An Efficient EMD-Based Reversible Data Hiding Technique Using Dual Stego Images
Princess Sumaya University for Technology, Amman, 11941, Jordan
* Corresponding Author: Ahmad A. Mohammad. Email:
Computers, Materials & Continua 2023, 75(1), 1139-1156. https://doi.org/10.32604/cmc.2023.035964
Received 12 September 2022; Accepted 08 December 2022; Issue published 06 February 2023
AbstractExploiting modification direction (EMD) based data hiding techniques (DHTs) provide moderate data hiding capacity and high-quality stego images. The overflow problem and the cyclic nature of the extraction function essentially hinder their application in several fields in which reversibility is necessary. Thus far, the few EMD reversible DHTs are complex and numerically demanding. This paper presents a novel EMD-based reversible DHT using dual-image. Two novel 2 × 4 modification lookup tables are introduced, replacing the reference matrix used in similar techniques and eliminating the numerically demanding search step in similar techniques. In the embedding step, one of the modification tables modifies a pixel in odd columns in the first cover image and keeps its image in the second cover image intact. The other modification table modifies a pixel in even columns in the second cover image and keeps its image in the first cover image intact. This embedding strategy enables direct reversibility at almost zero computational cost. This technique embeds one secret digit into each pixel in the first cover image and its image in the second cover image resulting in one bit per pixel (bpp) data embedding rate. The use of the numbering system enables direct and numerically efficient conversion of the binary secret message to the and vice versa. The advantages of the proposed algorithm are straightforward reversibility, simplicity, numerical efficiency, direct conversion of the binary secret message to and vice versa, and elimination of the need for the reference matrix by replacing it with two 2 × 4 lookup tables. Simulation results show that the embedding rate of the proposed technique is one bpp. It achieved more than 49 dB average Peak to Signal Noise Ratio (PSNR) over all test images.
In the era of the internet, the internet of things (IoT), and mobile communications, the interchange of vast amounts of digital data has become an easy task. Some of this information includes some secret data. Unintended recipients can easily access this data. Thus, there is a need for the development of means to protect the transmitted data. Thus far, researchers have developed two main methods to achieve data protection: Data encryption  and Data hiding [2–43]. Data encryption techniques share the data in a meaningless form and can be easily identified as confidential data, making it vulnerable to attacks . DHTs conceal the secret data in meaningful digital media (usually a digital image called a cover image). Depending on the nature of the hidden message, one can classify DHTs into two main categories: Digital watermarking, when the message is concerned with the cover, and steganography, otherwise . Steganographic techniques transparently conceal the secret message into a cover object such as a digital image. The main objective of these techniques is to hide the mere existence of the secret message in the cover object. However, the concealment of the secret message in the cover object produces a distorted stego object with reduced visual quality. Steganographic techniques should keep the distortion minimal and transparent so it cannot be detected. Another important objective in steganography is to increase the data hiding capacity, defined as the number of secret message bits embedded in the cover object. However, increasing data hiding capacity introduces more distortions in the cover image, reducing visual quality and destroying transparency. Thus, there is a need for a tradeoff between transparency and data-hiding capacity . DHTs may also be classified as irreversible [3–23] and reversible [24–43]. In irreversible DHTs, the distortions to the cover object are permanent, and one can’t restore the original cover object after extracting the secret message. In reversible DHTs, the distortions to the cover image are not permanent, and one can restore the original cover image after extracting the secret message. The reversibility of DHTs is necessary for some applications, such as military, law enforcement, and medical applications. DHTs are numerous and may be carried out in the spatial or transform domains . In the spatial domain, one can find several methods, such as Image interpolation, Difference expansion (D.E.), Histogram shifting (H.S.), Least significant bit substitution (LSB), and EMD-based methods .
The EMD-based methods have some attractive features: They provide high-quality stego images and can attain relatively high embedding rates. EMD-based techniques have witnessed extensive research efforts [4–36]. Reference  introduced the first EMD-based technique, partitioning the cover image into groups of -pixels; they embedded a secret digit into each group of -pixels by increasing or decreasing the value of only one pixel in the group by one. The stego image quality is relatively low. Reference  proposed an Improved EMD (IEMD) method in which they were able to increase the embedding rate up to 1.5 bits per pixel (bpp) while maintaining good-quality stego images. Reference  introduced the diamond encoding technique in which they embed a secret digit into a pair of cover pixels. The embedding parameter k controls the embedding rate. The image quality becomes low as goes high. Reference  proposed a fully EMD technique (FEMD), in which they utilized a new extraction function and increased the data hiding capacity up to 4.5 bpp. However, the image quality is low. Reference  proposed a formula FEMD (FFEMD) in which they eliminated the need for the matrix function table needed in the FEMD method. However, numerical complexity is increased. Reference  combined difference expansion and FFEMD to improve the data hiding capacity and image quality. They have also proposed a solution to the overflow problem in EMD-based techniques. Reference  proposed a method to embed an secret digit into each group of n pixels obtaining up to a four bpp embedding rate. This method is similar to the IEMD scheme, with a slight improvement in image quality. The above-mentioned EMD-based algorithms are irreversible and numerically demanding. Due to the extraction function’s cyclic nature, most EMD-based DHTs are irreversible, and one can find a few reversible EMD-based DHTs [24–43]. Reference  proposed the first reversible EMD-based DHT using two stego images. They insert A secret digit into two consecutive pixels of the two cover images. The algorithm is somewhat numerically demanding, provides an embedding rate of one bpp, and high-quality stego images of 45 dB PSNR values. Reference  proposed an EMD-based reversible DHT using two stego images. They insert A secret digit into one pixel in the first cover image and its dual image in the second cover image. The algorithm is numerically demanding and provides an embedding rate of 1.55 bpp and stego images of 38 dB PSNR values. Reference  proposed an adaptive EMD-based reversible DHT using two stego images. Using a numeral system, they inserted two secret digits into two adjacent pixels in the first and second cover images. The algorithm is numerically demanding, provides an embedding rate of 1.55 bpp, and stego images quality of 43 dB PSNR values. Reference  proposed an EMD-based reversible DHT using two stego images. They insert a secret digit into one pixel in the first cover image and its dual image in the second cover image. The algorithm is numerically demanding and provides an embedding rate of 1.55 bpp and stego images of 38 dB PSNR values. Reference  proposed an EMD-based reversible DHT using two stego images. Using a numeral system, they inserted two secret digits into two adjacent pixels in the first and second cover images. The algorithm is numerically demanding and provides an embedding rate of 1.55 bpp and stego image quality of 43 dB PSNR values. Reference  proposed a reversible DHT using an orientation combination of dual stego images. Using a numeral system, they inserted a secret digit into two adjacent pixels in each stego image. The algorithm is numerically demanding and provides an embedding rate of 1.14 bpp and stego image quality of 50 dB PSNR values. Reference  proposed an adaptive EMD-based reversible DHT using two stego images. Using a numeral system, they inserted two secret digits into two adjacent pixels in the first and second cover images. The algorithm is numerically demanding and provides an embedding rate of 1 bpp and stego image quality of 50 dB PSNR values. Reference  proposed an RDHT using an orientation combination of dual stego images. Using a numeral system, they inserted two secret digits into two adjacent pixels in the two stego images. The algorithm is numerically demanding and provides an embedding rate of 1.58 bpp and stego images quality of 45 dB PSNR values. Reference  proposed a numerically efficient RDHT using two shadow images. Using a combination of and numeral systems, they obtained an embedding rate of 1.56 bpp and stego images quality of 43 dB PSNR values. However, the algorithm needs the reference matrix. Reference  proposed an RDHT using three shadow images. Using a combination of and numeral systems, they obtained an embedding rate of 1.13 bpp and stego images quality of 48 dB PSNR values. However, the algorithm is somewhat complex and requires the transmission of side information to achieve reversibility. Reference  proposed an adaptive turtle-shell-based RDHT using dual images. Using an numeral system, they obtained an embedding rate of 1.25 bpp and stego images quality of 45 and 49 dB PSNR values. Again, the algorithm is numerically demanding. All these methods use two or more cover images and improve security via secret key sharing .
We summarize our findings by the following points:
• Due to the cyclic nature of the extraction function and the overflow problem, EMD-based DHTs are not directly reversible
• EMD-based DHTs can provide high embedding rates. However, increasing the embedding rate decreases the stego image quality
• In general, increasing embedding rates above two bpp results in significant distortions in the stego image, making it vulnerable to detection
• Researchers proposed the use of dual images to develop reversible EMD-based DHTs
• The use of dual images increases the security of the secret data via secret key sharing
• Thus far, existing dual image-based EMD-based reversible DHTs are numerically demanding, require complex search algorithms, and need to generate a reference matrix
This paper presents an EMD-based reversible DHT using dual stego images. The contributions of our technique are:
• Simple and numerically efficient embedding and extraction algorithms
• Relatively high data-hiding capacity
• Direct and numerically efficient conversion of the binary secret message to and vice versa
• Elimination of the need for the reference matrix used in similar techniques by replacing it with novel two lookup tables, and elimination of the need for the complex and numerically demanding search step required by similar techniques
• An embedding strategy that enables straightforward reversibility with almost zero computational cost
• Increased Security by using secret key sharing and the equal high-quality of the two stego images
The rest of this paper proceeds as follows. In Section 2, we give a summary of some related work. Section 3 outlines the proposed algorithm, the data-hiding step, and the data extraction and image recovery step. Section 4 gives experimental simulation results. Finally, Section 5 presents the discussion, conclusion, and future work.
This section presents the fundamentals of EMD-based DHTs and gives details of some related techniques.
2.1 Fundamentals of EMD-Based DHTs
These schemes conceal an secret digit in a group of -pixels of the cover image I. The first step in these schemes is to segment I into groups of -pixels. The extraction function takes the form where s and l are positive integers. The extraction function is cyclic and has a range of values. We refer to these values as characteristic values. Using the vector of pixels as the origin of an -coordinate system, we search for a vector within a neighborhood-set . The new vector (stego vector) should satisfy the condition ; where is the secret digit in the number system. We define the neighborhood-set as the set of vectors that satisfy a distance condition of the form , or minimize a cost function . We choose k such that each characteristic value occurs at least once over the set . The distortion in should be minimal. Thus, we pick a small neighborhood set to satisfy: 1) There are precisely l elements in , and 2) The values of over are mutually exclusive. Increasing n usually decreases the distortion and data hiding capacity. For this, the most commonly used value is . Fig. 1 shows some examples (patterns) of the neighborhood set and the corresponding characteristic values. The second step is to embed an secret digit into the vector so that the distance between the original cover vector and the resulting stego vector is minimal. In the extraction step, the stego image pixels are segmented into a group of vectors in the same order as that in the embedding step. The resulting secret message is: .
This section presents two closely related DHTs proposed by [8,31].
2.2.1 EMD Algorithm 
This scheme is the basis of EMD-based DHTs. It embeds a secret digit in a sequence of -cover pixels. The algorithm starts by segmenting the cover image into blocks of -non-overlapping pixels. In order to conceal the secret digit, only one of the -pixels is increased or decreased by one. The extraction function used in the embedding and extraction phases is:
; where represents the pixel value, n is the number of pixels, and is the modulus operation. To embed a secret digit into a group of pixels , we find the difference and obtain the modified Block as follows:
At the receiver end, we extract the secret digit as . The hiding capacity for this scheme is bpp. The maximum hiding capacity occurs when with a 1.161 bpp embedding rate. This scheme is irreversible.
2.2.2 Reversibility Data Hiding 
Reference  proposed an EMD-based reversible DHT scheme using dual images. The extraction function is: , where are two adjacent pixels in the original cover image I. Fig. 2a shows the reference matrix M. Fig. 2b shows a block centered at pixel pair . They divide the block pixels into two distinct groups: and . Each block contains nine positions labeled as: , as shown in Fig. 2b. consists of the positions , while consists of the locations . The algorithm embeds two secret digits into each pair of pixels , resulting in two stego images and . The algorithm starts by scanning the reference matrix M and obtaining the block corresponding to the pixel pair . Then it locates the position of in and the position of in . Denote the pixel values in these locations as and . Let be the square of the length of the line connecting these positions and find the stego pixels values using the following algorithm:
Embedding Algorithm 
If () or () or () or (), then
Do not embed the secret message. Set and
ElseIf or () or , then
ElseIf , then Set .
If , then
ElseIf the line segment from to is clockwise, then
Secret message extraction and original image recovery 
Given the two stego pixel pairs and , resulting from the embedding step, find the distance . Next, recover the original pixel pair and extract the secret digits according to the following cases:
Case 1: ;
If ( or , or , or ), then
Set and . Do not extract any secret digits.
Set and . Extract the two secret digits from the reference matrix M corresponding to and .
Case 2 ;
Set and . Extract the two secret digits from the reference matrix M corresponding to and
Case 3 ;
Set and . Extract the two secret digits from the reference matrix M corresponding to and .
Case 4 ;
If , then
Extract the two secret digits from the reference matrix M corresponding to and .
Case 5 ;
Set . Set equal the center pixel that forms a clockwise rotation with the line segment from to . Extract the two secret digits from the reference matrix M corresponding to and .
Case 6 ;
Set Set the pixels equal to the pixel that is adjacent to pixels and (. The line segment connecting the points () and () forms a counterclockwise rotation around . Extract the two secret digits from the reference matrix M corresponding to and .
The proposed algorithm is a reversible EMD-based algorithm that uses the general extraction function: . In this paper, we set and . Choosing means we need to convert the message into the number system, which requires almost zero computational cost because we can convert two or eight bits of the binary message stream instead of converting the whole message. Thus, the proposed scheme uses the extraction function . The neighborhood-set in Fig. 3 shows how the extraction function generates the complete number system set. The shaded row in Fig. 3 results from decrementing x by one and incrementing it by 0, 1, or 2 while keeping y intact. The shaded column in Fig. 3 results from decrementing y by one and incrementing it by 0, 1, or 2 while preserving the value of x. To understand the proposed algorithm, consider a cover image consisting of two neighboring pixels ; we generate two cover images , as shown in Fig. 4. A secret message digit is embedded by substituting and in the extraction function and modifying . The embedding process finds the distance , locates this value in the shaded row of the neighborhood set and adjusts in the first cover image accordingly to obtain the first stego image . Another secret message digit is embedded using and . To do this: find the value and the distance , locate this value in the shaded column of the neighborhood set , and modify in the second cover image accordingly to obtain the second stego image . Note that the shaded pixels remain intact and can be combined to recover the original cover image achieving reversibility.
In the proposed algorithm, we facilitate the embedding process by defining two novel modification tables to eliminate the search step in the neighborhood set and modify the cover images systematically in a numerically efficient manner. In Fig. 4, the changes to pixels needed to embed the differences and are given in the first modification matrix given by Eqs. (1) and (2) below.
Eq. (3) finds the stego pixels ():
We calculate the stego pixels () using Eq. (4).
For extraction, we find the two secret digits and . One should notice that the embedding step may change the original embeddable pixel value by a maximum absolute value . The change in pixel value may result in overflow or underflow. As outlined in the following subsection, we adjust the border embeddable pixels’ values to prevent this problem.
3.1 The Proposed Data Embedding Algorithm
Given the gray level original cover image and a binary secret message W with length ; the embedding procedure consists of the following steps:
Step 1: Obtain the dual cover images and as: .
Step 2: Classify the dual cover images’ pixels into two classes: embeddable pixels and non-embeddable pixels. Embeddable pixels in the first cover image are the pixels . Embeddable pixels in the second cover image consist of pixels . The White pixels (even columns in and odd columns in ) in Fig. 5 show the embeddable pixels for the dual cover images, while the shaded pixels (even columns in and odd columns in ) are the non-embeddable pixels. The union of the non-embeddable pixels in the two stego images gives the original cover image .
Step 3: Prevent overflow and underflow problems by modifying each embeddable pixel in the cover images and using Eq. (5).
where denotes the absolute maximum change to any embeddable pixel. In our algorithm, .
Step 4: Sequentially scan the cover images from left to right and top to bottom and embed one secret digit using steps 5–8.
Step 5: Get two bits from the binary secret message and find its value .
Step 6: Let the current decimal values of the pixels in the preprocessed cover images be: and . Use Eq. (6) to find the value of the extraction function as:
Step 7: Use Eq. (7) to find the distance d as:
Step 8: Find and as defined by Eqs. (1) and (2), and use Eq. (8) to find the new values of the stego pixels and .
The result of this process is the two stego images and . We repeat steps 4–8 until all secret message bits are embedded.
3.2 The Proposed Data Extraction and Image Recovery Algorithm
Given the dual stego images and we recover the original cover image and extract the secret message using the following steps:
Step 1: Classify the stego images’ pixels into two classes: embeddable and non-embeddable pixels, in the same manner as in the embedding step. Fig. 6 shows this classification where the white pixels are the embeddable pixels while the shaded pixels are the non-embeddable pixels.
Step 2: Recover the original cover image by combining the non-embeddable pixels in the two stego images and .
Step 3: Refer to Fig. 6 and let the current decimal values of the stego images’ pixels in and be and . Extract the secret message digit using Eq. (9).
Step 4: Find the binary value of and concatenate it to the extracted binary message. Extract all secret message digits by repeating steps 3 and 4.
We conclude this section by giving a numerical example to illustrate the operational steps of the proposed reversible DHT. Fig. 7 shows a 1 × 2 original cover image and the corresponding dual cover images and . The shaded pixels in Fig. 7 are non-embeddable, while the white pixels are embeddable. The preprocessing step replaces the value 255 in the cover image by 253. Fig. 8 shows the resulting preprocessed cover images.
Given the binary secret message W = , convert every two binary bits into its format. The resulting secret message is S = . The digits of the secret message S are embedded in the dual cover images as follows:
Assign the value of the first pixel in as: and assign the value of the first pixel in as → Use Eq. (6) to find the value of the extraction function, → Get the first secret digit → Use Eq. (7) to find the distance . → use Eq. (8) to find the stego images pixels as:
Assign the value of the next pixel in as: → assign the value of the first pixel in as: .→ use Eq. (6) to find the value of the extraction function → Get the next secret digit → Use Eq. (7) to find the distance . → use Eq. (8) to find the stego images pixels as:
Fig. 9 shows the resulting dual stego images.
For extraction, starting with dual stego images in Fig. 9, the original cover image is obtained by combining the shaded columns in , and . We extract the hidden message as follows: Assign the value of the first pixel in as: and assign the value of the first pixel in as: . → Use Eq. (9) to find the value of the first hidden digit as → Convert to its two bits binary equivalent → Concatenate this to the extracted binary message. Assign the value of the next pixel in as: and assign the value of the next pixel in as: . → use Eq. (9) to find the value of the next hidden digit as → Convert to its two-bit binary equivalent → Concatenate this to the extracted binary message. The resulting extracted message: Sext = [S1dS2d] =  is equivalent to the embedded message W.
This section presents the simulation results of the proposed technique. To measure the quality of the stego images, we use the PSNR values of the stego images. The PSNR between an original cover image I and the stego image , is given by Eq. (10) :
We use the embedding rate as a measure of the data hiding capacity. The embedding rate R is given by Eq. (11).
where represents the total number of the embedded binary bits.
We used the CPU time as a measure to evaluate the numerical efficiency. In our simulations, we used twelve grayscale cover images. The secret message is a randomly generated binary secret message using MatLab. Fig. 10 shows the twelve test images. We performed our simulations on a personal computer with an Intel(R) Core(T.M.) i7-6700 CPU @ 3.40 GHz and 8 G.B. memory.
4.1 Performance Evaluation of the Proposed Technique
Fig. 11 shows the stego images at the maximum embedding rate. The high quality of the stego images is clear, and the human eye can detect no traces of any artifacts.
Table 1 gives the main features of our technique. The high quality of the stego images is evident via an average PSNR value of more than 49 dB. In addition, Table 1 also shows that the average absolute distortion in the pixels is about 0.5/pixel. Moreover, Table 1 also shows that, at the maximum embedding rate, our algorithm achieved a high average PSNR value of 49 dB. We conducted another experiment to investigate the PSNR values of the stego images against different embedding rates. Fig. 12 shows the results for the Lena image, where the PSNR values for both stego images ranged from 80 dB when the embedding rate is 1/16 bpp to 49 dB when the embedding rate is one bpp. An essential feature of our algorithm is that the PSNR values are practically identical for both stego images. This feature improves security. In addition, the PSNR values are almost identical for all test images. This shows that the performance of our technique does not depend on the cover image. Our technique embeds one digit (two binary bits) in each pixel pair in the cover images resulting in an embedding rate of one bpp, as Table 1 shows. Table 2 demonstrates the numerical efficiency of our technique. The numerical efficiency of our technique resulting from using the modification tables and eliminating the search step is evident in the short embedding and extraction times shown in Table 2.
4.2 Performance Comparisons with Related Work
Table 1 also compares our technique with  in terms of PSNR values, average absolute error, and embedding rates. The average embedding rate of our technique is one bpp, while that of  is 0.9859 bpp. In obtaining these results, we converted each byte of the binary message to its value in  and value in our technique. The PSNR values for both techniques are close. However, The PSNR values of our two stego images are almost identical, while that of  are slightly different. Fig. 12 gives the PSNR values vs. Embedding rate for both techniques and shows that the PSNR values of our two stego images lie between the two stego images’ PSNR values in . Another feature of our technique is numerical efficiency and simplicity. Instead of using a neighborhood matrix in the embedding and extraction phases in , we only need two lookup tables, resulting in significant savings in storage and numerical calculations. Although the execution time depends on the status of the operating system and may vary from one run to another, Table 2 shows that the embedding CPU time of our technique is one-fifth of that of . Table 2 also shows that the extraction CPU time of our technique is 0.8 that of . Table 3 compares the PSNR values of our technique with that of [24,31,42,43]. These techniques have maximum embedding rates of 1, 1, 1.25, and 1 bpp, respectively. For a fair comparison, Table 3 gives PSNR values at the same embedding rate of 1 bpp. Our technique and  give equal PSNR values for both stego images making them less susceptible to detection by intruders. The PSNR value of the two stego images in [31,42,43] by 1, 5, and 5 dB, respectively, making it more vulnerable to detection. Overall, our technique outperforms other techniques in terms of image quality, simplicity, and numerical efficiency.
This paper presented a new reversible RDHT based on EMD. The proposed technique is simple and numerically efficient. It limits image distortions and increases stego image quality and transparency. Each pixel pair of the two stego images (one pixel from each stego image) hides one secret digit by modifying only one pixel with a maximum distortion of two resulting in an embedding rate of one bpp. Using the two novel lookup tables highly reduces numerical calculations by eliminating the numerically demanding search step in similar techniques. The embedding strategy enabled direct reversibility at almost zero computational cost. Increased security is achieved by transmitting the two stego images on separate channels so that only the receiver with both stego images and the embedding sequence can extract the secret message. Experimental results demonstrate the high quality of the two stego images with more than 49 dB average PSNR values. Experimental results also show that the embedding CPU time of our technique is one-fifth of that of . The extraction CPU time of our technique is 0.8 that of .
Funding Statement: The author received no funding for this study.
Conflicts of Interest: The author declares that he has no conflicts of interest to report regarding the present study.
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