Watermarking Images in the Frequency Domain by Exploiting Self-Inverting Permutations

In this work we propose efficient codec algorithms for watermarking 
images that are intended for uploading on the web under intellectual property 
protection. Headed to this direction, we recently suggested a way in which an 
integer number w which being 
transformed into a self-inverting permutation, can be represented in a two 
dimensional (2D) object and thus, since images are 2D structures, we have proposed 
a watermarking algorithm that embeds marks on them using the 2D representation 
of w in the spatial 
domain. Based on the idea behind this technique, we now expand the usage of 
this concept by marking the image in the frequency domain. In particular, we 
propose a watermarking technique that also uses the 2D representation of 
self-inverting permutations and utilizes marking at specific areas thanks to 
partial modifications of the image’s Discrete Fourier Transform (DFT). Those 
modifications are made on the magnitude of specific frequency bands and they 
are the least possible additive information ensuring robustness and 
imperceptiveness. We have experimentally evaluated our algorithms using various 
images of different characteristics under JPEG compression. The experimental 
results show an improvement in comparison to the previously obtained results 
and they also depict the validity of our proposed codec algorithms.


Introduction
Internet technology, in modern communities, becomes day by day an indispensable tool for everyday life since most people use it on a regular basis and do many daily activities online [1].This frequent use of the internet means that measures taken for internet security are indispensable since the web is not risk-free [2,3].One of those risks is the fact that the web is an environment where intellectual property is under threat since a huge amount of public personal data is continuously transferred, and thus such data may end up on a user who falsely claims ownership.
It is without any doubt that images, apart from text, are the most frequent type of data that can be found on the internet.As digital images are a characteristic kind of intellectual material, people hesitate to upload and transfer them via the internet because of the ease of intercepting, copying and redistributing in their exact original form [4]. Encryption is not the problem's solution in most cases, as most people that upload images in a website want them to be visible by everyone, but safe and theft protected as well.Watermarks are a solution to this problem as they enable someone to claim an image's ownership if he previously embedded one in it.Image watermarks can be visible or not, but if we don't want any cosmetic changes in an image then an invisible watermark should be used.That's what our work suggests a technique according to which invisible watermarks are embedded into images using features of the image's frequency domain and graph theory as well.
We next briefly describe the main idea behind the watermarking technique, the motivation of our work, and our contribution.

Watermarking
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 own-ership 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 w I and, thus, producing a new object w I , such that can be reliably located and extracted from w w I even after w I 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.

Motivation
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.

Contribution
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 repre-sented 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 .w 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 I by first mapping the elements of matrix  and then using the information stored in A  to mark specific areas of image I in the frequency domain resulting the watermarked image w I .We also propose an efficient algorithm for extracting the embedded self-inverting permutation   from the watermarked image w I by locating the positions of the marks in w I ; 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 w w I 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].

Road Map
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.

Theoretical Framework
In this section we first describe discrete structures, namely, permutations and self-inverting permutations, and briefly discuss a codec system which encodes an integer number into a self-inverting permutation w  .
Then, we present a transformation of a watermark from a numerical form to a 2D form (i.e., 2D representation) through the exploitation of self-inverting permutation properties.

Self-Inverting Permutations
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 n ; in this way we think of the permutation as a rearrangement of the elements of the set 9 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 .A self-inverting permutation (or, for short, SiP) is a permutation that is its own inverse: .

Encoding Numbers as SiPs
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 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   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 Y ; we then construct the binary sequence of length 9; we flip the elements of and construct the sequence   0, 0, 0, 0,1,1, 0, 0,1 B  12 w ; the binary number 1100 is the integer  .

2D Representations
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
A as follows: .Note that, there is one label in each row and in each column, so each cell in the matrix A 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 n n  A  as follows:  the cell at row i and column i  is marked by a specific symbol, for each 1, 2, , i n   ;  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 repre- sentation of  is constructed from the corresponding 2D representation, there is also one symbol in each row and in each column of the matrix A  .
We next present an algorithm which extracts the permutation  from its 2DM representation matrix.More precisely, let  be a permutation over n and let be the 2DM representation matrix of  (see, Figure 1(b)); given the matrix A  , we can easily extract  from A  in linear time (i.e., linear in the size of matrix A  ) by the following algorithm: Output: the permutation  ; Step 1: , and for each column of matrix j A  1 j , , 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 A  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 A has the following property: Thus, the corresponding matrix Based on this property, it is also easy to see that the resulting permutation  can be also taken by reading the matrix A  from left column to right column and write down the positions of its marked cells.
Hereafter, we shall denote by   n a SiP and by  the number of elements of   .

The Discrete Fourier Transform
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.

 
is an image of size we use the following formula for the Discrete Fourier Transform: for values of the discrete variables u and in the ranges Typically, in our method, we are interested in the magnitudes of DFT coefficients.The magnitude   F , u v of the Fourier transform at a point is how much frequency content there is and is calculated by Equation (1) [14].

Previous Results
Recently, we proposed a watermarking technique based on the idea of interfering with the image's pixel values in the spatial domain.In this section, we briefly describe the main idea of the proposed technique and state main points regarding some of its advantages and disadvantages.Recall that, in the current work we suggest an expansion to this idea by moving from the spatial domain to the image's frequency domain.

Method Description
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 Next, the algorithm computes the size of the input image I and according to its size, covers it with an imaginary grid C which divides the image into Then the algorithm rst to each grid-cell ij C , lo  , , , the lumn j of the one t has the gr ce between the twelve neighboring and the five cross pixels, 1 ,  i j n tha e n    ; then, the element i   is set equal to j .

Main Points
for images with general to deliver good re g inte-

The Frequency Domain Approach
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 sults 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.
Having described an efficient method for encodin gers as self-inverting permutations using the 2DM representation of self-inverting permutations, we next describe codec algorithms that efficiently encode and decode a watermark into the image's frequency domain [14][15][16][17].
We next describe the embedding algorith posed technique which encodes a self-inverting permutation (SiP)   into a digital image I .Recall that, the permutation   is obtained over the set n N  , where 2 1 n n    and n is the length of the binar epresen- n integer w which actually is the image's watermark [12]; see, Subsection 2.2.
The watermark w , or equivalentl y r tation of a y the self-inverting permutation   , is invisible and it is inserted in the frequency doma of specific areas of the image in I .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:  Step 2: N ula ext, calc te the size N M  of the input image I and cover it with an imaginary grid C with Step 4: F ea ll or ch DFT c ij e F , compute its magnitude ij M and phase ij P matrices which are both of size Step 5: Then, the algorithm of the n n takes each  , and plac imaginary ellipsoidal n d as "Red" and "Blue", in the matrix ij es two uli, den an ote M (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 ij M magnitude matrix and have widths The areas covered by the "Red" and the "Blue" ellipsoidal annuli determine two groups of magnitude values on ij M (see, Figure 2).S 6: For each magni tep tude matrix ij M , 1 , i j n    , co in the area mpute the average of the values tha e s covered by the "Red" and the "Blue" ellipsoidal annuli; let ij t ar AvgR be the average of the magnitude values belonging to the "Red" ellipsoidal annulus and ij AvgB be the one of the "Blue" ellipsoidal annulus.
Step 7: Then r each row i of the ma matrix inverting permutation (SiP)   from a watermarked digital image w I , which can b later represented as an integer w .
The s f-i e el nver rmutation ting pe   is obtained from the frequency domain of specific areas of the watermarked image w I .More precisely, using the same two "Red" and "Blue" ellipsoidal annuli, we detect certain areas of the watermarked image w I 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 Step 2: Then, again for , 1 , i j n For each DFT cell, compute its magnitude m ing the Fast Fourier Transform (FFT) t t Fourier Transform (DFT) resulting a n n    grid of DFT cells.
Step 3: ge atrix ij M and phase matrix ij P which are both of Step 4: For each ma atrix ij gnitude m M , 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 ing great da es 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 max c greater than a specific value, we only need a value that definitely enables the extracting algorithm to successfully extract the watermark.
We next desc turned by the function f ; note that, the parameter b P and r P of our implementation are fixed with the lues 2 nd 2, respectively.The main steps of this computation are the following: (1) Check if the extracting algorithm ing their value as that minimizes the additive information to the image and, thus, assures minimum drop to the image quality.

Experimental Evaluation
In this section we present the experim proposed watermarking method which we have implemented using the general-purpose mathematical software package Matlab (version 7.7.0)[9].
We experimentally evaluated our gital color images of various sizes and quality characteristics.Many of the images in our image repository where taken from a web image gallery [18] and enriched by some other images different in sizes and characteristics.Our experimental evaluation is based on two objective image quality assessment metrics namely Peak Signal to Noise Ratio (PSNR) and Structural Similarity Index Metric (SSIM) [19].
There are three main arking: fidelity, robustness, and capacity [5].Our watermarking method appears to have high fidelity and robustness against JPEG compression.We tested our codec al color images of various sizes (from 200 130  up to 4600 3700  ) and various quality charact plementation we set both of the para r and b P equal to 2; see, Subsection 4.1.Recall that, 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 increas s using mainly the additive parameter opt c c  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 im s a numbe s i portance 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, Q factor) i r that determines the degree of los n 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 90 The quality function f returns the factor c, which ha c ext te e n watermarked image hu n s the minimum value opt that allows the extracting algorithm to successfully ract the watermark.In fact, this value opt c 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 c .Thus, for the tested images we compute the appropria values for the parameters of the quality function f ; this computation can be efficiently done by using th algorithm described in Subsection 4.3.
To demonstrate the differences o man visual quality, with respect to the values of the additive factor c , we watermarked the original images Lena and Baboo and we embedded in each image the same watermark with In order to evaluate the watermarke tained 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 refere d 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 I of size w w 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: The highest value of SSIM is 1, and it is achieved when the original and watermarked images, that is, I and w I , are identical.Ou watermarked images have excellent PSNR and SS 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 c that enables robust marking under qualities 90, 75 Q  nd 60 does not result in a significant image distor as bles 1 and 2 suggest; see also the watermarked images on Figure 5.
In closing, we mention that Lena and Baboon images of

Other Experimental Outcomes
In the following host images and our embedding algorithm justify them by providing explanations on the observed outcomes.

The Additive Value Influences
As the experimental results show the PSNR and SSIM values decr ages with lower quality index in its JP see, Tables 1 and 2. That happens since our embedding algorithm adds more information in the frequency of marked image parts.By more information we mean a greater additive factor c ; see, Equation (3).We next discuss an important issue concerning the additive value opt c c  returned by function f ; see, Subsection 4.3.In Table 3, we show a sample of our results demonstrating for each JPEG quality the respective values of the additive factor opt c .The figures show that the opt c value increases as the quality factor of JPEG compression decreases.It is obvious that the embedding algorithm is image dependent.It is worth noting that opt c values are small for images of relatively small size while they increase as we move to images of greater size.
Moving beyond the sample images in order to show the behaviour of additive value opt c under different image sizes, we demonstrate in Figure 6 the average c pt values of all the tested images grouped in three different sizes.We decided to select three representative groups for small, medium, and large image sizes, that is, 200 × 200, 500 × 500 and 1024 × 1024, respectively.For 2) Then, quantization of the DCT coefficients tak place.This is done by simply dividing each component of the DCT coefficients matrix by the corresponding constant from the sam en rounding to the nearest integer.
3) In the third step, it's entropy coding which involves arranging the image components in a "zigzag" order employing run-length encoding (RLE) algorithm that groups  similar frequencies together, inserting length coding zeros, and then using Huffman coding on what is left.Focusing on the second step, we should point out that images with higher compression (lower quality) make use of a Quantization matrix which typically has greater values corresponding to higher frequencies meaning that information for high frequency is greatly reduced as it is less perceivable by human eye.
As we mentioned our method marks images in the higher frequency domain which means that as the compression ratio increases marks gradually become weaker and thus opt c increases to strengthen the marks.Furthermore, someone ma ice that c also in- In this paper we propose a method for embedding invisiprove y not opt creases for larger images.That is because regardless of the image size the widths of the ellipsoidal annuli remain the sam he larg e meaning that t er the image the less frequency amplitude is covered by the constant sized annuli.That makes marks less robust and require a greater opt to strengthen them.

Frequency Domain Imperceptiveness
It is worth noting that the marks made to embed the watermark in the image are not just invisible in the image itself but they are also invisible in the image's overall Discrete Fourier Transform (DFT).More precisely, if someone suspects the existence of the watermark in the frequency domain and gets the image's DFT, it is impossible to d ct something unusual.This is also d ete emonstrated in Figure 7, which shows that in contrast with using the ellipsoidal marks in the whole image, using them in specific areas makes the overall DFT seem normal.in numerical form, transformed into self-inverting permutations, and embedded into an image by partially marking the image in the frequency domain; more precisely, thanks to 2D representation of self-inverting permutations, we locate specific areas of the image and modify their magnitude of high frequency bands by adding the least possible information ensuring robustness and imperceptiveness.We experimentally tested our embedding and extracting algorithms on color JPEG images with various and different characteristics; we obtained positive results as the watermarks were invisib they didn't affect the images' quality and they were extractable despite the JPEG  It is worth noting that the proposed algorithms are robust against cropping or rotation attacks since the watermarks are in SiP form, meaning that they determine the embedding positions in specific image areas.Thus, if a part is being cropped or the image is rotated, SiP's symmetry property may allow us to reconstruct the watermark.Furthermore, our codec algorithms can also be modified in the future to get robust against scaling attacks.That can be achieved by selecting multiple widths concerning the ellipsoidal annuli depending on the size of the input image.

Concluding Remarks
Finally, we should point out that the study of our quality function f remains a problem for further investigation; indeed, f could incorporate learning algorithms [24] so that to be able to return the c accurately and in a very short computational time.

Figure 1 .
Figure 1.The 2D and 2DM representations of the self-inverting permutation .
In a similar manner, if we have the transform u v i.e the image's fourier representation we can use the Inverse Fourier Transform to get back the image   , pix oss pixels), a omputes the difference between the brightness of the central pixel 0 ij p and the average brightness of twelve neighboring ixels around the cross pixels, and stores the resulting value in the variable dif   0 p .Finally, it computes the maximum absolute value of all n n

4. 1 .
Embed Watermark into Image m of our pro-First we should mention that

rP 1 R and 2 R 1 :
, the width of the "Red" ellipsoidal a nnulus, oidal  b P , the width of the "Blue" ellipsoidal annulus,  , the radiuses of the "Red" ellips annulus on y-axis and x-axis, respectively.The algorithm takes as input a SiP   and a digital image I , in which the user embeds the watermark, and returns the watermarked image w I ; it consists of the following steps.Algorithm Embed_SiP-to-Image the host image Input: the watermark w Compute first the 2DM pr re esentation of the permutation   , i.e., construct an array A  of size
r P and b P , respectively;  the radiuses of the "Red" ellipsoi nulus are 1 R dal an (on the y-axis) and 2 R (on the x -axis), while the "Blue" ellipsoidal annulus radiuses are computed in accordance to the "Red" ellipsoidal annulus and have values  the inner perimeter of the "Red" ellipsoidal annulus widths of the two ellipsoidal annuli coincides to the outer perimeter of the "Blue" ellipsoidal annulus;
embedding and extracti al hms, let us next describe the function f which returns the additive value opt c c  (see, Step of the embedding algorithm Embed -Image).iP-toBased on our ma amplifies the marks in the "Red" ellipsoidal annulus by adding the output of the function f .What exactly f does is returning the optimal value that allows the etracting algorithm under the current requirements, such as JPEG compression, to still be able to extract the watermark from the image.The function f takes the image and the parameters 1 R , 2 R , b P , and r P of our proposed marking model ee tep 5 of e -bedding algorithm and Figure2), and returns the minimum possible value opt c that added as c to the values of the "Red" ellipsoidal annulus enables extracting (see, Step 8 of the embedding algorithm).More precisely, the function f initially takes the interval  c for marking the "Red" ellipsoidal ann s it allows ext acting, and computes the opt c in  Figur wed e 4. The original images of Lena and Baboon follo by their watermarked images with different additive values c, i.e., the optimal value and a relatively large value c; both images are marked with the same watermark (6,3,2,4,5,1).

2 C
the mean luminances of the l an rm origina d wate arked image I respectively,   I  is the standard deviation of I ,   w I  ard is the stand deviation of w I , whereas 1 Care constants to avoid null den inator.We se a me SSIM (MSSIM) index to evaluate the overall image quality over the and u an om M sliding windows; it is given by the following formula:

Figure 5 .Figure 5 ,
Figure 5. Sample images of three size groups for JPEG quality factor Q = 75 and their corresponding water

Figure 6 .
Figure 6.The average optimal c values for the tested images grouped in three deferent sizes under the JPEG quality factors = 90, 75 and 60 Q .
ble watermarks into images and their intention is to the authenticity of an image.The watermarks are given (a) (b)atermarked image marked partially with our technique. opt

Figure 7 .
Figure 7. (a) The DFT of a watermarked image marked on the full image's frequency domain.(b) The DFT of a w and stores it in the variable Maxdif   of the imaginary grid C to find among the n  grid-cells 1 2 i