A Formula Adaptive Pixel Pair Matching Steganography Algorithm

Pixel pair matching (PPM) is widely used in digital image steganography. As an important derivation, adaptive pixel pair matching method (APPM) offers low distortion and allows embedded digits in any notational system. However, APPM needs additional space to store, calculate, and query neighborhood set, which needs extra cost. To solve these problems, a formula adaptive pixel pair matching (FAPPM) method is proposed in this paper. The basic idea of FAPPM is to use the formula to get the stego image pixel pair without searching the neighborhood set for the given image pixel pair. This will allow users to embed secret message directly without storing and searching the look-up table. Experimental results and analysis show that the proposed method could embed secret data directly without searching the neighborhood sets by using a formula and it still maintains flexibility in the selection of notional system, high image quality, and strong anti-steganalysis ability.


Introduction
Information hiding is a technology of embedding secret data into the media for covert communication [1].With the rapid development of Internet, a large number of data are transmitted over the Internet.At present, the main media using for data hiding includes images, audio, and video, where digital image is the most widely used media [2].Researchers have shown a great interest in image steganography for the last decade [3].LSB replacement [4] is one of the most commonly used steganographic techniques, which makes full use of the characteristics that the human visual system is not sensitive to small changes in pixels and the negligible contribution of the low bit plane of the pixel to the image quality.However, this method can only add 1 or remain unchanged for the even pixels and can only decrease 1 or remain unchanged for the odd pixels.Therefore, this unbalanced embedding distortion leads to the histogram attack to the images [5,6].Chan et al. [7] proposed an optima pixel adjustment process (OPAP) method, which adjusted the pixels to reduce the distortion caused by least significant bit (LSB) embedding.The LSB and OPAP methods both employed one pixel as an embedding unit to embed secret message.As the development of steganography, methods using two or more pixels as a basic unit for B-ary secret information embedding were put forward.This kind of stenographic algorithm can improve the embedding capacity and image quality by subtle modifying the pixel.
In 2006, Miekikainen [11] proposed a LSB matching method.It employed two pixels as embedding unit.In this method, when payload was 1 bit per pixel, the mean square error (MSE) is 0.375, while the MSE of LSB [4] was 0.5.Zhang and Wang [12] proposed exploiting modification direction (EMD) method, which added and subtracted 1 in one pixel and embedded 2n + 1-ary secret message in n pixels.When n = 2, a quinary number was embedded in each pair of pixels.The capacity can reach the maximum (1/2)log 2 5 = 1.161 bit per pixel (bpp).Chao et al. [13] extended this method and proposed a diamond encoding (DE) method.It can embed 2k 2 +2k+1-ary information to each pair of pixels and achieve high embedding efficiency by adding and subtracting 1 operation in n pixels.In [8], the author used a codebook to  improve the EMD scheme, and one secret (2 n+x -1)-ary digit was hidden in a group of pixels in an image as a modified secret digit.In [9], the authors proposed a method to modify a group of pixels by ±1 to embed a secret digit, but it is only applicable to 3 n -ary notational system.Kuo et al. [14] proposed a formula diamond encoding (FDEMD) data hide scheme, and it could conceal a digit in (2k 2 +2k+1)-ary system.It simplified the embedding procedure and embedded secret data without storing and calculating characteristic value matrix.Hong et al. [10] designed a new extraction function and new neighborhood set of two pixels called adaptive pixel pair matching (APPM).It allowed embedding digits in arbitrary notational system and the distortion caused by embedment using APPM was minimized; therefore the resultant marked image quality could be well preserved [15].In [16], secure adaptive pixel pair matching (SAPPM) was proposed to hide multiple data types such as text, image, and audio which incorporated cryptography along with steganography.A transformed version of adaptive pixel pair matching (APPM) was used for image steganography to get lower distortion [17].However, APPM need to calculate, store, and query the modified neighborhood set Print "Error" End Switch End if End While Algorithm 1 modified neighborhood set table, and it can realize the data hiding in any notional system.

A Review of Adaptive Pixel Pair Matching (APPM)
The APPM method [10] used a pair of pixels (, ) as a coordinate, where an extraction function   (, ) was designed.Then a neighborhood set Φ(, ) of (, ) was established.
where (, ) and Φ(, ) satisfied the following three conditions: (i) In the neighborhood set Φ(, ), there are exactly  pairs of coordinates.
(ii) In the neighborhood set Φ(, ), the extracted function values for each coordinate are mutually exclusive.
(iii) According to (, ) and Φ(, ), a digit can be embedded in any notional system.
The way to find the extraction function coefficient   and Φ(, ) can be converted to find the following optimal solution: Minimize According to the above,   and Φ(, ) can be calculated with different B-ary.For APPM proposed by Hong [10],   corresponding to B-ary is listed in Table 1.Meanwhile, parts of Φ(, ) corresponding to B-ary are illustrated in Figure 1.
Compared with DE and EMD method, APPM has the flexibility to choose a better notational system for data embedding to decrease the image distortion.The selection    of B-ary system is determined by the size of the cover image C. Given the size of C is M×N, B is the minimum value satisfying ⌊ × /2⌋ ≥ |  |.However, it needed to calculate, store, and query the neighborhood set as shown in Figure 1.

The Proposed Formula Adaptive Pixel Pair Matching Method (FAPPM)
In order to solve the above shortcomings, this paper puts forward a formula adaptive pixel pair matching embedding method to find the stego-pixel pair without a neighborhood set.
3.1.Embedding Procedure.In the embedding procedure, four vectors at most are produced.Two vectors are calculated when D>0, and the other two vectors are calculated when D<0.In Algorithm 1, i represents vectors 1 to 4 in turn.
Figure 2 shows the embedding process overview.

Extraction
Procedure.Through extraction function, secret digits can be extracted from the stego image.The detailed process is given in Algorithm 2.

Experimental Results.
The experiments are performed using Matlab R2013a, and eight 512 × 512 grayscale images are used as shown in Figure 3.The stego images are shown in Figure 4, where B=27.As seen from Figures 3 and 4, the difference between the cover images and the corresponding stego images is very little and can not be distinguished by human's eyes.It illustrated the good imperceptibility of the proposed method.
As message embedding, it will introduce the distortion in the image.Peak signal-to-noise ratio (PSNR) is usually used to measure the quality of image.The definition of PSNR is as follows: where MSE is the mean square error between the cover image and stego image; it is defined as follows: Here, the symbols  , and   , represent the pixel values of the cover image and stego image in the position (, ), respectively, and  and  are the width and height of the original image.
As the proposed method can embed secret digit in any notional system, experiments are done to test the relationship between embedding payload and image quality, and the results are shown in Figure 5.It can be found that the PSNR is decreased as the embedding capacity is increased.However, the PSNR still achieved a high value when the embedding capacity reached 1%.

Comparison with Other Methods.
Here EMD [8], EMD-3 [9], APPM, and FAPPM are compared from six aspects: the embedding method, the national system, payload, capacity, PSNR, and the storage space.The results are listed in Table 2.As seen from Table 2, FAPPM method uses a mathematical method to embed secret data and it does not need any space to store neighbor table; furthermore, it does not affect the capacity and image quality.
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Analysis of the Security.
Anti-steganalysis is one of the most important criteria to measure the performance of a steganographic method.In this paper, a detection method based on histogram differential statistics analysis proposed by Zhao [18] is used to test the security of the FAPPM method.Normally, in an image with no hiding message, the horizontal difference histogram Ĥℎ and the vertical difference histogram ĤV are coincident.But, when the message is embedded in a pair of pixels, its Ĥℎ and ĤV will be changed.The distance between Ĥℎ and ĤV is used to construct a statistical detector to detect the variation between histograms.The distance is defined as follows: where  is a predefined threshold and  represents the difference between Ĥℎ and ĤV .The larger the  is, the greater the difference between Ĥℎ and ĤV is.That is, the probability that the image contains secret information is high.Here experiments are done to compare the histogram variation of FAPPM and FDEMD under high payload.Both FAPPM and FDEMD methods are used to generate 100 stego images, respectively.Ĥℎ , ĤV , and their average value are calculated, respectively.The parameters are B=53, B=211, and T=20.All the test images were fully embedded.The experiment results are shown in Figure 6.It can be seen that there is almost no difference between Ĥℎ and ĤV for FAPPM, while that for FDEMD is significant, which indicates the probability that the successful steganalysis for FDEMD is higher than that of the proposed method.
The RS attack method can detect LSB secret data embedding in grayscale or color images.Each pixel block is classified into the regular group , the singular group , and the unusable group  by a flipping function and mask ., , and  denote the number of , , and , respectively.For inverse mask -, -, -, and U- denote the number of , , and , respectively.When no information is embedded,  --m≈0 and  --m≈0.The RS attack results are shown in Figures 7 and 8.It can be seen that the algorithm of this paper can guarantee  --m≈0 and  --m≈0, and the existence of secret information cannot be detected by RS steganalysis method.

Conclusion
This paper proposed a simple and convenient data embedding method based on APPM.Compared with the APPM method, it has the advantage of no needing to compute and store the neighborhood set.Compared with the FDEMD method, the secret data of any notional system is realized by the FAPPM method, which makes the embedding notational system selection more flexible.The experimental results showed that FAPPM method has high image quality and the strong anti-steganalysis ability.Our future work will be concentrated on the use of the formula method of the adjacent three pixels as the embedding unit.

Figure 3 :
Figure 3: The eight gray cover images.

Figure 6 :
Figure 6: Comparison of the averaged vertical and horizontal difference histograms of FAPPM and FDEMD.

Figure 7 :
Figure 7: The difference of Rm and R-m for RS attack.

Figure 8 :
Figure 8: The difference of Sm and S-m for RS attack.

Table 1 :
Extraction Function Coefficient   of APPM.

Table 2 :
Comparison of results.