Abstract
Reversible data hiding (RDH) is a method that allows a cover image to be completely recovered from its corresponding stego image without distortion after the embedded secret messages have been extracted. Prediction-error expansion (PEE), as a classic RDH scheme, has been studied extensively due to its high quality of stego images. Based on prediction errors, threshold values, and the relative distances between each bin and zero bin, we present a bidirectional shift and double-way prediction strategy to solve the multiple embedding problem. Compared with the original algorithm, this scheme only takes a little more time and reduces the PSNR slightly, but it improves the embedding capacity significantly and allows for reversible data hiding. When both threshold values of and are equal to 2, the average ER value of 108 test images is 1.2 bpp which is ideal for medium data payload. At the same time, the PSNR is above 30 dB, making embedded information visually imperceptible. These data, together with other experimental results, show that the method proposed in this paper has obvious advantages in image quality and embedding capacity.
1. Introduction
The concept of steganography was first recorded and used in 1499 by Johannes Trithemius in his Steganographia, which was placed on the Index Librorum Prohibitorum in 1609 [1] and removed in 1900 [2], but the practice of secret communication has existed since ancient times. There are stories from Roman times concerning how slaves could hide messages and send the messages to recipients even though they were closely watched. Currently, in the computer age, camouflaged information is still processed to make it unrecognizable to the human eye, but it can possibly be revealed by computer visual recognition. In recent years, with the increasing emphasis on information security, data hiding technology has been used extensively for copyright protection and the authentication of the content of digital multimedia [3–5]. The content of an original image is altered after some data have been embedded in the image, and the changes are visually imperceptible. But there is an inherent shortcoming of the traditional data hiding scheme, which is that the cover image cannot be restored completely after the receiver extracts the secret information [6–8]. For some specific scenarios, such as military and medical information and the preservation of artwork, even slight distortions in the images are intolerable since it is always essential to recover the original raw data. Therefore, reversible data hiding techniques have been proposed and developed that enable the receiver to extract the secret information, as is done in the traditional scheme, as well as to reconstruct the entire original image without distortion. From the perspective of its application, RDH can be considered a covert communication channel for the transmission of secret information. Thus, it can be seen that its application is quite extensive, such as embedding patients’ private information into corresponding medical images and providing lossless authentication watermark for satellite images.
The first reversible data hiding (RDH) scheme was proposed in 1997 by Barton in his patent [9]. Barton’s scheme compresses some of the alternative superimposed bits, adds a stream of bits, and embeds them into data blocks. Since then, RDH has attracted the attention of many scholars, and related works have been published frequently. In 2001, an improved algorithm [10] focusing on lossless recovery was proposed by Honsinger et al. One year later, Fridrich et al. [11] developed a high-capacity scheme based on embedding message bits in the status of pixel groups. In 2002, Celik et al. presented a lossless image compression algorithm [12] by using a prediction-based conditional entropy coder. This method uses some feature sets S of the image to form saved space by lossless compression and then utilizes them to hide information. However, as EC (embedding capacity) increases, more bit planes must be compressed, resulting in a sharp increase in distortion, which prevents the method from achieving satisfactory performance.
Thus, as can be seen, it has become the focus of scholars to determine how to reduce distortion while considering as much secret storage capacity as possible. Subsequently, expansion-based RDH schemes have been proposed to achieve the desired effectiveness, which is defined by three basic classifications, i.e., histogram shifting (HS), difference expansion (DE), and prediction-error expansion (PEE). The first HS-based reversible data hiding was proposed by Ni et al. [13] in 2003. The scheme is based on the concept of histogram of an image, which normally refers to the histogram of grayscale values of all pixels in the image, which shows the number of pixels at different grayscale values found in that image. For an 8 bit grayscale image, there are 256 possible grayscale values. The histogram will display the distribution of the pixels among those grayscale values between 0 and 255, as shown in Figure 1. For instance, the red mark in Figure 1 shows image “Baboon” has 2759 pixels with a grayscale value of 121. The scheme is implemented by shifting a certain histogram bin, usually the peak bin, which has the maximum number of pixels in the image. But HS techniques do not work well when the histogram of an image is equal. Although multiple pairs of peaks and minimums can be used for embedding, the embedding capacity is still insufficient. In 2003, with peak and zero bins to be chosen from the middle and the edge part of the histogram, respectively, a special case of HS, called the DE-based scheme, was first proposed by Tian [14], and it was based on the difference expansions of pixel pairs. Later, several improved techniques for DE-based embedding [15] were proposed, and they included integer transformation, pixel difference [16, 17], and prediction-error expansion (PEE). PEE is the focus of this paper, which was first proposed by Thodi and Rodriguez [18], and it extended the pixels and predicted differences for embedding data. Compared with the methods based on DE and HS, prediction-error histogram distribution is sharper and more centralized, so PEE has better performance. In addition to this, unlike DE, which only considers the correlation of two neighbouring pixels, and also unlike HS, which only considers the current pixel itself, PEE uses more adjacent pixels to obtain better performance. Existing PEE-based RDH methods are mainly based on the modification of the one- or two-dimensional prediction-error histogram (PEH). The approach we propose using the PEH to form a method with a bidirectional shift and double-way prediction strategy for RDH and the details associated with its use are covered in the next section.
In this paper, we propose an improved RDH method based on the study [19] with double-way prediction and bidirectional shifting. The difference between the current value and the prediction value is used to form a prediction-error histogram. This histogram will be used to hide secret messages by expanding and shifting, and the embeddable bins are called peak (peak point) bins. Through outward shifting, they will be used to insert secret bits into the vacant bins reserved by other unembeddable bins that are also known as outer region bins. This process will be accomplished through two stages of prediction and corresponding processing based on the predicted values. The process is repeated for all pixels of the entire image to produce the stego image. During the extraction procedure, to draw out all of the secret bits and recover the original image, the receiver end can utilize the auxiliary information, which includes the position information of zero bins, threshold values, and pixels in the last row and the last column that have been kept intact. First, we calculate the prediction values of pixels in neighbourhood in the last row and the last column in the stego image. And the difference, named the prediction error (PE), between the prediction value and the stego-pixel value can be worked out at the same time. According to the PE, the quantity of the pixel value changes and the hidden information can be determined. Thus, the original pixel value can be worked out. Then pixel by pixel, the prediction value of the currently processed pixel can be determined using the pixels in its neighbourhood that have been recovered before it. Finally, all the pixel values of the original image and secret message are recovered. Therefore, the algorithm proposed in this paper is an RDH scheme. In addition, the experimental results show that the proposed method can achieve higher embedding capacity and better image quality than the previous scheme.
The contributions of this paper are as follows:(1)We apply bidirectional histogram shifting to improve the hiding capacity of a secret message. During the shifting procedure, we make two shift directions opposite to each other to reduce distortion.(2)We utilize a proposed double-way prediction strategy to get more embedding capacity.(3)Through the experimental data of 108 images, it is found that when , the algorithm effectively and significantly improves the information hiding capacity by 65% but only reduces the PSNR value about 6%, and the PSNR value remains above 30 dB.(4)The method can completely recover the image without any error; therefore, it is an RDH scheme with high hiding capacity.
The rest of this paper is organized as follows: In Section 2, Wang et al.’s method is reviewed, and the proposed method is described in Section 3. In Section 4, the experimental results are presented and discussed, and the conclusions are given in Section 5.
2. Review of Wang et al.’s Method
In this section, we introduce Wang et al.’s method [19], which is based on a double-way prediction error and right-left shift. To further decrease distortion of the stego image, the authors presented an extension of the traditional histogram shifting technique. The prediction error is produced by the difference between the value of the current pixel and the average value of the sum of its three adjacent pixels, i.e., the pixels in the right, bottom right, and bottom. Their technique is based on the principles that (1) the neighbouring pixels have a strong correlation and (2) the prediction-error distribution of adjacent pixels has a prominent maximum, that is, most of the prediction errors are expected to be very close to zero. The distribution of prediction errors has a zero mean value, similar to the Laplace distribution, which can be concluded from papers [20–23]. The authors utilize this characteristic of the distribution in combination with right-left shifting to improve the embedding capacity. To further introduce the algorithm, some basic concepts are provided in the next paragraph, and an example is given to illustrate the embedding and extraction procedures.
2.1. Definition and Production of Prediction-Error Histogram
To discuss the details of Wang et al.’s method, we use a grayscale image I of size H × W, where H and W are the height and width of an image, respectively. Assume that the upper left coordinate position of I is (1, 1) and each pixel of I is assigned a coordinate. The coordinates of all pixels are successively increased from left to right and from top to bottom of the image. One of them is referred to as I(i, j), as shown in Figure 2.
The scheme first predicts all pixels to form the prediction values P(i, j) of a pixel I(i, j) using the following equation:where fix(•) is a function, of which the input is a real number and the output is an integer. It rounds the input to the nearest integer toward zero. For example, fix(3.2) = 3 and fix(8.6) = 8. Then, the PE (prediction-error) value Pe(i, j) of I(i, j) is obtained by the following equation:
The pixels in the last row and the last column of an image are the reference pixels for recovery and thus cannot be predicted; the original values will be maintained. The preservation of these original data will play an important role in the extraction and recovery process.
Then, Pe(i, j) is produced by the formulation as formulated in equation (2). The prediction-error histogram (PEH) is generated by h(Pe) function, which counts the number of occurrences of a prediction-error (PE) value. It contains the peak bins, zero bins, and other bins, which are distributed symmetrically with values in the range of [−255, 255] for an 8 bit grayscale image. The PEH of “Barbara” is shown in Figure 3(a) as an example, where h(0) = 19209, h(81) = 337, and h(−68) = 100, which mean there are 19,209 pixels with a prediction-error value of “0,” 337 pixels with a prediction-error value of “’81,” and 100 pixels with a prediction-error value of “−68.” Traditionally, the peak bin, which is also called the embeddable bin, is the bin that is located in the middle of the histogram. Here, the concept of peak bin is extended to all of the central bins within the threshold called TH, and the number of peak bins is defined as PK = 2 × TH + 1, which consists of the number of traditional peak bins and the bins on the left and right sides of it within the distance of TH, as shown in Figure 3(b).
During the embedding procedure, if the secret bit to be embedded is “1,” the corresponding peak bin will be expanded by 1 unit; otherwise, it remains unchanged. The outer bins will shift outward to make room for peak bins. Zero bins, which are located on two edges of the outer sides of the PEH, are denoted as Zrn and Zlm, respectively, acting as buffers for shifting. Here, zero means there are no PE values equal to that bin value. For example, in “Barbara,” Zr1 = 125, which means that there are no PE values equal to 125. Zr2 = 126, Zr3 = 129, Zr4 = 130, Zr5 = 132 and Zl1 = −122, Zl2 = −129, Zl3 = −131 are the other zero bins, as shown in Figures 3(c) and 3(d), which are the enlarged version of Figure 3(a). In addition, subscripts n and m record the number of zero bins between the current bin and peak bins. For example, when the value of the current bin is 128, the largest Zrn less than 128 is Zr2, where n is 2, which means there are 2 zero bins between peak bins and 128. When right shifting is needed during the embedding procedure, these 2 zero bins can be occupied to reduce two moves and thus to reduce distortion. For instance, if we assume that all bins must shift to the right direction by Δbins = 5, then bin 128 will move to 131 rather than 133 because 128 + Δbins − n = 128 + 5 − 2 = 131, and the occupancy of these 2 zero bins is achieved by subtracting 2, as shown in Figure 3(e). The same thing happens for a bin value of −130; that is, the smallest Zlm larger than −130 is Zl2, where m is 2, and when left shifting, these 2 zero bins can be occupied to reduce the quantity of moves by 2 and lessen distortion.
2.2. Embedding Procedure
The secret information to be hidden is denoted as a binary string S, with elements s1, s2, si,…, sj,…, sL of {0, 1}, , where L is the length of S. We set TH = 2 and take image “Barbara” as an example. The number of peak bins of “Barbara” is 5, as shown in Figure 3(a). According to the distribution of prediction errors of “Barbara,” Figure 4 is provided, and the details about the figure demonstrate right and left shifting of peak bins and the whole prediction-error histogram of “Barbara.” Because the image being discussed is a grayscale image, the histogram of prediction errors would be distributed in the interval [−255, 255]. Due to the similarity of the values of neighbouring pixels, the peak bins were concentrated on value 0 and its adjacent values. It is this feature that the authors used to hide information. There are two phases of movement of the PEH in the algorithm, which are first shifting to right and later shifting to left. The first line in Figure 4 is the original PEH of “Barbara,” the second line is the PEH after right shifting, and the third line is the PEH after left shifting.
When right shifting (RS), bin value −2 remains as −2 to hide the secret bit “0,” and it changes to −1 to hide the secret bit “1.” In a similar way, bin value −1 moves to 0 to embed the secret bit “0,” and it shifts to 1 to hide the secret bit “1.” One bin moved after another, bin value 2 will move to 6 and 7 to hide “0” and “1.” But the bins on the right and left sides of peak bins will not be used to hide secret bits. Due to the movement of peak bins, the bins on the right side will have to shift outward to make room for peak bins. Thus, bin 3 will move to 8, bin 4 will move to 9, etc. To reduce the distortion, every zero bin can be used to buffer the amount of shifting by reducing the movement to the right side by one bin. Therefore, bin value 127 will move to 130 because there are 2 zero bins, i.e., Zr1 = 125 and Zr2 = 126, between bin value 127 and peak bins. The amount of moved units is PK − 2 =5–2 = 3. Those bins with values less than −2 will not be affected and maintain their original values.
When left shifting (LS), PKl is 10, which is twice as large as the PK value. It is because after right shifting, bins of the peak region have already been expanded to twice the number of the original peak region. This time, all of the bin values that are larger than 8 remain unchanged, but those less than 8 will move. Concerning bin value 7, it remains unmoved to hide the secret bit, “0,” while it shifts to 6 to hide “1.” In a similar way, bin by bin, bin value −2 moves to −11 to embed the secret bit “0” and shifts to −12 to hide the secret bit “1.” The bins on left sides of PKl will not be used to hide secret data but must be moved to make room. For example, bin value −3 will shift to −13 and bin value −120 will shift to −130. Instead of moving to the left side by PKl = 10, bin value −123 moves by 10 − 1 = 9 units because there is a zero bin Zl1 = −122 between −123 and peak bins, which is used to reduce one unit of movement.
To ensure better understanding, three examples are provided to explain how shifting and expanding work during the right-left shifting phase:
Example 1. The red values in Figure 4 are PE values of peak bins according to the definition of TH. In the cover image “Barbara,” the original pixel value I(2, 2) is 198, the corresponding prediction value P(2, 2) is 196 (calculated from equation (1)), and the prediction-error value is 2 (calculated from equation (2)), which is one of the peak bin values. In the RS (right shifting) stage, the amount of shifting is 5, which is the result of 7 − 2, where the secret bit si is assumed to be “1” here. Thus, the value of the marked pixel is 198 + 5 = 203. In the LS (left shifting) stage, the amount of shifting is 0, where the secret bit sj is assumed as “0.” After the above procedure, the pixel value I(2, 2) of the cover image is changed from 198 to 203, which is the final value of its corresponding stego pixel, with embedding two secret bits.
Example 2. The green values in Figure 4 are bin values between 3 and 133. Number 3 is the smallest prediction error greater than TH, and number 133 is the greatest nonzero bin value. In the RS stage, these bin values will add a relative positive integer according to the subtraction calculation of PK − n, where n is the total quantity of zero bins less than the current bin value. In the LS stage, these bin values will remain unchanged. For example, I(371, 457) = 228, P(371, 457) = 97, and Pe(371, 457) = 228 − 97 = 131 from equation (2), and the amount of shifting is PK − n = 5–4 = 1, where n is 4 because the largest zero bin less than 131 is Zr4 (Figure 3(d)), which means there are 4 zero bins less than 131. Thus, bin 131 moves to 132, and the marked pixel value is 228 + 1 = 229. In the second-stage hiding procedure, the shifting of bin 132 is not needed, so the value of the final stego image pixel of I(371, 457) is 229.
Example 3. The purple values in Figure 4 are bin values between −3 and −137. Number −3 is the largest prediction error less than −TH, which is −2, and number −137 is the smallest nonzero bin value. In the RS stage, these bin values will remain unchanged because they are less than −PK. In the LS stage, these bin values will subtract a relative positive integer according to the subtraction calculation of PKl − m, where m is the total quantity of zero bins greater than the current discussing bin value. For example, I(378, 453) = 51, P(378, 453) = 188, and Pe(378, 453) = 51 − 188 = −137. In the RS stage, bin value −137 remained unchanged; thus, the amount of shifting of bin value −137 is 0, which means pixel value 51 will remain unchanged. In the LS stage, the amount of shifting is PKl − m = 10 − 8 = 2, where m is 8 because the smallest zero bin on the right side of bin −137 is Zl8 (Figure 3(c)). At last, the final stego-pixel value of I(378, 453) is 51 − 2 = 49.
2.3. Extracting Procedure
After the embedding procedure, TH, the zero bins Zrn and Zlm, and the stego image, in which the last row and the last column are original and intact, will be sent to the receiver. Thus, the predicted pixel values can be calculated from the pixel values in the last row and the last column. To be specific, according to equation (1), for an image with a size of 512 × 512, the prediction value P(511, 511) is first calculated, and after that, the difference between the prediction value and the stego-pixel value of P(511, 511) is worked out. Then, based on the prediction error and how it relates to TH, the zero bins Zrn and Zlm, the original value of I(511, 511), and the embedding bit value (if it had been hidden before) are solved. Now as a given message, the original value of I(511, 511) can be used to determine I(511, 510). Thus, pixel by pixel, the whole secret information can be extracted and the original image can be recovered in the reverse processing order of the embedding procedure.
3. Proposed Scheme
To improve the quality of the stego image and increase the amount of secret information that can be hidden, we use bidirectional PEH shifting in double-way prediction based on Wang et al.’s method [19]. The process of generating the stego image is divided into two stages: embedding process 1 using the raster scan order and embedding process 2 using the inverse raster scan order. First of all, pixels in the last row and the last column are kept and used as reference pixels. The first prediction is made in a raster scan order, i.e., from the first pixel in the upper left corner of an image, top to bottom, and left to right, and the top-left pixel is predicted by the other three pixels in its square-shaped adjacent context. The calculation process is shown in equation (1), and the prediction error, P (i, j), is calculated by equation (2); the second-stage prediction is processed in the reverse raster scan order from the pixel in the bottom right corner, right to left, and bottom to top, and the lower-right pixel is predicted by the other three pixels as follows, respectively:where I1 and I1(i, j) are the stego image and its stego pixel after embedding process 1. In the process of information hiding in these two stages, PE (prediction-error) values are determined separately, and they are named and , respectively. In addition, during embedding process 1, we first expand or shift bins of the PEH to the right direction and afterward to the left direction. During embedding process 2, to avoid severe distortion which may occur, the movement is set to the opposite direction from the first embedding process. That is to say, during embedding process 2, we first expand or shift bins of the PEH to the left direction and then to right. The number of moving units depends on the interval in which the value of the bin falls. After the shifting of all the pixels has been finished, the original prediction error (PE) is altered to the modified prediction error (MPE). The difference of the MPE and PE, called Δbins, is added to the original pixel to create the value of the stego pixel.
The notations of this scheme are presented in Table 1, and the subsequent discussion explains the algorithm process in detail, including the pseudocode of the hiding and extracting information procedure.
The details of embedding process 1 are explained by the pseudocode presented in Algorithm 1.
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Figure 5 is an illustration of embedding process 1. We assume TH = 2, and there is a pixel P (i, j) with its PE value Pe (i, j) = 0, which is a value belonging to bin value 0 shown in Figure 5(a). During right shifting of the PEH, bin value 0 is shifted to bin value 2, with “0” embedded, as shown in Figure 5(c). That is to say, Δbin (i, j) = Pe (i, j) + TH +si = 0 + 2 + 0 = 2. Then, I (i, j) is changed to I1r (i, j), where I1r (i, j) = I (i, j) + Δbin (i, j) = I (i, j) + 2. Then, left shifting is used to embed “1” in I1r (i, j), which changes bin value 2 to −4. Per (i, j) = Pe (i, j) + Δbin (i, j) = Pe (i, j) + 2; Δbinl (i, j) = Per (i, j) − TH − PK − sj = Pe (i, j) + 2 − 2 − 5 − 1 = −6. Finally, I1 (i, j) = I1r (i, j) + Δbinl (i, j) = I (i, j) + 2 − 6 = I (i, j) − 4. After that, pixel I (i, j) has been embedded with “0” and “1” with pixel value adding−4. Thus, with varied quantity of movements of peak region bins in the PE histogram, secret data are hidden. Moving the outer region makes room for the peak region to move and hide secret information. Those pixel values whose prediction-error values are located in the right outer region are increased. Those pixel values whose prediction-error values are located in the left outer region are decreased. In addition, with help of zero bins, the amount of movement in the rear of the outer region can be reduced to some extent. Of all these, the maximum change is 2 × PK.
Next, the current stego image I1 will be predicted with equation (3), and the order of prediction processing is in the reverse order from the prediction in embedding process 1. Then, the PEH shifting direction is toward left and then right which is the opposite order of embedding process 1 also. Compared with Wang et al.’s scheme [19], the distortion will be a little bit larger due to the increase in the hiding capacity of secret information. The pseudocode of embedding process 2 is given in Algorithm 2.
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The overflow/underflow problem occurs when the values of some pixels, called boundary pixels, are changed from 255 to greater than 255 or from 0 to negative values. For these pixels, according to reference [24], we use a different approach from Wang et al.’s. The values that will exceed the boundary after Δbins are added will not be allowed to participate in the addition; for them, the original values are kept unchanged and called pseudovalues. The pixel values, whose addition result values are equal to the corresponding pseudovalues, are called genuine values. A boundary array can be recorded to distinguish genuine values from pseudovalues to avoid the confusion. Each member of the array corresponds to a boundary pixel in the stego image, with genuine “0” and pseudo “1,” respectively.
3.1. An Example of Embedding Process
As shown in Figure 6, pixel I (31, 12) of “Baboon” is identified as I (31, 12), and its pixel value is 119. Based on equation (1), Pe (31, 12) = −2 satisfies the condition , where TH = 0. Thus, Δbin value in right shifting of embedding process 1 of I (31, 12) is 0, I1r (31, 12) = I (31, 12) = 119, and Per (31, 12) = −2. In left shifting of embedding process 1, due to Per (31, 12) < TH = 0, Δbinl (31, 12) = −PKl = −2; therefore, I1 (31, 12) = I1r (31, 12) + Δbinl (31, 12) = 119 + (−2) = 117, which is the result of embedding process 1. Next comes embedding process 2. The new prediction value is generated according to equation (3), and the prediction error is obtained. Now comes left shifting of embedding process 2. According to the given threshold value, i.e., = 0, we find that . Assume that the current information to be hidden is sj = 1. Thus, Δbin∗ (31, 12) = −1, which expands bin value 0 to −1 to hide the secret bit “1”; thus, , and. Then, we proceed with right shifting of embedding process 2. satisfies the condition , and we assume that the current information to be hidden is so that (31, 12) = 1 hides “1.” Finally, the stego pixel is . At this point, the embedding processes are finished.
3.2. An Example of Extraction Process
Now, we assume that the receiver receives the stego image, threshold TH, , zero bins Zrn, Zlm, Z∗rn∗, and Z∗lm∗, and a boundary array of over/underflow pixels trying to extract the secret and recover the original image. First, based on the first line and first column pixels that have been preserved, the receiver can determine the prediction value of pixel (2, 2) and, then, its I1 (2, 2) pixel value. Then pixel by pixel, in the raster scan order, all of the ones before I1 (31, 12) will be recovered. Then, calculating by equation (3) and using , , and that have been determined, the prediction value is determined, that is, P∗ (30, 12) = 117. And the difference between P∗ (30, 12) and stego pixel , which is , can then be figured out. Based on the difference value and = 0, we can determine that = −1 and = 1. Thus, , sj = 1, and . Thus, I1 (31, 12) = 117. Through the same process, we can determine every I1 (i, j), pixel by pixel. Subsequently, the first phase of the extraction process is completed. And, based on the values of I1 and all of the values in the last column and last row of the image that have maintained their original values from the cover image, we can determine, pixel by pixel, the prediction value and prediction errors of all the pixels. The difference between I1 (31, 12) and P (30, 12), called Pel (31, 12), is −4; thus, Per (31, 12) = −2, , and I1r (31, 12) = I1 (31, 12) − Δbinl (31, 12) = 117 + 2 = 119 because of the given threshold value TH = 0; also, Δbin (31, 12) = 0 and Pe (31, 12) = 0. Thus, I (31, 12) = I1r (31, 12) − Δbin (31, 12) = 119, which is the original pixel value. The process is shown in Figures 6 and 7, in which the blue arrow points out the embedding process and the orange arrow indicates the extraction process.
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4. Experimental Results and Analysis
The experimental platform we use consists of an Intel 3.41 GHz i7–6700 CPU, 32 GB RAM, and the Windows 10 Professional 64 bit operating system. Our experiments are implemented by MATLAB. The performance of the proposed scheme is estimated by some criteria, i.e., embedding capacity (EC), peak signal-to-noise ratio (PSNR), and execution time (s: seconds). Tables 2–5 show some figures that are determined by using standard 512 × 512 grayscale common, medical, texture, and aerial images, respectively.
EC can be controlled by quantity of peak bins, and it can be calculated by equation (6). The embedding rate (ER) represents the percentage of the embedded secret bits in the whole pixels of the cover image. ER is defined as in (7). Here, H × W is the size of the cover image:where PK is the quantity of peak bin values we use to hide the secret messages, the value of which, in fact, will change in different embedding phases according to the TH value we use.
In general, the peak signal-to-noise ratio (PSNR) is used to measure the visual quality of the reconstructed image or the stego image. The PSNR is defined aswhere MSE is the mean square error between the cover image and the stego image. For a grayscale cover image of size H × W, the MSE is defined aswhere pij represents the pixel value in row i and column j of the original image and is the corresponding stego-pixel value of pij.
As a common sense, the difference between the cover image and the stego image is invisible to naked eyes when the value of the PSNR exceeds 30 dB. Thus, to control the imperceptible distortion of a stego image, the embedding process will always be stopped when the PSNR value is less than 30. In our scheme, to avoid huge distortion, we only provide the experimental data with both of TH and values less than 2.
Compared with the original algorithm of Wang et al.’s [19] that made only one-way prediction, our algorithm uses double-way predictions to generate two PEHs in order to hide more information. To minimize PSNR growth, during the second prediction, histogram shifting will be conducted from left to right, which is contrary to the shift order of the first prediction phase. Moreover, the original algorithm subtracted 255 when pixel values were greater than 255, or simply added 255 when pixel values were less than zero. The solution of the underflow/overflow problems they used was inefficient. In this paper, we solve the problems in an efficient way based on the study [24] by avoiding some boundary pixels from performing huge transition changes, such as from white to black or from black to white. Increased embedding capacity (ΔEC) in bits and the EC increasing rates (IR), as well as the PSNR and its decreasing rate (DR), for the eight images in Figure 8 are shown in Table 2. Comparing our proposed algorithm with the algorithm proposed by Wang et al., Table 2 shows that, by using our proposed algorithm, there are a huge increase in EC and a slight decrease in PSNR, and the decrease rate of PSNR is significantly smaller than the growth rate of EC.
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Table 2 indicates that, by using the proposed double-way prediction, the embedding capacity is increased for all the images we studied. However, the increment of each image is different, and the increase range of the same image under different TH and conditions is also different. Moreover, PSNR values of all of the images decreased, and the amounts of the decreases are different. It is worth noting that the increase rate of EC is far greater than the decrease rate of PSNR. And an example is used to illustrate this finding. For image “Lena,” when TH = 0 and = 0, ∆EC is 35,188 bits, which means that compared with the original algorithm, the proposed algorithm can embed 35,188 more bits in the image, improving the embedding capacity by 67.3%; for the same conditions, the PSNR value maintains the ideal value of 40.9, with a decrease rate of only 8.1%; time cost increased by only 1.74 seconds. Thus, compared with the original algorithm, the proposed algorithm sacrifices a little bit of PSNR value and time cost in order to hide a lot more secret information. This phenomenon is even more pronounced for image “Baboon.” The amount of hidden information is increased greatly by more than 85%, while the PSNR value decreased by slightly less than 9% when TH = 0 and = 0. Similar results are obtained for the other images. Therefore, we come to the conclusion that, for common images, the proposed scheme provides a very large increase in information hiding capacity with only a small reduction in PSNR. Next, we use three groups of images, i.e., medical images, texture images, and aerial images, and each group has eight images to illustrate the relevant issues.
Figure 9 shows eight medical images with sizes of 512 × 512, corresponding to these pictures; Table 3 shows us the payloads (bits), PSNR, and embedding rate (ER) with condition values of TH = 2 and = 2. Table 3 indicates that the hiding capacities of all eight medical images are increased by about 95%, and their bpp (bits per pixel) values generally are greater than 3.8. In fact, most of the decrease rates in PSNR values are about 13%, which means that compared with the original algorithm [19], the proposed scheme provides significantly increased EC values, while PSNR values are reduced slightly. It is concluded that the proposed double-way prediction algorithm significantly increases storage capacity for secret messages and guarantees ideal ER and good PSNR values of stego medical images. Thus, the scheme is particularly suitable for medical images as cover images.
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The texture images shown in Figure 10 illustrate the related information in Table 4. Since the structure of the texture of each image is different, the information hiding capacity of each image is also different. Some images, e.g., Figure 10(e), have large smooth blocks that make the histogram sharper and thus allow more information to be hidden, and its ER value is 2.95. However, Figure 10(h) has a complex and nonsmooth texture, and it produces a flatter histogram, thereby dramatically reducing the information that can be hidden, and its ER value is as low as 0.11. Thus, it is apparent that the proposed scheme will allow images with more smooth areas to hide much more information.
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For the eight aerial images shown in Figure 11, the payload, PSNR, and ER of them are given in Table 5. From the experimental data, we can get to the conclusion that all of the increasing rates of payload are greater than 50% and all of the decreasing rates of PSNR are about 10%. But the ER value here is a little bit low, being within 1.
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From Figure 12, we can hardly see the visual differences of the stego images from common, medical, texture, and aerial categories with different ER and PSNR.
We use 108 images randomly selected from SIPI and Dicom image databases to get the experimental data, and from Figure 13, we can see that ER is improved while the PSNR is decreased, but the increasing rate of ER is greater than the decreasing rate of PSNR obviously. Among them, Wang et al.’s algorithm has only one parameter, TH, and the proposed algorithm has two parameters, TH and . Therefore, ER and PSNR value curves of Wang et al.’s are not affected by the second parameter as the proposed scheme. Furthermore, the ER value of the proposed scheme is steadily increased, and when TH = 2, = 2, the average ER value of those 108 images is about 1.2 which is ideal for medium data payload. At the same time, the PSNR value is 29.7, making embedded information visually imperceptible.
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5. Conclusions
In this paper, we proposed an improved, bidirectional shift-based reversible data hiding scheme using a double-way prediction strategy. The embedding process consists of two phases, i.e., (1) creation of the first histogram of prediction-error and right-left shifting with embedding and (2) creation of the second histogram of prediction-error and left-right shifting with embedding. Both of the extraction of the embedded message and the recovery of the original image can be realized exactly using the stego image and relevant information. After experiments on common, texture, medical, and aerial images, we found that the embedding rate of medical images increases the most. Whether it is medical images or other types of images, we can all come to the conclusion that the proposed scheme is significantly superior to the previous scheme in embedding capacity, with a little bit of time loss and PSNR decline, such that when , the algorithm effectively and significantly improves the information hiding rate by 65% but only reduces the PSNR value about 6%, and the PSNR value remains very close to 30 dB. In our future work, we will further improve the visual quality with the increasing hiding rate.
Data Availability
No external data were used to support this study. All derived data sets have been generated in our infrastructure.
Disclosure
This research was performed as part of the employment of the authors.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Soft Science Foundation of Fujian Province, China (Grant no. B19085).