Model-Based Non-Independent Distortion Cost Design for Effective JPEG Steganography

Venue: acmmm_org_ACMMM_2024_Conference
Published: 2024-04-09 11:34:48

Abstract: Recent achievements have shown that model-based steganographic schemes hold promise for better security than heuristic-based ones, as they can provide theoretical guarantees on secure steganography under a given statistical model. However, it remains a challenge to exploit the correlations between DCT coefficients for secure steganography in practical scenarios where only a single compressed JPEG image is available. To cope with this, we propose a novel model-based steganographic scheme using the Conditional Random Field (CRF) model with four-element cross-neighborhood to capture the dependencies among DCT coefficients for JPEG steganography with symmetric embedding. Specifically, the proposed CRF model is characterized by the delicately designed energy function, which is defined as the weighted sum of a series of unary and pairwise potentials, where the potentials associated with the statistical detectability of steganography are formulated as the KL divergence between the statistical distributions of cover and stego. By optimizing the constructed energy function with the given payload constraint, the non-independent distortion cost corresponding to the least detectability can be accordingly obtained. Extensive experimental results validate the effectiveness of our proposed scheme, especially outperforming the previous independent art J-MiPOD.

Main paper: 
 introduction: 
 Digital steganography [7] is an important branch of information hiding that utilizes the data redundancy inherent in various types of digital media, as well as the physiological and psychological characteristics of human perception organs, to embed secret messages (a.k.a., payload) into the public digital media with certain coding, and then transmit the digital media containing the secret messages to achieve covert communication. As opposed to traditional encryption, steganography is concerned with concealing the communication behavior, making the communication process imperceptible or even undetectable.As one of the most commonly used digital media, digital image has always been the popular cover choice for steganography, and has yielded numerous impressive results in recent years [26, 31] . Leaving aside the current new paradigm of coverless generative steganography, the content-adaptive steganography based on the distortion minimization framework [5] has been the mainstream steganographic paradigm. And with the advent of Syndrome-Trellis Codes (STCs) that can closely achieve the rate-distortion bound [6] , the main effort in content-adaptive image steganography has focused on designing efficient distortion cost functions. To ensure the security of covert communication, it is required that the distortion cost function be designed to least possibly change the statistical distributions of the cover source for a given payload.For the time being, the design of distortion cost functions can be divided into three categories, i.e., heuristic-based, deep learning (DL)-based, and model-based. The first type relies on heuristic principles, whose guideline is to assign high costs to smooth areas and low costs to textured regions, based on the consensus that textured regions are difficult to model while smooth areas are easy, typically schemes are WOW [14] , S-UNIWARD [16] , HILL [22] in the spatial domain and UERD [13] , GUED [30] , J-UNIWARD [16] , DCDT [29] in the JPEG domain. Despite the impressive achievements of these empirical schemes, they are limited from the methodological point of view, since they can not provide accurate explanations as to why the given cost functions work. The second one, DL-based approaches, which have achieved excellent security performance, are generally designed as an automatic cost learning framework via utilizing the architectures of generative adversarial networks and reinforcement learning, typical schemes are UT-GAN [39] , JS-GAN [40] , SPAR-RL [36] , JEC-RL [37] , PICO-RL [23] , Steg-GMAN [18] . These methods use dynamic feedback from the discriminator (i.e., steganalyzer) to iteratively facilitate the generator to output better distortion cost to defeat the given "adversary". However, such methods are end-to-end and lack mathematical explanation as well. In addition, they rely heavily on well-designed networks and require both parties to form a Nash equilibrium. What's more, they are computationally demanding and data-dependent. The last type is the model-based, which builds the steganographic distortion cost based on mathematical principles, including minimizing the difference of model distribution between cover and stego (e.g., MG [8] , MVGG [25] , GMRF [28] ), and minimizing the power of optimal detector (e.g., MiPOD [24] , J-MiPOD [3] ). Not only does this type of scheme sound more mathematical, but they can provide a complete mathematical explanation, and for this reason, this paper will continue to explore such type in depth.MG [8] is the first practical attempt to design the distortion function based on the mathematical principle. The distortion costs are determined with the embedding change probabilities, which are obtained by minimizing the Kullback-Leibler (KL) divergence between the statistical distributions of cover and stego images, when the cover image is modeled as a sequence of independent multivariate Gaussian random variables with heterogeneous local variances. And the security performance is further improved in MVGG [25] by incorporating an improved variance estimator and adopting the pentanary symmetric embedding with a thicker-tail model. Developing upon these two works, an alternative approach, i.e., Minimizing the Power of Optimal Detector (MiPOD) [24] , has been proposed subsequently, with a performance close to the state-of-the-art compared to S-UNIWARD and HILL. In MiPOD, a closed-form expression for statistical detectability is also provided, allowing the design of so-called detectability-limited sender, i.e., controls the size of secure payload for a given image to not exceed a target detectability level. As for the acquisition of the embedding change probabilities associated with the distortion costs, it is derived by minimizing the power of the optimal likelihood ratio test (LRT). In light of recent research [9, 33] showing that DCT coefficients can be modeled individually, Cogranne et al. follow MiPOD and build a heteroscedastic model for the DCT coefficients, i.e., modeling DCT coefficients as independent but not identically distributed multivariate Gaussians. This extension of MiPOD for the JPEG images, which we referred to as J-MiPOD [3] , is also competitive as compared to the state-of-the-art J-UNIWARD.Despite the theoretical guarantees of security of the model-based MiPOD and J-MiPOD, their assumption of independence of the pixels/DCT coefficients is not very accurate. In this regard, some works have focused on developing more accurate possible statistical models of images by taking the correlations between pixels/DCT coefficients carefully into account. For example, [28] introduces a Gaussian Markov Random Field (GMRF) model with four-element cross-neighborhood to capture the dependencies among spatially adjacent (i.e., horizontal and vertical) pixels, and attains the embedding change probability for each pixel by minimizing the KL divergence in terms of a series of pairwise cliques between cover and stego within the GMRF model, which achieves superior security performance than the previous independent MiPOD. As for the exploration of the correlations between DCT coefficients for JPEG steganography, some recent works [11, 12] based on the complete and exact knowledge of the acquisition and processing pipeline (a.k.a., side-information in RAW image) build more accurate statistical models for JPEG images. Although it can provide significant performance improvements, it is too conditionally demanding and often difficult to achieve in practice. Therefore, giving solutions in the broader context of not providing side-information of images and using only a single compressed JPEG image deserves further investigation.To address this issue, in this paper, we propose a novel modelbased steganographic scheme, using the pairwise Conditional Random Field (CRF) model with four-element cross-neighborhood to capture the dependencies among DCT coefficients for JPEG steganography, denoted as CRF. Note that for the sake of simplicity, similarly to the previous art [28] , only ternary symmetric embedding is investigated in this paper, and the impact brought by the direction of embedding modification is not considered for the time being. Different from the construction of the previous art [28] , our proposed CRF model is not built on the spatially adjacent DCT coefficients, but rather on the same DCT mode in adjacent DCT blocks, since the correlation between spatially adjacent DCT coefficients inside one DCT block can hardly be obtained for a single compressed JPEG image. The proposed CRF model with four-element cross-neighborhood is characterized by the delicately designed energy function, which is defined as the weighted sum of a series of unary and pairwise potentials since the effects of each unary and pairwise potential on steganographic security are different. In our proposed scheme, the unary and pairwise potentials associated with the statistical detectability of steganography are defined as the KL divergence between the statistical distributions of cover and stego. With the aid of the proposed CRF model, secure JPEG image steganography is finally formulated as the minimization of the energy function, and the optimal embedding change probabilities of each DCT coefficient are attained by minimizing the total potentials (i.e., KL divergence) under the payload constraint. With the given embedding change probability, the corresponding steganographic distortion cost can be easily determined. Extensive experiments are carried out to verify the effectiveness of the proposed scheme (known as CRF) using CC-JRM [19] , DCTR [15] , GFR [27] and SCA-GFR [4] , on BOSSbase [1] and ALASKAv2 [2] database. Numerous results show that the proposed method can not only surpass the SOAT heuristic-based UERD and JUNIWARD, but also outperform the advanced model-based J-MiPOD by a clear margin. 
preliminaries 2.1 notations and basic concepts: 
 In this paper, matrixes and vectors are represented by capital and lowercase boldface symbols, respectively, sets are denoted by the calligraphic font, and elements within a matrix are indicated by italic fonts with subscript indices. A T is the transpose of matrix A, and Pr is the probability measure.For ease of presentation, in this paper, the JPEG gray-scale cover and stego images with size ℎ × 𝑤 are denoted as X = 𝑥 𝑘,𝑙 EQUATIONAs stated above, this paper focuses on the broader context where there is only a single JPEG image without any side-information about the cover source. In exploring the correlations between DCT coefficients for JPEG steganography, we will only consider the correlations of DCT coefficients from the same mode in horizontally or vertically adjacent blocks, since its correlations are the most significant, as evidenced by [35] . In this way, as illustrated in Figure 1 (𝐵 𝑚,𝑛 is the (𝑚, 𝑛)-th 8 × 8 DCT block), the JPEG image with size ℎ ×𝑤 can be split into 64 sub-images with size ℎ 8 × 𝑤 8 , depending on the DCT modes. Due to the formulation of considered correlations being identically constituted in each sub-image, the derivation will be performed in this paper using only one arbitrary sub-image for brevity.Mode DC Mode (7, 7) Mode(0,1) Other modes Without loss of generality, we denote the (𝑘, 𝑙)-th sub-image by S, omitting the index (𝑘, 𝑙) in the subsequent theoretical derivation for the sake of simplicity and readability. Specifically, to incorporate the considered correlations, we first define a Conditional Random Field (S, 𝜷) on an undirected graph S = (V, E). S is the observed sub-image, 𝜷 is a set of embedding change probabilities, and each 𝛽 𝑚,𝑛 ∈ 𝜷 corresponds to the one of (𝑚, 𝑛)-th DCT coefficient 𝑥 𝑚,𝑛 in S. As shown in Figure 1 , S consists of numerous nodes and edges, wherein V = {(𝑚, 𝑛)|1 ≤𝑚 ≤ℎ/8, 1 ≤ 𝑛 ≤𝑤/8} is a set that collects the node index for the grid-like structure of DCT coefficients, E is a set that collects the edges connecting horizontally or vertically adjacent nodes. Since only the correlations of horizontally or vertically adjacent nodes are considered in this paper, thus we will use the four-element cross-neighborhood N 4 (see Figure 1 ) to characterize S. Inside this four-element cross-neighborhood system, there are two types of cliques, i.e., one-element (unary) cliques and twoelement (pairwise) cliques. Formally, according to the Hammersley Clifford theorem [21] , the posterior distribution Pr(𝜷 | S) of any possible assignment to 𝜷 conditioned on S is a Gibbs distribution and defined as:1, 1 m n B   1, 1 m n B   1, 1 m n B   1, 1 m n B   ... ..EQUATIONwhere C is the set of all unary and pairwise cliques in S, 𝜓 𝑐 (𝜷 𝑐 | S) is the potential function defined on clique 𝑐, and Λ is a normalizing constant known as the partition function. In a CRF model, the corresponding energy function 𝐸 (𝜷 | S) is defined as:EQUATIONand the optimal assignment 𝜷 * , i.e. the maximum posterior (MAP) probability of the random field B, is given by:EQUATIONIt manifests that the problem of finding the optimal embedding probabilities for secure JPEG image steganography can be formulated as the minimization of a well-designed energy function.Concretely, the energy function in Eq. ( 4) corresponding to the proposed CRF model with four-element cross-neighborhood can be formulated as the sum of two kinds of local potentials 𝜑 𝑚,𝑛 and 𝜓 𝑐 :EQUATIONHerein, 𝜑 𝑚,𝑛 is referred to as the unitary potential, employed to characterize the detectability of embedding messages in individual DCT coefficient 𝑥 𝑚,𝑛 in V with embedding change probability 𝛽 𝑚,𝑛 , and 𝜓 𝑐 is called the pairwise potential, used to characterize the detectability of embedding messages in the correlated DCT coefficient pair 𝒙 𝑐 in E with embedding change probabilities 𝜷 𝑐 . In this way, maximizing the security of JPEG image steganography is then formulated as minimizing the potentials in Eq. ( 6).As for the characterization of steganographic detectability, there is a well-recognized metric, supported by a rigorous mathematical theory, namely KL divergence, which has already been used in recent works [8, 10, 28] . It shows that the smaller the KL divergence between the statistical distributions of cover and stego, the higher the security of steganography, which is exactly consistent with the relations between potentials in Eq. ( 6) and steganographic security. Based on this observation, in this paper, we will formulate both the unary and pairwise potentials in Eq. ( 6) as KL divergence between the statistical distributions of cover and stego. 
the proposed jpeg steganographic distortion cost design using crf model 3.1 overview: 
 For considering the correlations between DCT coefficients for JPEG steganography with symmetric embedding, we propose a Conditional Random Field (CRF) model with four-element cross-neighborhood to capture the correlation between DCT coefficients for nonindependent steganographic distortion cost design, which has not been considered in the prior art J-MiPOD [3] . With the aid of the proposed model, the JPEG steganographic distortion cost design is then formulated as minimizing the total potentials (all unary and pairwise) under the payload constraint. In this section, we first develop the statistical models for both JPEG cover and stego images, which are used to characterize the correlations between DCT coefficients. Next, define the unary and pairwise potentials using KL divergence between the statistical distributions of cover and stego. Finally, formulate the distortion cost design as the minimization of total potentials and provide an iterative optimization approach to obtain the optimal embedding probabilities, associated with distortion cost. ] in S as an instance in the subsequent derivation.As stated in [12, 32, 34, 35] , the DCT coefficients in a JPEG image, which are typically corrupted by Gaussian noise developed from the RAW image acquisition up to the JPEG compression, are feasible to be modeled as multivariate Gaussian random variables. Based on this insight, we move one step forward by modeling pairwise coefficients 𝒙 𝑐 in a single compressed JPEG image as a bivariate Gaussian random variable quantized to integer. Under the assumption of fine quantization limit (refer to [8, 25, 28] ), the Probability Mass Function (PMF) of 𝒙 𝑐 is defined as P Σ 𝑐 = (𝑝 (𝑡 1 , 𝑡 2 )) 𝑡 1 ,𝑡 2 ∈ I with:EQUATIONwhere 𝒕 𝑐 = [𝑡 1 , 𝑡 2 ] and 𝝁 𝑐 = [𝜇 𝑚,𝑛 , 𝜇 𝑚,𝑛+1 ] are the realization and expectation of 𝒙 𝑐 , respectively. Σ 𝑐 and Σ -1 𝑐 are the corresponding 2 × 2 covariance matrix and its inverse matrix, respectively:EQUATIONEQUATIONHere, 𝜚 𝑐 is the correlation coefficient between 𝑥 𝑚,𝑛 and 𝑥 𝑚,𝑛+1 , 𝜎 2 𝑚,𝑛 and 𝜎 2 𝑚,𝑛+1 are the variance of 𝑥 𝑚,𝑛 and 𝑥 𝑚,𝑛+1 , respectively. Note that, all these parameters have to be estimated from a single JPEG image, which will be described later in Section IV.In addition, according to [38] , for a multivariate Gaussian random variable, its marginal variables are also Gaussian distributed, i.e., 𝑥 𝑚,𝑛 still follows a Gaussian distribution. Thereby, the PMF of 𝑥 𝑚,𝑛 can be given by P 𝜎 𝑚,𝑛 = (𝑝 (𝑡 1 )) 𝑡 1 ∈ I with EQUATIONThen, the PMF of 𝒚 𝑐 , i.e., Q Σ 𝑐 ,𝜷 𝑐 = 𝑞 𝜷𝑐 (𝑡 1 , 𝑡 2 ) 𝑡 1 ,𝑡 2 ∈ I can be accordingly attained withEQUATIONSimilarly, the PMF of the corresponding stego coefficient 𝑦 𝑚,𝑛 , i.e., 𝛽 𝑚,𝑛 = 𝑞 𝛽𝑚,𝑛 (𝑡 1 ) 𝑡 1 ∈ I can be obtained with 𝑞 𝛽𝑚,𝑛 (𝑡 1 ) = Pr(𝑦 𝑚,𝑛 =𝑡 1 ) EQUATIONis the steganographic Fisher information, indicating the impact of information hiding on the cover model. The derivation of Eq. ( 14) is based on the conclusion that for small embedding change probabilities, the KL divergence is wellapproximated by its leading quadratic term of Taylor expansion, the detailed derivation process can refer to the prior arts [8, 28] . Eventually, in conjunction with Eq. ( 14) and Eq. ( 15), the unary potential 𝜑 𝑚,𝑛 (𝛽 EQUATIONwhere I 𝑐 is the 2 × 2 Fisher information matrix (FIM):EQUATIONEQUATIONin whichEQUATIONEQUATIONEQUATION)One can refer to the prior art [28] for the detailed derivations of FIM, which is dedicated to spatial image steganography. Finally, combining Eq. ( 17) -Eq. ( 21), the pairwise potential 𝜓 𝑐 (𝜷 𝑐 |𝒙 𝑐 ) is formulated as𝜓 𝑐 (𝜷 𝑐 |𝒙 𝑐 ) = 𝛽 2 𝑚,𝑛 𝜎 4 𝑚,𝑛 1-𝜚 2 𝑐 2 + 𝛽 2 𝑚,𝑛+1 𝜎 4 𝑚,𝑛+1 1-𝜚 2 𝑐 2 + 2𝜚 2 𝑐 𝛽 𝑚,𝑛 𝛽 𝑚,𝑛+1 𝜎 2 𝑚,𝑛 𝜎 2 𝑚,𝑛+1 1-𝜚 2 𝑐 2 .(22) Referring to Eq. ( 22), it can be seen that the correlation between DCT coefficients 𝑥 𝑚,𝑛 and 𝑥 𝑚,𝑛+1 has been taken into account by the introduction of correlation coefficient 𝜚 𝑐 . In addition, a closer look at Eq. ( 16) and Eq. ( 22) reveals that both unary and pairwise potentials are inversely proportional to the variance of DCT coefficient, implying that modifying the DCT coefficients with small variances are more detectable than in those with large variances. On the other hand, we know that the DCT coefficients with small variances are generally located in the smooth regions or high-frequency modes of JPEG images, and these DCT coefficients have been confirmed by previous studies [3, 13, 16, 29, 30] to be really unsuitable for embedding modifications. In this reagrd, we can claim that the construction of potentials in both Eq. ( 16) and Eq. ( 22) for secure JPEG steganography is theoretically feasible.Once the detailed expression of unary and pairwise potentials are determined (see Eq. ( 16) and Eq. ( 22)), the energy 𝐸 (𝜷 | S) can be accordingly obtained by summing all the unary and pairwise potentials in the sub-image S. Note that S is only one of the 64 mutually independent sub-images, then we expand the derived results to the entire image X, and haveEQUATIONwhere 𝜷 = 𝛽 𝑘,𝑙 𝑚,𝑛 ℎ×𝑤 are the embedding change probabilities, 𝐸 𝑘,𝑙 (𝜷 | S) is the energy of the (𝑘, 𝑙)-th subimage S and rewritten as:EQUATIONHowever, a closer look at Eq. ( 23) and Eq. ( 24) reveals that all the DCT coefficients are treated indistinguishably, which is inconsistent with domain knowledge in steganography. Previous arts [17, 29, 30] have shown that image regions with different complexity, as well as different frequency modes, can afford different levels of antidetectability. Given this, we further modify Eq. ( 23) and Eq. ( 24) by incorporating the domain knowledge in steganography as follows:EQUATIONIn the revised Eq. ( 25), we introduce three types of weights, i.e., the mode weight 𝑤 𝑘,𝑙 , the block weight 𝑤 𝑚,𝑛 and the pairwise weight 𝑤 𝑐 . The mode weight 𝑤 𝑘,𝑙 possesses a larger value on high-frequency modes than low ones, thus widening the difference between the corresponding energies of different frequency modes associated with the steganography detectability. As for the setting of 𝑤 𝑘,𝑙 , we first scan the DCT modes with Zig-Zag order and then set the last 𝑝 DCT modes with 𝑤 𝑘,𝑙 = 5 and 𝑤 𝑘,𝑙 = 1 for the other ones. For QF = 75 and QF = 95, 𝑝 are 16 and 3, respectively. The block weight 𝑤 𝑚,𝑛 assigns smaller values to the unary potentials in the DCT blocks of more complex regions. To achieve this, the classic spatial steganographic scheme S-UNIWARD [16] is applied to obtain the costs 𝝆 ′ = (𝜌 ′ 𝑖,𝑗 𝑚,𝑛 ) ℎ×𝑤 of the decompressed spatial image Z, and then summed 𝝆 ′ over blocks and make it as the block weight 𝑤 𝑚,𝑛 , i.e.,EQUATIONwhere 𝜌 ′ 𝑖,𝑗 𝑚,𝑛 is the distortion cost of the (𝑖, 𝑗)-th pixel in the (𝑚, 𝑛)th decompressed spatial block. As for the pairwise weight 𝑤 𝑐 , it is dedicated to pairwise potentials w.r.t. two adjacent DCT blocks, which allows us to simply compute 𝑤 𝑐 by merging the two involved block weights, e.g., 𝑤 𝑐 = (𝑤 𝑚,𝑛 + 𝑤 𝑚,𝑛+1 )/2.To this end, the energy of the entire image X associated with the statistical detectability has been eventually determined so that we can obtain the optimal embedding probabilities 𝜷 * with the given payload constraint by minimizing the Ê (𝜷 | X), corresponding to the minimum detectability, i.e.,              min 𝜷 Ê (𝜷 | X)) =where 𝐻 (𝑥) is the information entropy that 𝐻 (𝑥) = -𝑥𝑙𝑜𝑔(𝑥) -(1 -2𝑥)𝑙𝑜𝑔(1 -2𝑥), 𝐿 is the given embedding payload.Notably, the optimization of Eq. ( 27) is convex and consequently has a global minimum since all its 𝐸 𝑘,𝑙 (𝜷 | S) terms are convex functions regarding 𝛽 𝑘,𝑙 𝑚,𝑛 . To deal with this optimization, the classical Lagrange multiplier method is employed on the Eq. ( 27), and have 𝜕 𝜕𝛽 𝑘,𝑙 𝑚,𝑛 𝐹 (𝜷) -EQUATIONwhere 𝛾 > 0 is the Lagrange multiplier. After some simple arithmetic, we can obtain:EQUATIONwhere U 𝑘,and N 𝑘,𝑙 𝑚,𝑛 = {(𝑚-1, 𝑛), (𝑚+1, 𝑛), (𝑚, 𝑛-1), (𝑚, 𝑛+1)} is the set of the indexes of the four-element cross-neighborhood of 𝑥 𝑘,𝑙 𝑚,𝑛 , (𝜎 𝑘,𝑙 𝑚,𝑛 ) 2 is the variance of 𝑥 𝑘,𝑙 𝑚,𝑛 , 𝜚 𝑎,𝑏 𝑚,𝑛 is the correlation coefficient between 𝑥 𝑘,𝑙 𝑚,𝑛 and its neighbor 𝑥 𝑘,𝑙 𝑎,𝑏 . Once (𝜎 𝑘,𝑙 𝑚,𝑛 ) 2 and 𝜚 𝑎,𝑏 𝑚,𝑛 for each of the DCT coefficients are given, the embedding probabilities can be easily determined by numerically solving the equation (29) .To quickly solve Eq. ( 29) for all DCT coefficients, the inverse function to 𝑓 (𝑥) = 𝑥 log(𝑥-2) was tabulated, and a binary search is applied to find the corresponding Lagrange multiplier 𝛾. Note that there is a quantity, i.e., 𝛽 𝑘,𝑙 𝑎,𝑏 /𝛽 𝑘,𝑙 𝑚,𝑛 , in Eq. ( 31) that will hinder the rapid implementation of a lookup table in the solving process. To cope with this, an iterative optimization approach is developed for solving Eq. ( 29) until it converges. Specifically, in each new iteration, the quantity 𝛽 𝑘,𝑙 𝑎,𝑏 /𝛽 𝑘,𝑙 𝑚,𝑛 is firstly calculated by the 𝜷 obtained in the last iteration. As for the first iteration, the quantity can be set to 1 for brevity. In addition, a trick is also used for fast solving the Eq. ( 29) that dynamically adjusts the search range of the Lagrange multiplier according to the results obtained in the last iteration. In this way, our scheme converges in only three iterations, and the 𝛽 * corresponding to the least detectability can be accordingly obtained.Subsequently, it can be converted to the steganographic distortion cost by 𝜌 𝑘,𝑙 𝑚,𝑛 = log 1/𝛽 𝑘,𝑙 𝑚,𝑛 -2 .(32) 
parameter estimation: 
 Recall Eq. ( 30) and Eq. ( 31), determining the (𝜎 𝑘,𝑙 𝑚,𝑛 ) 2 and 𝜚 𝑎,𝑏 𝑚,𝑛 is the key to solving Eq. (27) . To estimate the variances of DCT coefficients, in J-MiPOD [3] , Rémi et al. extended the pixel variance estimator proposed in MiPOD to JPEG images with effective modifications, achieving low computational complexity and respectable empirical security. Given these benefits, we adopt this improved estimator to compute the DCT coefficient variance (𝜎 𝑘,𝑙 𝑚,𝑛 ) 2 , which is briefly summarised as follows: step 1: Decompress the JPEG image X to the corresponding spatial image Z using Eq. (2). step 2: Subtract the estimated pixel expectations, which are obtained by using a Wiener filter with a window size 2 × 2, from Z to acquire image residuals. As for the estimation of the correlation coefficients between the DCT coefficients from the same mode over the adjacent blocks, we propose a simple yet efficient estimation method. Overall, we foremost compute a block-level correlation coefficient between two adjacent blocks and then assign this block-level estimation to the 64 pairs of DCT coefficients with the same mode in these two adjacent blocks to serve as their respective correlation coefficients. Formally, the block-level correlation coefficients are efficiently computed by applying the Pearson product-moment correlation coefficient formula to the estimated noises obtained in the variance estimation. To this end, the correlation coefficient 𝜚 𝑎,𝑏 𝑚,𝑛 in ( 30) and ( 31) is formulated as: EQUATION 
time complexity: 
 For time complexity evaluation, we randomly selected 1000 512 × 512 images from BOSSBase to measure the average time-consuming in embedding distortion cost acquisition under QF=75 and QF=95 at 0.4 bpnzac. Not only the proposed CRF is evaluated, but also the other competitors are involved for comparison. The time complexity evaluation is performed with Matlab 2015b on a 3.20 GHz Intel CPU Xeon E-2836 with 64GB of memory running a 64-bit Windows 10 without parallel computing. As shown in Table 3 , UERD owns the minimum time-consuming, and J-UNIWARD consumes far more time than the other three methods. Numerically, the time complexity of J-UNIWARD is nearly 180 times of UERD and about 3 times of CRF. Although the time complexity of our CRF is 3 times of J-MiPOD, it is still affordable. 
conclusion: 
 In this paper, we propose a novel model-based JPEG steganographic scheme called CRF, wherein the Conditional Random Field (CRF) model with four-element cross-neighborhood is built to capture the mutual impact of embedding change probabilities by leveraging the correlation of DCT coefficients of the same mode in horizontally or vertically adjacent blocks. Overall, the CRF model is characterized by energy function defined as the weighted sum of a series of unary and pairwise potentials. Both kinds of potentials, in our scheme, are formulated as the KL divergence between the statistical distributions of cover and stego. Following this way, secure JPEG steganography is then formulated as the optimization problem of minimizing that energy function. The optimal embedding change probabilities corresponding to the least detectability can be eventually obtained by optimizing the constructed energy function with the given payload constraint after some necessary parameter estimation, based on which, the steganographic distortion cost can be converted accordingly. Numerous results show that the proposed CRF can not only surpass the SOAT heuristic-based UERD and J-UNIWARD, but also outperform currently the most advanced model-based J-MiPOD by a clear margin with affordable time complexity.In the current work, we built a CRF model to capture the mutual impact of embedding change probabilities, in the future, we will go a step further to directly capture the mutual impact of embedding changes to further improve JPEG steganography, which will lead us to investigate the model-based asymmetric steganography embedding.  
 
 Appendix: (a) Cover image (b) 𝜷: J-MiPOD (c) 𝜷: CRF 


Decision: As pointed out by two reviewers, the novelty of this work is rather limited. Many of the proposed techniques are slightly improved version of the existing ones, making the technical contributions marginal. These aspects should be addressed adequately in the CR version. Given the review comments and the final review ratings, I recommend paper acceptance.

Reviews: review: paper_summary: paper_summary: This paper introduces a novel model-based steganographic scheme to improve the security of JPEG steganography. Using the pairwise Conditional Random Field (CRF) model with a four-element cross-neighborhood, the proposed method effectively captures the dependencies among DCT coefficients. Extensive experimental results validate the effectiveness of the proposed scheme, which significantly outperforms the previous methods, especially J-MiPOD.
strength_weakness: strength_weakness: The State-of-the-Art section provides a detailed introduction to distortion cost functions, highlighting their tutorial implications. 

Figure 1. clearly describes the DCT modes splitting method, offering a visual and explanatory aid to understand the process. 

Comprehensive experiments have been conducted to thoroughly demonstrate the superiority of the CRF method.
limitations: limitations: The validity of the claim that CRF's time complexity is affordable is uncertain, as there is no clarification on what constitutes affordable.

Consider providing the rationale for why QF of 75 and 95 were chosen for evaluating performance gains, as this can further enhance the interpretation of the results.

Consider incorporating additional image comparisons alongside Figure 2., this would allow for a more comprehensive assessment of embedding change probability for J-MiPOD and CRF.
paper_summary: paper_summary: This paper proposes a novel model-based steganographic scheme for JPEG images when the side-information of images is unavailable. It leverages a Conditional Random Field (CRF) model to capture dependencies among Discrete Cosine Transform (DCT) coefficients. This model aims to enhance security in JPEG steganography by minimizing the Kullback-Leibler (KL) divergence between the statistical distributions of cover and stego images. The proposed scheme, characterized by its delicately designed energy function, outperforms previous methods, including the independent J-MiPOD, in empirical evaluations.
strength_weakness: strength_weakness: Highly explainable: The proposed method is based on complete mathematical derivations. By defining the energy function as a weighted sum of unary and pairwise potentials, and optimizing it under a payload constraint, the authors ensure a robust theoretical foundation for the proposed scheme​​.
Addresses real-world constraints: The scheme is designed to work without any side information of images, making it applicable to situations with only a single compressed JPEG image. This enhances its practicality in a wider range of real-world scenarios.
Superior Performance: The experimental results indicate that the proposed method consistently outperforms existing methods and achieves lower detection error rates across four steganalysis features (CC-JRM, DCTR, GFR, SCA-GFR), especially under higher payload conditions.
limitations: limitations: Computational Complexity: The CRF model, despite its superior performance, has a higher computational complexity compared to some existing methods like UERD and J-MiPOD. Although the paper mentions that the time complexity is affordable, it is still significantly higher than other methods, which could limit its practicality in real-time applications​​.
Limited Scope of Embedding Strategies: The paper focuses solely on ternary symmetric embedding. This limitation might restrict the applicability of the proposed method to other embedding strategies or more complex steganographic scenarios. Moreover, This paper only consider the correlations of DCT coefficients from the same mode in horizontally or vertically adjacent blocks. Does such a consideration have certain limitations?
Limited comparative models: To evaluate the security performance of CRF, the authors chose two heuristic-based schemes and one model-based scheme. However, they did not use the current state-of-the-art deep learning-based steganography algorithms for comparison.
paper_summary: paper_summary: This paper proposes a model-based steganographic scheme using the conditional random field, which uses four-element cross-neighborhood to capture the dependencies among DCT coefficients. The conditional random field model is designed by a energy function, which is defined as the weighted sum of a series of unary and pairwise potentials. Experiments show the effectiveness of the proposed steganography, compared with previous methods.

The proposed method demonstrates superior performance compared to the baseline methods. However, the contribution is limited, and the baselines and steganalysis methods are obsolete.
strength_weakness: strength_weakness: 1) The paper in general is well written and well organized.

2) The paper is technically sound, and the method description is succinct yet clear.
limitations: limitations: 1) Lack of significant contribution. The contributions of this paper are marginal. The work presented is merely an improvement over existing literature, without introducing substantial novel insights or breakthroughs.

2) Outdated comparative methods. The comparative methods used in this paper are outdated, primarily relying on techniques that are over a decade old. The paper should compare its results with more recent methods, especially the methods with cost designs based on learning mechanisms, to provide a more fair comparison.

3) Outdated steganalysis methods. The steganalysis techniques employed in this paper are also quite dated. The authors should utilize the latest deep learning-based steganalysis methods to assess the steganographic security, which would offer a more accurate and contemporary evaluation.

4) The visual comparison between Figure 2 (b) and (c) is not clear. The claimed differences related to steganographic embedding are not visually evident. The authors need to provide a clearer illustration of these differences to support their claims effectively.

5) The theoretical analysis presented in the paper is somewhat overly complicated and could benefit from simplification.
paper_summary: paper_summary: This paper covers an interesting topic about model-based JPEG image steganography. It proposes a novel steganographic scheme utilizing the Conditional Random Field (CRF) model with the four-element cross-neighborhood to capture the dependencies among DCT coefficients for JPEG steganography with symmetric embedding.
strength_weakness: strength_weakness: The techniques used appear to be correct. Moreover, this paper is well-written, clear, and easy to understand.
limitations: limitations: 1) The innovation is limited, the pairwise conditional random field model used in this paper already has the related method [1] in JPEG domain steganography. 
2) There is a lack of comparison with some of the most advanced model-based JPEG steganography methods, such as references [1, 2].
3) The steganalyzers used in the security experiments are all handcrafted-based. It is suggested to add CNN-based steganalyzers.
4) It is interesting to see how the hiding affects the different DCT frequency modes.
5) It is suggested to evaluate comprehensively the complexity of the algorithm through multiple indicators.

[1] Pan et al., Efficient JPEG Image Steganography Using Pairwise Conditional Random Field Model. SP 2024.

[2] Li et al., Quaternary Quantized Gaussian Modulation With Optimal Polarity Map Selection for JPEG Steganography, IEEE TIFS, 2023.