Steganography in 3D Geometries and Images by Adjacent Bin Mapping
© H.-T.Wu and J.-L. Dugelay. 2009
Received: 31 July 2008
Accepted: 6 February 2009
Published: 23 March 2009
A steganographic method called adjacent bin mapping (ABM) is presented. Firstly, it is applied to 3D geometries by mapping the coordinates within two adjacent bins for data embedding. When applied to digital images, it becomes a kind of LSB hiding, namely the algorithm. In order to prevent the detection using a metric named histogram tail, the hiding is performed in a pseudorandom order. Then we show that the steganalytic algorithms based on histogram characteristic function (HCF) can be prevented by implementing the algorithm on subsets of pixels having the same neighbor values. The experimental results show that important high-order statistics of the cover image are preserved in this way while little distortion is introduced to 3D geometric models with an appropriate bin size.
Steganography, the art of covert communication by hiding the presence of a message typically in multimedia content, has attracted the interests of researchers (e.g., [1–4]). Although the early steganographic methods can imperceptibly embed data into a cover object, traces of data embedding can be found within the characteristics of the stego objects. In the last decade, the technique of steganalysis (e.g., ) has been developed for the detection of hidden data. It has been shown by the novel steganalytic algorithms and detection-theoretic analysis that several hiding methods are detectable. Therefore, how to prevent the hidden message from being detected is a central topic of steganography research.
Most of the steganalytic algorithms (e.g., [6–21]) exploit statistical characteristics of the stego objects to detect the existence of hidden message. For instance, the (chi-squared) technique  and Provos' stegdetect  calculate the number of pixels whose values differ only in the least significant bit (LSB) to detect random LSB hiding. Furthermore, the occurrence of a pair of spatially adjacent pixels is counted for steganalysis of random LSB hiding in the regular/singular (RS) scheme  and more theoretical sample pair analysis (SPA) . By modeling the hiding process as additive noise, histogram characteristic function (HCF) is introduced in  to detect LSB, spread spectrum, and discrete cosine transform (DCT) hiding methods. Two ways of applying HCF are further proposed in  to detect the LSB matching steganography in gray-scale images. The detection-theoretic analysis for steganalysis can be found in [12, 13] for the block-based embedding in the Gaussian random covers and by modeling the cover as a Markov chain, respectively. Moreover, features such as image quality metrics  and the high-order statistics [15–17] are used through supervised learning to detect the arbitrary hiding scheme.
To avoid being detected by the steganalytic algorithms, quite a few algorithms are designed to preserve the statistics of the cover object. An early attempt is the F5 algorithm , in which some characteristics in the histogram of DCT coefficients are preserved to prevent attack . However, it is broken by the detector designed by Fridrich et al.  by estimating the cover histogram from the suspected image for comparison. In Provos' Outguess , part of JPEG coefficients are used to repair the modified histogram due to data embedding. But the changes at the block boundaries can be used for detection because the embedding is performed in the blockwise transform domain . A method attempting to preserve the histogram after LSB hiding is further presented by Franz , where a message that mimics the imbalance between the adjacent histogram bins is embedded. But the asymmetric embedding process determined by a cooccurrence matrix can be exploited for steganalytic attack, as shown in . Similarly, Eggers et al. propose a histogram-preserving data-mapping (HPDM) method  by embedding a message with the same distribution as the cover object. However, it is shown by Tzschoppe et al.  that HPDM can be detected by Lyu and Farid's steganalytic method  because higher-frequency components have not been separately treated from lower-frequency ones. So a histogram restoration algorithm is proposed in  without embedding in the low-probability region, and further adopted to preserve some second-order statistics in .
The model-based steganography  provides a new perspective by generating a stego object with a given distribution model. However, due to the lack of a perfect model, the steganographic algorithm using generalized Cauchy distribution can be broken by using the first-order statistics, that is, the measures without considering the interdependencies between observations, such as mean and variance . In our preliminary work , a new steganographic method is proposed to preserve the marginal distribution of a cover inherently, which is called adjacent bin mapping (ABM) hereinafter. In this paper, we apply ABM method to three-dimensional (3D) geometric models by mapping the coordinates within two adjacent bins for data embedding. When applied to digital images, it becomes a sort of LSB hiding, namely, the algorithm. For image steganography, we analyze one case that the algorithm is detectable by defining a high-order metric named histogram tail. And we try to prevent the detection by performing the hiding in a pseudorandom order. To prevent SPA steganalysis , the algorithm has been implemented on subsets of pixels having the same four neighbor values (left, right, up, and down), as shown in . In this paper, we show that the steganalytic algorithms in  to detect LSB matching steganography can be prevented by performing the algorithm on subsets of pixels having the same five neighbor values (i.e., left, right, up, down, and up-right, denoted by 5-N in short). The experimental results show that several important statistics of a cover image are preserved in this way, while little distortion is introduced to the virtual reality modeling language (VRML) models with an appropriate bin size.
The rest of this paper is organized as follows. In the next section, the ABM method is reviewed, and its application to geometry steganography is proposed. In Section 3, the algorithm is presented, and we try to prevent the histogram tail detection and the steganalytic algorithms based on HCF, respectively. The experimental results are given in Section 4. Finally, a conclusion is drawn in Section 5.
2. Adjacent Bin Mapping for Steganography
In this section, the data mapping method proposed in  is reviewed, which is called adjacent bin mapping (ABM) hereinafter. One important property of the ABM method is that it preserves the marginal distribution of a cover inherently. Other properties include the applicability to a variety of cover objects (e.g., represented by integers, floating or fixed point numbers) as well as the relative simplicity of both encoding and decoding.
2.1. The Adjacent Bin Mapping Method
The decoding process is much simpler: given the same scanning order as in the embedding process, the bit values can be extracted from the element positions (i.e., in the left or right bin) one by one. The extracted bit value will be 0 if an element is located in the left bin, or 1 if it is in the right one. For each embedding unit, once all elements in one bin (left or right) have been used up, the extraction process is finished. For example, the bit values that can be extracted from the Unit in Figures 2(b) and 3(b) are not "10011010011" but "100110100". Since the embedding and extraction operations in one unit do not interfere with those performed in other units, the operations in every embedding unit can be carried out in parallel. So both encoding and decoding processes can be performed according to the scrambled indices of all elements with a secret key shared by the sender and receiver.
The hiding rate is maximized if the maximum number of 0s or 1s are embedded. A parameter can be used to adjust the hiding rate, that is, the embedding process stops once the number of embedded bits reaches a fraction of the amount originally in one bin (left or right). Accordingly, the same value of should be used in the extraction process. Suppose there are and elements in the two bins of an embedding unit. Without loss of generality, we assume that is always inferior to , then the minimum and maximum amount of bits that can be embedded are and . With the parameter , the low and upper bounds of capacity in that unit will be and bits, where represents the ceil function. So the hiding rate can be adjusted with the parameter , which should be shared by the sender and receiver.
2.2. Steganography in 3D Geometries Using the ABM Method
In literature, a majority of steganography research has been conducted on digital images for their popularity. With the development of 3D scanning and modeling techniques, more and more 3D models have been used for geometry representation. With the dissemination such as using the virtual reality modeling language (VRML)  to represent 3D graphics on the Web, 3D models have become potential covers for covert communication. In the following, the ABM method is applied to 3D geometry with coordinates.
Given a gray-scale image, its histogram is calculated by counting the pixels with the same value, that is, the amount of pixels within every bin. Since the operations in one embedding unit are independent from those in the other units, we only discuss the operations in an arbitrary unit. In the normal LSB hiding, a string of bit values are used to replace the LSBs of pixel values. The histogram of cover image is probably changed due to the randomness of embedded data. Obviously, the histogram will be preserved if the amount of pixels within each bin is unchanged. So we constrain the replacement operations in the algorithm. As discussed previously in the general method, the key idea is that the number of embedded 0s and 1s should not exceed the original ones in the LSBs. Suppose that there are and pixels originally in the left and right bins of a unit, the time of embedding 0 should be no more than , and the time of embedding 1 should not exceed , respectively. Once there are 0s (or 1s) having been embedded, all the rest LSBs should be replaced with 1s (or 0s). In this way, the amounts of 0s and 1s in the LSBs are unchanged by data embedding. In the decoding process, the embedded bits are extracted one by one in the same order as in the embedding process. The extraction process is finished as soon as all LSBs in one bin (either left or right) have been extracted. Since part of the LSBs are used to repair the cover histogram, a portion of capacity is sacrificed.
3.2. The Histogram Tail Detection
3.3. Preventing the Steganalytic Algorithms Based on HCF
where is the HCF, , and is the DFT length. For gray-scale images, . Since the algorithm does not change the cover histogram, the HCF and COM of cover image are both preserved. Therefore, the steganalytic algorithms that are simply based on the COM of HCF (HCF-COM) are prevented.
In , two ways of applying the HCF are further proposed to detect the LSB matching steganography in the gray-scale images. The first algorithm downsamples a suspected image by a factor of two in both dimensions using an averaging filter. Then the downsampled image is used to calibrate the HCF-COM of the full-sized image. It is observed that for the presence of LSB matching steganography, the HCF-COM of the full-sized image is more affected than the one of the downsampled image. As for an image without the hidden data, HCF-COMs of the downsampled and full-sized images are roughly the same. In the second algorithm, the two-dimensional adjacency histogram is used instead of the standard one for steganalysis by considering one horizontal neighboring pixel. Since the adjacent pixels tend to have close intensities, the adjacency histogram is sparse off the diagonal.
To preserve the adjacency histogram as suggested in , the left and right neighbor values of every pixel in a selected subset should be the same. If the two-dimensional adjacency histogram is calculated vertically, the pixel values up and down the current one should also be the same. So we perform the hiding on the subsets of pixels having the same five neighbor values (left, right, up, down, and up-right, denoted by 5-N in short) as shown in Figure 10, where the pixels marked in black are chosen as the neighbors of others, that is, only the light-colored pixels are grouped into a subset if they have the same five neighbor values. As for the light-colored pixels in the leftmost column and in the bottom row, only four neighbor values are considered so that they are separately treated, respectively.
By implementing the algorithm in the 5-N way, the histograms of cover image and its downsampled version, the adjacency histogram of cover image, are all preserved. As a result, HCF-COMs of the full-sized and downsampled images, the two-dimensional COM based on the adjacency histogram, are unchanged by the hidden data. So the steganalytic algorithms in  to detect the LSB matching steganography and the SPA steganalysis in  to detect the random LSB hiding are prevented in principle. Moreover, all the steganalytic algorithms using the first-order statistics of cover image are not efficient because the marginal distribution is inherently preserved by the algorithm.
4. Experimental Results
4.1. Steganography in 3D Geometries
With the ABM method, steganography in the cover object represented by floating point numbers is enabled, such as 3D geometrical models with coordinates. Since the previous steganalysis archives are mainly dedicated to images, techniques to detect the hidden data in the other multimedia content are still rare. A secret key shared by the sender and receiver can be used to scramble the element indices to perform the hiding in a pseudorandom order. Since the bin size can be adaptively chosen for the cover object represented by the floating point numbers, it can also be used as a secret key to decode the hidden message from the stego object.
4.2. Steganography in Images
As shown in Table 2, the PSNRs of the stego images are all above 60 (dB) when the algorithm is implemented in the 5-N way with . Not surprisingly, the PSNR is higher when less bits are hidden in a stego image. From the experimental results, it can be seen that the capacity varies from one image to another. For a cover image consisting of many pixels having the same neighbor values, the hiding rate is high. Otherwise, for a cover image such as "Fall" in which this is hardly the case, only a few bits can be embedded. As shown in Figure 10, only one out of four pixel values is possible to be modified if the algorithm is implemented in the 5-N way. In our experiments, the hiding rate is normally no more than 0.06 bit/pixel. Compared with applying the algorithm in the 4-N way (left, right, up, and down) , the capacity in the 5-N way is lower because the requirement on the neighbor values of pixels within a selected subset is stricter, as shown in Table 2.
Meanwhile, the adjacency histogram was also preserved by applying the algorithm in the 5-N way, so that the steganalytic algorithms in  and the SPA steganalysis in  are both prevented. Furthermore, histogram tail of the cover image in the raster order was rarely changed. For the six images listed in Table 2, the experimental results show that the histogram tail in the raster order was unchanged by the hidden message. However, it is not yet possible to claim that the proposed algorithm is practically secure before other steganalysis algorithms using the high-order statistics would have been tested. Recently, high-order statistical features have been used by supervised learning for steganalysis; our future work includes to investigate if the proposed algorithm can resist those blind learning-based algorithms (e.g., ).
In this paper, we have presented the adjacent bin mapping (ABM) method for steganography and applied it to 3D geometrical models. By choosing an appropriate bin size, little distortion has been introduced to the VRML models to hide a secret message. Therefore, how to detect the secret message hidden in 3D geometries should be further investigated as well as in other covers represented by floating point numbers.
When applied to the gray-scale images, the ABM method becomes a kind of LSB hiding, namely, the algorithm. The histogram tail has been defined to detect the hiding in the raster order, and we have avoided the detection by performing the hiding in a pseudorandom order. To prevent the steganalytic algorithms in  to detect the LSB matching steganography, the pixels with the same five neighbor values (i.e., left, right, up, down, and up-right) have been grouped into each subset. It has been shown that several high-order statistics are preserved by applying the algorithm on the selected subsets of pixels. Our future work is to investigate if the proposed algorithm also resists to the blind learning-based steganalysis (e.g., ).
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