 Research
 Open Access
Efficient blind decoders for additive spread spectrum embedding based data hiding
 Amir Valizadeh^{1}Email author and
 Z Jane Wang^{1}
https://doi.org/10.1186/16876180201288
© Valizadeh and Wang; licensee Springer. 2012
Received: 16 July 2011
Accepted: 23 April 2012
Published: 23 April 2012
Abstract
This article investigates efficient blind watermark decoding approaches for hidden messages embedded into host images, within the framework of additive spread spectrum (SS) embedding based for data hiding. We study SS embedding in both the discrete cosine transform and the discrete Fourier transform (DFT) domains. The contributions of this article are multiplefold: first, we show that the conventional SS scheme could not be applied directly into the magnitudes of the DFT, and thus we present a modified SS scheme and the optimal maximum likelihood (ML) decoder based on the Weibull distribution is derived. Secondly, we investigate the improved spread spectrum (ISS) embedding, an improved technique of the traditional additive SS, and propose the modified ISS scheme for information hiding in the magnitudes of the DFT coefficients and the optimal ML decoders for ISS embedding are derived. We also provide thorough theoretical error probability analysis for the aforementioned decoders. Thirdly, suboptimal decoders, including local optimum decoder (LOD), generalized maximum likelihood (GML) decoder, and linear minimum mean square error (LMMSE) decoder, are investigated to reduce the required prior information at the receiver side, and their theoretical decoding performances are derived. Based on decoding performances and the required prior information for decoding, we discuss the preferred host domain and the preferred decoder for additive SSbased data hiding under different situations. Extensive simulations are conducted to illustrate the decoding performances of the presented decoders.
Keywords
 digital watermarking
 additive spread spectrum embedding
 optimum decoding
1. Introduction
The growing use of Internet has enabled the users to easily access, share, manipulate, and distribute the digital media data, and digital media has profoundly changed our daily life during the past decade. This proliferation of digital media data creates a technological revolution to the entertainment and media industries, brings new experience to users, and introduces new Internet concepts. However, the massive production and use of digital media also pose new challenges to the copyright industries and raise critical issues of protecting intellectual property of digital media, since current media sharing makes unauthorized copying and illegal distribution of the digital media much easier.
One popular technology for digital right protection is digital watermarking [1], where a specific signal (e.g., the ownership information) is embedded into the host media content without significantly degrading the perceptual quality of the original media data. In contrast to traditional encryption techniques, watermarked media data can still be used while remaining protected, and thus watermarking can provide postdelivery protection of digital media. It is worth mentioning that, despite the popularity of watermarking techniques, effective digital right protection is extremely challenging and currently there is no commonly accepted technical solution which is practically unbeatable when deployed to practical user settings. At any sense, watermarking techniques should only be considered as one important component of an overall protection system.
Amongst the proposed schemes for watermark embedding, spread spectrum (SS) and quantization based methods [2, 3] are the two main broad categories. In SS embedding, an additive or multiplicative watermark is added into the host signal. The quantization based schemes are implemented by quantizing the host signal to the nearest lattice point. In this article, we focus on spread spectrum embedding schemes originally proposed by Cox et al. [4]. At the receiver side, a blind detection scheme is employed, since the original image is generally not available and thus is treated as a noise source. There are two main approaches of SS embedding: the additive spread spectrum watermarking and the multiplicative spread spectrum (MSS) watermarking. In additive SS [5, 6], the watermark is spread over the host signal uniformly while in MSS [7, 8], the watermark spreads according to the content of the host signal. In order to reduce the noise effect of the host signal in additive SS, Malvar and Florencio [9] proposed the improved spread spectrum (ISS), a new modulation technique exploiting the side information at the encoder to reduce the effect of host signal and improve the decoding performance [10]. Recently, the authors have proposed an embedding scheme incorporating the SS and ISS schemes which employs the correlation between the host signal and the signature code to improve the decoding performance [11].
As summarized in [12], depending on different purposes, there are two main types of watermarking schemes: In one type, the embedded watermark is used to communicate a specific hidden message (e.g., binary identification numbers used for image tracking and for video distribution, or a secret hidden message represented by binary sequences) which must be extracted with sufficient decoding accuracy. In the other type of systems, the goal is only to verify whether a specific embedded watermark (e.g., representing copyright information) is presented or not, and the embedded watermark normally does not communicate a secret message that needs to be accurately decoded. It is important to emphasize that the above two problems are formulated differently and different detection (decoding) approaches are desired to serve different performance criteria. References [13–18] explicitly have pointed out this distinction in their works.
Based on the above two types of watermarking schemes, current researches on watermark extraction can be categorized into two broad topics: watermark decoding [12, 13, 19] for the case of decoding the hidden message and watermark detection [16–18, 20–22] for the case of detecting the presence of a specific watermark. Although watermark detection and decoding problems seem to be similar from the hypothesis testing point of view, they actually serve different goals and thus different criteria are used. In watermark decoding, the embedded hidden message should be decoded accurately at the receiver side and therefore the bit error rate is usually used as the performance criterion to measure the accuracy of the decoder in extracting the hidden message, and the watermark decoding problem can be formulated as minimizing the bit error rate. In watermark detection, the goal is to determine whether a specific watermark exists or not, and the detection criteria are mainly based on NeymanPearson Theorem (i.e., maximizing the probability of detection for a given probability of false alarm). Performance criterion such as the false alarm probability and the true detection probability are used for evaluating the watermark detector performance. To our knowledge, the majority of the current literature has been focused on watermark detection and many algorithms have been proposed. For instance, in [23] a watermark based on the host content is added and the detection is accomplished with the NeymanPearson criterion. In [24], a new perceptual masking is proposed and a correlation based detector is studied for watermark detection. In [25], a class of watermark detectors, including the generalized likelihood ratio, Bayesian, and Rao test detectors are proposed. In this article, we focus on the topic of watermark decoding, since we are particularly interested in communicating hidden message. Since in practice the original host image is generally not available at the decoder side, we focus on blind watermark decoding.
The very first decoder used for watermark decoding in SS embedding is the traditional correlator proposed by Cox. This decoder extracts the embedded information using the correlation between the signature code and the received data. Utilizing the probability density function (PDF) of the host signal could help enhancing the performance of watermark decoding. An optimum ML decoder for additive SS in the DCT domain was proposed by Hernandez et al. [26]. The optimal decoder for multiplicative SS in the DFT domain was investigated in [13]. Regardless of the above referred literatures, compared with the research works on watermark detection, watermark decoding is less studied, and a thorough analytical study of watermark decoding is still required. It is worth emphasizing that, since the watermark decoding problem is formulated as different hypotheses testing problem from the watermark detection problem (i.e., with H_{0} being the noiseonly hypothesis), a specific watermark detector does not necessarily mean a specific watermark decoder. For instance, the local optimum (LO) test (which is based on the derivative of the likelihood) will yield different forms for the LO detector and the LOD. Also, even though ML criterion has been used for both watermark detection and watermark decoding, it is derived differently and has different meanings (i.e., the ML watermark decoder is a Bayesian approach to minimizes the probability of bit error when assuming the equal prior probability of the bit information and thus assuming the threshold to be 1; while for watermark detection, the ML solution is the likelihood ratio test (LRT) detector based on the NeymanPearson theorem, where the LRT exploits the probability of false alarm to set the detection threshold). We would also like to emphasize that since different performance criteria are desired in watermark detector and watermark decoding, a specific type of efficient watermark detector does not necessarily mean an efficient watermark decoder.
The common objective of communicating hidden message using watermarking is to successfully embed and decode an imperceptible watermark which can be resistant against distortions and attacks. In order to reduce the performance degradation under certain attacks such as geometric attacks and to take advantage of the properties of certain transform domain, the message embedding can be performed in different domains such as the discrete cosine transform (DCT) domain [27], the discrete Fourier transform (DFT) domain [28–31], and the discrete wavelet transform (DWT) domain [32, 33].
In this article, our main purpose is to provide a rigorous watermark decoding framework for data hiding using spread spectrum embedding in the DCT domain and the DFT magnitude domain. In the literature of additive SS, there is lack of investigation on the optimal and suboptimal decoders using the additive SS in the DFT magnitude domain and we will fill this gap in this article. We will show that the conventional SS scheme could not be applied directly in the DFT magnitude domain and thus we will propose a modified SS embedding scheme. To further provide a guidance on the preferred domain for information hiding using additive SS embedding, based on the derived decoders, we will discuss which domain is preferred under different circumstances. We present a theoretical framework of optimal decoders for additive SS and improved SS in the DCT and DFT magnitude domains. Embedding in the DFT domain has its own advantages and it motivates us to develop optimal watermark decoding schemes for this domain. We note that optimal decoders using ISS provide better decoding performances than the traditional additive SS. As the optimum ML decoder requires the distribution parameters of the host image and the watermark strength information, to address this concern, we also investigate several suboptimal watermark decoders. By invoking the Taylor series, the LOD is proposed by relaxing the requirement on watermark strength. We derive the generalized maximum likelihood (GML) decoder for information hiding in the DFT magnitude domain. Further, due to simplicity and good performance, we employ the linear minimum mean square error (LMMSE) criterion and derive the LMMSE decoders. We derive the theoretical performance analysis of the proposed ML, LOD, and LMMSE decoders, where the theoretical performance of the ML decoder is served as the performance upper bound of watermark decoding schemes. The main contributions of this article are summarized as follows:

Proposed modified SS and ISS embedding schemes in the DFT magnitude domain.

Derive the ML and GML decoders for SS and ISS in the DFT magnitude domain; Derive the ML decoder for ISS embedding in the DCT domain.

Derive the LOD decoders for SS embedding in the DCT and the DFT magnitude domains, and derive the LOD decoders for ISS embedding in the magnitude of the DFT domain.

Derive the LMMSE decoders for SS and ISS embedding in both the DCT and the DFT magnitude domains.

Provide the theoretical biterrorrate performance analysis of the above decoders.
The rest of this article is organized as follows. In Section 2, the traditional additive SS and ISS embedding schemes are briefly reviewed for data hiding and communicating hidden message. Host probability distribution functions for DCT and DFT domains are described in Section 3. The optimal ML decoders are derived in Section 4 and the corresponding biterrorrate analyses are presented. In Section 5, the suboptimal decoders, including LOD, GML, and LMMSE decoders are presented and their theoretical performance analyses will be provided. The simulation results are demonstrated in Section 6 to validate the analysis. Finally, the discussions and concluding remarks are given in Section 7.
2. Additive SS and ISS embedding procedure
Suppose a host image $\mathbf{I}\in {\mathcal{M}}^{m\times n}$ is supposed to be watermarked, where means the image alphabet, e.g., $\mathcal{M}=\left\{0,1,...,255\right\}$ for a gray scale image, and m and n represent the size of the image in the pixel (spatial) domain. Here for simplicity we assume m = n, though the results could be extended to the general case of having unequal m and n. The additive SS embedding procedure for a host image is summarized as follows. First, the image I is partitioned into $\frac{m}{p}\times \frac{m}{p}$ sub blocks of size p × p. Then, each block is usually transformed to a domain which is insensitive to tampering, i.e., $\mathcal{T}\left(\mathbf{I}\right)\in {\mathcal{R}}^{p\times p}$ where denotes the transform function transforming the host image into the new domain called host domain. For the total $q={\left(\frac{m}{p}\right)}^{2}$ sub blocks, each of them conveys one hidden information bit b ∈ {±1} for onemessage embedding or multiple hidden information bits for multimessage information embedding. A perfect transform should remove the imperceptible part of the data and should be insensitive against operations such as translation, lowpass filtering, compression, and other standard signal processing manipulations. In the context of image processing, two popular transform domains include the DCT and DFT which are of interest of this article.
For each block of size p × p in the transformed domain, a subset of host coefficients with length l ≤ p^{2} is selected to be the carrier vector for embedding. Such selected vectors ${\mathbf{x}}_{i}\in {\mathcal{R}}^{l}$, i = 1, 2,..., q, are used for information embedding. A signature code s = [s_{1}, s_{2},..., s_{ l }]^{ T }with length l can be employed for one bit embedding, and using multiple signature codes can allow us to embed multiple bits simultaneously. Usually in decoding problem, the signature code coefficients are from the values +1 and 1.
2.1. Additive SS embedding scheme
which can be easily shown equals to A^{2} in SS embedding. The point that should be taken into account is that different domain host signals may affect the procedure of adding the information. Using the DCT domain makes no restriction on the additive SS watermarking. Employing the DFT and explicitly embedding the information in the magnitude of DFT coefficients limit the set of coefficients to be watermarked, since the watermarked DFT coefficients are required to be positive. We will discuss about the embedding scheme in the magnitude of the DFT domain more precisely when the optimal decoders are explained in Section 4.
2.2. ISS embedding scheme
At the receiver side the hidden information needs to be decoded. Since the optimal decoders require the distribution of the host signal, different distributions for the DCT coefficients and the magnitude of DFT coefficients will be discussed in the next section.
3. Data hiding in the DCT and the DFT magnitude domains
and σ_{ x }means the standard deviation of the host signal and Γ(.) means the Gamma function defined as $\mathrm{\Gamma}\left(x\right)={\int}_{0}^{\infty}{t}^{x1}{e}^{t}dt$. The power exponent c is the shape parameter where its smaller value leads to the more impulsive shape and heavier tail. The scale parameter β and the shape parameter c can be estimated from the host signal [34].
where u(.) determines the step function which returns one where its argument is positive and returns zero when its argument is negative. Moreover, the parameters η > 0 and γ > 0 represent the scale and the shape parameters of the Weibull distribution.
4. The ML optimal decoders
As discussed in Section 3, the PDF of the host signal can be different depending on the transform domain. In practice, due to different desired properties, different transform domains could be used for data hiding. Derivation and performance analyses of the ML decoder for SS embedding require the distribution of the host signal in a specific domain. In the following subsections, the ML decoders for SS and ISS embedding schemes in DCT and DFT domains are derived. It is worth mentioning that the ML decoder for the SS scheme in the DCT domain has been already proposed in [26].
4.1. ML decoders in the DFT domain
The modified SS embedding scheme (11) and the vector e defined in (12) reveal that, for those coefficients where e_{ i }> 0, the watermarked signal becomes r_{ i }= x_{ i }, meaning that the coefficient r_{ i }does not convey information directly. However, because of the structure of the optimal decoder, which will be derived shortly, such r_{ i }'s still could help for decoding. We also note that, for the modified SS scheme (11), by increasing the watermark amplitude A, the number of coefficients with e_{ i }> 0 increases and consequently the number of watermarked coefficients decreases. Generally, the goal of this modified SS embedding scheme is to make all the watermarked coefficients positive.
Investigating the test statistic of the ML decoder in the DFT magnitude domain reveals that the bit information amplitude as well as the PDF parameters should be provided at the receiver side. Now, we proceed to show that the decoding procedure is error free for two cases. For one case that b = +1 and there is one coefficient with r_{ i }+ s_{ i }A < 0 at the decoder side, we can see that the test statistic in (16) goes to infinity and thus the decoder (15) definitely decides $\widehat{b}=+1$. More precisely, in this case, ln(u(r_{ i }s_{ i }A)) is positive and ln(u(r_{ i }+ s_{ i }A)) goes to minus infinity, and thus the test statistic in (16) goes to infinity. Similarly, for the other case that b = 1 and at the decoder side there is one coefficient with r_{ i } s_{ i }A < 0, the test statistic in (16) goes to minus infinity and thus the decoder (15) definitely decides $\widehat{b}=1$.
As mentioned earlier, these coefficients which do not convey information directly could help decoding indirectly. To explain this better, let assume that b = +1 and x_{ i }+ s_{ i }A < 0, thus the corresponding coefficient becomes r_{ i }= x_{ i }at the embedding side. At the decoder side we will have r_{ i }+ s_{ i }A < 0, which based on the above discussion, leads to the decision $\widehat{b}=+1$. Similarly, let assume that b = 1 and x_{ i } s_{ i }A < 0, thus the corresponding coefficient becomes r_{ i }= x_{ i }at the embedding side. At the decoder side we will have r_{ i }+ s_{ i }A < 0, which leads to decision $\widehat{b}=1$. In both cases, the decoder performance would be error free and therefore, even some coefficients do not convey hidden information directly, they still could contribute to accurate decoding indirectly.
If there is a host signal coefficient x_{ i }< 2A, according to the earlier discussion, the probability of error would equal to zero. Therefore, the theoretical error probability of the modified SS scheme is expressed as (19) when the mean and variance could be achieved using expressions (20) and (21).
We will apply the modified ISS scheme in (28) for embedding, and use the decoder in (26) for extracting the hidden information. So far, as a summary, the modified ISS embedding scheme (28) in the DFT magnitude domain has been proposed in order to make all the watermarked coefficients positive and to make the decoder (26) meaningful.
4.2. ML decoders in the DCT domain
We can see that the ML decoder requires knowledge of the bit information amplitude and the shape parameter as well as the signature code. For practical implementation, the receiver should either have these prior information or estimate them. One way to avoid estimating the shape parameter is to use a general value for all images, hoping it could describe the distribution of the DCT coefficients relatively well [35].
Similar to the ISS embedding in the magnitude of the DFT domain, the parameter k could be determined using the constrained maximization (35) taking into account the mean and variance defined in (42) and (43).
5. Suboptimal decoders
As shown in Section 4, the ML decoder requires the host distribution parameters as well as the watermark amplitude. Assuming low distortion due to watermark, we could estimate the host signal parameters using the received signal, while estimating the watermark amplitude is not easy because of the complex structure of the embedding scheme. Therefore, to reduce the dependency on such prior information, in this section, we will investigate two suboptimal decoders [37]. In addition, since it was shown that the ML decoder for embedding in the magnitude of the DFT domain is sensitive to watermark amplitude, we hope that the suboptimal decoders in this domain could decrease this sensitivity and lead to good performances in the presence of additional noise.
5.1. Local optimum decoder
where $\stackrel{\u0301}{f}\left(a\right)$ is the first order derivative of f(.) at the point x = a.
The provided above decoder expression reveals that it is independent of the watermark amplitude and appropriate for the cases which there is no access to the watermark amplitude.
Since the LODs are approximations of the ML decoders, the degraded decoding performances from the optimal ML ones are expected. LOD has the advantage that the additional information of the watermark amplitude at the decoder side is not required.
It is worthy mentioning that, since LOD is independent of the watermark amplitude A, decoding performance degradation is observed in LOD compared with ML, especially for the high watermark amplitude cases. As discussed in Section 4, though those watermarked coefficients with x_{ i }s_{ i }A < 0 or x_{ i }+ s_{ i }A < 0 do not convey information directly, they do help accurate decoding in the ML decoder in (16). From the LOD in (53), it is clear that with no access to the watermark amplitude information, all the received coefficients are exploited for extracting the hidden information even though some of them do not convey any information. This is the source of the decoding performance degradation from ML.
The main goal is to estimate b from $\widehat{\mathbf{y}}$. From the expression of $\widehat{\mathbf{y}}=\mathbf{s}Ab+\mathbf{e}$, it is clear that the correlator $\widehat{b}=\mathsf{\text{sign}}\left\{{s}^{T}\widehat{\mathbf{y}}\right\}$ is the solution. It should be pointed out that since the GML decoder does not have access to the watermark amplitude information, the GML decoder's performance degrades from that of ML. However, when compared with LOD, GML yields less errors for high watermark amplitudes and thus provides better decoding performance.
Therefore, the GML decoder for ISS is $\widehat{b}=\mathsf{\text{sign}}\left\{{s}^{T}\widehat{\mathbf{y}}\right\}$ where $\widehat{\mathbf{y}}$ is defined as in (58).
5.2. LMMSE decoder
In Section 4, we focused on the optimal ML decoders, which require the PDF parameters as well as the watermark strength and signature code. Since providing the watermark strength information to the decoder is not always possible, the LOD and GML decoders were proposed in Section 5 to make the decoders independent of this information. However, the PDF parameters are still required by these decoders. This motivated us to develop suboptimal decoders which depends neither on PDF parameters nor on the watermark strength information. Here, we introduce the LMMSE decoder which requires only the signature code as prior information at the decoder side.
where the autocorrelation matrix R_{r} defined as R_{ r }= E{rr^{ T }}, can be estimated at the receiver side. Therefore, from former expression, we can see that only the signature code is required at the receiver side. Although the LMMSE decoder has the same structure for information embedding in the DCT and magnitude of the DFT, its performance varies in these host domains. As explained earlier for LOD in the magnitude of the DFT domain, all the coefficients do not convey the information. On the other hand, the autocorrelation matrix is estimated using all the coefficients of the received signal and thus it causes degradation in the decoding performance.
and R_{ q }= E{(e + q)(e^{ T }+ q^{ T })}.
6. Experimental results
In this section, simulations on real images are conducted to illustrate the performance of the proposed watermark decoders for decoding hidden message. A set of testing images, such as [13], with size 512 × 512 is employed for information embedding which includes "Boat", "Peppers", "Baboon", "Lena", and "Barbara" to represent almost a wide range of images.
For information embedding in the DCT domain, for each 8 × 8 block of the image, the DCT coefficients are calculated and all coefficients except the dc one are used as the host signal to convey the hidden information, therefore, 63 coefficients are used for conveying of one bit of information. For information hiding in the DFT domain, since the coefficients should remain conjugate symmetric, 31 coefficients are employed. For the DCTdomain data hiding, determining an appropriate value of the shape parameter is important, though the details are out of scope of this article. One approach could be using the ML estimation [39, 40]. In practice, to reduce the computational complexity, an alternative way is to use a constant value regardless of the specific image under analysis. One such constant value was suggested [35] as c = 0.8, and we use this value in our simulations for avoiding additional estimations. Our results are based on 100 simulation runs with using different signature codes, and since each block with size 8 × 8 is used for hiding one bit, the total number of embedded bits is 512^{2}/8^{2} = 4096 in each test image.
In summary, some useful observations can be concluded from the experimental results: with the watermark amplitude information available at the receiver side, the DFT magnitude domain data hiding could result in better performances when the ML decoder is employed; With no access to the watermark amplitude information, the information embedding in the DCT domain data hiding is preferred, and the LMMSE decoder is preferred; When considering additional noise, data hiding in the DCT domain with ISS is preferred than the DFT magnitude domain ISS embedding. However, for SS embedding, the LMMSE decoder in the magnitude of the DFT domain provides fairly comparable performances to that of LMMSE in the DCT domain.
7. Conclusion
In this article, the optimal and suboptimal decoders for additive spread spectrum data hiding were investigated. Overall, we presented a rigorous decoding analysis framework of additive spread spectrum and ISS data hiding when the information bit is embedded into the DCT and the magnitude of the DFT domains, respectively. Generalized Gaussian distribution and Weibull distribution were used for deriving the ML decoders in the different domains. To improve the accuracy of the extracted hidden message, we employed ISS embedding and presented the optimal ML decoders. The theoretical error analyses of SS and ISS embedding in the DCT domain and in the magnitude of the DFT domain were derived. Simulation results showed that, when the watermark amplitude is available at the decoder side, data hiding in the magnitude of the DFT domain could yield better decoding performances than that of the DCT domain.
Though theoretically the ML decoder achieves the decoding performance upper bound, it requires additional prior information such as the watermark amplitude. To relax the requirements on such prior information, the LOD and LMMSE decoders were derived for practical data hiding applications in the DCT domain. The LOD decoder is independent of the watermark amplitude, though it still requires the host signal parameters. The LMMSE decoder provides a linear decoder in terms of the received signal which is independent of the watermark amplitude. The LOD and LMMSE decoders yield performances close to that of the ML decoder, with the LMMSE being slightly better than the LOD, especially at low DWR. For the proposed suboptimal decoders, we also provided the theoretical analysis of the bit error rate decoding performances.
The suboptimal LOD was also proposed in the DFT magnitude domain. However LOD does not provide close performances to that of the ML decoder, probably because that LOD uses all the received coefficients for decoding. In order to address this issue, the GML decoder was proposed to provide an estimate of the watermark amplitude and the bit information. Although GML could tackle the LOD deficiency at low DWR, its performance is much worse than that of ML in the absence of any additional attack/distortion. The LMMSE decoder in the DFT magnitude domain shows better performance than that of the LOD and GML decoders.
The simulation results suggest that, with no access to the watermark amplitude information at the decoder side, the suboptimal decoders in the DCT domain are more reliable than their counterparts in the DFT magnitude domain. Among the proposed suboptimal decoders, overall the LMMSE decoders are preferred. As expected, the ISS embedding scheme outperforms SS in both the DCT and the DFT magnitude domains, and thus is preferred. Simulations in the presence of additional noise showed that ISS embedding in the DCT domain is preferred. The GML and LMMSE decoders are preferred in the presence of additional noise than the ML one for data hiding in the magnitude of DFT domain.
Declarations
Acknowledgements
The work was supported by a SPG grant from the Natural Sciences and Engineering Research Council of Canada (NSERC).
Authors’ Affiliations
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