A Steganographic Method Based on Pixel-value Differencing and the Perfect Square Number

The pixel-value differencing (PVD) scheme uses the difference value between two consecutive pixels in a block to determine how many secret bits should be embedded. There are two types of the quantization range table in Wu and Tasi's method. The first was based on selecting the range to provide high imperceptibility. Most of the related studies focus on increasing the capacity using LSB and the readjustment process, so their approach is too conformable to the LSB approach. There are very few studies focusing on the range table design. Besides, it is intuitive to design it by using the width of the power of two. This work designs a new quantization range table based on the perfect square number to decide the payload by the difference value between the consecutive pixels. Our research provides a new viewpoint that if we choose the proper width for each range and use the proposed method, we can obtain better image quantity and higher capacity. In addition, we offer a theoretical analysis to show our method is well defined. The experiment results also show the proposed scheme has better image quantity and higher capacity.


Introduction
The pixel-value differencing (PVD) [1] scheme provides high imperceptibility to the stego image by selecting two consecutive pixels and designs a quantization range table to determine the payload by the difference value between the consecutive pixels.Besides, it offers the advantage of conveying a large number of payloads, while still maintaining the consistency of an image characteristic after data embedding.
In recent years, several studies have been proposed to improve the PVD method.Wu et al. 's [2] presented a method combining pixel-value differencing and the LSB replacement method.Yang and Weng [3] proposed a multipixel differencing method that uses three difference values in a fourpixel block to determine how many secret bits should be embedded, and Jung et al. 's [4] proposed an image data hiding method based on multipixel differencing and LSB substitution.Liu and Shih [5] proposed two extensions of the PVD method, the block-based approach and Haar-based approach, and Yang et al. proposed an information hiding technique based on blocked PVD.Liao et al. 's [6] proposed a four-pixel differencing and modified LSB substitution, and Yang et al. 's [7] proposed a data hiding scheme using the pixelvalue differencing in multimedia images.
Some studies focused on increasing the capacity [3,5,8] using LSB [2,4] or a readjusted process [6,7] to improve the embedding capacity or image quantity.Few studies focus on the range table design.Besides, it is intuitive to design it using the width of the power of two.
In this work, we design a new quantization range table based on the perfect square number to decide the payload by the difference value between the consecutive pixels.It differs from the design of Wu and Tsai's scheme, in which the quantization range table is based on the range width of the power of two.The perfect square number provides an elegant mathematical model to develop a new quantization range table, which divides each range into two subranges for embedding different numbers of secret bits.
The remainder of this paper is organized as follows.Section 2 briefly describes Wu and Tsai's PVD approach.Section 3 presents our scheme on how to create a new quantization table based on the perfect square number, how the embedding procedure works, and how to extract the secret data from the stego image.Section 4 offers a theoretical analysis and shows the experiment results.Finally, Section 5 concludes this paper.

Review of Wu and Tsai's PVD Approach
The gray-valued cover image is partitioned into nonoverlapping blocks of two consecutive pixels, states   and  +1 .From each block, we can obtain a difference value   = |  −  +1 |; then   ranges from 0 to 255.If   is small, then the block is located within the smooth area and will embed less secret data.Otherwise, it is located on the edge area, and it can embed a greater amount of secret data.The quantization range table is designed with  contiguous ranges, and the range table ranges from 0 to 255.The number of secret bits hidden in two consecutive pixels depends on the quantization range table.
The embedding algorithm is described as follows. Step Repeat Steps 1-5 until all secret bits are embedded and the stego image is produced.
In the extracting phase, the same Steps 1 and 2 in the embedding algorithm are used.The difference    = |   −   +1 | is computed for each two consecutive pixels in the stego image, and then the same quantization range table is searched to find   .Compute  =    −  , and transform  into the binary stream.Repeat until all secret data is completely extracted.

Proposed Scheme
In this section, the proposed scheme is described in three parts: the new quantization range table is based on the perfect square number, embedding procedure, and extraction procedure.1.For each pixel value  ∈ [0, 255], choose the nearest perfect square number  2 (we will define the nearest perfect square number later), then we have range  2 −  ≤  2 <  2 +  for  ∈ [1,16].The width of this range is ( 2 + ) − ( 2 − ) = 2, and the embedding bit length is  = ⌊log 2 2⌋.For each range [ 2 − ,  2 + ), if the width of this range is larger than 2  , then we divide this range into two subranges:

The New Quantization Range
For example, if the pixel value is 34, the nearest perfect square number is 36; then we have range: [30,41].The width of this range is 12, and the embedding bit length is  = ⌊log 2 12⌋ = 3.Since 12 > 2 3 , divide this range into two subranges [30, 33] and [34,41].
By the definition of subranges, if the to-be-embedded + 1 secret bits equal one of the  + 1 LSB bits in the first subrange, then we claim it can embed  + 1 secret bits.Otherwise, the second subrange's width is always 2  , and it can embed  secret bits.Therefore, we can guarantee one of the continuous series numbers equals the  bits secret data which we want to embed.
There are two important concepts we want to emphasize here.First, if the difference value is located in the first subrange, there is no modification needed, so this design does not violate the basic concept of PVD and HVS (Human Visual System).Second, we notice almost the difference values belonging to range [56, 255] are used to embed the same size of data, 4 bits of secret data.Our design in Table 1 still coincides with the basic concept of PVD-embedding a lower amount of secret data in the smooth area and a greater amount of secret data in the edge area.

Embedding
Procedure.Before embedding secret data, the function Nearest PerfectSquare() is defined to find the nearest perfect square number for difference value , where  is the difference value of two consecutive pixels.
The function Nearest PerfectSquare() returns the nearest perfect square number , and  is the range number of .According to range number , the secret data is embedded into the cover image by the embedding procedure.The  is the perfect square number;  is the length of embedding bits.
embedding procedure of proposed method is summarized as follows.
The proposed embedding procedure is as follows.
Input.The grayscale cover image pixel value (), where  is a pixel index.LSB(, ) is  bits LSB binary stream for pixel value .Secret() represents  bits binary secret data.
Step 2. Find the nearest perfect square number  by function Nearest PerfectSquare(), and  is the range number of  in Table 1.
We illustrated the embedding examples in Figure 2.

Extraction Procedure.
The extraction procedure of the proposed method is summarized as follows.
The proposed extraction procedure is as follows.
Input.The grayscale stego image pixel value ().LSB(, ) is a decimal number transform from  bits LSB binary stream for pixel value .Secret() represents  bits binary secret data.
Output.Secret data.4, and the experiment results are shown in Table 5.
From Table 5, we found the experiment results have larger capacity and better PSNR than those of the theoretical analysis.The capacity and PSNR seem to be affected by the secret data, with more pixel-value difference falling in the first subranges and matching the secret data; we can obtain more capacities and less distortion.

Conclusions
This work designs a new quantization range table based on the perfect square number.In particular, we propose a new technology to design the range table.The width of the range is no longer a power of two, and if the difference value is located in the first subrange, there is no modification needed.Therefore, this design has not violated the basic concept of PVD and HVS (Human Visual System).If we choose a proper width for each range and use the proposed method as mentioned above, we can obtain better image quantity and higher capacity.The theoretical analysis shows the proposed scheme is well defined and has larger capacity and higher PSNR than those of Wu and Tsai's second type range table design.The experiment results also show the proposed scheme provides large capacity and high imperceptibility.In addition, our study ingeniously uses the perfect square number to achieve the goal.
1. Calculate the difference   = |  −  +1 | for each block of two consecutive pixels   and  +1 .Search the quantization range table for   to determine how many bits will be embedded.Obtain the range   in which   = [  ,   ], where   and   are the lower bound and the upper bound of   , and  = ⌊log 2 (  −   )⌋ is the number of embedding bits.
Table Based on Perfect Square Number.We design a new quantization range table based on the perfect square number in Table

Table 1 :
The quantization range table based on the perfect square number.

Table 2 :
Distributions of pixel-value difference, average payload, and average MSE for images using the proposed method.

Table 3 :
The distributions of pixel-value difference, payload, and MSE for images using Wu and Tsai's method.

Table 4 :
Comparison between Wu and Tsai's method and the proposed method by theoretical analysis.

Table 5 :
The experiment results use Figure4as the cover image.We also use the same test images as the real test shown in Figure