve the security of JPEG image steganography, the steganographer often embeds secret messages into the quantized DCT coefficients scrambled pseudorandomly. And because there are a lot of quantized DCT coefficients with value of 0 in JPEG images, if the steganographer embeds messages into these coefficients, the doubtful artificial clue will be found by steganalyzer. Thus, many JPEG image steganography methods do not embed message bits into these coefficients and do not embed message bits into the coefficients whose values would be changed to be 0. These JPEG image steganography methods can be described as follows.
Input: a cover JPEG image C = c_{1}c_{2}…c_{N}, a secret message bit sequence M = m_{1}m_{2}…m_{L} and a stego key K.
Output: a stego JPEG image.
Steps:

1.
Scramble the quantized DCT coefficients in the cover JPEG image C according to the stego key K, to generate the scrambled coefficient sequence C^{′} = Scr(C, K), where \( {C}^{\prime }={c}_1^{\prime }{c}_2^{\prime}\dots {c}_N^{\prime } \) denotes the scrambled coefficient sequence and Scr(C, K) is the scrambling function.

2.
Embed the secret message bit sequence M into the scrambled coefficient sequence C^{′}.

2.1.
Assign the initial index of the secret message bit as 1, viz. i = 1, and assign the initial index of the scrambled coefficient as 1, viz. j = 1.

2.2.
Take the ith message bit m_{i} from the secret message bit sequence M.

2.3.
Take the jth coefficient \( {c}_j^{\prime } \) from the scrambled coefficient sequence C^{′}.

2.4.
If the value of coefficient \( {c}_j^{\prime } \) cannot carry a message, for example, the value of coefficient \( {c}_j^{\prime } \) is 0, go to step 2.8.

2.5.
Embed the ith message bit into the jth coefficient \( {c}_j^{\prime } \).

2.6.
If the embedding changes the value of coefficient \( {c}_j^{\prime } \) to be the value which cannot carry a message, for example, F5 steganography changes the coefficient value 1 to be 0, assign the index of the scrambled coefficient as j + 1, viz. j = j + 1. If j > N, return 0, otherwise go to step 2.3.

2.7.
Assign the index of the secret message bit as i + 1, viz. i = i + 1. If i > L, go to step 3.

2.8.
Assign the index of the scrambled coefficient as j + 1, viz. j = j + 1. If j > N, return 0, otherwise go to step 2.2.

2.1.

3.
Inverse scramble the coefficient sequence after embedding according to the stego key K;

4.
Encode the obtained coefficient sequence to a stego JPEG image, and return the generate stego JPEG image.
3 Methods—Payload location based on optimal estimation of cover cofrequency subimage
3.1 Principle
When the secret messages are embedded into the pseudorandomly scrambled coefficients as described in Section 2, if the investigator possesses T stego images S_{1}, S_{2}, ⋯, S_{T} embedded along the same embedding path, then either of the following two cases may happen to the coefficients S_{1}(i, j), S_{2}(i, j), …, S_{T}(i, j) in the same position (i, j) of T stego images:

1)
If the position (i, j) is a stego position, the steganographer will determine whether to embed the message bit into the coefficient in this position according to whether the coefficient is available. Thus, any coefficient of S_{1}(i, j), S_{2}(i, j), …, S_{T}(i, j) is either an unavailable coefficient or a stego coefficient containing a message bit.

2)
If the position (i, j) is a nonstego position, the steganographer will not embed the message bit into the coefficient in this position regardless of whether the coefficient is available. Thus, no coefficients of S_{1}(i, j), S_{2}(i, j), …, S_{T}(i, j) contain a message bit.
Let C_{1}, C_{2}, …, C_{T} denote the corresponding cover images of the stego images S_{1}, S_{2}, …, S_{T}. A residual r_{t}(i, j) of the coefficient in the position (i, j) of the tth stego image is defined as
Let \( \overline{r}\left(i,j\right) \) denote the mean of all r_{t}(i, j) over T stego images in the position (i, j).
If the position (i, j) is a nonstego position, \( \overline{r}\left(i,j\right) \) must equal to 0, viz. \( \overline{r}\left(i,j\right)=0 \). If the position (i, j) is a stego position, \( \overline{r}\left(i,j\right) \) must be larger than or equal to 0, viz. \( \overline{r}\left(i,j\right)\ge 0 \), where the equal sign only holds in the case of that all of the coefficients C_{1}(i, j), C_{2}(i, j),…, C_{T}(i, j) are not modified. When one possesses enough stego images, the probability that none of the coefficients C_{1}(i, j), C_{2}(i, j),…, C_{T}(i, j) is modified is small. Thus, the investigator should be able to distinguish the stego positions from the nonstego positions according to the means of residuals if he can obtain the cover images.
However, the investigator often cannot know the cover JPEG images. In this case, if the investigator can estimate the cover images, which are denoted by \( {\hat{C}}_1,{\hat{C}}_2,\dots, {\hat{C}}_T \), he can compute the mean of the estimated residuals in the same position (i, j) of different stego images as follows:
If the investigator possesses enough stego images embedded along the same path and can estimate the covers of them accurately enough, he may also be able to distinguish the stego positions from the nonstego positions with a success rate higher than a random guess based on the averaged estimated residuals as follows:
where f(i, j) = 1 denote that the position (i, j) is determined as a stego position, f(i, j) = 0 denote the position (i, j) is determined as a nonstego position, and Thr is a decision threshold.
Certainly, the more accurately the cover JPEG images are estimated, the higher the accuracy of payload location is. Therefore, in the following subsection of this section, a method is proposed to estimate the optimal cover cofrequency subimages, then combine them to estimate the cover JPEG image.
3.2 Optimal cover JPEG image estimation
In [22], Quach et al. considered the strong correlation between neighboring pixels of spatial image and used the maximum a posterior (MAP) probability algorithm to estimate the optimal cover image corresponding to the stego image of LSB replacement and LSB matching steganography, which was used to locate the hidden information of LSB replacement and LSB matching steganography. In JPEG compression, the DCT transformation of pixel values greatly reduces the correlation between adjacent coefficients. And in order to improve the efficiency of JPEG compression, the DCT transformation is performed on each nonoverlapping pixel block with a size of 8 × 8. Since the coefficients in the same position represent the magnitude of energy in the same frequency and the adjacent blocks in an image still have strong similarity, the coefficients in the same position of adjacent blocks still have a strong correlation. According to the property, this section will use the same method in [27] to divide the given JPEG images into 64 cofrequency subimages, then use the maximum a posterior probability algorithm to estimate the optimal cover cofrequency subimages, and combine them to get the optimal estimation of cover JPEG image.
3.2.1 Markov model of cofrequency subimage
Let \( {S}_t^d \) and \( {C}_t^d \) denote the cofrequency subimages composed of the dth quantized DCT coefficients in all 8 × 8 blocks of the tth stego image and its cover image, d = 1, 2, …, 64. In a statistical sense, the optimal estimation of cover cofrequency subimages corresponding to \( {S}_t^d \) should be the cover cofrequency subimage estimation \( {\hat{C}}_t^d \) with the maximum a posterior probability, that is
Then, the optimal cover cofrequency subimage estimation is transformed into a problem of maximum a posterior probability estimation.
Similar to [22], the following two assumptions are set:
where k is a given positive integer. Eq. (5) indicates that each quantized DCT coefficient in the stego cofrequency subimages is only related to the corresponding quantized DCT coefficient in the cover cofrequency subimages, while Eq. (6) indicates that the cover cofrequency subimage \( {C}_t^d \) is modeled with a korder Markov model.
For a given steganography algorithm, one can calculate the probabilities that the quantized DCT coefficient value changes to different possible values under a specific embedding rate α, viz. the transition probability in assumption (5). Besides, the prior probability in (6) can be computed from a large number of cover images.
After dividing all quantized DCT coefficients into 64 cofrequency subimages, each subimage is scanned by four modes as shown in Fig. 1 to calculate the cooccurrence matrices of the adjacent elements.
In JPEG image, the distributions of coefficient values in different cofrequency subimages show obvious differences. As shown in Fig. 2, the absolute values of coefficients in the low frequencies (corresponding to the upper left positions) are usually larger and equal to zero with the lowest probabilities, and most of the absolute values of coefficients in the high frequencies (corresponding to the lower right positions) equal to zero. Figure 3 presents the frequencies of zero coefficient in the different subimages, where 10,000 images with a size of 512 × 512 in Bossbase 1.01 (http://agents.fel.cvut.cz/stegodata/) are JPEG compressed with a quality factor of 75. The abscissa is the index of the position in the 8 × 8 block from left to right and top to bottom. It can be seen that the relative frequencies of zero coefficient in the subimages corresponding to the lower right positions are close to 1.
3.2.2 Optimal cover JPEG image estimation based on firstorder Markov model
In theory, we should compute the probabilities for all possible covers and search the cover which satisfies Eq. (4). But there are too many possible coefficient values in the cover image to search the whole possible space. Fortunately, the cofrequency subimage can be modeled by the hidden Markov model, and the Viterbi algorithm is a common method to solve the problem of the hidden Markov model. It has been used in cover image estimation of spatial steganography such as LSB replacement and LSB matching in [22]. Therefore, The Viterbi algorithm will also be adopted to search the optimal cover cofrequency subimage. The Viterbi algorithm first computes the scores of the possible values of the first cover element as follows:
Then, the scores of the possible values of the subsequent cover elements are computed as follows:
where c_{k, i} is possible value of the kth cover element in the ith image.
Take a stego cofrequency subimage with four quantized DCT coefficients S = (2, 0, −1, 1) of the typical F5 steganography as example, where the embedding ratio is 0.5. According to the embedding rule of F5 steganography, the possible values of the four cover coefficients are c_{1} ∈ {2, 3}, c_{2} ∈ {−1, 0, 1}, c_{3} ∈ {−1, −2}, and c_{4} ∈ {1, 2}. Figure 4 shows the trellis for Viterbi algorithm, which takes the possible values of four cover coefficients as nodes. The Viterbi algorithm first computes the scores of nodes in the first column of the trellis, where the value of p(c_{1}) can be obtained by statistics of a large number of cover JPEG images. For ease of understanding, it is assumed that the values of p(c_{1}) are as shown in the second column of Table 1. When the embedding ratio of F5 steganography is q, the coefficient value transition probability of F5 steganography is as follows:
Then the scores of the subsequent nodes are computed in sequence by Eq. (8), and each node is connected with the previous node which maximizes its score. The values of p(c_{k} c_{k − 1}) also can be obtained by statistics of a large number of cover JPEG images. It is assumed that the values of p(c_{k} c_{k − 1}) are as shown in the last column of Table 1.
Finally, take the coefficient values in the path ending at the node with the largest score in the last column as the optimal estimation of the cover coefficients, as shown by the gray node in Fig. 4. It can be seen that when the embedding ratio is 0.5, the optimal estimation of the cover coefficient sequence of S = (2, 0, −1, 1) is \( \hat{\mathrm{c}}=\left(3,1,2,2\right) \).
After the optimal estimation of each cover cofrequency subimage is obtained by the Viterbi algorithm, one can place the coefficients of all estimated cover cofrequency subimages at the original positions of them to combine the optimal estimation of the cover JPEG image. The whole process is shown in Fig. 5, which is described in Algorithm 1.
In theory, each cover cofrequency subimage may be estimated more precisely by the firstorder Markov model in the corresponding frequency. However, in many frequencies, there are a large number of coefficients with value of 0 which result in that the statistical significance of nonzero coefficient is not significant. Thus, in follows the firstorder Markov model merged over different positions is used to estimate the cover cofrequency subimages.
4 Payload location algorithm for F5 steganography without Matrix Encoding
The F5 steganography algorithm improves F4 by using shuffling. In F5 steganography, the positive odd and negative even represent the bit 1, while the positive even and negative odd represent the bit 0, and the DCT coefficients with value of 0 and DC coefficients do not carry secret information. The coefficient value transition probability of F5 steganography is shown by (9). When T stego JPEG images of F5 steganography are given, we can adopt the existing quantitative steganalysis algorithms to estimate the embedding ratios and then use the proposed Algorithm 1 in Section 3 to estimate the corresponding cover JPEG images. For each given stego JPEG image, we can scan it by 4 different modes as shown in Fig. 1, and then 4 estimated cover JPEG images can be obtained by Algorithm 1.
After that, the residuals between the given stego image and the estimated cover JPEG images are computed as follows:
which is slightly different from the previous residual calculation Eq. (1). For each position, 4T residuals can be computed from the given T stego JPEG images and 4T estimated cover JPEG images by (10), and then be averaged. The averaged value will be used to determine whether this position is a stego position. The detailed steps of the payload location for F5 steganography are given in Algorithm 2.
5 Results and discussion
5.1 Experimental setup
In total, 10,000 PGM images with a size of 512 × 512 were downloaded from the BOSSbase1.01 and converted to cover JPEG images with a quality factor of 75. Nine thousand images were randomly selected from the generated cover JPEG images to count the firstorder Markov model of cover cofrequency subimage. The remaining 1000 images were used to test the performance of the proposed algorithm. A pseudorandom path was generated by scrambling the integer sequence 1, 2,…, 512 × 512. Then along the generated path, the pseudorandom message bits were embedded into the remaining 1000 images by F5 steganography (without matrix encoding) with ratio q = 0.5.
5.2 Markov model selection
From Algorithm 1 and 2, it can be found that the payload location accuracy is highly affected by the adopted firstorder Markov model. In Section 3, we suggest to merge the Markov models over different frequencies to estimate the cover cofrequency subimage more precisely. Thus, we tried to merge proper Markov models.
Firstly, the 64 Markov models m_{1}…m_{64} counted from subimages corresponding to 64 positions in 8 × 8 matrix were applied to estimate the cover JPEG images separately, and the Markov model m_{i} with the highest payload location accuracy was selected. Then, each of the remaining 63 models was merged to m_{i} to obtain 63 new merged modes m_{i1}…m_{i63}, and the merged Markov model m_{ij} with the highest payload location accuracy was selected. This operation was repeated until all models were merged. The merged model with the highest payload location accuracy was selected as the final model.
One thousand test stego JPEG images with embedding ratio 0.5 were used to select the proper merged Markov model. Table 2 presents the location correctness of each cofrequency subimages with the single corresponding Markov model, namely, 64 cofrequency subimage models are used for the corresponding subimages respectively. Table 3 shows the results when the optimal merged Markov model was used.
In Tables 2 and 3, the correctness in the most upper left is not shown because the DC coefficients are not changed by F5 steganography. Comparing Table 2 with 3, we can see that for most positions, the location accuracy by using the optimal merged Markov model is much higher than that by using the individual model. Especially, the algorithm with the optimal merged Markov model can rightly distinguish the stego positions in low frequencies with accuracy close to 90%, even close to 95%. For the highfrequency positions, because there are very few available coefficients, it is still hard to distinguish the stego positions.
5.3 Performance analysis of location proposed algorithm for F5 steganography
Figure 6 shows the payload location accuracy of MAPF5 with the optimal merged Markov model for different numbers of stego images when the embedding ratio is 0.5. It can be seen that the more the number of stego images, the higher the accuracy. As the number of images increases, the fluctuation of the residual means becomes smaller, and the residual means are closer to the change caused by information embedding. Therefore, the number of stego images is very important for locating the stego positions.
Figure 7 compares the accuracies of the proposed algorithm and the payload location algorithm based on cofrequency subimage wavelet filtering (CSWF5 )[27]. The 1000 stego images are generated with the same embedding path and the embedding ratio of 0.5. In the upper left corner of 8 × 8 block where the number of the 0 coefficient is relatively small, MAPF5 obtains better results than CSWF5. In practice, the results of the two payload location algorithms can be further combined.
6 Conclusion
This paper proposes a payload location method based on optimal estimation of cover cofrequency subimage. The proposed method divides each given stego JPEG image into 64 cofrequency subimages, then estimates the optimal cover JPEG image by applying the maximum a posterior probability algorithm to the cofrequency subimages, and finally determines the stego positions according to the averaged residuals between given multiple stego images embedded along the same path and the estimated cover images. The proposed method is applied to the payload location for F5 steganography without matrix encoding and the experimental results show that the proposed algorithm can locate the stego positions with higher accuracy than prior works.
However, the proposed payload location method cannot work for the modern adaptive JPEG image steganography, JUNIWARD, UERD, and GUED. Therefore, in future, we will try to adapted the proposed cover JPEG image estimation method for the modern adaptive JPEG steganography. Besides, we will also try to improve the performance by using unsupervised learning to cluster the image blocks with similar contents [28].
Availability of data and materials
Please contact the author for data requests.
Abbreviations
 JPEG:

Joint photographic experts group
 LSB:

Least significant bit
 QPBO:

Quadratic pseudobinary optimization
 MLSB:

Multiple least significant bits
 DNN:

Deep neural network
 DCT:

Discrete cosine transform
 MAP:

Maximum a posterior
 DC:

Direct current
 CSW:

Cofrequency subimage wavelet filtering
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Acknowledgements
Thanks to all those who have suggested and given guidance for this article
Funding
This work is supported by the National Natural Science Foundation of China (Grant Nos. 61872448, 61772549, U1804263) and Natural Science Basic Research Plan in Shanxi Province of China (No. 2018JM6017).
Author information
Authors and Affiliations
Contributions
All authors took part in the discussion of the work described in this paper. The author Jie Wang carried out the experiments of the paper and wrote the paper. The author Chunfang Yang designed the algorithms of this work and revised the paper. The author Ma Zhu helped conduct the experiments. All authors read and approved the final manuscript.
Authors’ information
Chunfang Yang is currently an associate professor of Zhengzhou Science and Technology Institute. He received his MA and PhD degrees in computer science and technology from Zhengzhou Information Science and Technology Institute, Zhengzhou, China, in 2008 and 2012, respectively. His research interest includes image steganography and steganalysis technique.
Jie Wang is currently a master degree candidate of Zhengzhou Science and Technology Institute. His research interest includes image steganography and steganalysis technique.
Ma Zhu is currently an associate professor of Zhengzhou Science and Technology Institute. She received her MA degree in computer science and technology from University of Electronic Science and Technology of China, Chengdu, China, in 2007. Her research interest includes computer network and multimedia security technique.
Xiaofeng Song is currently an associate professor of School of Information and Communication, National University of Defense Technology, Xi’an, China. He received his MA degree in computer science and technology from Xidian University, Xi’an, China, in 2009 and received his PhD degrees in computer science and technology from Zhengzhou Information Science and Technology Institute, Zhengzhou, China, in 2016. His research interest includes image steganography and steganalysis technique.
Yuan Liu is currently an associate professor of Huanghe S & T University, Zhengzhou, China. She received her MA degree in computer science and technology from Harbin Institute of Technology, Harbin, China, in 1992 and received her PhD degrees in computer science and technology from Zhengzhou Information Science and Technology Institute, Zhengzhou, China, in 2005. Her research interest includes information security technique.
Yuemeng Lian is currently an engineer of Henan Huizhi Scientific & Technical Development Co., Ltd. Zhengzhou, China. Her research interest includes information security technique.
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Wang, J., Yang, C., Zhu, M. et al. JPEG image steganography payload location based on optimal estimation of cover cofrequency subimage. J Image Video Proc. 2021, 1 (2021). https://doi.org/10.1186/s13640020005422
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DOI: https://doi.org/10.1186/s13640020005422