Reversible data hiding scheme based on pixel-value differencing in dual images

Abstract

In this article, a new reversible data hiding scheme using pixel-value differencing in dual images is proposed. The proposed pixel-value differencing method can embed more secret data as the difference value of adjacent pixels is increased. In the proposed scheme, the cover image is divided into non-overlapping blocks and the maximum difference value is calculated to hide secret bits. On the sender side, the length of embeddable secret data is calculated by using the maximum difference value and the log function, and the decimal secret data are embedded into the two stego-images after applying the ceil function and floor function. On the receiver side, the secret data extraction and the cover image restoration can be performed by using the correlation between two stego-images. After recovering the cover image from two stego-images, the secret data can be extracted using the maximum difference value and the log function. The experimental results show that the proposed scheme has a higher embedding capacity and the proposed scheme differs in embedding the secret data depending on the characteristics of the cover image with less distortion. Also, the proposed scheme maintains the degree of image distortion that cannot be perceived by the human visual system.

Keywords

Reversible data hiding dual images PVD steganography

Introduction

Recently, most of important information is transmitted over the Internet using advanced computer performance and Internet technology. Internet is convenient for information exchange because of easy access and various conveniences. However, the Internet is an open space, so important information can be deceived and altered by malicious attackers. Therefore, many data hiding schemes that can transmit information securely over the Internet have been proposed. In general, a data hiding scheme generates the stego-image by embedding the secret data into a cover image. Many data hiding schemes have been proposed based on least significant bits (LSB) replacement and pixel-value differencing (PVD) in irreversible data hiding methods.^1–5 The LSB scheme embeds or extracts the secret data in the lower bit region in each pixel or pixel group that is based on the fact that it is very difficult to be detected by the human visual system even if the lower bits of a pixel change. The LSB-based schemes uniformly embed the secret data as same quantity into all pixels of a cover image. However, there are edge and smooth areas in an image and it means that different amount of secret bits can be embedded on edge and smooth areas. It is based on the fact that the alteration of edge areas cannot be distinguished well, but the alteration of smooth areas can be distinguished easily in human visual system. In other words, an edge area can be hidden more secret data than a smooth area. The PVD was first proposed by Wu and Tsai to distinguish edge and smooth areas. In the PVD method, two pixels are located in an edge area if the difference value between the two consecutive pixels is large, then more secret data can be hidden depending on the range table. On the other hand, if the difference value between the two consecutive pixels is small, less secret data can be hidden since the two pixels are located in a smooth area. Data hiding schemes using PVD method are proposed continuously.^6–8 Although the PVD method can embed more secret data and guarantee high imperceptibility, the cover image cannot be restored because the original image is damaged after extracting the secret data from the stego-image. It could cause serious problems in the military and medical fields that are sensitive to the distortion of the original image, so reversible data hiding schemes have been proposed to overcome restoring the cover image.^9–12 The reversible data hiding scheme can extract the secret data from the stego-image and restore the original cover image perfectly without loss of cover image. In general, reversible data hiding schemes generate a single stego-image by embedding secret data into a single cover image, but recently, dual image–based reversible data hiding schemes have been proposed that generate two stego-images after embedding secret data into a cover image. Dual image–based reversible data hiding scheme was first proposed by Chang et al.¹³ in 2007. This scheme embeds and extracts secret data based on the quinary notation and improves the extraction function of exploiting modification direction (EMD) scheme¹⁴ to increase the embedding capacity. A modulus function is designed, and two stego-pixel values are defined according to the value of the quinary secret data. In 2009, Lee et al. proposed a dual image–based reversible data hiding scheme using four directions based on the central point.¹⁵ This scheme uses the relationship between two images to determine if secret data can be embedded in the second image. In the embedding process, two stego-pixel values are determined according to the combination of two binary secret bits. Qin et al. proposed a scheme for performing different embedding processes on two images,¹⁶ where the EMD scheme was applied to the first image and three rules based on the first image to embed the secret data was used to the second image. In this scheme, the distortion of the second stego-image is larger than the first stego-image. In 2015, Lu et al. proposed a dual image–based reversible data hiding scheme to reduce loss of original image by applying center folding strategy (CFS) method.¹⁷ The CFS method reduces the distortion of the image by adjusting the range of secret data. In this scheme, secret data are embedded by using the ceil function and the floor function to generate two stego-images. In 2017, Yao et al.¹⁸ proposed a dual image–based reversible data hiding scheme using the selection strategy of shiftable pixels’ coordinates to improve the Lu et al.’s scheme. Yao et al. used the diagram to improve the scheme of Lu et al., and the selection strategy of the shiftable pixels’ coordinates was used to increase the embedding capacity. In this scheme, secret data are classified into even and odd numbers, and different embedding methods are applied. In 2019, Shastri and Thanikaiselvan proposed a scheme to improve the image quality of Yao et al.’s scheme. The CFS method, shiftable pixel coordinate selection strategy method, and lookup table were used to improve the image quality.¹⁹ In this article, we propose a dual image–based reversible data hiding scheme using the PVD method. The proposed scheme divides a cover image into sub-block and uses the maximum difference value in each sub-block to decide the length of embeddable secret data. Therefore, the proposed scheme can embed more secret data when the difference value of pixel values is larger in each sub-block.

The rest of the article is organized as follows. The PVD method and dual image–based reversible data hiding schemes are described in the “Related works” section. In the “The proposed scheme” section, the proposed embedding scheme is described and extracting and restoring scheme is explained in detail. The experimental results are explained and analyzed to demonstrate the superiority of the proposed scheme. The “Conclusion and discussion” section concludes the article.

Related works

PVD

Wu and Tsai proposed a PVD scheme that adjusts the size of secret data that can be embedded according to the difference between two neighboring pixels. In the grayscale cover image, two consecutive pixels are used as one block. The image is divided by blocking. When the pixels of each block are denoted by $C_{i}$ and $C_{i + 1}$ , the difference value $d_{i}$ is obtained using $d_{i} = C_{i + 1} - C_{i}$ . If the lower and upper range values in the block are $u_{i}$ and $l_{i}$ , the number n of bits that can be embedded in each block is calculated using equation (1)

n = \log_{2} (u_{i} - l_{i} + 1)

(1)

If a value obtained by converting n bits into a decimal form is given as b, a new difference value $d_{i}'$ is calculated using equation (2)

d_{i}' = {\begin{matrix} l_{i} + b for d \geq 0, \\ - (l_{i} + b) for d < 0 \end{matrix}

(2)

When the new difference value is $d_{i}'$ and the difference value of the block is denoted as $d_{i}$ , the difference value $m_{i}$ between two pixels is calculated using equation (3)

m_{i} = d_{i}' - d_{i}

(3)

Suppose that two consecutive pixels in a grayscale stego-image are denoted by $S_{i}$ and $S_{i + 1}$ , the stego-image pixels can be calculated using equation (4)

\begin{matrix} f ((g_{i}, g_{i + 1}), m) = (g_{i}', g_{i + 1}') \\ = {\begin{matrix} (g_{i} - ⌈ \frac{m}{2} ⌉, g_{i + 1} + ⌊ \frac{m}{2} ⌋) \\ if d_{i} is an odd number; \\ (g_{i} - ⌊ \frac{m}{2} ⌋, g_{i + 1} + ⌈ \frac{m}{2} ⌉) \\ if d_{i} is an even number \end{matrix} \end{matrix}

(4)

Zhang et al.’s scheme

In 2019, Zhang et al. proposed reversible data hiding scheme using histogram shifting for high embedding capacity. This scheme used additive homomorphic encryption and block permutation for image encryption and consists of encryption, secret data embedding, secret data extraction, and image restoration phase.

The encryption phase consists of additive homomorphic encryption and Block permutation. The cover image is divided into non-overlapped blocks, and a key stream R is generated to apply additive homomorphic encryption. When s is defined as the value of the original image and r is an element of the key stream R, the value x of the encrypted image $X'$ is defined by equation (5)

s_{t} (i, j) = (s_{t} (i, j) + r_{t}) \mod N

(5)

Then, block permutation is performed to enhance security. Block permutation uses the Logistic chaotic function to generate a key sequence and performs block permutation based on this key sequence to create a fully encrypted image X.

The secret data embedding step consists of a prediction step, a histogram expansion, and a shifting step. In the prediction phase, three sets are used. In the case of set 0, the predicted value is calculated using MED. When $x_{i, j}$ is located at $(i, j)$ of the image, the predicted value $p_{x}$ is calculated using equation (6)

p_{x} = {\begin{matrix} \min (x_{i, j + 1}, x_{i + 1, j}) if x_{d} \geq \max (x_{i, j + 1}, x_{i + 1, j}) \\ \max (x_{i, j + 1}, x_{i + 1, j}) if x_{d} \leq \max (x_{i, j + 1}, x_{i + 1, j}) \\ x_{i, j + 1} + x_{i + 1, j} - x_{i + 1, j + 1} otherwise \end{matrix}

(6)

In case of set 1, the predicted value $p_{x}$ is calculated using the rhombus predictor of equation (7). When $x_{i, j}$ is located at $(i, j)$ of the block, adjacent pixels are $x_{i, j - 1}$ , $x_{i - 1, j}$ , $x_{i + 1, j}$ , and $x_{i, j + 1}$ , and [] is a rounding function

p_{x} = [\frac{x_{i, j - 1} + x_{i - 1, j} + x_{i + 1, j} + x_{i, j + 1}}{4}]

(7)

The case of set 2 is similar to set 1. The $p_{x}$ value in set 2 is the domain mean value of the vertical and horizontal directions. When the predicted value $p_{x}$ is calculated, the prediction error d is calculated using the following equation

d (i, j) = x (i, j) - p_{x} (i, j)

(8)

After the prediction error value d is calculated, the histogram right expansion and shifting steps are performed. When the range of this step is defined $[t_{n 1}, t_{n 2}]$ , the number of histograms extended to the right is calculated using the following equation

t_{\exp 1} = t_{n 2} - t_{n 1} + 1

(9)

The prediction error $d_{r} (i, j)$ is calculated using equation (10)

d_{r} (i, j) = {\begin{matrix} d (i, j) if d (i, j) < t_{n 1} \\ 2 d (i, j) - t_{n 1} if t_{n 1} \leq d (i, j) \leq t_{n 2} \\ 2 d (i, j) + t_{\exp 1} if d (i, j) > t_{n 2} \end{matrix}

(10)

If the secret data are $s_{r}$ , the original pixel value is $x (i, j)$ , and the prediction error is $d (i, j)$ , then $xrm (i, j)$ with the secret data embedded is calculated using equation (11)

x_{rm} (i, j) = {\begin{matrix} x (i, j) if d (i, j) < t_{n 1} \\ x (i, j) + d (i, j) - t_{n 1} + s_{r} if t_{n 1} \leq d (i, j) \leq t_{n 2} \\ x (i, j) + t_{\exp 1} if d (i, j) > t_{n 2} \end{matrix}

(11)

To generate a stego-image, a similar process is performed for histogram left expansion and shifting.

Lu et al.’s scheme

Lu et al.’s scheme used CFS method to reduce image distortion in embedding of secret data. Lu et al.’s scheme selects k bits of binary secret data and converts it into decimal form. The secret data are d, and the secret data range for k is given as R. When the data to which CFS can apply are $\bar{d}$ , equation (12) is used to calculate $\bar{d}$

\bar{d} = d - 2^{k - 1}

(12)

In order to generate two stego-images, the d value is divided into two types using the following equation

\bar{d_{1}} = ⌊ \frac{\bar{d}}{2} ⌋, \bar{d_{2}} = ⌈ \frac{\bar{d}}{2} ⌉

(13)

When a pixel value of the cover image is defined as $x_{i, j}$ , the first stego-image pixel is defined as $x_{i, j}'$ , and the second stego-image pixel is defined as $x_{i, j} ″$ . And, equation (14) is used to embed the secret data into the two stego-images

x_{i, j}' = x_{i, j} + \bar{d_{1}}, x_{i, j} ″ = x_{i, j} - \bar{d_{2}}

(14)

Yao et al.’s scheme

In 2017, Yao et al. improved Lu et al.’s scheme by using selection strategy of shiftable pixel coordinates. Yao et al. improved the embedding capacity of secret data by improving Lu et al.’s scheme as shown in Figure 1, where Figure 1 shows a diagram of the embedding of secret data for both schemes. As shown in Figure 1, Yao et al.’s scheme increased the embedding capacity since the number of cases of embedding is larger than that of Lu et al.’s scheme. If the number of bits of the binary secret data that can be embedded is k, then the k binary secret bits are converted into a decimal secret data d. If d is equal to $2^{k} - 1$ , additional bits can be embedded. The pixel value $(x_{i}, x_{i})$ of the image changes into $(x_{i} - 1, x_{i} + 1)$ when the additional bit is 0 and $(x_{i}, x_{i})$ is converted into $(x_{i} + 1, x_{i} + 1)$ when the additional bit is 1.

Figure 1.

Comparison of (a) Lu et al.’s scheme and (b) Yao et al.’s scheme.

If d is not equal to $2^{k} - 1$ , then the secret data are embedded by following two cases. When each pixel of two stego-images is called $x_{i}'$ and $x_{i} ″$ , d is embedded by using equation (15).

Case 1: d is an even number

{\begin{matrix} x_{i}' = x_{i}^{+} ⌊ \frac{d}{4} ⌋ \\ x_{i} ″ = x_{i}' + (\frac{d}{2}) \end{matrix}

Case 2: d is an odd number

{\begin{matrix} x_{i}' = x_{i}^{-} ⌊ \frac{d}{4} ⌋ \\ x_{i} ″ = x_{i}' + ⌈ \frac{d}{2} ⌉ \end{matrix}

(15)

Shastri and Thanikaiselvan’s scheme

In 2019, Shastri and Thanikaiselvan proposed a scheme to improve the quality of the image while maintaining the amount of secret data capacity in Yao et al.’s scheme. This scheme uses the CFS method and the shiftable pixel coordinate selection strategy method and creates a lookup table to improve the PSNR of the image. Three-bit binary secret data are converted to decimal secret data d. If the value of the converted decimal secret data is 7, one additional bit (0 or 1) can be embedded. CFS method is applied to decimal secret data using equation (12) to generate $\bar{d}$ . As shown in Table 1, trinary mapping is performed on $\bar{d}$ to obtain the values of $m_{1}$ and $m_{2}$ .

Table 1.

Trinary mapping.

$\bar{d}$	Trinary mapping
	$m_{1}$	$m_{2}$
−4	−1	−1
−3	−1	0
−2	−1	1
−1	0	−1
0	0	0
1	0	1
2	1	−1
3	1	0
4	1	1

Initial shifts are calculated using equation (16).

Case 1: d is an even number

δ_{1} = - m_{1} \frac{⌈ \bar{d} / 3 ⌉}{2}, δ_{2} = - m_{2} \frac{⌊ \bar{d} / 3 ⌋}{2}

Case 2: d is an odd number

δ_{1} = - m_{1} \frac{⌊ \bar{d} / 3 ⌋}{2}, δ_{2} = - m_{1} \frac{⌈ \bar{d} / 3 ⌉}{2}

(16)

The final shifts $s'$ and $s ″$ are calculated using equation (17).

Case 1: $d < 7$

\begin{matrix} s' = ⌊ δ ⌋ \\ s ″ = {\begin{matrix} - ⌈ δ_{1} + δ_{2} ⌉, d is even \\ [- δ_{1} + δ_{2}], d is odd \end{matrix} \end{matrix}

Case 2: $d \geq 7$

\begin{matrix} s' = - ⌊ δ ⌋ \\ s ″ = {\begin{matrix} ⌈ δ_{1} + δ_{2} ⌉, d is even \\ [δ_{1} + δ_{2}], d is odd \end{matrix} \end{matrix}

(17)

The created lookup table, Table 2, is as follows.

Table 2.

Lookup table.

Trinary mapping		Initial shifts		Final shifts
$m_{1}$	$m_{2}$	$δ_{1}$	$δ_{1}$	$s^{'}$	$s^{″}$
−1	−1	−0.5	−1	−1	1
−1	0	−0.5	0	−1	1
−1	1	0	0.5	0	−1
0	−1	0	0	0	0
0	0	0	0	0	0
0	1	0	−0.5	0	−1
1	−1	−0.5	0	−1	0
1	0	−0.5	0	1	−1
1	1	−1	−0.5	1	−1

Equation (18) is used to generate two stego-image pixels $s'$ and $s ″$ . The $C_{i}$ is the cover image pixel

\begin{matrix} S_{i}' = C_{i}^{+} s' \\ S_{i} ″ = C_{i}^{+} s ″ \end{matrix}

(18)

The proposed scheme

The proposed scheme uses PVD method to dual image–based reversible data hiding scheme for high embedding capacity. The cover image divides into sub-blocks with non-overlapping and the maximum difference value is calculated between the pixels in each sub-block. The maximum difference value and log function are used to determine the number of embedding secret bits. The binary secret bits that can be embedded into each pixel are converted into decimal secret data. The decimal secret data are embedded by using the floor function and ceil function into the cover image to generate two stego-images. In the extraction process, the two stego-images are divided into non-overlapping blocks. The cover image is recovered using the average value of the pixel values of the two stego-images. The maximum difference value is calculated within each block of the cover image, and secret data are extracted using the maximum difference value and the log function. In this section, we describe a new dual image–based reversible data hiding scheme that embeds secret data according to the difference value of the pixels in the block for high embedding capacity.

Embedding algorithm

The detailed embedding algorithm is as follows. Each pixel in the block calculates the difference value with the other pixels in the block and embeds the decimal secret data based on the maximum difference value.

Algorithm 1. The embedding algorithm

Input: A cover image with width W and height H and the binary secret data

Output: The two stego-images of $W \times H$

Step 1. Divide a cover image into non-overlapping blocks with $M \times M$ size.

Step 2. Define the $i th$ block as $B_{i}$ , and each pixel in $B_{i}$ as $c_{i}^{j, k} (j, k = 0, 1, \dots, M)$ . And, calculate the max difference value $d_{i}^{j, k} (j, k = 0, 1, \dots, M)$ using equation (19). The max function returns the maximum value

\begin{matrix} d_{i}^{0, 0} = \max (| c_{i}^{0, 0} - c_{i}^{0, 1} |, \dots, | c_{i}^{0, 0} - c_{i}^{0, M} |, | c_{i}^{0, 0} - c_{i}^{1, 0} |, \dots, | c_{i}^{0, 0} - c_{i}^{M, 0} |, | c_{i}^{0, 0} - c_{i}^{1, 1} |, \dots, | c_{i}^{0, 0} - c_{i}^{M, M} |) \\ . . . . . . . \\ d_{i}^{j, k} = \max (| c_{i}^{j, k} - c_{i}^{0, 0} |, \dots, | c_{i}^{0, 0} - c_{i}^{0, k - 1} |, | c_{i}^{j, k} - c_{i}^{1, 0} |, \dots, | c_{i}^{j, k} - c_{i}^{j - 1, 0} |, | c_{i}^{j, k} - c_{i}^{1, 1} |, \dots, | c_{i}^{j, k} - c_{i}^{j - 1, k - 1} |) \end{matrix}

(19)

Step 3. Apply the log function to the maximum difference value $d_{i}^{j, k}$ of each pixel in the block for obtaining embedding size $l_{i}^{j, k} (j, k = 1, 2, \dots, M)$ by the following equation

l_{i}^{j, k} = ⌊ lo g_{2} d_{i}^{j, k} ⌋

(20)

Step 4. Convert the $l_{i}^{j, k}$ size binary secret bits to decimal secret data $s_{i}^{j, k}$ , and the value to be embedded is defined using the following formula. $T_{i}^{j, k}$ and $t_{i}^{j, k}$ are temporary values for embedding secret data

T_{i}^{j, k} = ⌊ \frac{s_{i}^{j, k}}{2} ⌋, t_{i}^{j, k} = ⌈ \frac{s_{i}^{j, k}}{2} ⌉

(21)

Step 5. Calculate the following equation to combine the temporary values and cover image pixel values to generate two stego-images. When two stego-images are defined as $S_{1}$ and $S_{2}$ , the pixels in each stego-image are called $S_{1} p_{i}^{j, k}$ and $S_{2} p_{i}^{j, k}$

S_{1} p_{i}^{j, k} = c_{i}^{j, k} + T_{i}^{j, k}, S_{2} p_{i}^{j, k} = c_{i}^{j, k} - t_{i}^{j, k}

(22)

Step 6. Repeat the above steps for all blocks.

An example of the embedding algorithm is as follows. The cover image is divided into 2×2 size blocks. The first block is $B_{1}$ and the pixels of $B_{1}$ are $c_{1}^{0, 0} = 120$ , $c_{1}^{0, 1} = 140$ , $c_{1}^{1, 0} = 125$ , and $c_{1}^{1, 1} = 112$ . And, the maximum values $d_{1}^{0, 0} = 20$ , $d_{1}^{0, 1} = 28$ , $d_{1}^{1, 0} = 15$ , and $d_{1}^{1, 1} = 28$ are calculated using equation (19). The size of the secret data $l_{1}^{0, 0} = 4$ , $l_{1}^{0, 1} = 4$ , $l_{1}^{1, 0} = 3$ , and $l_{1}^{1, 1} = 4$ is calculated by equation (20). If the binary secret bit stream is $101100111011111_{2}$ , then the secret bits are brought according to the size of $l_{i}^{j, k}$ . The binary secret bits are then converted to decimal secret, such as $s_{1}^{0, 0} = 11$ , $s_{1}^{0, 1} = 3$ , $s_{1}^{1, 0} = 5$ , and $s_{1}^{1, 1} = 15$ . Each pixel defines temporary pixels, $T_{1}^{0, 0} = 5$ , $t_{1}^{0, 0} = 6$ , $T_{1}^{0, 1} = 1$ , $t_{1}^{0, 1} = 2$ , $T_{1}^{1, 0} = 2$ , $t_{1}^{1, 0} = 3$ , $T_{1}^{1, 1} = 7$ , and $t_{1}^{1, 1} = 6$ . Each temporary pixel generates $S_{1} p_{1}^{0, 0} = 125$ , $S_{2} p_{1}^{0, 0} = 114$ , $S_{1} p_{1}^{0, 1} = 141$ , $S_{2} p_{1}^{0, 1} = 138$ , $S_{1} p_{1}^{1, 0} = 127$ , $S_{2} p_{1}^{1, 0} = 122$ , $S_{1} p_{1}^{1, 1} = 119$ , and $S_{2} p_{1}^{1, 1} = 104$ using equation (22). This embedding example is shown in Figure 2.

Figure 2.

Embedding example.

Extraction and recovering algorithm

The detailed extraction algorithm is as follows. In the recovering process, the cover image is restored by using the average value between the two stego-images. The secret data are extracted using the maximum difference value among the pixels in the restored cover image.

Algorithm 2. The extraction and recovering algorithm

Input: The two stego-images of $W \times H$

Output: A cover image with width W and height H and the binary secret data

Step 1. Divide the two stego-images into $M \times M$ size blocks.

Step 2. Restores cover image pixel $c_{i}^{j, k}$ using pixels in the same position of two stego-images

c_{i}^{j . k} = ⌈ \frac{(S_{1} p_{i}^{j . k} + S_{2} p_{i}^{j . k})}{2} ⌉

(23)

Step 3. Calculate the maximum difference value $d_{i}^{j, k}$ for each pixel using equation (19).

Step 4. Calculate the number of bits embedded into the pixel using the following equation. The number of secret bits in each pixel is defined as $l_{i}'^{j, k}$

l_{i}'^{j, k} = ⌊ lo g_{2} d_{i}^{j, k} ⌋

(24)

Step 5. Extract the embedded secret data $s_{i}^{j, k}$ using $l_{i}'^{j, k}$ and the following equation

s_{i}^{j, k} = | S_{1} p_{i}^{j, k} - c_{i}^{j, k} | + | S_{2} p_{i}^{j, k} - c_{i}^{j, k} |

(25)

Step 6. Repeat the above steps for all blocks.

An example of the extraction and recovering algorithm is as follows. The stego-image is divided into 2×2 size blocks. In the first block, the cover pixels $c_{1}^{0, 0} = (125 + 114) / 2 = 120$ , $c_{1}^{0, 1} = (141 + 138) / 2 = 140$ , $c_{1}^{1, 0} = (127 + 122) / 2 = 125$ , and $c_{1}^{1, 1} = (119 + 104) / 2 = 112$ are restored using equation (23). And, the maximum values $d_{1}^{0, 0} = 20$ , $d_{1}^{0, 1} = 28$ , $d_{1}^{1, 0} = 15$ , and $d_{1}^{1, 1} = 28$ are calculated using equation (19). The size of the secret data $l_{1}'^{0, 0} = 4$ , $l_{1}'^{0, 1} = 4$ , $l_{1}'^{1, 0} = 3$ , and $l_{1}'^{1, 1} = 4$ is calculated by equation (24). And, the secret data $s_{1}^{0, 0} = | 125 - 120 | + | 114 - 120 | = 11$ , $s_{1}^{0, 1} = | 141 - 140 | + | 138 - 140 | = 3$ , $s_{1}^{1, 0} = | 127 - 125 | + | 122 - 125 | = 5$ , and $s_{1}^{1, 1} = | 119 - 112 | + | 104 - 112 | = 15$ are extracted using $l'_{i}^{j, k}$ and equation (25). This extraction and recovering example is shown in Figure 3.

Figure 3.

Extraction and recovering example.

Experiment results and analysis

The reversible data hiding scheme measures the image quality and embedding capacity to evaluate the performance of the proposed algorithm. The embedding capacity of the secret data means the amount of secret data that can be inserted into a cover image. The visual image quality means the degree of distortion of the stego-image. The PSNR is used to evaluate the quality of an image in general. If the PSNR is more than 30 dB, image distortion cannot be detected easily by the human visual system. Therefore, most reversible data hiding schemes aim to keep the PSNR value above 30 dB. There is a trade-off relationship between the embedding capacity and the PSNR. In the data hiding scheme, the goal is to increase the embedding capacity while maintaining good image quality. The PSNR is defined as equation (26)

PSNR = 10 \times lo g_{10} \frac{255^{2}}{MSE}

(26)

When the pixels of the cover image are $p_{i}$ and the pixels of the stego-image are defined as $p'_{i}$ , the mean squared error MSE is defined in equation (27)

MSE = \sum_{i = 1}^{W \times H} \frac{{(p_{i} - {p'}_{i})}^{2}}{W \times H}

(27)

In the proposed method, two 512×512-sized stego-images are generated using a cover image of 512×512 sizes as shown in Figures 4 –6. In 2002, Wang and Bovik²⁰ proposed a method to measure the quality index of images using correlation loss, luminance distortion, and contrast distortion

Q = \frac{4 δ_{xy} \bar{p_{x}^{2}} \bar{p_{x}^{2}}}{(δ_{x}^{2} + δ_{y}^{2}) (\bar{p_{x}^{2}} + \bar{p_{y}^{2}})}

(28)

Figure 4.

Cover images: (a) Lena, (b) Peppers, (c) Baboon, (d) Boat, (e) Barbara, (f) Airport, (g) Elaine, (h) Man and woman, (i) Crowd, (j) Tank, (k) Bridge, and (l) Truck.

Figure 5.

First stego-images: (a) Lena (39.1330 dB), (b) Peppers (39.1375 dB), (c) Baboon (31.1448 dB), (d) Boat (36.6100 dB), (e) Barbara (33.4702 dB), (f) Airport (34.5950 dB), (g) Elaine (39.1330 dB), (h) Man and woman (36.1232 dB), (i) Crowd (37.0450 dB), (j) Tank (39.4645 dB), (k) Bridge (33.1101 dB), and (l) Truck (39.6765 dB).

Figure 6.

Second stego-images: (a) Lena (38.2845 dB), (b) Peppers (38.2407 dB), (c) Baboon (30.7269 dB), (d) Boat (35.9289 dB), (e) Barbara (32.9912 dB), (f) Airport (34.0230 dB), (g) Elaine (37.9787 dB), (h) Man and woman (35.5408 dB), (i) Crowd (36.3602 dB), (j) Tank (38.4224 dB), (k) Bridge (32.6074 dB), and (l) Truck (38.6272 dB).

Each element of the equation is given in equations (29)–(31)

\bar{p_{x}} = \frac{1}{wh} \sum_{i = 0}^{wh - 1} p_{i}, \bar{p_{y}} = \frac{1}{wh} \sum_{i = 0}^{wh - 1} p_{i}'

(29)

δ_{x}^{2} = \frac{1}{wh - 1} \sum_{i = 0}^{wh - 1} {(p_{i} - \bar{p_{x}})}^{2}, δ_{y}^{2} = \frac{1}{wh - 1} \sum_{i = 0}^{wh - 1} {(p_{i}' - \bar{p_{y}})}^{2}

(30)

δ_{xy} = \frac{1}{wh - 1} \sum_{i = 0}^{wh - 1} (p_{i} - \bar{p_{x}}) (p_{i}' - \bar{p_{y}})

(31)

The exponent is defined as the combination of loss of correlation, luminance distortion, and contrast distortion, and is redefined as equation (32)

Q = \frac{δ_{xy}}{δ_{x} δ_{y}} \cdot \frac{2 \bar{p_{x}^{2}} \bar{p_{y}^{2}}}{\bar{p_{x}^{2}} + \bar{p_{y}^{2}}} \cdot \frac{2 δ_{x} δ_{y}}{δ_{x}^{2} + δ_{y}^{2}}

(32)

$δ_{xy} / δ_{x} δ_{y}$ is the correlation coefficient between the two images. The second element $2 \bar{p_{x}^{2}} \bar{p_{y}^{2}}$ / $\bar{p_{x}^{2}} + \bar{p_{y}^{2}}$ measures the luminance between the two images, and the third element $2 δ_{x} δ_{y}$ / $δ_{x}^{2} + δ_{y}^{2}$ compares the two images to measure the similarity of the two images.

Visual image quality

The first stego-images with PSNR values are shown in Figure 5, and the second stego-images with PSNR values are shown in Figure 6. Figures 5 and 6 show that all images of the proposed scheme maintain a PSNR value over 30 dB.

Table 3 compares the PSNR of the proposed scheme and other schemes when the embedding capacity is about 500,000 bits. In Table 3, the average PSNR of the proposed scheme is 50.9679 dB. Yao et al.’s average PSNR is 51.1431 dB, and Shastri and Thanikaiselvan’s average PSNR is 53.0077 dB. Shastri and Thanikaiselvan’s PSNR is about 2 dB higher than the proposed scheme. However, the proposed scheme keeps the PSNR value more than 30 dB, and it is difficult to discriminate the distortion of the image in the human visual system. Table 4 compares the PSNR of the proposed scheme and other schemes when the embedding capacity is about 800,000 bits. In Table 4, the average PSNR of the proposed scheme is 42.2817 dB. The average PSNR of Lu et al.’s scheme is 45.1003 dB, the average PSNR of Yao et al.’s scheme is 46.3038 dB, and the average PSNR of the Shastri and Thanikaiselvan’s scheme is 50.2927 dB. When the embedding capacity is 800,000 bits, the PSNR of other schemes is higher than the proposed scheme. However, the proposed scheme keeps the PSNR value more than 42 dB, so there is no problem.

Table 3.

Comparison of PSNR values (embedding capacity: 500,000 bits).

Images	Embedding capacity: about 500,000 bits
	PSNR(dB)
	Lu et al.’s scheme		Yao et al.’s scheme		Shastri and Thanikaiselvan’s scheme		Proposed scheme
	Stegoimage 1	Stegoimage 2	Stegoimage 1	Stegoimage 2	Stego image 1	Stegoimage 2	Stegoimage 1	Stegoimage 2
Lena	51.1411	51.1411	51.1411	51.1411	54.6650	51.6514	54.3882	49.0005
Peppers	51.1410	51.1410	51.1410	51.1411	54.1421	51.1411	53.7174	48.9405
Boat	51.1411	48.1477	51.1412	51.1411	54.6392	51.6491	53.9046	49.1195
Airport	51.1410	51.1411	51.1410	51.1411	54.6863	51.6446	52.6487	48.2680
Elaine	51.1411	51.1411	51.1411	51.1411	54.6563	51.6495	54.1089	49.1711
Man and woman	45.7249	45.7250	51.1429	51.1429	54.2411	51.1011	49.2490	46.6389
Crowd	51.1411	51.1411	51.1411	51.1411	54.6648	51.6525	50.3859	47.5465
Tank	51.1411	51.1411	51.1411	51.1411	54.6542	51.6470	55.6880	49.9683
Bridge	28.0364	28.0364	51.1606	51.1600	54.3411	51.0411	52.7151	48.3978
Truck	51.1411	51.1411	51.1411	51.1411	54.6455	51.6422	55.6468	49.8563
Average	48.2889	47.9896	51.1432	51.1431	54.5335	51.4819	53.2452	48.6907

Table 4.

Comparison of PSNR values (embedding capacity: 800,000 bits).

Images	Embedding capacity: about 800,000bits
	PSNR(dB)
	Lu et al.’s scheme		Yao et al.’s scheme		Shastri and Thanikaiselvan’s scheme		Proposed scheme
	Stegoimage 1	Stegoimage 2	Stegoimage 1	Stegoimage 2	Stego image 1	Stegoimage 2	Stegoimage 1	Stegoimage 2
Lena	46.3669	44.1555	46.3806	46.3777	50.6910	49.8983	41.9185	40.8975
Peppers	46.2210	44.1201	46.2510	46.1211	50.6812	49.8921	41.6474	40.6505
Boat	45.1355	43.3782	46.3821	46.378	50.6754	49.8839	41.7740	40.7788
Airport	46.3753	44.1496	46.3825	46.3812	50.7011	49.8971	40.4291	39.5992
Elaine	46.3632	44.1528	46.3775	46.3674	50.6918	49.8900	44.6722	43.1341
Man and woman	46.7249	44.1350	46.1229	46.0519	50.6956	49.8900	41.5937	40.6720
Crowd	46.4022	44.1653	46.3630	46.3663	50.6892	49.9021	39.8101	39.0848
Tank	46.3639	44.1586	46.3833	46.3782	50.6848	49.8997	44.9897	43.4184
Bridge	45.1264	44.0162	46.1626	46.1201	50.7123	49.9110	45.8635	43.7844
Truck	46.3605	44.1366	46.3611	46.3689	50.6795	49.8892	46.6001	44.3163
Average	46.1439	44.0567	46.3166	46.2910	50.6901	49.8953	42.9298	41.6336

Embedding capacity

The maximum embedding capacity of the proposed scheme and other schemes is compared in Table 5. The average embedding capacity of the proposed scheme is at least about 100,000 bits higher than Lu et al.’s scheme and Yao et al.’s scheme on average. And, the average embedding capacity of the proposed scheme is at least about 600,000 bits higher than Shastri and Thanikaiselvan’s scheme on average. Shastri and Thanikaiselvan’s scheme aims to improve the image quality when the embedding capacity is not large, so the maximum embedding capacity is low. As a result, the proposed scheme can embed more secret data than other schemes while maintaining the degree of image distortion that cannot be detected by the human visual system. Figure 7 compares the maximum embedding capacity using a histogram. The histogram shows that the proposed scheme has higher embedding capacity than other schemes in most images. And, the embedding capacity of secret data is almost constant in other schemes. However, the proposed scheme can embed the secret data up to 1,800,000 bits depending on the characteristics of the image. In the case of Baboon image and Bridge image, the embedding capacity of secret data is high because of the large difference between adjacent pixels.

Table 5.

Comparison of maximum embedding capacity.

Images	Embedding capacity (bits)
	Lu et al.’s scheme	Yao et al.’s scheme	Shastri and Thanikaiselvan’s scheme	Proposed scheme
Lena	1,310,720	1,310,928	844,386	1,193,320
Peppers	1,352,538	1,310,220	844,740	1,272,538
Baboon	1,310,480	1,318,738	844,735	1,893,914
Boat	1,310,640	1,319,012	844,429	1,471,024
Barbara	1,310,720	1,318,893	844,237	1,472,104
Airport	1,310,690	1,318,866	844,373	1,532,374
Elaine	1,310,715	1,318,877	844,709	1,478,328
Manand woman	1,310,525	1,318,742	844,609	1,460,710
Crowd	1,310,576	1,318,826	844,783	1,138,294
Tank	1,310,720	1,318,960	844,462	1,424,864
Bridge	1,310,220	1,318,220	844,611	1,818,296
Truck	1,310,720	1,318,937	844,869	1,335,548
Average	1,314,105	1,317,435	844,579	1,457,610

Figure 7.

Comparison of embedding capacity of the proposed scheme and other schemes.

Quality index

Figure 8 shows the quality index values for each cover image. The quality index measures the loss of correlation, luminance distortion, and contrast distortion between the cover image and the stego-image. The proposed scheme maintains a similar quality index to Lu et al.’s scheme, Yao et al.’s scheme, and Shastri and Thanikaiselvan’s scheme. Therefore, the proposed scheme has a higher embedding capacity than other schemes, and the image quality is similar to other schemes.

Figure 8.

Comparison of the quality index about four images.

PSNR and embedding capacity

Figure 9 shows PSNR and embedding capacity of the six images of the proposed scheme in the graph. The proposed scheme embedded more than 1,300,000 bits secret data while keeping PSNR more than 40 dB in tank image. In the case of truck image and Elaine image, the proposed scheme embedded more than 1,300,000 bits secret data while keeping PSNR about 39 dB. In other images, the PSNR can be maintained at more than 34 dB while maintaining an embedding capacity of more than 1,300,000 bits. The proposed scheme has lower PSNR than other schemes when the embedding capacity is low. However, when the embedding capacity is increased, PSNR is higher than other schemes. In the case of embedding capacity is 1,300,000 bits, PSNR is higher than other schemes for most images.

Figure 9.

Comparison of embedding capacity and PSNR of dual stego-images: (a) Tank, (b) Truck, (c) Boat, (d) Airport, (e) Barbara, and (f) Elaine.

Conclusion and discussion

In this article, a dual image–based reversible data hiding scheme using PVD has been proposed. The cover image was divided into blocks of size M×M, and the maximum difference value between the pixels in each block was calculated to decide the length of the embeddable bits. The proposed scheme increased the embedding capacity by using the maximum difference value in the sub-block. Two types of temporary values were generated by applying the ceil function and floor function to secret data. The temporary values have been combined with the pixel values of the cover image to generate two stego-images. The correlation of two stego-images was used to restore the cover image. The secret data could be extracted using the reconstructed cover image, where the maximum difference value and log function were used. Experimental results showed that the proposed scheme maintains the high embedding capacity and good image quality that has less distortion by the human visual system than the previous schemes. Also, the proposed scheme had the different embedding capacity depending on the characteristics of the cover image that makes robust to some attacks. In future work, we will plan to adopt interpolation techniques on dual images to increase the embedding capacity with good image quality.

Footnotes

Acknowledgements

We thank the anonymous reviewers for their valuable suggestions that improved the clarity of this article.

Handling Editor: Ru Zhang

Author contributions

All authors discussed the contents of the manuscript and contributed to its preparation. P.-H.K. conducted all experiments and wrote the manuscript. K.-W.R. designed and conducted the literature review. K.-H.J. proposed the concept of algorithm and wrote the manuscript.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A1A09081842) and Korea Research Fellowship Program through the NRF funded by the Ministry of Science and ICT (2019H1D3A1A01101687).

ORCID iD

Ki-Hyun Jung

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