A New Reference Pixel Prediction for Reversible Data Hiding with Reduced Location Map

Abstract

In this paper, a new reversible data hiding method based on a dual binary tree of embedding levels is proposed. Four neighborhood pixels in the upper, below, left and right of each pixel are used as reference pixels to estimate local complexity for deciding embeddable and non-embeddable pixels. The proposed method does not need to record pixels that might cause underflow, overflow or unsuitable for embedment. This can reduce the size of location map and release more space for payload. Experimental results show that the proposed method is more effective in increasing payload and improving image quality than some recently proposed methods.

1. Introduction

Data hiding is a technique which embeds additional information into images, music, and other media for delivery to specific destination. A digital image is a suitable data hiding carrier since it is a common type of content on the internet. Image data hiding schemes can be divided into two kinds, namely, non-reversible ones [1]-[5] and reversible ones [6]-[18]. A non-reversible hiding scheme has high payload but it sacrifices image quality. A reversible hiding scheme preserves the original image but it is restricted to low payload.

Tian [9] proposed a reversible scheme to embed one bit of information by expanding the difference of two neighboring pixel pair to achieve a high payload of 0.5 bits per pixel (bpp). However, larger pixel difference expansion will increase distortion. Ni et al. [10] proposed a reversible data hiding method based on histogram-shifting. Information is embedded by shifting histogram between the peak and zero points. Since only one grayscale value per pixel is modified, the image quality is preserved. However, the payload is restricted by the histogram peak height. Thodi and Rodriguez [11] proposed the shifting of prediction error scheme by combining the pixel value prediction and histogram-shifting. Since the prediction errors in Thodi and Rodriguez’s method are smaller in comparison to the difference expansion, their scheme offers higher image quality than that of Tian’s method.

Tsai et al. [12] proposed a reversible method based on the modification of pixel differences. The image is divided into N blocks where every block is a group of n×n pixels. The center pixel of each block is not modified to ensure image reversibility. A total of N×((n×n)-1) pixel difference values for the blocks are calculated and used to build a difference histogram. The difference values are either positive or negative making up the two peaks used for embedding secret data. Tai et al. [13] also proposed a reversible method based on histogram shifting. This method makes use of the inverse s-order to scan the image, and the previous scanned pixel is the reference pixel. The absolute difference values between the current scanned pixels and the reference pixels are calculated, and are used to generate the difference histogram for embedding data. Tai et al.’s method also makes use of a binary tree with embedding level L to offer an adjustable payload. However, Tai et al.’s method shifts the histogram of absolute prediction errors. The shifted negative prediction errors resulted in unnecessary image distortion.

Hong et al. [14] also proposed a reversible histogram shifting method similar to Tsai et al.’s method. The image was partitioned into blocks of n×n pixels, and the center pixels of blocks are not only selected as reference pixels but are also used to estimate the local variance. The variance is then used to control the payload size and image quality. This method outperforms Tsai et al.’s method in terms of payload and image quality. In 2012, Hong [15] proposed a reversible method by using the dual binary tree (DBT) and error energy estimator [19] to improve Tai et al.’s method. This method avoids unnecessary pixel modifications and uses the median edge detection (MED) [20] predictor to improve Tai et al.’s method with higher payloads and image quality. However, the choice of reference pixels in this method is not fully explored, and the size of the location map can be further reduced. Both improvements resulted in higher payloads and image quality.

In previous research on reversible data hiding using prediction error shifting, one or more reference pixels are used to calculate prediction errors [6]-[7],[11]-[15]. The to-be-predicted pixels are half enclosed by the reference pixels. For example, Hong’s method [15] only uses the left, upper and upper left pixels as reference pixels to predict the current pixel, whereas Tai et al.’s method only uses previously visited pixel to predict the current pixel. However, it is known that using the full neighborhood pixel information (termed as full enclosed) is more precise than the half enclosed partial neighborhood pixels in terms of predicting pixel values. In addition, some pixels after embedding may result in underflow or overflow situations. Therefore, a location map for recovering the image is used to keep records of the pixels. Full enclosed reference pixels provide more information to eliminate the requirement of recording the location map. Since the location map occupies space, reducing the length of the location map is also important.

In this paper, we propose a full enclosed prediction method that outperforms previous related works. The full enclosed prediction is based on top, below, left and right neighborhood pixels, known as reference pixels. Before a secret message is embedded, the variance of the reference pixel is calculated. The aim is to prevent embedding in complex pixels so as to maintain a high image quality. Furthermore, the reference pixel is also used to predict the current pixel for possible overflow or underflow which would then be excluded from embedding. Finally, the location map records only pixels that are used in embedding. This decreases the size of the un-compressed location map. The size of the location map is smaller than the one in Hong’s MED prediction method [15] which requires the map to be the size of the image.

 

2. Related Work

Hong’s [15] method makes use of prediction error shifting with a dual binary tree (DBT) of embedding levels for hiding data (see Fig. 1). The tree node represents the prediction error. For example, if the embedding level is L=1, the peaks used for embedding are -2,-1, 0, 1 and 2. The prediction errors are classified into three types, namely embeddable, shiftable and non-embeddable.

Fig. 1.Dual binary tree

The error energy estimator predicts the energy of prediction error. When the error energy is greater than a pre-defined threshold, the prediction error is considered as non-embeddable and no embedding is performed. Otherwise, the prediction error is either embeddable or shiftable. The error energy is calculated as in Eq. (1):

where Δr,c is the error energy of pixel Ir,c located at(r,c). er-1,c is the prediction error of pixel at (r-1,c). dv and dh are defined in Eq.(2) and Eq.(3), respectively.

Here is the detailed procedure of the embedding method. Suppose an 8-bit gray cover image Ic of size N×N is to be embedded using the embedding level L and threshold TH. To avoid underflow and overflow, pixel values larger than 255-2L or smaller than 2L are shifted by 2L units. A 2-dimensional location map B is created with 0 and 1 representing un-shifted and shifted pixel locations. The JBIG-2 coder [21] is used to compress B. The compressed location map and secret data S are concatenated to form a bitstream M. Let the shifted image be I.

We scan I using the raster scan order and use Eq.(1) to calculate the error energy Δ for each visited pixel. If Δ≤TH, this means that the corresponding prediction error is either embeddable or shiftable; then MED is used to predict the current pixel values and to obtain the prediction errors e. If the prediction error e fulfills -2L≤e<2L, then e is embeddable. In this case, e is modified to 2e+m, where m is the message bit extracted from M. If the condition fails then e is shiftable and has to be shifted by 2L. If Δ>TH, then the corresponding prediction error is non-embeddable and is excluded from embedding. Repeat this procedures until all the bits in M are embedded. Finally, we get the stego image I'.

To extract the secret data and to recover the cover image Ic from I', we scan the stego image in raster-scan order and calculate the error energy Δ as well as prediction error e. If Δ≤TH and , a message bit m can be extracted by calculating m=mod(e,2). Once the message bits are all extracted, pixels previously modified in the embedding stage are shifted to obtain the pre-shifted image I. Based on the lengths of M and S, we extract the compressed location map from I' and then decompress it into B. Based on B, we transform I back to Ic

 

3. The Proposed Method

Hong [15] proposed the method to use DBT with embedding level L and MED to predict pixel values. The MED prediction is taken from half enclosed pixels, namely, the left side, upper and left upper pixels. The half enclosed prediction provides less accurate results in comparison to full enclosed one. Error energy was used to reject non-embeddable prediction error from unnecessary modification. Similar to MED, the estimation used in [15] of error energy is also half enclosed. Moreover, to avoid underflow and overflow, a 2-dimensional array is used to record the location map. The location map is compressed and hidden together with the secret data. The size of the location map before compression is the size of the image. However, for most natural images, underflow and overflow occurred in few pixels with values less than 2L and greater than 255-L. As a result, it is inefficient to record these pixels in the location map.

Our proposed method is an improvement of [15] which also uses DBT with embedding level L to hide data. However, the proposed method is a full enclosed prediction (FEP) which makes use of the upper, below, left and right pixels as reference pixels. The largest and smallest pixels are discarded and the medium two pixels are averaged and used in the prediction. Both FEP and MED performed well when predicting pixels near the edge. However, FEP has a higher accurate prediction since the full enclosed reference pixels are used to decide whether a prediction error is non-embeddable or embeddable. Furthermore, the reference pixel can be used to accurately predict pixels that might underflow or overflow, and these pixels will be skipped in embedding stage and will not be recorded in the location map. Therefore, the location map is smaller for not having to record unnecessary pixels whereby leaving more space for payload.

3.1 Full Enclosed Predictor

The full enclosed predictor (FEP) uses by the upper, below, right and left reference pixels. Suppose Ir,c is the pixel located at r-th row and c-th column of the image I. The reference pixels are Ir-1,c, Ir+1,c, Ir,c-1 and Ir,c+1. The four reference pixels are sorted in descending order to obtain the sorted result Dr,c, as in Eq. (4).

where Next, we ignore the largest and smallest reference pixels and use the average of second and third largest pixels. Eq. (5) calculates the predicted value pr,c of Dr,c.

where round(x) function rounds x to the nearest integer. The prediction error for Ir,c is er,c=Ir,c-pr,c.

3.2 Variance of Reference Pixel

Variances of the full enclosed reference pixels are used to estimate whether a pixel is embeddable or non-embeddable. The variance is calculated by

where is average value of reference pixels of Ir,c defined by

Suppose vr,c is greater than a predefined threshold TH, then Ir,c is excluded for data embedding because it is likely to carry no data but has to be shifted.

3.3 Predicting Underflow and Overflow (PUO)

Empirically, there are few pixels with possible underflow or overflow after embedment. Therefore, we use the reference pixels to predict pixels that might result in underflow or overflow. These pixels will then be excluded from embedment to prevent the downgrading of image quality. Suppose max(Dr,c)≥255-2L or min(Dr,c)≥-2L, then we assume that Ir,c could result in either underflow or overflow. Therefore, no embedment is done for Ir,c will, therefore, be excluded from the location map.

3.4 Prevent Underflow and Overflow with Location Map

In order to avoid underflow and overflow after data embedment, one possibility is to modify these pixels and record their locations in the location map. The location map will be compressed and then embedded with the secret information.

Our proposed method do not record the overflow and underflow locations but instead records the decisions made based on reference pixels. We need only to sequentially record the properties of the processed pixels as in Eq.(8).

In Eq.(8) Bb represents the bth value of location map B starting from 0. Every time Ir,c satisfies Eq. (8), then b is increased by 1. After being recorded the location map, pixel values are shifted by 2L unit to prevent underflow and overflow. After modifying all pixels which might result in either underflow or overflow, B is compressed to be B' using JBIG-2 coder [21] and concatenated to the secret data to be embedded in the image. The size of the uncompressed location map of the proposed method is equal to the number of embeddable and shiftable prediction errors, which is always smaller than the location map used in [15], where the size of the uncompressed location map is equal to the size of the cover image.

3.5 Embedding Procedure

Let the cover image Ic be an 8-bit gray image of size N×N. The stego image I' is the cover image with embedded secret data S and B is the location map. First we divide Ic into two disjoint sets, S1 and S2. Suppose (r,c) is the location of pixel where 2≤r≤N-1 and 2≤c≤N-1, If mod(r,2)=mod(c,2) then (r,c)∈S1, otherwise(r,c)∈S2. The embedding process for S1 and S2 are similar and thus we describe only the embedding process for set S1. If more message bits are to be embedded, then perform the embedding for set S1 followed by S2.

3.6 Extraction and Recovery Procedure

In our proposed method, the image is divided into two groups for embedding. In the extraction and recovery process, the group embedded last will be the first to be extracted. For example, if S1 is embedded first followed by S2 then extraction and recovery starts from S2 followed by S1. Both S1 and S2 require the same procedure to extract and recover. The procedure for extraction and recover of S1 is as the following:

 

4. Experimental Results

In this section, experimental tests will be performed using eight 512×512 gray-scale images, namely, Lena, Baboon, Aerial, Boat, House, Jet, Sailboat, and Truck (refer to Fig. 2). These images were downloaded from the USC-SIPI image database [22]. The peak-to-noise ratio (PSNR) is used to evaluate the quality of an image, and the embedded information will be generated by a pseudo random number generator (PRNG). In our proposed embedment we may embed in one group (either S1 or S2) or in both groups. Unless specifically mentioned, all experiments will embed in both groups.

Fig. 2.Eight test images.

4.1 Payload and Quality Analysis

First, we will compare our method with Hong’s method [15] using all eight images for the highest payload at different embedding levels. The results are as shown in Table 1 for embedding levels L = 0 to L = 4. Results show that the proposed method has higher payloads than that of [15]. The reason is that the FEP predictor used in the proposed method provides better performance in comparison to MED predictor used in [15].

Table 1.Maximum payload comparison between the proposed method and [13]

Fig. 3 shows comparison results between the proposed method and two other research, namely, Hong [15] and Tai et al. [13], in terms of payload size and image quality. The experimental images Lena and Baboon are used. As illustrated, Tai et al.’s method has low payload and low image quality. The proposed method has the highest payload capacity while maintaining high image quality. In Lena, Hong’s method and the proposed method maintain the same image quality for the payloads between 0.035 bpp and 0.085 bpp. On the other hand, the proposed method has the highest payload and image quality for image Baboon. The reason for the differences is that Lena has a smoother gradient than Baboon. The proposed method maintains high image quality for high payload due to the high prediction precision of FEP. And smaller location map. Overall the proposed method has better capacity and image quality in comparison to both Hong’s and Tai et al.’s methods.

Fig. 3.Compare payload and image quality

Next, comparisons will be made between Tai et al.’s [13], Hong’s [15], Hong and Chen’s[14], and Hu et al.’s [18] methods for pure payload and image quality. The experimental images used are Lena and Baboon. For [13], [15] and the proposed method, the embedding levels are set at L=0 to 4. The block size used in [11] is set to 3×3 with repeated embedding to achieve high payload. Fig. 4 shows that for the same payload, our proposed method outperformed the other four methods. For example, when Lena’s payload is 0.6 bpp, the image quality of our proposed method is 41 dB while methods [15] and [18] are less than 40 dB, method [14] is 37.5 dB, and method [13] is only 36 dB. For Baboon at payload 0.6 bpp, our proposed method is 30 dB, methods [15] and [18] are around 29 dB, while method [13] is 28 dB, and method [14] is only 25 dB.

Fig. 4.High payload comparison

4.2 Comparing the Location Map

Table 3 shows the comparison results for the size of the compressed location map between Hong’s [15] and the proposed method. The results show that when the embedding level increases, more pixels will result in underflow or overflow. This will increase the size of the location map in both methods. Note that in Table 3 when L=3 and L=4 , the sizes (marked in *) of the location map for three experimental images of the proposed method are slightly larger than those of Hong’s. This is a natural phenomenon in smooth images where fewer pixels are excluded for underflow or overflow. However, on average, the proposed method has smaller location maps than that of Hong’s. For example, at L=5, the size of the averaged location map of the proposed method is only 6,696 bits, but Hong’s method requires 18,267 bits.

Table 3.Comparison of the size of the compressed location map

Table 4 shows the size of the compressed location map with and without excluding those pixels that might underflow or overflow. The threshold is set to TH = ∞ to maximize the payload. When embedding level increases, pixels that might underflow or overflow also increase which means the size of location map will also increases. From the table, Aerial and Boat show obvious increase in the size of the location map. However, by excluding pixels that might overflow and underflow can effectively decrease the location map size and thus increase the payload.

Table 4.Location map comparison with and without excluding those pixels that might underflow or overflow

 

5. Conclusion

The proposed reversible data hiding method makes use of the predictive FEP, variance control, and a modified location map recording techniques. FEP makes use of four neighborhood pixels as reference pixels. The largest and smallest ones are discarded while the medium two pixels are averaged to be used for prediction. This method eliminates predicting pixels on the edge and avoids the bigger predictive error between the biggest and the smallest reference pixel, and offers precise predictive error to increase payload. The pixel variance is used to predict non-embeddable pixel, and to exclude pixels of underflow or overflow. The modified location map only records pixels that are used in embedding. This decreases the length of the location map. Experimental results show that our proposed method is effective in increasing image payload and image quality, and has better results compared to four other similar methods.