In general, watermarks are symbols which are placed into physical objects such as documents, photos, etc. and their purpose is to carry information about objects’ authenticity [5].

A digital watermark is a kind of marker embedded in a digital object such as image, audio, video, or software and, like a typical watermark, it is used to identify ownership of the copyright of such an object. Digital watermarking (or, hereafter, watermarking) is a technique for protecting the intellectual property of a digital object; the idea is simple: a unique marker, which is called watermark, is embedded into a digital object which may be used to verify its authenticity or the identity of its owners [6,7]. More precisely, watermarking can be described as the problem of embedding a watermark into an object and, thus, producing a new object, such that can be reliably located and extracted from even after has been subjected to transformations [7]; for example, compression, scaling or rotation in case where the object is an image.

In the image watermarking process the digital information, i.e., the watermark, is hidden in image data. The watermark is embedded into image’s data through the introduction of errors not detectable by human perception [8]; note that, if the image is copied or transferred through the internet then the watermark is also carried with the copy into the image’s new location.

Intellectual property protection is one of the greatest concerns of internet users today. Digital images are considered a representative part of such properties so we consider important, the development of methods that deter malicious users from claiming others’ ownership, motivating internet users to feel safer to publish their work online.

Image Watermarking, is a technique that serves the purpose of image intellectual property protection ideally as in contrast with other techniques it allows images to be available to third internet users but simultaneously carry an “identity” that is actually the proof of ownership with them. This way image watermarking achieves its target of deterring copy and usage without permission of the owner. What is more by saying watermarking we don’t necessarily mean that we put a logo or a sign on the image as research is also done towards watermarks that are both invisible and robust.

Our work suggests a method of embedding a numerical watermark into the image’s structure in an invisible and robust way to specific transformations, such as JPEG compression.

In this work we present an efficient and easily implemented technique for watermarking images that we are interested in uploading in the web and making them public online; this way web users are enabled to claim the ownership of their images.

What is important for our idea is the fact that it suggests a way in which an integer number can be represented with a two dimensional representation (or, for short, 2D representation). Thus, since images are two dimensional objects that representation can be efficiently marked on them resulting the watermarked images. In a similar way, such a 2D representation can be extracted for a watermarked image and converted back to the integer.

Having designed an efficient method for encoding integers as self-inverting permutations, we propose an efficient algorithm for encoding a self-inverting permutation into an image by first mapping the elements of into an matrix and then using the information stored in to mark specific areas of image in the frequency domain resulting the watermarked image. We also propose an efficient algorithm for extracting the embedded self-inverting permutation from the watermarked image by locating the positions of the marks in; it enables us to recontract the 2D representation of the self-inverting permutation.

It is worth noting that although digital watermarking has made considerable progress and became a popular technique for copyright protection of multimedia information [8], our work proposes something new. We first point out that our watermarking method incorporates such properties which allow us to successfully extract the watermark from the image even if the input image has been compressed with a lossy method. In addition, our embedding method can transform a watermark from a numerical form into a two dimensional (2D) representation and, since images are 2D structures, it can efficiently embed the 2D representation of the watermark by marking the high frequency bands of specific areas of an image. The key idea behind our extracting method is that it does not actually extract the embedded information instead it locates the marked areas reconstructing the watermark’s numerical value.

We have evaluated the embedding and extracting algorithms by testing them on various and different in characteristics images that were initially in JPEG format and we had positive results as the watermark was successfully extracted even if the image was converted back into JPEG format with various compression ratios. What is more, the method is open to extensions as the same method might be used with a different marking procedure such as the one we used in our previous work. Note that, all the algorithms have been developed and tested in MATLAB [9].

The paper is organized as follows. In Section 2 we present an efficient transformation of a watermark from an integer form to a two dimensional (2D) representation through the exploitation of self-inverting permutation properties. In Section 3 we briefly describe the main idea behind our recently proposed image watermarking algorithm, while in Section 4 we present our contribution with this paper. In Section 5 we show properties of our image watermarking technique and evaluate the performance of the corresponding watermarking algorithms. Section 6 concludes the paper and discusses possible future extensions.

Informally, a permutation of a set of objects is an arrangement of those objects into a particular order, while in a formal (mathematical) way a permutation of a set of objects is defined as a bijection from to itself (i.e., a map for which every element of occurs exactly once as image value).

Permutations may be represented in many ways. The most straightforward is simply a rearrangement of the elements of the set; in this way we think of the permutation as a rearrangement of the elements of the set such that “1 goes to 5”, “2 goes to 6”, “3 goes to 9”, “4 goes to 8”, and so on [10,11]. Hereafter, we shall say that is a permutation over the set.

Definition: Let be a permutation over the set, where. The inverse of the permutation is the permutation with. A self-inverting permutation (or, for short, SiP) is a permutation that is its own inverse:.

By definition, a permutation is a SiP (self-inverting permutation) if and only if all its cycles are of length 1 or 2; for example, the permutation is a SiP with cycles: (1,5), (2,6), (3,9), (4,8), and (7).

There are several systems that correspond integer numbers into permutations or self-inverting permutation [10]. Recently, we have proposed algorithms for such a system which efficiently encodes an integer into a self-inverting permutations and efficiently decodes it. The algorithms of our codec system run in time, where is the length of the binary representation of the integer, while the key-idea behind its algorithms is mainly based on mathematical objects, namely, bitonic permutations [12].

We briefly describe below our codec algorithms which in fact correspond integer numbers into self-inverting permutations; we show the correspondence between the integer and the self-inverting permutation by the help of an example.

Example W-to-SiP: Let be the given watermark integer. We first compute the binary representation of the number 12; then we construct the binary number and the binary sequence by flipping the elements of; we compute the sequences and by taking into account the indices of 0s and 1s in, and then the bitonic permutation on numbers by taking the sequence; since is odd, we select 4 cycles, , , of lengths 2 and one cycle of length 1, and then based on the selected cycles construct the self-inverting permutation.

Example SiP-to-W: Let be the given self-inverting permutation produced by our method. The cycle representation of is the following: (1,5), (2.6), (3,9), (4,8), (7); from the cycles we construct the permutation; then, we compute the first increasing subsequence and the first decreasing subsequence; we then construct the binary sequence of length 9; we flip the elements of and construct the sequence; the binary number 1100 is the integer.

We first define the two-dimensional representation (2D representation) of a permutation over the set, and then its 2DM representation which is more suitable for efficient use in our codec system.

In the 2D representation, the elements of the permutation are mapped in specific cells of an matrix as follows:

or, equivalently, the cell at row and column is labeled by the number, for each.

Figure 1(a) shows the 2D representation of the selfinverting permutation.

Note that, there is one label in each row and in each column, so each cell in the matrix corresponds to a unique pair of labels; see, [10] for a long bibliography on permutation representations and also in [13] for a DAG representation.

Based on the previously defined 2D representation of a permutation, we next propose a two-dimensional marked representation (2DM representation) of which is an efficient tool for watermarking images.

In our 2DM representation, a permutation over the set N_n is represented by an matrix as follows:

• the cell at row and column is marked by a specific symbol, for each;

• in our implementation, the used symbol is the asterisk, i.e., the character “^*”.

Figure 1(b) shows the 2DM representation of the permutation. It is easy to see that, since the 2DM representation of is constructed from the corresponding 2D representation, there is also one symbol in each row and in each column of the matrix.

We next present an algorithm which extracts the permutation from its 2DM representation matrix. More precisely, let be a permutation over and let be the 2DM representation matrix of (see, Figure 1(b)); given the matrix, we can easily extract from in linear time (i.e., linear in the size of matrix) by the following algorithm:

Algorithm Extract__from_2DM Input: the 2DM representation matrix of;

Output: the permutation;

Step 1: For each row of matrix, , and for each column of matrix, , if the cell is marked then;

Step 2: Return the permutation;

Remark 1. It is easy to see that the resulting permutation, after the execution of Step 1, can be taken by reading the matrix from top row to bottom row and write down the positions of its marked cells. Since the permutation is a self-inverting permutation, its 2D matrix has the following property:

• if, and

• otherwise,.

Thus, the corresponding matrix is symmetric:

• if, and

• otherwise,.

Based on this property, it is also easy to see that the resulting permutation can be also taken by reading the matrix from left column to right column and write down the positions of its marked cells.

Hereafter, we shall denote by a SiP and by the number of elements of.

The Discrete Fourier Transform (DFT) is used to decompose an image into its sine and cosine components. The output of the transformation represents the image in the frequency domain, while the input image is the spatial domain equivalent. In the image’s fourier representation, each point represents a particular frequency contained in the image’s spatial domain.

If is an image of size we use the following formula for the Discrete Fourier Transform:

for values of the discrete variables and in the ranges and.

In a similar manner, if we have the transform i.e the image’s fourier representation we can use the Inverse Fourier Transform to get back the image using the following formula:

for and.

Typically, in our method, we are interested in the magnitudes of DFT coefficients. The magnitude of the Fourier transform at a point is how much frequency content there is and is calculated by Equation (1) [14].

The algorithms behind the previously proposed technique were briefly based on the following idea.

The embedding algorithm first computes the 2DM representation of the permutation, that is, the array (see, Subsection 2.3); the entry of the array contains the symbol “^*”,.

Next, the algorithm computes the size of the input image and according to its size, covers it with an imaginary grid, which divides the image into

grid-cells of size,.

Then the algorithm goes first to each grid-cell, locates its central pixel and also the four pixels, , , around it, (these five pixels are called  cross pixels), and then computes the difference between the brightness of the central pixel and the average brightness of twelve neighboring pixels around the cross pixels, and stores the resulting value in the variable dif. Finally, it computes the maximum absolute value of all differences dif, , and stores it in the variable Maxdif.

The embedding algorithm goes again to each central pixel of each grid-cell, , and if the corresponding entry contains the symbol “^*”, then it increases the value of each one of the five cross pixels by Maxdif – dif +, where is a positive number used to make marks robust to transformations.

In a similar manner, the extracting algorithm is searching each line of the imaginary grid to find among the grid-cells the column of the one that has the greatest difference between the twelve neighboring and the five cross pixels,; then, the element is set equal to.

First we should mention that for images with general characteristics and relatively large size this method delivers optically good results. By saying “good results” we mean that the modifications made are quite invisible. Also the method’s algorithms run really fast as they simply access a finite number of pixels. Furthermore, both the embedding and extracting algorithms are easy to modify and adjust for various scenarios.

On the other hand, the method fails to deliver good results either for relatively small images or for images that depict something smooth which allows the eye to distinct the modifications on the image. Also we decided to move to a new method as there were also problems due to the fact that the positions of the crosses are centered at strictly specific positions causing difficulties in the extracting algorithm even for the smallest geometric changes such as scaling or cropping where we may lose the marked positions.

We next describe the embedding algorithm of our proposed technique which encodes a self-inverting permutation (SiP) into a digital image. Recall that, the permutation is obtained over the set, where and is the length of the binary representation of an integer which actually is the image’s watermark [12]; see, Subsection 2.2.

The watermark, or equivalently the self-inverting permutation, is invisible and it is inserted in the frequency domain of specific areas of the image. More precisely, we mark the DFT’s magnitude of an image’s area using two ellipsoidal annuli, denoted hereafter as “Red” and “Blue” (see, Figure 2). The ellipsoidal annuli are specified by the following parameters:

• , the width of the “Red” ellipsoidal annulus• , the width of the “Blue” ellipsoidal annulus• and, the radiuses of the “Red” ellipsoidal annulus on y-axis and x-axis, respectively.

The algorithm takes as input a SiP and a digital image, in which the user embeds the watermark, and returns the watermarked image; it consists of the following steps.

Algorithm Embed_SiP-to-Image Input: the watermark and the host image;

Output: the watermarked image;

Step 1: Compute first the 2DM representation of the permutation, i.e., construct an array of size such that the entry contains the symbol “^*”,.

Step 2: Next, calculate the size of the input image and cover it with an imaginary grid with

grid-cells of size,.

Step 3: For each grid-cell, compute the Discrete Fourier Transform (DFT) using the Fast Fourier Transform (FFT) algorithm, resulting in a grid of DFT cells,.

Step 4: For each DFT cell, compute its magnitude and phase matrices which are both of size

,.

Step 5: Then, the algorithm takes each of the magnitude matrices, , and places two imaginary ellipsoidal annuli, denoted as “Red” and “Blue”, in the matrix (see, Figure 2). In our implementation• the “Red” is the outer ellipsoidal annulus while the “Blue” is the inner one. Both are concentric at the center of the magnitude matrix and have widths and, respectively;

• the radiuses of the “Red” ellipsoidal annulus are (on the y-axis) and (on the -axis), while the “Blue” ellipsoidal annulus radiuses are computed in accordance to the “Red” ellipsoidal annulus and have values and, respectively;

• the inner perimeter of the “Red” ellipsoidal annulus coincides to the outer perimeter of the “Blue” ellipsoidal annulus;

• the values of the widths of the two ellipsoidal annuli are and, while the values of their radiuses are and.

The areas covered by the “Red” and the “Blue” ellipsoidal annuli determine two groups of magnitude values on (see, Figure 2).

Step 6: For each magnitude matrix, , compute the average of the values that are in the areas covered by the “Red” and the “Blue” ellipsoidal annuli; let be the average of the magnitude values belonging to the “Red” ellipsoidal annulus and be the one of the “Blue” ellipsoidal annulus.

Step 7: For each magnitude matrix, , compute first the variable as follows:

.

Then, for each row of the magnitude matrix, , compute the maximum value of the variables in row; let be the max value.

Step 8: For each cell of the 2DM representation matrix of the permutation such that “^*” (i.e., marked cell), mark the corresponding grid-cell,; the marking is performed by increasing all the values in magnitude matrix covered by the “Red” ellipsoidal annulus by the value

where. The additive value of is calculated by the function (see, Subsection 4.3) which returns the minimum possible value of that enables successful extracting.

Step 9: Reconstruct the DFT of the corresponding modified magnitude matrices, using the trigonometric form formula [14], and then perform the Inverse Fast Fourier Transform (IFFT) for each marked cell, , in order to obtain the image.

Step 10: Return the watermarked image.

In Figure 3, we demonstrate the main operations performed by our embedding algorithm. In particular, we show the marking process of the grid-cell of the  Lena image; in this example, we embed in the  Lena image the watermark number which corresponds to SiP.

In this section we describe the decoding algorithm of our proposed technique. The algorithm extracts a self-

inverting permutation (SiP) from a watermarked digital image, which can be later represented as an integer.

The self-inverting permutation is obtained from the frequency domain of specific areas of the watermarked image. More precisely, using the same two “Red” and “Blue” ellipsoidal annuli, we detect certain areas of the watermarked image that are marked by our embedding algorithm and these marked areas enable us to obtain the 2D representation of the permutation. The extracting algorithm works as follows:

Algorithm Extract_SiP-from-Image Input: the watermarked image marked with;

Output: the watermark;

Step 1: Take the input watermarked image and calculate its size. Then, cover it with the same imaginary grid, as described in the embedding methodhaving grid-cells of size.

Step 2: Then, again for each grid-cell, , using the Fast Fourier Transform (FFT) get the Discrete Fourier Transform (DFT) resulting a grid of DFT cells.

Step 3: For each DFT cell, compute its magnitude matrix and phase matrix which are both of size.

Step 4: For each magnitude matrix, place the same imaginary “Red” and “Blue” ellipsoidal annuli, as described in the embedding method, and compute as before the average values that coincide in the area covered by the “Red” and the “Blue” ellipsoidal annuli; let and be these values.

Step 5: For each row of, , search for the column where is minimized and set,.

Step 6: Return the self-inverting permutation.

Having presented the embedding and extracting algorithms, let us next describe the function which returns the additive value (see, Step 8 of the embedding algorithm Embed_SiP-to-Image).

Based on our marking model, the embedding algorithm amplifies the marks in the “Red” ellipsoidal annulus by adding the output of the function. What exactly does is returning the optimal value that allows the extracting algorithm under the current requirements, such as JPEG compression, to still be able to extract the watermark from the image.

The function takes as an input the characteristics of the image and the parameters, , , and of our proposed marking model (see, Step 5 of embedding algorithm and Figure 2), and returns the minimum possible value that added as to the values of the “Red” ellipsoidal annulus enables extracting (see, Step 8 of the embedding algorithm). More precisely, the function initially takes the interval, where is a relatively great value such that if is taken as for marking the “Red” ellipsoidal annulus it allows extracting, and computes the in.

Note that, allows extracting but because of being great damages the quality of the image (see, Figure 4). We mentioned relatively great because it depends on the characteristics of each image. For a specific image it is useless to use a greater than a specific value, we only need a value that definitely enables the extracting algorithm to successfully extract the watermark.

We next describe the computation of the value returned by the function; note that, the parameters and of our implementation are fixed with the values 2 and 2, respectively. The main steps of this computation are the following:

(1) Check if the extracting algorithm for validly obtains the watermark from the image; if yes, then the function returns;

(2) If not, that means, doesn’t allow extracting; then, the function uses binary search on and computes the interval such that:

• doesn’t allow extracting• does allow extracting, and

• ;

(3) The function returns;

As mentioned before, the function returns the optimal value. Recall that, optimal means that it is the smallest possible value which enables extracting

from the image. It is important to be the smallest one as that minimizes the additive information to the image and, thus, assures minimum drop to the image quality.

We tested our codec algorithms on various 24-bit digital color images of various sizes (from up to) and various quality characteristics.

In our implementation we set both of the parameters and equal to 2; see, Subsection 4.1. Recall that, the value 2 is a relatively small value which allows us to modify a satisfactory number of values in order to embed the watermark and successfully extract it without affecting images’ quality. There isn’t a distance between the two ellipsoidal annuli as that enables the algorithm to apply a small additive information to the values of the “Red” annulus. The two ellipsoidal annuli are inscribed to the rectangle magnitude matrix, as we want to mark images’ cells on the high frequency bands.

We mark the high frequencies by increasing their values using mainly the additive parameter because alterations in the high frequencies are less detectable by human eye [20]. Moreover, in high frequencies most images contain less information.

In this work we used JPEG images due to their great importance on the web. In addition, they are small in size, while storing full color information (24 bit/pixel), and can be easily and efficiently transmitted. Moreover, robustness to lossy compression is an important issue when dealing with image authentication. Notice that the design goal of lossy compression systems is opposed to that of watermark embedding systems. The Human Visual System model (HVS) of the compression system attempts to identify and discard perceptually insignificant information of the image, whereas the goal of the watermarking system is to embed the watermark information without altering the visual perception of the image [21].

The quality factor (or, for short, factor) is a number that determines the degree of loss in the compression process when saving an image. In general, JPEG recommends a quality factor of 75 - 95 for visually indistinguishable quality loss, and a quality factor of 50 - 75 for merely acceptable quality. We compressed the images with Matlab JPEG compressor from  imwrite with different quality factors; we present results for, and.

The quality function returns the factor c, which has the minimum value that allows the extracting algorithm to successfully extract the watermark. In fact, this value is the main additive information embedded into the image; see, formula (3). Depending on the images and the amount of compression, we need to increase the watermark strength by increasing the factor. Thus, for the tested images we compute the appropriate values for the parameters of the quality function; this computation can be efficiently done by using the algorithm described in Subsection 4.3.

To demonstrate the differences on watermarked image human visual quality, with respect to the values of the additive factor, we watermarked the original images Lena and Baboon and we embedded in each image the same watermark with and, where; the results are demonstrated in Figure 4.

In order to evaluate the watermarked image quality obtained from our proposed watermarking method we used two objective image quality assessment metrics, that is, the Peak Signal to Noise Ratio (PSNR) and the Structural Similarity Index Metric (SSIM). Our aim was to prove that the watermarked image is closely related to the original (image fidelity), because watermarking should not introduce visible distortions in the original image as that would reduce images’ commercial value.

The PSNR metric is the ratio of the reference signal and the distortion signal (i.e., the watermark) in an image given in decibels (dB); PSNR is most commonly used as a measure of quality of reconstruction of lossy compression codecs (e.g., for image compression). The higher the PSNR value the closer the distorted image is to the original or the better the watermark conceals. It is a popular metric due to its simplicity, although it is well known that this distortion metric is not absolutely correlated with human vision.

For an initial image of size and its watermarked image, PSNR is defined by the formula:

where is the maximum signal value that exists in the original image and MSE is the Mean Square Error given by

The SSIM image quality metric [19] is considered to be correlated with the quality perception of the HVS [22]. The SSIM metric is defined as follows:

where and are the mean luminances of the original and watermarked image respectively, is the standard deviation of, is the standard deviation of, whereas and are constants to avoid null denominator. We use a mean SSIM (MSSIM) index to evaluate the overall image quality over the sliding windows; it is given by the following formula:

The highest value of SSIM is 1, and it is achieved when the original and watermarked images, that is, and, are identical.

Our watermarked images have excellent PSNR and SSIM values. In Figure 5, we present three images of different sizes, along with their corresponding PSNR and SSIM values. Typical values for the PSNR in lossy image compression are between 40 and 70 dB, where higher is better. In our experiments, the PSNR values of of the watermarked images were greater than 40 dB. The SSIM values are almost equal to 1, which means that the watermarked image is quite similar to the original one, which proves the method’s high fidelity.

In Tables 1 and 2, we demonstrate the PSNR and SSIM values of some selected images of various sizes used in our experiments. We observe that both values, PSNR and SSIM, decrease as the quality factor of the images becomes smaller. Moreover, the additive value that enables robust marking under qualities and 60 does not result in a significant image distortion as Tables 1 and 2 suggest; see also the watermarked images on Figure 5.

In closing, we mention that Lena and Baboon images of Figure 4 are both of size. Lena image has PSNR values 55.4, 50.1, 46.2 and SSIM values 0.9980, 0.9934, 0.9854 for, respectively, while Baboon image has PSNR values 52.7, 46.2, 42.5

Table 1. The PSNR values of watermarked images of different sizes under JPEG qualities.

Table 2. The SSIM values of watermarked images of different sizes under JPEG qualities.

and SSIM values 0.9978, 0.9908, 0.9807 for the same quality factors.

In the following, based on our experimental results, we discuss several impacts concerning characteristics of the host images and our embedding algorithm, and also we justify them by providing explanations on the observed outcomes.