 Research
 Open Access
Noninteger expansion embedding techniques for reversible image watermarking
 Shijun Xiang^{1}Email author and
 Yi Wang^{1}
https://doi.org/10.1186/s136340150232z
© Xiang and Wang. 2015
 Received: 11 October 2014
 Accepted: 16 May 2015
 Published: 8 July 2015
Abstract
This work aims at reducing the embedding distortion of predictionerror expansion (PE)based reversible watermarking. In the classical PE embedding method proposed by Thodi and Rodriguez, the predicted value is rounded to integer number for integer predictionerror expansion (IPE) embedding. The rounding operation makes a constraint on a predictor’s performance. In this paper, we propose a noninteger PE (NIPE) embedding approach, which can proceed noninteger prediction errors for embedding data into an audio or image file by only expanding integer element of a prediction error while keeping its fractional element unchanged. The advantage of the NIPE embedding technique is that the NIPE technique can really bring a predictor into full play by estimating a sample/pixel in a noncausal way in a single pass since there is no rounding operation. A new noncausal image prediction method to estimate a pixel with four immediate pixels in a single pass is included in the proposed scheme. The proposed noncausal image predictor can provide better performance than Sachnev et al.’s noncausal doubleset prediction method (where data prediction in two passes brings a distortion problem due to the fact that half of the pixels were predicted with the watermarked pixels). In comparison with existing several stateoftheart works, experimental results have shown that the NIPE technique with the new noncausal prediction strategy can reduce the embedding distortion for the same embedding payload.
Keywords
 Reversible watermarking
 Noninteger prediction error
 Expansion embedding
 Noncausal prediction
1 Introduction

Type I algorithms use moduloarithmeticbased additive spread frequency techniques [1, 2], some of which provide robustness but often cause saltandpepper artifacts due to the many wrapped around pixel intensity. In this direction, a different approach proposed by Vleeschouwer et al. [3] can reduce the artifacts by using the circular interpolation of the bijective transform of image histogram. Finally, good results are reported by using the integertointeger waveletbased reversible watermarking in [4].

The direct method to reversible watermarking is to compress a set of selected features from an image in a lossless way and substitute the selected features with their compressed versions and the watermark data [5–8], referred to as type II algorithms in the literature. In order to ensure the watermark imperceptivity, the selected features are usually in the least significant bit area, such as the generalized LSB (gLSB) embedding algorithm [8], which is an extension of the work [5]. Theoretical analysis of reversible watermarking has been presented by Kalker and Willems in [7]. Compared with type I algorithms, type II ones can provide higher payload.

The third type of reversible watermarking algorithms can be classified as difference expansion (DE) embedding methods, in which a common feature is to use difference operators to create features with a small magnitude, and further expand these features in order to create vacancies for embedding the watermark and the auxiliary information. The DE embedding technique was originally developed by Tian [9] and improved in [10–13]. The capacity in Tian’s method is close to 0.5 bpp in a single pass by using two adjacent pixels as a group. By extending the DE to a generalized integer transform, the auxiliary information can be reduced with groups of three or four pixels [10]. In [11], the sorting step in the wavelet domain was introduced to expand those smaller pixel difference. The DE scheme was introduced for 2D vector map in [12], and the location map was reduced in [13].

A fruitful research direction, proposed by Thodi and Rodriguez [14], is predictionerror expansion (PE) embedding technique. Comparing with the DEbased methods, one of the advantages of the PE technique is that it significantly adds the number of the feature elements that expanded for data embedding. The other advantage is that a predictor generates feature elements that are often smaller in magnitude than the feature elements generated by a difference operator. With embedding into each pixel, the PE embedding techniques provided the maximal capacity up to 1 bpp in a single pass. So far, several versions of PE reversible watermarking algorithms have been proposed in [14–19] and others in order to improve the performance of the PE schemes by focusing on the reduction of the size of the auxiliary information (with the use of the sorting and histogram shifting techniques [15]), the reduction of the prediction error (by using multiple predictors [16], the prediction by flooring the average value of the four immediate pixels [15] and adaptive prediction [17]), and the reduction of the embedding distortion (by the pixel selection [17] and the low distortion transform by splitting the difference between the current pixel and its prediction context [18, 19]). These existing PEbased schemes share a common property, that is, the predicted value or its variety was rounded to integer value for expansion embedding.
Reversible watermarking algorithms have also been proposed for digital audio files by expansion embedding [20–22]. There are other novel reversible watermarking approaches which are worth mentioning, too. Data embedded into the histogram bins has been proposed by Leest et al. in [23]. Additive embedding strategy by combining histogram shifting technique [24] and bilinear interpolation prediction has been proposed by Chen et al. in [25, 26].
This paper proposes a noninteger PE (NIPE) embedding strategy, which can proceed noninteger prediction errors for data embedding by expanding only the integer element of a prediction error while keeping the fractional element unchanged. Furthermore, we propose a new noncausal image prediction method for the NIPE technique. In comparison with the classical PE technique [14], the NIPE strategy has lower embedding distortion for noninteger prediction values by using better predictor. The new predictor can provide better performance than existing causal predictors (such as difference predictor, MED predictor, and GAP) due to the fact that there is no rounding operation in the prediction phase. Comparing with Sachnev et al.’s noncausal doubleset prediction method [15], the proposed prediction method can further reduce the prediction errors by predicting data in a single pass (which avoids the distortion problem due to the fact that half of the pixels in the second set predicted with the watermarked pixels in the first set). Experimental results for some standard test images have shown that the proposed NIPE method with the noncausal image predictor can embed more payload for the same watermark distortion.
The outline of this paper is as follows. In the next section, the proposed NIPE embedding technique is introduced. This is followed by a description of a new prediction strategy and an image predictor. We then address the proposed reversible watermarking scheme and compare the proposed prediction method with existing typical predictors. Furthermore, the watermarking scheme’s performance is tested against other typical reversible image watermarking works. Finally, we draw the conclusions.
2 Predictionerror expansion embedding
PE embedding is a technique to expand a prediction error to create a vacant position and insert a bit into the vacant position, generally at the least significant bit (LSB). The PEbased scheme was originally developed by Thodi and Rodriguez [14] and later developed by other researchers (such as [16–19]). In these methods, causal predictive methods with only past pixels are often applied so that the predicted value or its variety can be rounded to integer value for integer predictionerror expansion (IPE) embedding.
In this section, we proposed a NIPE embedding technique, which really brings a predictor into full play to reduce the prediction errors in comparison with the IPE embedding technique. In the NIPEbased method, the predicted value is no longer rounded to integer number. This is beneficial to using more efficient predictor in the NIPE ^{1}. The basic principles of the IPE method [14] and the proposed NIPE method are described as follows.
2.1 Classical IPE embedding
Since y _{ w } is an integer number, \(\hat {y}\) and e are required to be integer values for expansion embedding. This is why we denote Thodi and Rodriguez’s method as the IPE embedding scheme in this paper. It is worth noting that the condition that \(\hat {y}\) should be integer makes an undesirable requirement, that is, the prediction context often contains only causal pixels so as to obtain the same predicted value \(\hat {y}\) in the decoder.
where mod(e _{ w },2) is the remainder on division of e _{ w } by 2 and \(\lfloor \frac {e_{w}}{2}\rfloor \) represents the greatest integer less that or equal to \(\frac {e_{w}}{2}\).
2.2 NIPE embedding technique
Sections 2.1 shows that the IPE embedding approach [14] is suffering from an undesired requirement, that is, the prediction value should be rounded to integer number for expansion embedding. This will restrain a predictor’s performance since the prediction context is restricted to only causal pixels (past pixels) in order to generate the same predicted value in the decoder. In this section, we propose a new expansion embedding approach, one of the advantages of which is able to deal with noninteger prediction values for data embedding. The other, more important, advantage of the approach is that the current pixel can be estimated in a single pass by using noncausal predictive way to improve prediction performance ^{2}.
From the expression \(y_{w}=\hat {y}+2\times e + w\) in Eq. (2), we find that in order to recover the marked pixel y _{ w }, not exact of an integer \(\hat {y}\) is needed, but of the sum of \(\hat {y}+2\times e\). Towards this direction, the basic idea of the proposed approach in this paper is to allow \(\hat {y}\) to take noninteger value but make sure that the combination of \(\hat {y}\) and e takes integer value for hiding a given bit w.
For a pixel in intensity y, the prediction error e is a noninteger value when its estimate \(\hat {y}\) takes noninteger number. In this case, split the noninteger error e into two parts: integer part ℓ (ℓ=f i x(e)) and fractional part δ (δ=e−ℓ). Here, f i x(.) is a function to strip off the fractional part of its argument and returns the integer part. The function does not perform any form of rounding or scaling, e.g., fix(−4.4)=−4 and fix(5.4)=5. The basic idea of the NIPE embedding technique expands only the integer part of a prediction error for data embedding while keeping the fractional part unchanged. The detail is described as follows.
where w is a binary bit, taking either 0 or 1. Equation (4) can make sure that the fractional element of e _{ w } is equal to that of e. This is beneficial to recover the watermark bit and the original pixel intensity in the decoder.
Equation (5) shows that even though \(\hat {y}\) and e take noninteger values, the watermarked pixel y _{ w } is an integer number.
where ℓ _{ w } is the integer element of e _{ w } and δ _{ w }=e _{ w }−ℓ _{ w }.
 Case 1
Let y=100, w=1, and e=100−101.4=−1.4 when \(\hat {y}=101.4\). In the encoder, e _{ w }=e+ℓ−w=−1.4−1−1=−3.4, ℓ _{ w }=fix(e _{ w })=−3, δ _{ w }=e _{ w }−ℓ _{ w }=−0.4, w=mod(ℓ _{ w },2)=1, \(y=\hat {y}\,+\, \text {fix}(\frac {\ell _{w}}{2}) + \delta _{w}= 101.4+ \text {fix}(\frac {3}{2})0.4=100\).
 Case 2
: Let y=100, w=1 and e=100−97.4=2.6 when \(\hat {y}=97.4\). In the encoder, e _{ w }=e+ℓ+w=2.6+2+1=5.6, ℓ _{ w }=fix(e _{ w })=5, δ _{ w }=e _{ w }−ℓ _{ w }=0.6, w=mod(ℓ _{ w },2)=1, \(y=\hat {y}+ \text {fix}(\frac {\ell _{w}}{2}) + \delta _{w}= 97.4+ \text {fix}(\frac {5}{2})+0.6=100\).
We can see from these two cases that the NIPE scheme can effectively deal with the noninteger prediction values for reversible watermarking.
Equations (4), (5), and (6) form the proposed NIPE embedding strategy, which can avoid the rounding operation in the IPE.
2.3 Distortion analysis of NIPE and IPE embedding
 Case 1
\(\hat {y}\) is integer: From (2) in the IPE, \(p'_{w}= y\hat {y} + w\). From (5) in the NIPE, \(p''_{w}= \text {fix}(y\hat {y}) + w= y\hat {y} + w\). In this case, p w′=p w″, indicating that the NIPE has the same embedding distortion as the IPE. For example, if y=100, w = 1, \(\hat {y}=102\). With NIPE, p w″=fix(100−102)+1=−1; with IPE, p w′=100−102+1= −1;
 Case 2
\(\hat {y}\) is noninteger and \(y\hat {y}>0\): In this case, \(p'_{w}= y\lfloor \hat {y}\rfloor + w\), \(p''_{w}=fix(y\hat {y}) + w= yfix(\hat {y}) 1 + w\). Since the estimate of a pixel \(\hat {y}\) is always positive, we have \(p''_{w}= yfix(\hat {y})1 + w=y\lfloor \hat {y}\rfloor 1 + w\), that is p w′=p w″+1, meaning that the IPE scheme will introduce more distortion in this case. For example, if y=100, w=1, \(\hat {y}=98.4\). With NIPE, p w″=f i x(100−98.4)+1=2; With IPE, p w′=100−98+1=3.
 Case 3
\(\hat {y}\) is noninteger and \(y\hat {y} < 0\): In this case, \(p'_{w}= y\lfloor \hat {y}\rfloor + w\). Referring to (5), \(p''_{w}= \text {fix}(y\hat {y}) w= y\text {fix}(\hat {y}) w = y\lfloor \hat {y}\rfloor w\). From \(y\lfloor \hat {y}\rfloor =p'_{w} w\), we have p w″=p w′−2w, indicating the distortion from the NIPE is higher when w=1. When w=0, the NIPE and IPE has the same embedding distortion. For example, if y=100, \(\hat {y}=101.4\). With NIPE, p w″=fix(100−101.4)−w=−1−w; with IPE, p w′=100−101+w= −1 + w. If w is 1, then p w″=−2 and p w′=0; If w is 0, then p w″=p w′=−1.
In statistics, the predictor error (\(e=y\hat {y}\)) is equal to the probability of positive or negative and the watermark bit (w) is also equal to the probability of 1 or 0. As a result, the NIPE has the same embedding distortion as the classical IPE scheme (e.g., the same PSNR value).
3 Pixel prediction
Image prediction is an important step in lossless compression coding applications [27–30]. For an image, each pixel is estimated from its neighborhood to generate the prediction error. Usually, the mean value and variance of the predicted errors are smaller than that of the original pixels. This is beneficial to improve coding efficiency. There are two main prediction ways: 1) causal prediction [27, 30] and 2) noncausal prediction [28, 29]. The main difference between causal and noncausal predictive ways is that the prediction context of the former is restricted to only past pixels while the latter uses past and future pixels. By using causal prediction, the predicted value can be rounded to integer number for enhancing coding efficiency since the prediction context of a pixel has been known, such as the median edge direction (MED) predictor in JPEGLS standard [30]. Noncausal predictive ways are beneficial to reduce the prediction errors for image vector quantization coding [28, 29] and speech compression [31].
3.1 Image predictors for reversible watermarking
The same predictor was also considered as the median of three simple linear predictors: x _{ t }, x _{ r }, and x _{ t }+x _{ r }−x _{ tr } [33], that is, \(\hat {y}=\frac {2x_{t}+2x_{r}x_{\textit {tr}}}{3}\).
Comparing with causal prediction ways, noncausal prediction approaches can improve the prediction precision by using more neighboring samples. The prediction method described in [15] was very different one, which can combine the IPE scheme and noncausal prediction with bilinear interpolation in a way that the image is divided into two sets (like a chess board, the black set, and the white set). In the first pass, the pixel in the black set is predicted with four immediate pixels in the white set to generate an integer difference for IPEbased expansion embedding. Then, the white set is predicted by using the marked version of the black set for the second pass of embedding. Similar idea has been used for additive PE embedding [26] which divided an image into four sets and further embedded the data into the sets one by one by using multipasses embedding. The problem in the noncausal predictors [15, 26] is that part of the pixels are predicted with the watermarked pixels instead of the original ones. This will introduce some unnecessary distortion since the distribution of the prediction errors estimated from the marked pixels has a bigger variance than that estimated from the original pixels.
In the following section, we will present a noncausal predictive model for the NIPE embedding scheme. Comparing with the noncausal prediction methods presented in [15, 26], the predictor proposed in this paper can predict all the pixels before watermarking. This is beneficial to reducing the distortion due to part of the pixels predicted by the modified pixels.
3.2 Proposed noncausal prediction model
In this section, a new method incorporating data prediction not restricted to only causal pixels is designed for the proposed NIPE technique. This noncausal predictive method can predict data in a single pass.
Equation (10) shows that the information of the first 2p pixels/samples {y _{1},y _{2},⋯,y _{2p }} is needed to be saved as part of the residual signal for the recovery of the original signal. From the data series {y _{1},y _{2},⋯,y _{2p }} and d _{ p+1}, the pixel/sample y _{2p+1} can be recovered. Then, y _{2p+2},y _{2p+3},⋯y _{ N } can be restored in sequential order.
The resulting signal above can be denoted by D={y _{1},y _{2},⋯,y _{2p },d _{ p+1},d _{ p+2},⋯,d _{ N−p }}. The data {y _{1},y _{2},⋯,y _{2p }} can be further predicted to generate the difference {e _{1},e _{2},⋯,e _{2p }} with the predictor described in Section 3.3. Denote d _{ i }=e _{ i+p },p<i<N−p+1. As a result, the original signal Y can be further predicted as E={e _{1},e _{2},⋯,e _{2p },e _{2p+1},e _{2p+2},⋯,e _{ N }}.
The above model shows that the noncausal prediction model can be performed in a single pass since there is no rounding operation on the predicted value \(\hat {y_{i}}\) in Eq. (8). For a twodimensional image, it can be mapped into the onedimensional form by using a scanning operation (e.g., in a raster scan or zigzag scan order).
3.3 Predictor with p = 1
Equation (13) indicates that when the first two pixels y _{1} and y _{2} are saved, the third pixel y _{3} can be recovered by referring to the prediction error d _{2} in Eq. (13), then recovering y _{4}, y _{5} and the other pixels in sequential order. Let e _{1}=y _{1}, e _{2}=y _{2}−y _{1} and e _{ i+1}=d _{ i }. Overall, the output of the predictor is denoted as E={e _{1},e _{2},e _{3},⋯,e _{ N−1}}. The predictor in the case of p=1 has been fully proven effective for digital audio in our earlier work [34].
3.4 Proposed noncausal prediction for image
In this section, we design a new image prediction method by referring to the proposed prediction model. Though this prediction method looks like the one described in [15], it is a different one with higher prediction accuracy. In [15], half of the pixels were predicted with the modified pixels instead of the original pixels. In the proposed method, all the pixels are predicted with the original pixels. This explains why the proposed NIPE watermarking scheme has better performance. The detail on the proposed image predictor is addressed as follows:
where I(i,j) is the pixel intensity in the ith row and the jth column, satisfying 1≤i≤R and 1≤j≤C. Denote the resulting 1D signal by Y={y _{1},y _{2},⋯,y _{ N }}.
3) In order to further reduce the prediction errors, the first 2C elements in D (the pixels in the first two rows) are predicted before data embedding. This can be done by raster scanning the pixels in the first two rows and predicting the 2×C pixels by using the noncausal predictor in the case of p=1, described in Section 3.3. The first 2C pixels are predicted and kept as E _{1}={e _{1},e _{2},⋯,e _{2C }}.
4) From steps 2 and 3, we have E={e _{1},e _{2},⋯,e _{2C },e _{2C+1},e _{2C+2},⋯,e _{ N }}, which is the output of the proposed image predictors in this paper.
where C<k≤N−C. Finally, the original image is restored from the residual signal E.
3.5 Comments
 1.
The data is predicted in a noncausal way that most of the pixels can be predicted with four immediate pixels. This improves prediction performance since some predictors only use causal pixels for data prediction, such as MED predictor.
 2.
All the pixels can be predicted in a single pass with original pixels as context. This is beneficial to avoid the the distortion problem due to part of the pixels predicted by the modified pixels, such as the predictor in [15].
 3.
In the literature, the IPE watermarking scheme [15] has a satisfactory performance. The NIPE technique with the predictor above can further reduce the embedding distortion for the same payload by avoiding part of the pixels predicted by the modified pixels.
4 Proposed watermarking scheme
The proposed watermarking scheme is a combination of existing techniques (histogram shifting in [14, 24]) and new techniques (NIPE embedding and noncausal image prediction).
4.1 Prediction expansion with histogram shifting
where ℓ _{ i } is integer part of the ith prediction error, e _{ i }, satisfying e _{ i }=ℓ _{ i }+δ _{ i }. The marked prediction error is denoted by \(e_{w_{i}}\) after the bit w _{ i } is inserted.
where ℓ _{ wi }=fix(e _{ wi }) is the integer element of e _{ wi } and δ _{ wi }=e _{ wi }−ℓ _{ wi }. It is worth noting that δ _{ wi }=δ _{ i } since the encoder only expands the integer part while keeping the fraction element unchanged. Finally, the original pixels are recovered from the original prediction errors by performing inverse prediction operation in Section 3.4.
The ratio between the sets S _{1} and S _{2} can be controlled by changing the embedding threshold T. The bigger the threshold value T, the higher the embedding payload, the more the embedding distortion is.
4.2 Capacity analysis
The marked images may suffer from overflow and underflow problems due to expansion embedding and histogram shifting operations. Towards this direction, an embedding testing step is first performed to pick up those pixels with overflow or underflow problems. The testing process has been addressed in [14, 15] in detail. For an image with mbit representation, when a watermarked pixel in intensity is not in the range [0,2^{ m }−1], labeled as a bad pixel. The bad pixels in position will be saved and embedded with the payload to indicate the expandable locations.
where C is the maximal embedding rate (bounded to 1), N _{1} the length of the expandable set S _{1}, N _{ p } the number of the bad pixels, and N the number of the cover image pixels.
5 Encoder and decoder
 1.
Referring to Section 3.4, predict the cover image Y to get the prediction errors E;
 2.
Find the bad pixels in position by using an embedding testing operation. The embedding testing step is similar to that in [14, 15]. Each bad pixel consumes 25 bits of payload to indicate the embedding position;
 3.
Referring to Section 2.2, embed the data (including P, T, and the bad pixels in position) into E to generate E _{ w };
 4.
Reconstruct the marked image Y _{ w } from E _{ w } by using the inverse prediction operation as described in Section 3.4, step 5).
In the decoder, the same prediction operations are performed on Y _{ w } to get E _{ w }. Then, the information of P and T is extracted from the LSB values of the first 32 prediction errors. Furthermore, the hidden data and the original prediction errors E are extracted from E _{ w }. Finally, the original image Y is completely recovered from E by using the inverse prediction operations.
6 Experimental testing and analysis
6.1 Comparison of typical predictors
The shape of the histogram of the predicted errors is often used to measure the performance of the embedding scheme. In general, distribution of the prediction errors obeys a Laplacian distribution. The shape of the distribution is determined by the absolute mean and variance. If the mean is close to zero, the variance essentially determines the shape of the histogram. The smaller variance value, the better performance can be achieved for the reversible watermarking scheme.
In the literature, the predictor proposed in [15] provides a satisfactory performance by dividing an image into two sets (like a chess board divided into black and white sets) to achieve noncausal prediction in a way that a pixel can be predicted with its four immediate pixels. This prediction method is suffering from a distortion problem. That is, half of the pixels were predicted with modified pixels instead of the original ones. As shown in Section 3.4, the proposed prediction method can predict a pixel with its four immediate pixels, and all the pixels can be predicted completely with original pixels as context. As a result, the proposed prediction method has better prediction accuracy. In order to better evaluate the effect of the rounding and watermarking operations, we have computed the absolute mean values and the standard deviations of all the 28 test images which are predicted with the proposed predictor in this paper (denoted by μ and σ), the proposed predictor with an integer rounding operation (μ _{ r } and σ _{ r }) and the prediction method in [15] (μ _{ rw } and σ _{ rw }).
Performance comparison of five predictors
Predictors  Lena  Baboon  Fishingboat  F16  

μ  σ  μ  σ  μ  σ  μ  σ  
4.6724  7.8454  18.4130  27.6633  7.2441  11.1876  5.2971  11.9235  
Integer MED predictor [14]  4.3407  6.9049  13.5220  19.6804  6.3836  9.3957  3.4273  6.2054 
Noninteger MED predictor [33]  4.3060  6.7608  13.3652  19.2567  6.5863  9.7152  3.6089  6.6339 
Sachnev’s predictor [15]  3.2063  4.8738  10.9786  15.6010  5.2220  7.7943  2.7578  4.8683 
Proposed predictor  3.2049  4.8565  10.9782  15.5957  5.2209  7.7847  2.7465  4.8515 
6.2 Comparison with other recent algorithms
The classical IPE embedding scheme, proposed by Thodi and Rodriguez [14], is a highcapacity reversible watermarking algorithm by developing PE embedding technique with MED predictor. We can see from Fig. 7 that the NIPE technique with the proposed noncausal predictor can provide higher embedding payload in comparison with the IPE scheme. The basic reason is that the proposed noncausal predictor has higher prediction precision than the MED predictor used in [14].
Coltuc [18,19] has also developed Thodi and Rodriguez’s work and has presented the results for the MED predictor. The basic idea of the approach is to embed the entire expanded difference not only into the current pixel but also its context. Then, the minimization of the square errors is considered to reduce the embedding distortion. When the parameter α is 0.25 (referred to [18], Eq. (4)), we can see from Fig. 7 that Coltuc’s embedding method has lower embedding distortion than the IPE for the same embedding rate. In [19], the embedding approach has been further generalized as a low distortion transform (LDT) for reversible watermarking. Comparing with the LDT technique with the MED predictor in [18], the proposed NIPE scheme can provide higher embedding payload or capacity for the same embedding distortion. The basic reason is that the proposed prediction strategy can better estimate the current pixel by incorporating data prediction not restricted to only causal pixels, as listed in Table 1.
Another important improvement on Thodi and Rodriguez’s work has been proposed by Sachnev et al. [15] by introducing a highprecision prediction strategy and sorting technique. Since a pixel can be estimated with its four immediate pixels, the IPEbased scheme can provide satisfactory performance in the literature. Figure 7 shows that the reversible watermarking approach proposed in this paper has lower embedding distortion at all embedding rates than Sachnev et al.’s double embedding scheme. The reason is that the predictor used for NIPE can predict pixels with four original immediate pixels when one used in [15] predicted half of pixels with modified pixels as context.
7 Conclusions
This paper presents a NIPE embedding technique for reversible watermarking. The NIPE technique can remedy a major drawback of Thodi and Rodriguez’s work (called IPE in this paper) that the predicted values should be rounded to integer number for data embedding. With the NIPE technique, the rounding operation in the prediction process (that often appears in IPEbased reversible watermarking algorithms to generate integer prediction errors) can be discarded. This is beneficial to use better predictor. In order to prove the advantage, we proposed a prediction model and designed an image predictor for the NIPE. The new predictor can predict pixels with four immediate pixels, and all pixels can be predicted with the original pixels. With the proposed NIPE and predictor, the embedding distortion is smaller than that in [15] at all embedding rates. Experimental results have shown that the predictor designed in this paper can provide the best performance than several existing typical prediction methods. In comparison with other typical reversible watermarking algorithms, the proposed scheme (combining the NIPE technique and new prediction method) performs better.
8 Endnotes
^{1}In the IPE, the prediction errors should be rounded to integer value. This rounding operation brings a constraint on a predictor’s performance.
^{2}In the literature [15,26], noncausal predictive methods have been used for the IPE by using multipasses prediction for multilayers embedding. Since part of the pixels were predicted by using the watermarked pixels instead of the original ones, some distortion has been introduced.
Declarations
Acknowledgements
This work was supported by NSFC (No. 61272414). The authors would like to thank the reviewers’ valuable comments.
Authors’ Affiliations
References
 W Bender, D Gruhl, N Morimoto, A Lu, Techniques for data hiding. IBM Syst. J. 35(3), 313–336 (1996).View ArticleGoogle Scholar
 B Macq, in Proc. the European Signal Processing Conf,. Lossless multiresolution transform for image authenticating watermarking (Tampere, Finland, 2000).Google Scholar
 C De Vleeschouwer, JE Delaigle, B Macq, Circular interpretation of bijective transformations in lossless watermarking for media asset management. IEEE Trans. Multimedia. 5(1), 97–105 (2003).View ArticleGoogle Scholar
 S Lee, CD Yoo, T Kalker, Reversible image watermarking based on integertointeger wavelet transform. IEEE Trans. Inf. Forensics Secur. 2(3), 321–330 (2010).View ArticleGoogle Scholar
 J Fridrich, M Goljan, R Du, in Proc. SPIE Photonics West, Security and Watermarking of Multimedia Contents III, 3971. Invertible authentication (San Jose, 2001), pp. 197–208.Google Scholar
 J Fridrich, M Goljan, R Du, Lossless data embeddinga̧ł paradigm in digital watermarking. EURASIP J. Appl. Signal Process. 2, 185–196 (2002).View ArticleGoogle Scholar
 AACM Kalker, FMJ Willems, in Proc. 14th Int. Conf. Digital Signal Processing, 1. Capacity bounds and constructions for reversible datahiding (Santorini, 2002), pp. 71–76.Google Scholar
 MU Celik, G Sharma, AM Tekalp, E Saber, Lossless generalizedLSB data embedding. IEEE Trans. Image Process. 14(2), 253–266 (2005).View ArticleGoogle Scholar
 J Tian, Reversible data embedding using a difference expansion. IEEE Trans. Circuits Syst. Video Technol. 13(8), 890–896 (2003).View ArticleGoogle Scholar
 AM Alattar, Reversible watermark using the difference expansion of a generalized integer transform. IEEE Trans. Image Process. 13(8), 1147–1156 (2004).MathSciNetView ArticleGoogle Scholar
 L Kamstra, H Heijmans, Reversible data embedding into images using wavelet techniques and sorting. IEEE Trans. Image Process. 14(12), 2082–2090 (2005).MathSciNetView ArticleGoogle Scholar
 X Wang, C Shao, X Xu, X Niu, Reversible datahiding scheme for 2D vector maps based on difference expansion. IEEE Trans. Inf. Forensics Secur. 2, 311–319 (2007).View ArticleGoogle Scholar
 Y Hu, HK Lee, J Li, DEbased reversible data hiding with improved overflow location map. IEEE Trans. Circuits Syst. Video Technol. 19(2), 250–260 (2009).View ArticleGoogle Scholar
 DM Thodi, JJ Rodriguez, Expansion embedding techniques for reversible watermarking. IEEE Trans. Image Process. 15, 721–729 (2007).MathSciNetView ArticleGoogle Scholar
 V Sachnev, HJ Kim, J Nam, S Suresh, Y Shi, Reversible watermarking algorithm using sorting and prediction. IEEE Trans. Circuits Syst. Video Technol. 19(7), 989–999 (2009).View ArticleGoogle Scholar
 HW Tseng, CP Hsieh, Predictionbased reversible data hiding. Inf. Sci. 179, 246–2469 (2009).View ArticleGoogle Scholar
 X Li, B Yang, T Zeng, Efficient reversible watermarking based on adaptive predictionerror expansion and pixel selection. IEEE Trans. Image Process. 20(12), 3524–3533 (2011).MathSciNetView ArticleGoogle Scholar
 D Coltuc, Improved embedding for predictionbased reversible watermarking. IEEE Trans. Inf. Forensics Secur. 6(3), 873–882 (2011).View ArticleGoogle Scholar
 D Coltuc, Low distortion transform for reversible watermarking. IEEE Trans. Image Process. 21(1), 412–417 (2012).MathSciNetView ArticleGoogle Scholar
 Veen van der M, F Bruekers, A van Leest, S Cavin, in Proc. SPIE Photonics West, Electronic Imaging 2003, Security and Watermarking of Multimedia Contents V, 5020. Highcapacity reversible watermarking for audio (San Jose, California, 2003), pp. 1–11.Google Scholar
 B Bradley, AM Alattar, in Proc. SPIE Photonics West, Electronic Imaging 2005, Security and Watermarking of Multimedia Contents VII, 5681. Highcapacity, invertible, datahiding algorithm for digital audio (San Jose, California, 2005), pp. 789–800.Google Scholar
 D Yan, R Wang, in International Conference on Intelligent Information Hiding and Multimedia Signal Processing. Reversible data hiding for audio based on prediction error expansion (Harbin, 2008), pp. 249–252.Google Scholar
 A Van Leest, M Van der Veen, F Bruekers, in Proc. IEEE Conf. Image Processing, 3. Reversible image watermarking (Barcelona, 2003), pp. 731–734.Google Scholar
 Z Ni, Y Shi, N Ansari, S Wei, in Proc. IEEE Int. Symp. Circuits and Systems, 2. Reversible data hiding (Bangkok, 2003), pp. 912–915.Google Scholar
 M Chen, Z Chen, X Zeng, Z Xiong, in Proc. 11th ACM Workshop Multimedia and Security. Reversible data hiding using additive predictionerror expansion (Princeton, 2009), pp. 19–24.Google Scholar
 L Luo, Z Chen, M Chen, X Zeng, Z Xiong, Reversible image watermarking using interpolation technique. IEEE Trans. Inf. Forensics Security. 5(1), 187–193 (2010).View ArticleGoogle Scholar
 Anil K Jain, Fundamentals of digital image processing (Prentice, Hall, Englewood Cliffs, NJ, 1989).MATHGoogle Scholar
 A Asif, JMF Moura, Image codec by noncausal prediction, residual mean removal, and cascaded vector quantization. IEEE Trans. Circuits Syst. Video Technol. 6(1), 42–55 (1996).View ArticleGoogle Scholar
 N Balram, JMF Moura, Noncausal predictive image codec. IEEE Trans. Image Process. 5(8), 1229–1242 (1996).View ArticleGoogle Scholar
 M Weinberger, G Seroussi, Sapiro G, The LOCOI lossless image compression algorithm: principles and standardization into JPEGLS. IEEE Trans. Image Process. 9(8), 1309–1324 (2000).View ArticleGoogle Scholar
 WR Gardner, BD Rao, Noncausal linear prediction of voiced speech. IEEE Asilomar Conference on Signals, Systems and Computers, (Pacific Grove, CA, Oct. 1992).Google Scholar
 Wu X, Memon N, Contextbased, adaptive, lossless image coding. IEEE Trans. Commun. 45(4), 437–444 (1997).View ArticleGoogle Scholar
 S Martucci, in Proc. IEEE Int. Symp. Circuits and Systems. Reversible compression of HDTV images using median adaptive prediction and arithmetic coding (New Orleans, 1990), pp. 1310–1313.Google Scholar
 S Xiang, in IH 2012, LNCS 7692. Noninteger expansion embedding for predictionbased reversible watermarking (SpringerBerkeley, 2013), pp. 224–239.Google Scholar
Copyright
This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.