We propose a new watermarking method based on quantization index modulation. A concept of initial data loss is introduced in order to increase capacity of the watermarking channel under high intensity additive white Gaussian noise. According to the concept some samples in predefined positions are ignored even though this produces errors in the initial stage of watermark embedding. The proposed method also exploits a new form of distribution of quantized samples where samples that interpret “0” and “1” have differently shaped probability density functions. Compared to well-known watermarking schemes, this provides an increase of capacity under noise attack and introduces a distinctive feature. Two criteria are proposed that express the feature numerically. The criteria are utilized by a procedure for estimation of a gain factor after possible gain attack. Several state-of-the-art quantization-based watermarking methods were used for comparison on a set of natural grayscale images. The superiority of the proposed method has been confirmed for different types of popular attacks.

Digital media have a great impact on many aspects of modern society. Some aspects assume that we deal with audio-visual data that relates to a person or an organization. Information about the relation quite often should be preserved. Watermarking approach is to insert the information in the media itself [

In the field of digital image watermarking (DIW) digital images are used as a media (or host). DIW incorporates many different techniques and one of the most popular among them is quantization index modulation (QIM). Methods that belong to QIM are widely used in blind watermarking where neither original media nor watermark is known to the receiver [

In most cases quantization is implemented to some coefficients rather than to signal samples. In order to obtain coefficients a transform is applied to a host signal. It has been shown that some transforms provide coefficients that are more robust to popular image processing algorithms like JPEG, geometric modifications, and so forth [

It is assumed that during quantization each of the original coefficient values belongs to one of equally spaced intervals. Further, inside each interval coefficients to interpret “0” and “1” are selected. The task of quantization is to separate coefficients that represent different bits inside each interval. The separation efficiency influences robustness and invisibility. The result of the separation can be characterized by the size of original interval, distribution of separated samples, and the distortion incurred by the separation.

However, all the known implementations of QIM are far from achieving the capacity limit under AWGN. The simplest scalar realization of QIM is to replace all the coefficient values from a certain interval by a single value equal either to the left or right endpoint depending on a bit of a watermark [

Some other modifications of QIM have emerged over the past years. Forbidden zone data hiding (FZDH) modifies only a fraction (controlled by

The main advantage of the techniques based on QIM with different kinds of compensation [

Many different approaches have been developed to improve robustness against GA of QIM related methods [

Estimation of the scaling factor requires modelling. Some feature that is unique for the watermarked and attacked signal might be described by a model [

For instance, a kind of GA and a constant offset attack followed by AWGN are assumed in [

The method of recovery after GA and AWGN is proposed in [

As an opposite for estimation, invariance to GA, in general, requires more complex transform of original signal (e.g., nonlinear,) to obtain coefficients. It is necessary to modify coefficients to embed a watermark. However, a model to estimate distortions of the host is more complex in that case. Distortions should be controlled which limits the choice of the kind of QIM to one that adds less complexity to a model of distortion. This, for example, might result in reducing the number of adjustable parameters of QIM. This is one of the reasons why invariant to GA approaches are more vulnerable to AWGN compared to DC-QIM.

Rational dither modulation (RDM) is one of the most popular watermarking methods invariant to GA [

In this paper, we propose our own scalar QIM-based watermarking approach that is beneficial in several aspects. The approach addresses the mentioned gaps in the literature: it both delivers higher capacity under AWGN and recovers after GA. In order to do this, host signal coefficients are separated in a way that the resulting distributions for coefficients that interpret “0” and “1” are different. This distinctive feature is used by a simple yet efficient procedure for estimation of a scaling factor under GA. A concept of initial data loss (IDL) is introduced in order to increase watermark channel capacity under low watermark to noise ratios (WNRs). According to IDL, some fraction of wrong watermark bits is accepted during embedding procedure.

The rest of the paper is organized as follows. In Section

In this section we define a new model of quantization. First, it is necessary to show that according to our model the separation of original coefficients is possible and we can embed information. Formal logic approach is used to define dependencies between several conditions that are important for the separation of original coefficients. Separation argument (SA) represents the model in a compact form yet has a clear structure which is sufficient to reason the intuition behind the dependencies. Second, it is necessary to assure conditions when SA is sound.

Symbol

Let us denote a watermark bit by

We will use SA to describe the quantizer

The intuition behind SA can be explained in the following way. Initially samples with labels “

Another consideration is that for any two

And, lastly, the condition for IDL is

An illustration of an example where SA is sound is given in Figure

Illustration of the process of separation.

Parameters of the pdfs

We propose such

Distribution of the quantized coefficients: (a) inside

Namely, the proposed truncated pdfs are a linear function and a constant:

The soundness of SA is guaranteed if it is possible to satisfy

We start from defining parameters of

Using (

Functions

According to (

In the experiment section of the paper the goal is to find the highest capacity for a given WNR. Different values of the parameters need to be checked for that purpose. Preserving (

For the proposed pdfs we can now define

From (

According to (

The model was proposed in the previous section. It was shown that it is suitable for coefficient separation and the conditions necessary for soundness of SA were defined. In this section we focus on efficiency of separation. The main characteristic that can be estimated analytically is the watermark channel capacity under AWGN. It is required to calculate such characteristic for different WNRs. First, we express WNR in terms of parameters of the quantization scheme. Second, we express error rates in terms of parameters of the quantization scheme. This makes it possible to include WNR in the expression for error rates (and capacity).

The variance

For the matter of convenience of the experiment it is better to use a single parameter (control parameter) that can be adjusted in order to provide the desired value of

The second distortion component

The total quantization distortion

For any combination of

Bit error rate (BER) and channel capacity can be calculated without simulation of watermark embedding procedure. It is important that the kind of threshold used to distinguish between “0” and “1” is suitable for analytic estimations. Further we assume that the position of the threshold remains permanent after watermark is embedded and does not depend on attack parameters. In Figure

The absolute value of quantized sample in any interval is

There are two cases when errors occur in non-IDL samples. An error in “0” is incurred by a noise if and only if the both following conditions are true:

An error in “1” occurs if and only if the following is true:

Two cases when errors occur in IDL samples can be presented with the following conditions for “0” and “1,” respectively:

The pdf of AWGN with variance

Now we can show that BER_{0} and BER_{1} can be calculated according to (

Let us first assume

Now let us assume

In this section we describe conditions, procedure, and results of two different kinds of experiments based on analytic estimation of capacity as well as simulations. The preferred index of attack severity is WNR (indexes

In this subsection of our experiment we use

We use two variants of the proposed quantization scheme with adjustable parameters: nonsymmetric QIM (NS-QIM) and nonsymmetric QIM with IDL (NS-QIM-IDL). Such a decision can be explained by a consideration that IDL is acceptable for some application, but other applications may require all the watermark data to be embedded correctly.

In Figure

Analytic-based estimation of capacity under AWGN.

Capacity is calculated analytically according to the description provided in the literature for DC-QIM and QIM. During the estimation, the subsets

Therefore, for such estimation we assume that quantized coefficients from the

As can be seen from Figure

The advantage of analytic estimation of error rates according to (

In case of experiments with real signals the parameters of the proposed watermarking scheme must satisfy some other constraints instead of (

Some lower limit of DWR has to be satisfied for watermarked host, which assures acceptable visual quality. DWR is calculated according to

Therefore, using (

In contrast to analytic based experiment,

After watermark is embedded and AWGN with

A variant NSC-QIM with constant (nonadjustable) parameters is also used in some experiments. The intention to adjust the parameters in order to maximize capacity is natural. However, maximization requires information about WNR to be known before watermark embedding and transmission. In some application areas level of noise (or severity of an attack) might change over time or remain unknown. Therefore watermark should be embedded with some constant set of parameters depending on expected WNR.

Different positions of the threshold can be used to extract a watermark. An optimal position of the threshold is not obvious. Placing the threshold in the middle of the interval might be inefficient because the distribution of quantized samples inside embedding interval is nonsymmetric. Two kinds of thresholding are proposed: permanent and nonpermanent. The permanent position is

The nonpermanent position of

The performance of the proposed method was evaluated using real host signals. For that purpose we used 87 natural grayscale images with resolution 512 × 512. Each bit of a watermark was embedded by quantizing the first singular value of SVD of 4 × 4 block. This kind of transform is quite popular in digital image watermarking and the chosen block size provides a good tradeoff between watermark data payload and robustness [

Capacity under AWGN for natural grayscale images.

It can be seen that the resulting capacity after AWGN attack is the highest for NS-QIM. The other two methods whose performance is quite close to NS-QIM are DC-QIM and FZDH. Compared to DC-QIM the advantage is more obvious for higher variance. However, for moderate variance the advantage is more obvious compared to FZDH.

Methods QIM and RDM do not have parameters that can be adjusted to different variance. Under some circumstances adjustment is not feasible for NS-QIM as well. We have chosen constant parameters

Other image processing techniques except additive noise are able to destroy a watermark and one of them is JPEG compression which is quite popular. The capacity of the proposed watermarking method was also compared with other methods and the procedure of embedding was the same as in AWGN case. However, this time JPEG compression with different levels of quality was considered as an attack. The results are plotted in Figure

Capacity under JPEG for natural grayscale images.

According to the plots in Figure

It has been demonstrated that for some popular types of attack the performance of NS-QIM is comparable or better than that of DC-QIM. The mentioned DC-QIM is considered to be one of the best quantization methods for watermarking, but it is extremely vulnerable to GA. On the other hand the performance of RDM is not as good under AWGN and JPEG attacks and is comparable to that of QIM. In this subsection, we propose a procedure for GA recovery in order to fill an important gap in the literature and introduce a watermarking method that provides high efficiency under AWGN as well as GA. The procedure utilizes features that are unique for the proposed approach and have not been discussed in the field of watermarking before.

We are proposing several criteria that will be used by the procedure to provide robustness against GA for NS-QIM. The criteria exploit nonsymmetric distribution inside embedding interval and help to recover a watermarked signal after the attack. It is presumed that a constant gain factor is applied to the watermarked signal (followed by AWGN) and the task is either to estimate the factor or the resulting length of embedding interval.

Let us denote the actual gain factor by

The core of the procedure of recovery after GA is the following. For each particular value

One of the following criteria is being applied to the random variable

The value of

The intuition behind the proposed procedure of recovery from GA is the following. The variance of the coefficients of the host signal is much larger than the length of embedding interval. Embedding intervals are placed next to each other without gaps and even small error in estimation of

Experimental results demonstrate high level of accuracy of the proposed procedure of recovery after GA. Grayscale image Lena.tif with dimension 512 × 512 was used as a host signal for that purpose. A random watermark sequence was embedded into the largest singular values of SVD of 4 × 4 blocks using NS-QIM with

Plots of criteria values toward guessed length of embedding interval: (a) criterion

Despite the fact that for the same

The embedding constraints for the current experiment are the same as described in Section

The watermark embedding domain was the same as in previous tests: first singular values of SVD of 4 × 4 blocks from 512 × 512 grayscale images were quantized,

However, during watermark extraction no information except initial guess interval

Capacity under GA followed by AWGN.

It can be seen from Figure

The capacity plots for NS-QIM, NSC-QIM, and RDM in case of JPEG attack are shown in Figure

Capacity under GA followed by JPEG compression.

From Figure

In the experiment section we have estimated the capacity of the proposed method in both analytical and empirical ways. Following both ways we can witness that the proposed method provides higher capacity compared to the other reference methods. In this section we are to discuss in more detail measures of watermarking efficiency, conditions of the experiments, and the reasons of superiority of NS-QIM-IDL.

Channel capacity

The largest singular values of SVD of

Among the reference (and state of the art) methods used for comparison no one performs better than the proposed watermarking methods simultaneously under both AWGN and GA. Hence, the proposed methods fill the gap existing in watermarking literature. This is thanks to several new advancements used for embedding and extraction of a watermark.

In the case when AWGN is applied at the absence of GA the benefit is caused mostly by IDL and the kind of thresholding during watermark extraction. From Figure

The proposed method is in advantageous position compared to RDM in the case when GA is used to attack the watermarked image. As one of its stages, GA assumes AWGN and this explains superiority of NS-QIM over RDM in general. The success of recovery is due to easy and efficient procedure that utilizes a unique feature introduced by the proposed methods. The feature is created during quantization and is a result of different quantization rules for “0” and “1.”

The proposed estimation of scaling factor in this paper has some advantages compared to other known retrieving procedures. For instance, a model of a host is used in [

The nonpermanent thresholding was proposed with the aim to avoid transmitting any additional information to the receiver. For example, different size of embedding interval

In the paper we do not consider a constant offset attack. In some other papers like [

The new watermarking method based on scalar QIM has been proposed. It provides higher capacity under different kinds of attacks compared to other existing methods. The proposed NS-QIM-IDL method is the most beneficial in case of GA and AWGN. The advantages of the method are due to its unique approach to watermark embedding as well as a new procedure of recovery and extraction.

The main features of the unique approach to watermark embedding are a new kind of distribution of quantized samples and IDL. In general there is no line of symmetry inside embedding interval for the new distribution of quantized samples. This feature is used to recover a watermark after GA. The feature of IDL can reduce distortions introduced to a host signal which are caused by watermarking. This is done by letting some watermark bits to be interpreted incorrectly at the initial phase of embedding and before any attack occurs. The proposed IDL is extremely beneficial for low WNRs under AWGN attack.

The new procedure of recovery after GA exploits the nonsymmetric distribution of quantized samples. One out of two different criteria might be chosen to serve as a goal function for the procedure. The criteria behave in a similar way despite the differences in realization. It has been demonstrated experimentally that the proposed recovery procedure estimates the original length of embedding interval with deviation of 0.02% even in case when WNR is quite low. Nonpermanent thresholding was proposed in order to avoid transmitting additional information to the site where watermark extraction is done. The technique is simple and establishes the threshold in the position of the median of the distribution inside embedding interval.

The mentioned advancements implied considerable performance improvement. Under conditions of AWGN and JPEG attacks (at the absence of GA) the capacity of the proposed method is at the same or higher level compared to DC-QIM. The most advantageous application of NS-QIM-IDL is under AWGN for WNRs around −12 dB where it performs up to 10^{4} times better than DC-QIM. Under the condition of GA followed by high level of AWGN the performance of the proposed method is up to 10^{3} times higher than that of RDM. For the case when GA is followed by JPEG with

The authors declare that there is no conflict of interests regarding to the publication of this paper.