Improved PHARM for JPEG Steganalysis: Making PHARM More Efficient and Effective

The PHase-Aware pRojection Model (PHARM) feature set, built as phase-aware histograms of quantized projections obtained by convolving residuals with random matrices, achieves competitive detection performance against modern adaptive JPEG steganography while having significant computational cost. In this paper, we propose three improvements to the original PHARM, making it more efficient and effective. First, we reduce the maximum projection matrix size to decrease the computational complexity of convolution and better capture steganographic embedding changes. Second, we select more than one phase pair per projection to compute phase-aware histograms, thus correspondingly reducing the number of projections for each residual. Third, the transposition symmetry is also taken into consideration to make our features more robust while preserving the feature dimensionality. The numerous experiments are given to demonstrate the efficiency and effectiveness of our improved PHARM.

SECTION I. Introduction

The purpose of steganography is to embed secret messages into cover objects without arousing a warder’s attention. Because of the wide usage of JPEG images, increasingly much attention has been paid to JPEG steganography. Recently, with the advent of the STCs (Syndrome-Trellis Codes) coding technique [1], some advanced adaptive JPEG steganographic schemes have been designed, such as UED (Uniform Embedding Distortion) [2] and J-UNIWARD (JPEG UNIversal WAvelet Relative Distortion) [3]. These adaptive schemes are difficult to detect because their embedding changes are constrained to complex regions which are hard to model.

Although the previous JPEG steganalysis features (e.g., PEV [4] and JRM [5]) can effectively detect early embedding schemes (e.g., F5 [6], OutGuess [7], and model-based steganography [8]), they are less effective for detecting modern adaptive JPEG steganography. To better detect those adaptive embedding schemes, some JPEG phase-aware feature sets are proposed, such as DCTR (Discrete Cosine Transform Residual) [9], GFR (Gabor Filter Residual) [10], and PHARM (PHase-Aware pRojection Model) [11]. The success of JPEG phase-aware features can be summarized as follows. First, unlike the previous JPEG features formed from quantized DCT coefficients, JPEG phase-aware features are built in the spatial domain. This is probably because the statistical characteristics captured in the spatial domain are more sensitive to those adaptive JPEG embedding algorithms [12]. Second, JPEG phase-aware features compute histograms from the subsets of an image split by their JPEG phase rather than directly from the whole one. This is effective because the statistical properties of pixels in a decompressed JPEG image depend on their positions within the $8\times 8$ pixel grid.

The DCTR and GFR features compute histograms from the subsets of noise residuals. The difference between them is that DCTR uses 64 DCT bases to generate noise residuals whereas GFR uses 256 Gabor filters. But unlike DCTR and GFR, PHARM projects a large number of random matrices on residuals before the histogram computation to diversify the features. Meanwhile, however, PHARM is so computationally expensive because of these complicated random projections.

In this paper, we review the design of PHARM and propose three improvements for less computation time and higher detection accuracy.

• First, the maximum size of projection matrices is reduced. To generate diverse projections in PHARM, residuals are convolved with many projection matrices of random sizes. Compared to large size projection matrices, moderate size ones not only reduce the computational complexity of convolution but also more effectively capture the embedding changes caused by adaptive JPEG steganography. In this paper, hence, compared to the original PHARM, the maximum projection matrix size decreases from eight to four.

• Second, for each projection, multiple phase pairs are selected to compute phase-aware histograms. The original PHARM computes phase-aware histograms in a wasteful manner, choosing only one phase pair to compute a histogram for each projection. That is, one projection corresponds to only one phase-aware histogram. To obtain the original 12600-dimensional PHARM, a large number of (900) projections have to be generated for each residual, which is very time consuming. In this paper, we randomly select three different phase pairs per projection to compute phase-aware histograms and correspondingly reduce the number of projections per residual to keep the same feature dimension, which greatly reduces the computation time while maintaining the detection accuracy.

• Third, the transposition symmetry is taken into account. The original PHARM applies three geometrical transformations (horizontal mirroring, vertical mirroring, and rotation by 180 degrees) to projection matrices for symmetrization. However, the transposed transformation is not considered. In this paper, besides the existing projection matrices, we also utilize their transposed versions to generate projections, and then merge the corresponding histograms based on the proper symmetries. This method further improves the detection accuracy while preserving the feature dimensionality.

The rest of this paper is organized as follows. In Section II, we briefly introduce the original PHARM to make this paper self-contained. In Section III, we describe the improved PHARM in detail. In Section IV, experiments are conducted to demonstrate the efficiency and effectiveness of our improved PHARM. Conclusions are drawn in Section V.

SECTION II. Original PHARM Features

In this section, we briefly describe how the original PHARM features are calculated to help readers better understand our proposed improvements. The calculation procedures of PHARM are described as follows (see Fig. 1).

1. Decompress a JPEG image into the spatial domain without rounding the pixel values to integers. The obtained decompressed JPEG image is denoted as $\mathbf {X} \in \mathbb {R}^{n_{1} \times n_{2}}$ .

2. Generate the linear residual $\mathbf {Z}$ by convolving the decompressed JPEG image $\mathbf {X}$ with one of seven linear high-pass filters $\mathbf {K}$ . $$\begin{equation*} \mathbf {Z} = \mathbf {X} \star \mathbf {K},\tag{1}\end{equation*}$$ where the symbol ‘$\star$ ’ denotes convolution.

3. Generate the projections $\mathbf {P}$ by convolving the residual $\mathbf {Z}$ with $\nu$ random matrices ${\Pi }$ . $$\begin{equation*} \mathbf {P}^{(i)} = \mathbf {Z} \star {\Pi }^{(i)}, ~i \in \{1, \ldots, \nu \},\tag{2}\end{equation*}$$ where ${\Pi }^{(i)} \in \mathbb {R}^{r \times s}$ , $r$ and $s$ are uniformly randomly selected from $\{1, \ldots, S\}$ , the elements of ${\Pi }^{(i)}$ are independent realizations of a standard normal random variable $\mathcal {N}(0,1)$ normalized to a unit Frobenius norm $\| {\Pi } \|_{2} = 1$ .

4. For each $i \in \{1, \ldots, \nu \}$ , select $u$ , $v \in \{0, \ldots, 7\}$ uniformly randomly and subsample the projection values $\mathbf {P}^{(i)}$ to $\mathbf {P}^{(i;u,v)} \triangleq \left ({p_{u + 8 \cdot k, v + 8 \cdot l}^{(i)}}\right)$ , $0\leq k \leq n_{1}/8-1$ , $0\leq l \leq n_{2}/8-1$ .

5. Compute the histogram $\mathbf {h}^{(i;u,v)}$ of the quantized values $\mathbf {P}^{(i;u,v)}$ , $$\begin{align*} \mathbf {h}^{(i;u,v)}_{t}=&\sum _{a,b}\left [{Q_{T}\left ({\left |{\left.{p_{a,b}^{(i; u,v)}}\right |}\right /q}\right)=t+1/2}\right],\qquad \\&\qquad \qquad \qquad \quad t \in \{0, \ldots, T-1\},\tag{3}\end{align*}$$ where $Q_{T}$ is a quantizer with $T$ centroids $\{1/2, \ldots, T-1/2\}$ , $T$ is the truncation threshold, $q$ is the quantization step, and [$P$ ] is the Iverson bracket equal to 0 when the statement $P$ is false and 1 when $P$ is true.

6. Enlarge the set $\mathbf {P}^{(i)}$ by adding to it projections obtained by convolving with three transformed versions of the matrix ${\Pi }^{(i)}$ , namely its horizontal flipping ${\Pi }^{(i)}_{\mathrm {flplr}}$ , vertical flipping ${\Pi }^{(i)}_{\mathrm {flpud}}$ , and rotation by $180^{\circ }~{\Pi }^{(i)}_{\mathrm {rot180}}$ .

7. Merge the histograms computed from the subsets of the projections obtained with ${\Pi }^{(i)}$ , ${\Pi }^{(i)}_{\mathrm {flplr}}$ , ${\Pi }^{(i)}_{\mathrm {flpud}}$ , and ${\Pi }^{(i)}_{\mathrm {rot180}}$ . The merging method is based on the symmetries between ${\Pi }^{(i)}$ and its geometrically transformed versions.

8. Concatenate all the symmetrized histograms to form the final feature set PHARM.

FIGURE 1.

The diagram of extracting the original PHARM.

The parameter setup for the original PHARM is illustrated as follows. The maximum projection matrix size $S = 8$ , the number of projections per residual $\nu = 900$ , the number of selected phase pairs per projection $N = 1$ , the quantization step $q = \frac {65}{4} - \frac {3}{20}QF$ , and the histogram threshold $T = 2$ . This setup gives PHARM the dimension of $7 \cdot T \cdot N \cdot \nu = 7 \cdot 2 \cdot 1 \cdot 900 = 12600$ .

SECTION III. Improved PHARM Features

The PHARM is effective for the detection of adaptive JPEG steganographic algorithms but has expensive computational cost. In this section, we elaborately modify its calculation process to reduce the computation time and further improve the detection accuracy.

A. Reducing Maximum Projection Matrix Size

In Step 3 of the PHARM calculation described in Section II, residuals are convolved with a large number of matrices of random sizes for diverse projections. The parameter $S$ determines the maximum width and height of each projection matrix and thus limits the range of interpixel dependencies that can be utilized for detection. In the original PHARM, the maximum projection matrix size $S$ is set to eight because it corresponds to the size of the JPEG blocks. In this paper, we decide to reduce the value of the parameter $S$ for two reasons.

On the one hand, the reduction of the maximum projection matrix size $S$ can reduce the computational complexity of convolution. In PHARM, a projection is obtained by convolving a residual with a projection matrix of random size. With an $r\times s$ projection matrix, the computation of each element of a projection requires $r\times s$ times multiplication and addition. That is, the computational complexity of convolution is directly related to the size of a projection matrix. Reducing the maximum projection matrix size can reduce the maximum times of multiplication and addition operations in convolution, thus accordingly decreasing the computational complexity. On the other hand, the reduction of the maximum projection matrix size $S$ can increase the detection accuracy. Some observations in steganalysis encourage us to select moderate size convolution matrices for residual generation. First, Zeng’s CNN [13] and Xu’s 20-layer CNN [14] respectively choose the $5\times 5$ and $4\times 4$ DCT filters to suppress image content in the preprocessing layer. Second, JPEG phase-aware features DCTRD and GFRD in [15] utilize a series of DCT and Gabor filters of size from $3\times 3$ to $6\times 6$ to generate residuals. These observations demonstrate that the embedding changes caused by adaptive steganographic schemes can be better captured by using moderate size convolution matrices. Hence, appropriately reducing the maximum projection matrix size can improve the detection accuracy against adaptive steganography to some extent.

Based on the above two reasons, we determine to reduce the maximum size of projection matrices $S$ in our PHARM. But it is not easy to choose its proper value. Next, we conduct a series of experiments to select a proper value for the parameter $S$ .

In the experiments of this subsection, the dataset used for performance evaluation is BOSSbase 1.01 [16] which contains 10000 grayscale images of size $512 \times 512$ . To reduce the time for parameter selection, we only select 7000 images from BOSSbase 1.01 and convert them into JPEG images with quality factor 75 as cover images. Then we generate stego images by using advanced adaptive JPEG steganographic scheme J-UNIWARD with a payload of 0.2 bpnzac. In each experiment, we randomly select 4500 pairs of cover and stego images for training and the rest for testing. The detection accuracy is quantified using the minimal total error probability under equal priors $$\begin{equation*} P_{\mathrm {E}} = \mathrm {min}_{P_{\mathrm {FA}}}\frac {1}{2}(P_{\mathrm {FA}}+P_{\mathrm {MD}}),\tag{4}\end{equation*}$$ where $P_{\mathrm {FA}}$ and $P_{\mathrm {MD}}$ are the false-alarm and missed-detection probabilities, respectively. The FLD (Fisher Linear Discriminant) ensemble classifier [17] is used in the training and testing stages. The $\overline {P}_{\mathrm {E}}$ is averaged over ten trials. Besides the detection accuracy, we also need to consider the effect of parameters on time consumed for feature extraction. In this paper, for a fair comparison, all feature extractors are implemented in MATLAB 2017b and run on our computer with Intel(R) Core(TM) Processor i5-3470 and 4 GB memory.

In Table 1, the detection accuracy and computation time for PHARM are given when the maximum projection matrix size $S$ ranges from two to eight. The values of other parameters are the same as in the original PHARM. That is, compared to the original PHARM, we only change the value of the parameter $S$ . From Table 1, it can be seen that compared to the original PHARM in which $S = 8$ , the modified PHARM with moderate values of the parameter $S$ has advantages in the detection accuracy and computation speed. Taking both the detection accuracy and computation time into consideration, the candidate values of the maximum projection matrix size $S$ in our PHARM are set as four and five.

TABLE 1 The Effect of the Maximum Projection Matrix Size $S$ on Detection Error $\overline{P}_{\mathrm{E}}$ and Computation Time for PHARM

B. Selecting Multiple Phase Pairs Per Projection

In Step 4 of the PHARM calculation described in Section II, for each projection, only one phase pair is randomly selected to compute a corresponding phase-aware histogram. That is, the original PHARM only computes one two-dimensional histogram for each projection. To construct the 12600-dimensional PHARM (concatenated histograms), each residual has to be convolved with 900 random matrices to generate enough projections. This is the main reason why PHARM is so demanding to compute.

The original PHARM obtains phase-aware histograms in such a wasteful manner. Being different from PHARM, DCTR and GFR both exploit all 64 phase pairs to compute phase-aware histograms. In DCTR and GFR, a residual is first subsampled according to 64 phase pairs, and then 64 phase-aware histograms are computed from the corresponding sub-residuals. Inspired by the calculation of DCTR and GFR, we believe that selecting multiple phase pairs does not have a serious adverse effect on steganography detection. With multiple phase-aware histograms per projection, the number of projections per residual correspondingly decreases to maintain the same dimension ($\nu = 900 / N$ ), thus dramatically reducing the computation time.

Hence, we decide to utilize more than one phase pair per projection to compute phase-aware histograms. Next, we experimentally decide how many phase pairs are selected for each projection and correspondingly determine the number of projections per residual.

In the experiments of Table 2, the training and testing sets are the same as those of Table 1. The detection accuracy and computation time are also measured in the same manner as in the experiments of Table 1.

TABLE 2 The Effect of the Number of Selected Phase Pairs Per Projection $N$ on Detection Error $\overline{P}_{\mathrm{E}}$ and Computation Time for PHARM

Table 2 shows the detection accuracy and computation time for PHARM with the different number of selected phase pairs per projection when the parameter $S$ is set as four and five. In Table 2, the number of selected phase pairs per projection $N$ ranges from one to five. To keep the same feature dimensionality, the number of projections per residual $\nu$ is correspondingly set as 900, 450, 300, 225, and 180. Except for the investigated parameters $S$ , $N$ and $\nu$ , the others are the same as in the original PHARM. From Table 2, it can be seen that using multiple phase pairs per projection does not bring a serious negative effect on the detection accuracy, even improving the accuracy in some cases. With more phase-aware histograms per projection, fewer projections are needed to compute accordingly, which reduces the computation time significantly. According to the results in Table 2, we select $S = 4$ , $N = 3$ , $\nu = 300$ in our final PHARM to balance the detection accuracy and computation speed.

C. Employing Transposition Symmetry

In Step 7 of the PHARM calculation described in Section II, the histograms of different projections are merged based on the symmetries between projection matrix ${\Pi }$ and its geometrically transformed versions ${\Pi }_{\mathrm {flplr}}$ , ${\Pi }_{\mathrm {flpud}}$ , ${\Pi }_{\mathrm {rot180}}$ . But their transposed versions are not used to further symmetrize the histograms.

The symmetrization is very useful for steganography detection. And the transposition symmetry is widely employed for symmetrization in various steganalysis features, such as PSRM [18], GFR-GSM [19], DCTRD, and GFRD.

In this paper, we attempt to employ the transposition symmetry for PHARM to further improve the detection accuracy. We first combine equations (1) and (2), and the projection $\mathbf {P}^{(i)}$ can be viewed as the result of convolution of the decompressed JPEG $\mathbf {X}$ and kernel $\mathbf {C}^{(i)}$ , and $\mathbf {C}^{(i)}$ is the convolution output of the linear high-pass filter $\mathbf {K}$ and the projection matrix ${\Pi }^{(i)}$ . $$\begin{equation*} \mathbf {P}^{(i)} = \mathbf {X} \star \mathbf {K} \star {\Pi }^{(i)} = \mathbf {X} \star \left ({\mathbf {K} \star {\Pi }^{(i)}}\right) = \mathbf {X} \star \mathbf {C}^{(i)}.\tag{5}\end{equation*}$$ Then, we can obtain the projection $\mathbf {P}'^{(i)}$ by convolving with the transposed version of kernel $\mathbf {C}^{(i)}$ , $$\begin{align*} \mathbf {P}'^{(i)}=&\mathbf {X} \star \left ({\mathbf {C}^{(i)}}\right)^{T} = \mathbf {X} \star \left ({\mathbf {K} \star {\Pi }^{(i)}}\right)^{T} \\=&\mathbf {X} \star \left ({{\Pi }^{(i)}}\right)^{T} \star \mathbf {K}^{T} = \mathbf {X} \star \mathbf {K}^{T} \star \left ({{\Pi }^{(i)}}\right)^{T}.\tag{6}\end{align*}$$ Due to the transposition symmetry, we can merge the histograms $\mathbf {h}^{(i;u,v)}$ and $\mathbf {h}^{\prime (i;v,u)}$ , which are computed from the $(u,v)$ th subset of the projection $\mathbf {P}$ and the $(v,u)$ th subset of the projection $\mathbf {P}'$ , respectively. Specifically, we can merge the histograms based on the symmetries shown in Fig. 2.

FIGURE 2.

Symmetries that can be utilized for a given $3\times 2$ matrix ${\Pi }$ and a given phase pair $(u, v) = (1, 0)$ . The residuals Z and $\mathrm {Z}'$ are generated by convolving the decompressed JPEG X with the high-pass filter K and its transposed version $\mathrm {K}^{T}$ . The large dot represents the upper-left element of ${\Pi }$ and shows how the matrix ${\Pi }$ is flipped, rotated, transposed to preserve the symmetries. The large dots of projection matrices ${\Pi }$ , ${\Pi }_{\mathrm {flplr}}$ , ${\Pi }_{\mathrm {flpud}}$ , ${\Pi }_{\mathrm {rot180}}$ are respectively located at the positions $(u, v)$ , $(u, 7-v)$ , $(7-u, v)$ , $(7-u, 7-v)$ within the $8\times 8$ grid of residual $Z$ . And the large dots of projection matrices ${\Pi }^{T}$ , ${\Pi }_{\mathrm {flplr}}^{T}$ , ${\Pi }_{\mathrm {flpud}}^{T}$ , ${\Pi }_{\mathrm {rot180}}^{T}$ are respectively located at the positions $(v, u)$ , $(7-v,u)$ , $(v, 7-u)$ , $(7-v, 7-u)$ within the $8\times 8$ grid of residual $\mathrm {Z}'$ . (a) Residual Z, (b) Residual $\mathrm {Z}'$ .

Next, we conduct some experiments to show that the proposed symmetrization strategy can further improve the detection accuracy of PHARM. In the experiments, 10000 images from BOSSbase 1.01 are converted into JPEG format with quality factors 75 and 95 as cover images and then stego images are generated by J-UNIWARD with a payload of 0.2 bpnzac. The $\overline {P}_{\mathrm {E}}$ is averaged over ten random 5000/5000 dataset splits. From Table 3, it can be seen that PHARM benefits from the transposition symmetry which indeed increases the robustness. In Table 3, both PHARM features adjust the values of the parameters $S = 4$ , $N = 3$ , $\nu = 300$ . The difference between them is that one employs the transposition symmetry, whereas the other does not utilize this useful symmetry.

TABLE 3 Comparison on Detection Error $\overline{P}_{\mathrm{E}}$ for PHARM Using and not Using Transposition Symmetry

D. Final Feature Design

We now summarize the computation procedures of our PHARM as follows (see Fig. 3).

1. Decompress a JPEG image into the spatial domain without rounding to obtain the decompressed JPEG image $\mathbf {X} \in \mathbb {R}^{n_{1} \times n_{2}}$ .

2. Generate the linear residuals $\mathbf {Z}$ and $\mathbf {Z}'$ by convolving the decompressed JPEG image $\mathbf {X}$ with the linear high-pass filter $\mathbf {K}$ and its transposed version $\mathbf {K}^{T}$ , respectively.

3. Generate the projections $\mathbf {P}$ by convolving the residual $\mathbf {Z}$ with $\nu$ random matrices ${\Pi }$ .

4. For each $i \in \{1, \ldots, \nu \}$ , select three different phase pairs $(u_{1}, v_{1})$ , $(u_{2}, v_{2})$ , $(u_{3}, v_{3})$ uniformly randomly and subsample the projection values $\mathbf {P}^{(i)}$ to $\mathbf {P}^{(i;u_{j},v_{j})} \triangleq \left ({p_{u_{j} + 8 \cdot k, v_{j} + 8 \cdot l}^{(i)}}\right)$ , $u_{j}, v_{j} \in \{0,1, \ldots, 7\}$ , $j \in \{1, 2, 3\}$ , $0\leq k \leq n_{1}/8-1$ , $0\leq l \leq n_{2}/8-1$ .

5. Compute the histogram $\mathbf {h}^{(i;u_{j},v_{j})}$ of the quantized values $\mathbf {P}^{(i;u_{j},v_{j})}$ by (3).

6. Enlarge the set $\mathbf {P}^{(i)}$ by adding to it projections obtained not only by convolving the residual $\mathbf {Z}$ with ${\Pi }^{(i)}_{\mathrm {flplr}}$ , ${\Pi }^{(i)}_{\mathrm {flpud}}$ , ${\Pi }^{(i)}_{\mathrm {rot180}}$ but also by convolving the residual $\mathbf {Z}'$ with $({\Pi }^{(i)})^{T}$ , $({\Pi }_{\mathrm {flplr}}^{(i)})^{T}$ , $({\Pi }_{\mathrm {flpud}}^{(i)})^{T}$ , $({\Pi }_{\mathrm {rot180}}^{(i)})^{T}$ .

7. Merge the histograms computed from the subsets of the projections obtained with ${\Pi }^{(i)}$ , ${\Pi }^{(i)}_{\mathrm {flplr}}$ , ${\Pi }^{(i)}_{\mathrm {flpud}}$ , ${\Pi }^{(i)}_{\mathrm {rot180}}$ and their transposed versions. The merging method shown in Fig. 2 is based on the symmetries between ${\Pi }^{(i)}$ and its flipped, rotated, and transposed versions.

8. Concatenate all the symmetrized histograms to form our improved PHARM.

FIGURE 3.

The diagram of extracting our PHARM.

All parameters in our PHARM except $S$ , $N$ and $\nu$ are the same as in the original PHARM. In our PHARM, the maximum projection matrix size $S = 4$ , the number of projections per residual $\nu = 300$ , the number of selected phase pairs per projection $N = 3$ , the quantization step $q = \frac {65}{4} - \frac {3}{20}QF$ , and the histogram threshold $T = 2$ . This parameter setup keeps the dimension of our PHARM, $7 \cdot T \cdot N \cdot \nu = 7 \cdot 2 \cdot 3 \cdot 300 = 12600$ , the same as that of the original PHARM.

SECTION IV. Experiments

In the previous section, we introduce our improved PHARM in detail. Unlike the original PHARM, the maximum size of projection matrices is reduced from eight to four ($S = 4$ ), three different phase pairs are randomly selected per projection to compute phase-aware histograms ($N = 3$ and $\nu = 300$ ), and the transposition symmetry is employed in our PHARM. In this section, our PHARM is subjected to experiments to demonstrate its efficiency and effectiveness. In these experiments, 10000 grayscale images of size 512 from BOSSbase 1.01 are converted into JPEG images with quality factors 75 and 95 as cover images. The adaptive steganographic schemes UED-JC and J-UNIWARD are used to generate stego images with different embedding rates. The detection accuracy is quantified by ${P}_{\mathrm {E}}$ (4), and the $\overline {P}_{\mathrm {E}}$ is averaged over ten random 5000/5000 dataset splits. The feature extraction time is measured on the same computer used in the previous experiments.

From Table 4, it can be seen that compared to the original PHARM, our new PHARM of the same dimension provides better detection performance regardless of quality factors, steganographic algorithms and embedding rates. Compared to GFR of larger dimensions, surprisingly, our improved PHARM also has higher detection accuracy for different conditions. For example, for quality factor 95 and J-UNIWARD at 0.4 bpnzac, compared to DCTR, PHARM and GFR, the detection accuracy of our improved PHARM can rise by 6.62%, 5.31%, and 1.44%, respectively. For quality factor 75 and UED-JC at 0.2 bpnzac, with respect to DCTR, PHARM and GFR, the improvements of our PHARM are 7.86%, 3.88%, and 2.41%, respectively. We also find that although the parameters of the improved PHARM are determined according to the detection performance for J-UNIWARD at 0.2 bpnzac and quality factor 75, they are effective for different conditions.

TABLE 4 Detection Error $\overline{P}_{\mathrm{E}}$ for UED-JC and J-UNIWARD for Quality Factors 75 and 95 When Steganalyzed With DCTR, Original PHARM, GFR, and Our Improved PHARM

The efficiency of feature extraction is also important to the practical steganalysis application. For a $512\times 512$ JPEG image, the time needed to extract the original and improved PHARM is 31.5 s and 20.1 s, respectively. The improved PHARM is more efficient than the original PHARM, only taking about two thirds of the computation time of the original PHARM.

To sum up, in contrast to the original PHARM, our improved PHARM not only offers better detection accuracy but also needs less calculation time.

SECTION V. Conclusion

In this paper, we modify the original PHARM feature set for less computation time and higher detection accuracy. There are three improvements in our proposed PHARM. First, the maximum projection matrix size is reduced from eight to four, which not only lowers the computational complexity but also better captures the slight changes in the adaptive steganograghy. Second, instead of only one phase pair, three different ones per projection are randomly selected to compute phase-aware histograms and the number of projections per residual correspondingly decreases from 900 to 300, which reduces the computation time dramatically. Third, the transposition symmetry is taken into consideration, which further improves the detection accuracy while preserving the feature dimensionality. With these three improvements, our improved PHARM takes roughly two thirds of the computation time of the original PHARM but achieves higher detection accuracy than DCTR, original PHARM, and even GFR of larger dimensions.

The future work will focus on the following several aspects. First, the knowledge of the selection channel can be incorporated within the improved PHARM to further boost the detection of adaptive steganography. Second, a novel weighted histogram scheme proposed in [19] can also be applied to our PHARM to make it more powerful. Third, random projections can be used for preprocessing in CNN based steganalysis. Fourth, we can try to use the linear filters introduced in [20]–​[22] for residual generation in PHARM. Fifth, in order to make our improved PHARM more usable in practice, we can implement it on a GPU device, similarly to GPU-PSRM [23], GPU-SRM and GPU-DCTR [24]. Sixth, as an efficient and effective scheme in JPEG image steganalysis, our improved PHARM can also inspire the design of novel steganalytic algorithms for HEVC and 3D-HEVC videos [25]–​[27].