I Introduction
Steganography is the science and art of covert communication without drawing suspicion from Warden [1]. With the rapid development of multimedia information technology, the image steganography technology and its applications [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] have also made great progress in the past decade. And among them, the contentadaptive JPEG image steganography is the most effective and practical one which conceals secret messages in quantized DCT coefficients.
With the emergence of breakthrough coding method–STCs (SyndromeTrellis Codes) [8], all the prevailing JPEG steganographic schemes are focused on the design of effective distortion function. Recalling the development of JPEG steganography in the past decade, it is observed that the mainstream design scheme for JPEG steganographic distortion function (additive distortion function), e.g., JUNIWARD [11], UERD [12], BET [7], GUED [6] and etc., can be summarized as the framework that =, where is the Intrablock distinguish Factor which indicates intrablock embedding costs for different modes, and is the Interblock distinguish Factor which suggests interblock embedding costs for different DCT blocks. The value of embedding cost determines the level of embedding priority, that is, smaller embedding cost will arise higher embedding priority, and vice versa. Based on this framework, numerous researchers have devoted themselves to designing better and .
In terms of the design for , all the mentioned steganographic distortion function adopt the quantization table (or its derivant) of JPEG cover. As far as the design for , it is the most striking difference among them, which can be summarized into two categories, i.e., design from DCT domain and design from spatial domain. For the first category, UERD is the typical one which adopts the block energy, i.e., the sum of the absolute value of dequantized DCT coefficient within the DCT block. However, this category method does not take into account the changes of spatial domain statistical characteristics after embedding, which can be easily detected by the JPEG phase feature sets based steganalyzers, e.g., DCTR [13] and GFR [14]. In view of this defect, the second category method that design from spatial domain came into being, such as JUNIWARD, GUED, BET and etc.. The JUNIWARD constructs utilizing the wavelet residuals obtained by filtering the decompressed cover using the Daubechies 8tap wavelet directional filter bank, and similarly, GUED designs
based on the Gabor residuals. Peculiarly, BET uses a domain transformation of embedding entropy scheme that transforms the block embedding entropy of the spatial domain into DCT domain, and then derives the JPEG embedding change probability based on a white noise embedding assumption, afterwards the corresponding
, i.e., JPEG block embedding cost, can be conversely derived according to the method in [15]. Experimental results show that BET is superior to UERD and JUNIWARD in resisting the detection of GFR. However, the fly in the ointment is that the procedure of BET is so complicate, including two rounds transformation for distortion cost, modification probability and entropy, which will limit its practical applicability.Motivated by the defect of BET, this paper proposes a domain transformation of embedding cost scheme. In contrast with BET, the proposed scheme no longer needs to calculate the mediums in the construction of distortion function of BET, i.e., embedding modification probability and embedding entropy, and instead, a embedding cost domain transformation function is introduced to directly transform the spatial embedding cost into JPEG domain by means of a linear weighted method. The proposed f comprises of the distinguish factors of intrablock and interblock , which are the pixel block embedding changes and the pixel block embedding cost, respectively. In order to further maintain the statistical undetectability of the stego in both DCT and spatial domains, an exponential model is then introduced to facilitate the construction of in terms of . Extensive experiments show that the proposed scheme has a more comprehensive performance improvement than UERD with the same computational complexity, and is superior to JUNIWARD and GUED in resisting the detection of GFR and SCAGFR [16]. Besides, it also rivals BET with an order of magnitude lower computational complexity.
The remainder of this paper is organized as follows. In the next Section, we mainly introduce the construction of embedding cost domain transformation function, which are followed by the experimental results and analysis in Section III. Finally, the paper is concluded in Section IV.
Ii The proposed block embedding cost transformation scheme
Iia The new interblock distinguish factor
As we know, the essence of and are to distinguish the embedding priorities of different DCT blocks and differentDCT modes within the same DCT block, respectively. Specifically, the smaller the value of distinguish factor is, the higher the corresponding embedding priority will be, and vice versa. In addition, the difference in security performance among different JPEG steganographic schemes under the same framework of minimal distortion embedding is due to the difference on the selection of DCT coefficients for embedding. Therefore, the following consensus can be easily obtained: 1) if the distinguish factors designed by different schemes have a high degree of similarity in evaluating the embedding priority, then they will be in the similar embedding security level, and vice versa, and 2) if the embedding security level of a steganographic scheme is higher, then the evaluation of embedding priority by this scheme will be more reasonable, and vice versa.
In our proposed scheme, we intend to replace the block embedding entropy in BET with pixel block embedding cost of the decompressed cover, which is referred to as the new Interblock distinguish Factor (). With regard to the method of calculating the spatial embedding cost, there are many choices, such as SUNIWARD [11], Hill [9], MiPOD [15] and etc.. While, to the best of our knowledge, the Hill may be the best one, due to its excellent security performance and minimal computational complexity ^{1}^{1}1We also test other spatial steganographic algorithms in BCT, while they are all inferior to Hill, which ulteriorly verifies the above consensus.. To verify the feasibility of the proposed in evaluating the DCT block embedding priority, we then make a simple experiment in the following.
Without loss of generality, we randomly select 2000 cover images from BOSSbaseJ75 and BOSSbaseJ95^{2}^{2}2Compress the images in BOSSbase ver1.01 [17] using JPEG Toolbox [18] with =75 and =95. separately, and then use UERD, JUNIWARD, GUED and Hill to calculate the embedding cost of each (DCT or pixel) block within each cover. It should be noted in this experiment that the block embedding cost under Hill is expressed by the sum of 64 pixels’ embedding costs within this block, while for JUNIWARD, the block embedding cost is expressed by the reciprocal sum of absolute value of wavelet filter residuals in three directions corresponding to this block. As referred before, the embedding priority is determined by the embedding cost, thus, the similarity of block embedding priority among different steganographic schemes can be indirectly obtained by calculating the similarity of their block embedding costs. As far as the choice of metric for similarity, we adopt the Spearman Correlation Coefficient (SCC) [19], which is one of the three popular statistical correlation coefficients, corresponding to the ‘corr’ Matlab command with type ‘Spearman’. The magnitude represents the degree of correlation (0 is irrelevant, 1 is completely linear relevant). Finally, the average Spearman Correlation Coefficients (SCCs) over 2000 cover images for Q75 and Q95 are summarized in Table I.
Different schemes  Q75  Q95 

SCC(UERD,Hill)  
SCC(JUNIWARD,Hill)  
SCC(GUED,Hill) 
Comparing the results of UERD with JUNIWARD, it is observed that JUNIWARD is closer to Hill than UERD in evaluating the block embedding priority along with higher security performance towards effective steganalyzer GFR. And similar situation also occurs in JUNIWARD and GUED. Furthermore, referring to the performance in BET [7] and GUED [6], it is then observed that BETHill is superior to GUED in resisting the detection of GFR, where the of BETHill (i.e., pixel block embedding entropy) is obtained by Hill. In this regard, it is convinced that if the evaluation of block embedding priority of a JPEG steganographic scheme is closer to Hill’s, then it would be more secure. Go for a step further, if we directly choose the block embedding cost calculated by Hill to evaluate the DCT block embedding priority, then it would yield optimal security performance in this case. Of course, if there are some other better spatial steganographic algorithms can substitute Hill, we believe that the similar conclusions will also be obtained as discussed with regard to Hill.
IiB The new intrablock distinguish factor
It is well known that when we arbitrarily modify a DCT coefficient (mode () in the () DCT block) in JPEG cover image, the corresponding spatial embedding changes can be easily derived by its inverse DCT transformation. Since JPEG compression is based on block DCT transformation, then the corresponding spatial embedding changes will just occur within its corresponding pixel block, which is only associated with its quantization step . In this way, the relationship between the DCT domain embedding modification and the spatial embedding changes can be expressed as:
(1) 
where represents the solitary modification on mode , and demonstrates the resultant corresponding spatial pixel block embedding changes, which is referred to as the new Intrablock distinguish Factor ().
IiC The construction of the proposed embedding cost domain transformation function
Since the proposed new is made to be the pixel block embedding costs, and is represented by the corresponding pixel block embedding changes when modifying each DCT mode, thereby, we consider it as a matter of course to use the to perform a linear weighted operation on the , and then accumulate the weighted results to obtain the JPEG embedding costs. Therefore, the proposed scheme can be referred to as Block embedding Cost Transformation (BCT) scheme, and the corresponding embedding cost domain transformation function is represented as:
(2) 
where represents the embedding cost of the pixel in the spatial block when modified by magnitude 1, indicates the resultant spatial embedding change magnitude of the pixel in the corresponding spatial block when modifying mode by magnitude 1, and is the JPEG embedding cost of mode in the DCT block.
It is noted that the spatial adaptive steganographic schemes, e.g., SUNIWARD, Hill, MiPOD and etc., are all prone to embed messages in texture regions of the cover, i.e., midtohigh frequency regions of the cover. Therefore, the proposed BCTHill scheme will have a tendency to modify more midtohigh frequency DCT coefficients in JPEG cover for data embedding. In other words, the proposed BCTHill scheme will migrate a part of embedding modifications to midtohigh frequency DCT coefficients of complex DCT block, so as to weaken the change of spatial statistical characteristics of cover as far as possible, while it should be moderate, otherwise terrible spatial embedding changes would be yielded, especially at small QFs. As thus, how to control the number of migration of embedding modifications will become an urgent problem to be solved. Since the in the proposed is totally determined by the quantization table, that the embedding priority of the lowfrequency mode is naturally higher than the one of the highfrequency mode, thus we can only anchor our hope on the proposed . As the proposed is constructed by the spatial block embedding costs , and we find that by reducing the relative magnitude of d in midtohigh frequency region, much more messages would be prone to be embedded in the corresponding DCT block, thereby increasing the number of modifications on the midtohigh frequency modes, and vice versa. In this regard, we then make the spatial embedding cost as the exponential form with parameter , and by which we desire to adjust flexibly the relative magnitude of , thus the proposed can be rewritten as:
(3) 
Iii Experimental results and analysis
Iiia Experiment setups
All the experiments in this Section are carried out on the image dataset BOSSbase ver1.01 [17] at =75 and =95, and for each, one half of them are used for training, while others for testing. Several stateoftheart universal and typical JPEG steganalyzers, including CCJRM22,510D [20], GFR17,000D [14], and the selectionchannelaware version of GFR (SCAGFR17,000D [16]
), are employed to evaluate the empirical security performance of the tested JPEG steganographic schemes, where the binary classifier is trained by the Fisher Linear Discriminants (FLD) ensemble
[21] with default settings. The classification error probability of FLD ensemble classifier, corresponding to the empirical security performance of the tested JPEG steganographic scheme, is reported by the mean value of the ensemble’s testing errors based on ten times of randomly testing and all the experiments are simulated at the corresponding payload distortion bound for relative payloads bpnzAC.IiiB Determining the optimal in BCTHill
Since the parameter in the exponential model can adjust the migration of embedding changes to the midtohigh frequency DCT coefficients, thus there should be an optimal setting for given steganalyzer, and relative payload. To determine the optimal , we randomly select 5,000 images with given , and of these 2,500 JPEG images are used for training, while others for testing. We set in range of [0.3,1.5] with step 0.1, and then intentionally search on this interval to find the corresponding to the maximum classification error probability at given relative payload for each of the three tested steganalyzers. Finally, the optimal parameters for =75 and =95 are summarized in Table II, which are irrespective of relative payloads.
optimal  Steganalyzer  

CCJRM  GFR  SCAGFR  
Q75  
Q95 
IiiC Security performance comparison and analysis
We then compare the security performance of the proposed BCTHill with UERD, JUNIWARD, GUED and BETHill under different relative payloads and on BOSSbaseJ75 and BOSSbaseJ95, which are summarized in Table III and IV, respectively. For brevity, all the data (except BCTHillpro) in Table III and IV are obtained under the optimal parameter setting for SCAGFR (i.e., =0.5 and =0.9 for Q75 and Q95, respectively.). This is because SCAGFR is the most effective steganalyzer and the performance of the proposed BCTHill won’t vary much for other steganalyzers, as has been validated by our experiments.
Referring to the results in Table III and IV, it is observed that under the detection of SCAGFR17000D, the proposed BCTHill shows an overall superior performance than UERD, JUNIWARD and GUED. In addition, BCTHill also consistently outperforms BETHill by a clear margin (improvements can reach 1.4%2.1% on average) under =75, and rivals BETHill under =95.
While for the detection of steganalyzer GFR, the situation is slightly different. Although our proposed BCTHill shows excellent security performance compared with the other tested schemes under =95, it doesn’t perform equally well under =75. We analyse that it may be due to the following reasons. The first is the suboptimal parameter setting for GFR. We then simulate BCTHill under the optimal parameter setting for GFR under =75 (i.e., =0.7), and the security performance is indeed improved as illustrated in Table III (BCTHillpro). The second may be the intuitive consideration in Eq. (3) that the spatial embedding distortion cost increase linearly with the magnitude of pixel embedding change.
Steganalyzer  Scheme  Relative payload (bpnzAC)  
0.1  0.2  0.3  0.4  0.5  
CCJRM  UERD  
JUNIWARD  
GUED  
BETHill  
BCTHill  
BCTHillpro  46.62  40.10  32.25  24.10  17.02  
GFR  UERD  
JUNIWARD  
GUED  
BETHill  
BCTHill  
BCTHillpro  41.30  29.90  19.61  12.06  6.80  
SCAGFR  UERD  
JUNIWARD  
GUED  
BETHill  
BCTHill 
Steganalyzer  Scheme  Relative payload (bpnzAC)  

0.1  0.2  0.3  0.4  0.5  
CCJRM  UERD  
JUNIWARD  
GUED  
BETHill  
BCTHill  
GFR  UERD  
JUNIWARD  
GUED  
BETHill  
BCTHill  
SCAGFR  UERD  
JUNIWARD  
GUED  
BETHill  
BCTHill 
In particular, the situation is dramatically different for the detection of CCJRM that both BCTHill and BETHill are inferior to JUNIWARD and GUED. We infer that it may be due to the more embedding modifications on midtohigh frequency modes than JUNIWARD and GUED, which would make the embedding traces in BCTHill and BETHill be exposed easier to steganalyzer CCJRM. To validate this inference, we remove the integral components of CCJRM, which are sensitive to the changes of the statistics of midtohigh frequency modes, and the resultant feature set is denoted as cropCCJRM. Immediately, applying the cropCCJRM17,270D to detect JUNIWARD, GUED, BETHill and BCTHill at 0.4 bpnzAC under =75 and =95, and the comparison results are collected in Table V. It shows that the security performance improvements of the proposed BCTHill can reach 2.42% and 2.85% at =75 and =95, respectively. And a similar circumstance also happens in BETHill. While for JUNIWARD and GUED, the improvements are marginal. Besides, similar to GFR, we then simulate BCTHill under the optimal parameter setting for CCJRM under =75 (i.e., =0.7), and the results show that although the security performance of BCTHill has been improved, as shown in Table III (BCTHillpro), it still inferior to JUNIWARD and GUED by a clear margin. To this end, an important remark can be drawn from these results: if we improve the JPEG steganography performance on resisting the detection of JPEG steganalyzers derived from spatial domain (e.g., GFR) through migrating the embedding modifications to midtohigh frequency DCT coefficients, then the risk of being detected by the steganalyzers derived from DCT domain (e.g., CCJRM) is likely to increase.
As a supplementary, we ulteriorly test the applicability of our proposed scheme for other datasets (other image dataset and QFs), and the results show that it is also applicable. Due to the limitation of paper length, we can not give out the specific experimental results here.
Scheme  CCJRM  cropCCJRM  

Q75  Q95  Q75  Q95  Q75  Q95  
JUNIWARD  
GUED  
BETHill  
BCTHill 
IiiD Practical evaluation of computational complexity
For this subsection, we aim at further evaluating the cost of our proposed BCTHill compared to UERD, JUNIWARD, GUED and BETHill in terms of computation time (CmpTime). Note that all the tested steganographic schemes are simulated under the same framework of minimal distortion embedding, thereby the main difference among them is the their distortion functions, so we only choose to test their practical computation times in calculation of distortion costs.
In our implementation, we calculate the average CmpTimes of the distortion costs for UERD, JUNIWARD, GUED, BETHill and BCTHill, over 2,000 JPEG images randomly selected from BOSSbaseJ75 and BOSSbaseJ95, respectively, using MATLAB 8.2 on a 3.0 GHz Intel Core i57400 CPU with 8GB memory. Finally, the results are listed in Table VI. It is observed that: 1) the CmpTime of BCTHill is greatly reduced than BETHill, GUED and JUNIWARD; 2) the computation of BCTHill could be implemented in a quite affordable time cost as little as UERD.
Average time consuming (s)  
UERD  JUNIWARD  GUED  BETHill  BCTHill  
Q75  
Q95 
Iv Conclusion
In this paper, a new JPEG steganographic scheme BCTHill using domain transformation of embedding cost is presented. The proposed BCTHill intends to transform the spatial block embedding cost in alignment with the DCT block into DCT domain by incorporating the proposed embedding cost domain transformation function . The proposed consists of the new proposed intrablock distinguish factor and interblock distinguish factor , which are obtained via the pixel block embedding changes and the pixel block embedding costs, respectively. For further maintaining the statistical undetectability of the stego, an exponential model is then introduced to facilitate the construction of in terms of . As for the determination of the parameter of exponential model, we search on a given interval to find the optimal which yields the maximum classification error probability for different and JPEG steganalyzers. Extensive experiments are carried out, which demonstrate that the proposed scheme has a more comprehensive performance improvement than UERD with the same computational complexity, and is superior to JUNIWARD and GUED in resisting the detection of GFR and SCAGFR. Moreover, the proposed scheme can rival BETHill with an order of magnitude lower computational complexity as well. Although the BETHill and BCTHill are inferior to JUNIWARD and GUED on resisting the detection of CCJRM, it gives us an important remark w.r.t JPEG steganography, as shown in Section IIIC. Incidentally, our proposed scheme broadens the application of spatial steganography, and by this way, not only the security of JPEG steganography would be improved based on a better spatial distortion function, but also the research of spatial steganography will become more meaningful.
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