Information Sciences 269 (2014) 94–105

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Information Sciences

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Robust watermarking against geometric attacks using partialcalculation of radial moments and interval phase modulation

0020-0255/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.ins.2013.11.020

⇑ Corresponding author.E-mail addresses: s.golabi@sutech.ac.ir (S. Golabi), ms_helfroush@sutech.ac.ir (M. S. Helfroush), danyali@sutech.ac.ir (H. Danyali), m.ow

sutech.ac.ir (M. Owjimehr).

Sasan Golabi, Mohammad Sadegh Helfroush ⇑, Habibollah Danyali, Mehri OwjimehrDepartment of Electrical and Electronics Engineering, Shiraz University of Technology, Shiraz, Iran

a r t i c l e i n f o

Article history:Received 25 December 2012Received in revised form 13 July 2013Accepted 19 November 2013Available online 28 November 2013

Keywords:Blind watermarkingRobustnessRadial momentInterval phase modulationPartial calculation of moment

a b s t r a c t

This paper presents a new image watermarking scheme which is blind and robust againstgeometric attacks. The proposed algorithm is based on the features of radial moments. Inthis algorithm the radial moments were first computed and then the watermark wasembedded into the differential phase of various blocks of the original image using a specialtype of PSK modulation called interval phase modulation (IPM). In order to embed thewatermark into the phase of moments, a particular method, namely the partial calculationof moments was utilized for computation. The implementation results show the improve-ment in the robustness and quality of the watermarked images in comparison with othermethods.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

The rapid growth of multimedia applications leads to an urgent need for adequate copyright protection techniques, espe-cially for image data. Robust watermarking can be used to trace copies or to implement copyright protection methods. Somewatermarking algorithms are published in order to satisfy this robustness requirement [2,7,13–15,18].

For robust watermarking schemes, the embedded watermark must be robust against a variety of possible attacks. Theseinclude robustness against signal processing attacks, such as filtering, additive noise, cryptographic, statistical and geometricattacks. While many methods have a good performance against signal processing attacks, they lack robustness to geometrictransformations. Rotation and scaling attacks are considered more challenging than other attacks. This is due to the fact thatchanging the image size or its orientation even slightly, could dramatically reduce the receiver ability to retrieve the water-mark. It has been proven that even very small geometric distortions can prevent the detection of a watermark [12,16,19,22].

The existing methods that can resist geometric attacks are classified into the exhaustive search, invariant domain, embed-ding template and feature-based methods. One concern in the exhaustive search is the computational cost in larger searchspace. Invariant domain methods usually suffer from implementation issues and are vulnerable to cropping. The embeddingtemplate-based techniques are vulnerable to template estimation attacks and cropping. By contrast, the feature-basedwatermarking techniques use image-dependent features to represent invariant reference points for both embedding anddetection. They are resistant to various attacks, including cropping and random bending attacks (RBA) by binding the water-mark synchronization with the image salient characteristics. These characteristics may be the whole image, some local re-gions, or feature points. This group of the watermark synchronization techniques, also known as the second generation

jimehr@

S. Golabi et al. / Information Sciences 269 (2014) 94–105 95

watermarking [22], has the worthwhile properties of invariance to noise, geometrical transformation and localization. Thisgroup of techniques can be divided into three sub-categories: moment-based, histogram-based and feature point-based.

Histogram-based watermarking techniques [3] use histograms to solve the geometric invariance problem. The histogramdistribution of an image is approximately invariant under geometric attacks. Xiang et al. [19] propose an invariant imagewatermarking in the low-frequency domain by using the histogram shape and mean in the Gaussian filtered low-frequencycomponent of images. The method proposed in [3] developed a geometrically robust image watermarking scheme by using ahistogram in a certain range to embed a watermark in circular regions centered on the Harris–Laplace feature points. Themajor limitation of the histogram-based methods is their incapability to resist local transformation.

Feature point-based watermarking methods use the feature points to form local regions for the embedding and extractingof the watermark. In [17], first, disks which represent the features of the image will be derived and 16 watermark bits will beembedded in each of them by an indirect method. For example, the number of disks extracted for the image Lena is 8 and forthe pepper is 4. For watermark extraction, it is required that the exact location of the feature disks be identified. Due to com-putational errors (the interpolation errors in the normalization process and the sampling errors in converting formulas fromcontinuous to discrete ones), the disks cannot be located accurately. Hence, this method is usually used only to detect thepresence of the watermark rather than extract it. In [4], two blind and non-blind watermarking methods are provided. Inthe blind method, the desired watermark will be changed with respect to the normalized original image. Then an inversenormalization operation is applied to the created watermark. Therefore, the watermark image and the host image will beof the same size. Now, the watermark is added to the host image. Due to applying the normalization to the watermark image,the normalization effects on the original image will be canceled. However, in these methods, watermark extraction involvesfinding the exact position of regions and because of the interpolation and normalization errors, this could not be performedaccurately.

Due to their ability to represent the global properties of the image, moments have been used in many application fields ofimage processing. Geometric moments are mainly used to capture global features of images. In [9], the absolute values of theZernike moments which are extracted from the normalized image are used for watermark embedding. These features areinvariant to geometric changes. The watermark will be changed and embedded into the host image using these features.In [5], to achieve robustness against affine transformation, the watermark is embedded into a moment-based normalizedimage. In [1,20], Zernike moments are used as geometrically robust image watermarks. Zhang et al. in [21] propose a geo-metric invariant blind image watermarking by using invariant Tchebichef moments and independent component analysis(ICA). In [10], a geometrically invariant watermarking method is proposed based on the orthogonal moments obtained bypolar harmonic transform (PHT). The magnitude of these moments is rotation invariant. However, they are not strictly trans-lation and scale invariant. To achieve these properties, the image needs to be resized to a domain (x, y) e [�11] � [�11]which increases the numerical and interpolation errors. Moreover, they have used direct modification of moments whichis less desirable in comparison with change indirectly the spatial pixels. A compensation image is proposed to solve thereconstruction problem. Not only, it raises the complexity; but also, there is a limitation for the increasing of the PSNR.

Invariant moment-based methods have low computational costs and do not need the complete normalization of the im-age. As the normalization process is one of the main sources of numerical errors, the features extracted by moments are moreaccurate and robust against common geometric and signal processing attacks than other methods. However, moment-basedtechniques are highly fragile against cropping attacks.

In this paper, a new robust watermarking scheme, which can effectively resist common geometric attacks, is proposed.The phase of adaptive radial moments, which in this paper is computed in a unit circle, is used for watermark embedding.According to the new phases after watermark embedding, pixels will be changed in the spatial domain. These phases arerobust against geometric attacks and therefore no normalization procedure is needed. Therefore, the computational costis low; moreover, the numerical accuracy will be high. Using a special type of phase modulation, called ‘‘interval phase mod-ulation (IPM)’’, increases robustness against compression attacks. The results of implementation show the improved qualityof the watermarked image and more capacity, as compared with other common geometric invariant methods.

The rest of this paper is organized as follows: in Section 2, moments, particularly, radial moments and their invariant fea-ture extraction will be reviewed. In Section 3, the watermark embedding and extraction procedures are explained. Imple-mentation and results are described in Section 4 and finally Section 5 provides the conclusion for this work.

2. Moments

The most common moments, are geometric moments which can be calculated as:

Mpq ¼ZZ

Rf ðx; yÞxpyq dxdy ð1Þ

where f(x, y) P 0 is a real bounded function with support on a finite region R; Mpq is the geometric moment of the order(p + q). In the discrete case, for a digital image of size N � N we have:

Mpq ¼XN

x

XN

y

f ðx; yÞxpyq ð2Þ

96 S. Golabi et al. / Information Sciences 269 (2014) 94–105

In the above formula, xpyq indicates the base of moment calculation, thus moment computation can be seen as the map-ping of image onto these basis. In the pattern recognition applications, the most important moments are invariant moments.These moments are actually those from which the features are extracted and these features are robust against some geomet-ric changes. For example, the amplitude of Zernike and Legendre moments is rotation invariant. In general, rotation invariantfeatures can be achieved from any complex moments. To have translation and scale invariant features, the image normali-zation can be used [8].

In this paper, the radial moments are used to achieve invariant features. For an image f(x, y), the radial moment Rpq oforder p and repetition q can be computed as:

Rpq ¼Z 2p

0

Z 1

0rpe�iqhf ðr cos h; r sin hÞrdrdh; i ¼

ffiffiffiffiffiffiffi�1p

ð3Þ

where f(r, h) is the projection of the image on Polar coordinates over a unit disk, p = 1,2, . . . ,1 and q can be any positive ornegative integer. According to the above equation, radial moments are defined in terms of polar coordinates (r, h) over a unitdisk. On the other hand, the intensity function of the image is defined in Cartesian coordinates (x, y). Therefore, an appro-priate image mapping is necessary. There are two main approaches for coordinate mapping from Cartesian to Polar. Inthe first approach, the square image is mapped onto a unit disk where the center of the image is assumed to be the originof coordinates. In this approach, all pixels outside the unit disk are removed, resulting in a loss of some image information. Inthe second approach, the square image is mapped inside the unit disk where the center of the image is assumed to be thecoordinate origin. Since the second method shows no loss of information, in this paper the second square-to-circle mappingapproach is applied (Fig. 1).

In the discrete case, for a digital image of size N � N the integrals in the above formula are replaced by summations andthe image is normalized inside the unit disk using the second mapping transformation. Thus, the discrete radial moments aredefined as:

Rpq ¼ kp

XN�1

j¼0

XN�1

k¼0

rpjke�iqhjk f ðj; kÞ jrjkj 6 1 ð4Þ

where kp is the number of pixels which have |rjk| 6 1.This equation has two sources of numerical and geometrical errors. The numerical error is due to approximating the inte-

gral by summations and the geometrical error is made by square to circular mapping transformation.To avoid these errors, the method described in [8] is used. According to geometric moment calculations and with some

mathematical computations, radial moments can be expressed in terms of geometric moments:

Rpq ¼Xp�q

2

k¼0

Xq

j¼0

p�q2

k

� �qj

� �ðiÞjMp�2k�j;2kþj ð5Þ

It can be shown that the radial moments after a degree image rotation are changed as:

R0pq ¼ e�iqaRpq ð6Þ

where R0pq is the radial moment of the rotated image. As can be seen from (6), the absolute value of radial moments is con-stant under image rotation and therefore the amplitudes of radial moments can be defined as rotation invariant features.

jR0pqj ¼ jRpqj ð7Þ

Fig. 1. Mapping of square image inside the unit disk [8].

S. Golabi et al. / Information Sciences 269 (2014) 94–105 97

However, as a large amount of image information is stored in the phase of moments by selecting only the amplitude, welose them. To provide radial coefficients with the rotation–invariant phase, a method that has been presented in [11] is usedto cancel the influence of the rotation on the phase of coefficients. This method has been performed by combining the phasecoefficients of different orders and repetitions to form the complex valued moments with rotation invariant phases:

R00pq ¼ Rpqe�iq\Rp0 ;1 p0 2 1;3;5;7; . . . ; 2� pmax

2

l m� 1

� �n oð8Þ

The phase of R00pq stays unchanged under the image rotation. Suppose that a is the rotation angle; R00pq and R00rpq represent theoriginal radial feature and the rotated version, respectively, we have:

\R00pq ¼ \Rpq � q\Rp0 ;1 ð9Þ

After rotation, the phase of R00pq is computed using the following equations:

\R00rpq ¼ \Rrpq � q \Rr

p0 ;1

� �¼ ð\Rpq � qaÞ � qð\Rp0 ;1 � aÞ ¼ \Rpq � q\Rp0 ;1 ¼ \R00pq ð10Þ

Therefore, the phase angle \R00rpq of the rotated image is the same as the phase angle \R00pq of the non-rotated image. The valueof order p0 could be any odd number. However, as mentioned in [11], the coefficients with lower orders will be more robustto noise and distortions and since R11 is zero for normalized images, by setting p0 = 3, (8) becomes:

R00pq ¼ Rpqe�iq\R31 ð11Þ

As can be seen from Eq. (3), the scale changing of an image does not change the radial moment phase. Hence, by using\R00pq, the features that are invariant against rotation and scaling are obtained. The translation invariance can usually beachieved by using the central moments instead of the geometric ones:

lpq ¼XN�1

x¼0

XN�1

y¼0

ðx� �xÞpðy� �yÞqf ðx; yÞ ð12Þ

where �x ¼ M10=M00 and �y ¼ M01=M00. Central moments are actually the moments which are invariant against image trans-lation. However, in the proposed method pixel distances from the image origin are used instead of the coordinate distancesand therefore, translation normalization can be avoided. As we know, normalization is a source of interpolation and numer-ical errors. Therefore, the proposed method is invariant against rotation and scaling and translation changes and moreover,the computational accuracy will be high.

3. The proposed method

Fig. 2 shows the diagram of the proposed watermark embedding scheme, which denotes embedding the watermark in\R00pq. To set the phase of the invariant phase-radial moment to the desired values, the values of specified pixels are changed.The approach to computing the phase of radial moments and changing the pixel values will be described in the followingsubsections.

3.1. Computation of the phase of radial moments

In the proposed method, the phase of extracted features explained in Section 2 will be used for watermark embedding.For this purpose, a method which is called here as ‘‘partial computation of moments (PCM)’’ will be used. This methodcomputes the radial moments by using geometric moments. Experimental results show that R42 results in the best invariant

Watermarked image

Combining the partitions

Changing the pixels of each partition to have

new phase

Changing the phase using interval modulation according

to the watermark bits

Input image

Image partitioning

Partial calculation of radial features

Finding the phase of features

Watermark

Fig. 2. Schematic diagram of watermark embedding.

98 S. Golabi et al. / Information Sciences 269 (2014) 94–105

feature and therefore, according to Eq. (8), we choose p = 4 and q = 2 as shown in Eq. (13). The radial moments are computedas shown in Fig. 3.

R0042 ¼ R42e�2i\R31 ð13Þ

As can be seen in Fig. 3, the image matrix is divided into two matrices. In these matrices, gray blocks represent the zeropixels and the white ones show the gray scale pixels. The goal of this conversion is computing, separately, the geometricmoments for four pixels which are symmetrical and have the same radial distance from the origin.

This figure shows that the geometric moment Mpq for the original matrix is the sum of Mpq of matrix 1 and Mpq of matrix 2.Notice that Fig. 3 is only an example and there are various options for matrix partitioning; however, the pixels of one of them haveto be symmetric to the origin point. The desired watermark bits will be embedded into the phase of features. From (13) we have:

\R0042 ¼ \R42 � 2\R31 ð14Þ

And from Eq. (5) we can have:

R42 ¼ ðM40 �M04Þ þ 2iðM31 þM13Þ; R31 ¼ ðM30 þM12Þ þ iðM21 þM03Þ ð15Þ

Now, for matrix 2, the phase computation for R0042 leads to:

\R0042 ¼ tg�1 2ðM31 þM13ÞM40 �M04

� �� 2tg�1 M21 þM03

M30 þM12

� �ð16Þ

For more simplicity, the following notations are defined:

ðM40 �M04Þ � A1; ðM31 þM13Þ � A2ðM30 þM12Þ � B1 ðM21 þM03Þ � B2

ð17Þ

To compute the geometric moments in matrix 1, the method presented by Hosny is used [8]. In this method as the pixelshave the same radial distance, for the computing of Mpq from Eq. (2) the numerical value of xpyq depends on the values of pand q, whether they are even or odd.

Based on the results obtained in [8], the geometric moments can then be evaluated according to the following four cases:Case 1: p and q are both even

Mpq ¼ bpqðf ð1;1Þ þ f ð1;2Þ þ f ð2;1Þ þ f ð2;2ÞÞ ð18Þ

Case 2: p is even and q is odd

Mpq ¼ bpqðf ð1;1Þ � f ð1;2Þ þ f ð2;1Þ � f ð2;2ÞÞ ð19Þ

Case 3: p is odd and q is even

Mpq ¼ bpqð�f ð1;1Þ � f ð1;2Þ þ f ð2;1Þ þ f ð2;2ÞÞ ð20Þ

Case 4: p and q are both odd

Mpq ¼ bpqð�f ð1;1Þ þ f ð1;2Þ þ f ð2;1Þ � f ð2;2ÞÞ ð21Þ

where f(i, j) is the intensity function of the pixel point (i, j), and bpq is a coefficient that depends on the distance of the pixelsfrom the origin. Since the pixels have the same radial distance from the origin, we have bpq = bqp. Before these calculations,the original image defined in the square 512 � 512 is mapped to be inside the unit circle, where the coordinate origin is thecenter of the circle. As a result, the value of bpq is too small.

f(2,1)

f(1,2)

= +

matrix1 matrix2

f(2,2)

f(1,1)

Initial matrix

Fig. 3. Block partitioning for partial calculation of moments.

S. Golabi et al. / Information Sciences 269 (2014) 94–105 99

Therefore, in matrix1 we have:

\R0042 ¼ tg�1 2ð�f ð1;1Þ þ f ð1;2Þ þ f ð2;1Þ � f ð2;2ÞÞ0

� �� tg�1 f ð1;1Þ � f ð1;2Þ þ f ð2;1Þ � f ð2;2Þ

�f ð1;1Þ � f ð1;2Þ þ f ð2;1Þ þ f ð2;2Þ

� �ð22Þ

Eq. (22) can be expanded to m �m matrix shown in Fig. 4 as:

\R0042 ¼ tg�1 2ð�f ð1;1Þ þ f ð1;mÞ þ f ðm;1Þ � f ðm;mÞÞ0

� �� 2tg�1 f ð1;1Þ � f ð1;mÞ þ f ðm;1Þ � f ðm;mÞ

�f ð1;1Þ � f ð1;mÞ þ f ðm;1Þ þ f ðm;mÞ

� �ð23Þ

Finally, the phase of R0042 is computed as:

\R0042 ¼ \fA1þ i2ðA2þ b1½�f ð1;1Þ þ f ð1;mÞ þ f ðm;1Þ � f ðm;mÞ�Þg � 2\fB1þ b2½�f ð1;1Þ � f ð1;mÞ þ f ðm;1Þþ f ðm;mÞ� þ iðB2þ b2½f ð1;1Þ � f ð1;mÞ þ f ðm;1Þ � f ðm;mÞ�Þg ð24Þ

where b1 = b31 + b13 and b2 = b21 + b03.

3.2. Watermark embedding

In order to partition the original image for the calculating of the phase of radial moments, we generate a mask image thathas the same size as the original image with a 508 � 508 matrix of 0’s inside and the pixel values of the original image onfour sides in two rows and two columns corresponding to original image.

At first we consider a 4 � 4 matrix using the corresponding pixels of original image around the origin of the mask imageand embed a watermark bit by changing the radial feature of this matrix. The size of the inner matrix is enlarged by addingtwo columns and two rows on both sides and construct a 8 � 8 matrix to embed the next bit. At each step, we add twocolumns and two rows on each side of the previous inner matrix and construct a new one to embed the next watermarkbit into it. This process is continued to the final matrix with size 508 � 508. Steps of partitioning the host image are shownin Fig. 5.

Using the phase for watermark embedding is the same as phase modulation. For some reasons, such as using summationinstead of integration and the created interpolation after image normalization, the values and features mentioned above willaccompany some errors. Consequently, in this paper we use a particular type of phase modulation called interval phase mod-ulation (IPM) for more robustness against numerical errors. The phase partitioning diagram is illustrated in Fig. 6. In this fig-ure, it can be observed that the phase of R0042 is first computed. Then the 0–360� range is divided into 30� intervals.

Corresponding to the embedding bit, the angle of R0042, will change at its own interval as below:

� Embedding bit = 0

\R0042new¼

\R0042old30iþ 5 6 \R0042old

6 30iþ 10 i ¼ f0;1; . . . ;11g30iþ 7:5� otherwise

(ð25Þ

� Embedding bit = 1(

\R0042new

¼\R0042old

30iþ 20 6 \R0042old6 30iþ 25 i ¼ f0;1; . . . ;11g

30iþ 22:5� otherwiseð26Þ

In the embedding process, according to Fig. 5 and Eq. (24), the value of each symmetric pixel can be modified to changethe phase of \R0042. This can be considered as a key to embedding and extracting the watermark bits. In this paper, the value off(1,1) and f(m,1) are changed. In Eq. (24), it can be seen that k=2 changes in f(1,1) and f(m,1) only changes the last part of theabove equation by b2k. As mentioned before, the value of b2 is too small and therefore a great change needs to be made inpixel values to have a specified change in feature angle. This leads to great damage in image quality. In order to avoid this, weadd b2k=2 instead of k=2, which has a small value and its effect on image quality becomes much weaker. For a desired phaseof R0042, b2k is given by:

+

f(1,m)

=

matrix2

f(1,1)

f(m,1)

matrix1

f(m,m)

Fig. 4. Block partitioning for the partial calculation of moments in m �m matrix.

…

Fig. 5. Partitioning the original image. Gray color shows zero pixels and white ones represent original gray scale pixels.

Fig. 6. Interval phase modulation diagram.

100 S. Golabi et al. / Information Sciences 269 (2014) 94–105

b2k ¼ tg\R42 � \R0042

2

� �B01� B02

� �� �ð27Þ

with

B01 ¼ B1þ b2½�f ð1;1Þ � f ð1;mÞ þ f ðm;1Þ þ f ðm;mÞ�B02 ¼ B2þ b2½f ð1;1Þ � f ð1;mÞ þ f ðm;1Þ � f ðm;mÞ�

The watermark embedding procedure:

(1) Partition the original image by using the generated mask image as mentioned in this section.(2) At the first step of partitioning (4 � 4 matrix), calculate \R0042 according to Eq. (24).(3) Examine the computed phase at the 30� intervals according to the embedding bit as mentioned in Eqs. (25) and (26)

and change this phase.(4) Find b2k according to the changes in phase and Eq. (27).(5) Change the value of pixels f(1,1) and f(4,1) by b2k=2.(6) Perform steps (1)–(5) for partitioning at the next level (8 � 8 matrix) to the last level (508 � 508 matrix) to embed 127

bits.

3.3. Extraction process

The extraction process is formulated as follows:

(1) Construct a 512 � 512 binary matrix in which the first and the last two rows and columns are 1 and put a matrix of 1sbased on the steps of the embedding process (Fig. 5) at its center. This matrix is shown in Fig. 7a.

(2) Compute the rotation angle h based on the method which is shown in Fig. 7c and estimate the scaling factor s by com-paring the size of the watermarked image and rotated image.

111

1

2.5

2.5

y

x

x

ytgrotation 1)( −=

Rotated image

a b cFig. 7. Matrices of extraction process. White color represents 1 and gray color shows 0.

S. Golabi et al. / Information Sciences 269 (2014) 94–105 101

(3) Rotate and scale the constructed binary matrix by h and s respectively and multiplied the watermarked image by it, tohave the masked watermarked image.

(4) Construct a new 512 � 512 zero matrix but with different pixel values as shown in Fig. 7b and compute b2 by com-puting M21 + M03 for this matrix. Rotate and scale it by the angle h and the factor s.

(5) Compare this rotated and scaled matrix with the result of Step (3) to find the location of f(1,1) and f(m,1) at each step.The floating part of f(1,1) and f(m,1) is b2k=2.

(6) Compute the value of k based on the result of steps (4) and (5).(7) Add k to the pixel values f(1,1) and f(m,1) of the masked watermarked image and compute \R0042 of this matrix.(8) Extract the watermark bit as:

watermark bit ¼0 30i < \R0042 6 30iþ 15 i ¼ f0;1; . . . ;11g1 30iþ 15 < \R0042 6 30iþ 30 i ¼ f0;1; . . . ;11g

(ð28Þ

A schematic diagram of this process is illustrated in Fig. 8.As can be seen from (24), \R42 remained unchanged by the proposed method and therefore, it was used directly to com-

pute b2k: It is clear that various signal attacks change the pixel values and thus the angle and amplitude of moments will bechanged. However, these attacks affect the phase of moments much lower than their amplitudes. Also, these changes can bereduced more using the proposed embedding method.

4. Implementation results

In this section, the performance of the proposed method against geometric transformation has been illustrated and eval-uated. The proposed algorithm has been tested on different types of images of size 512 � 512, namely Lena, Baboon and Pep-per as shown in Fig. 9(a1, b1, c1). A pseudorandom sequence of size 127-bits is used as the watermark pattern. Theperformance of the proposed watermarking scheme is evaluated in two aspects: imperceptibility and robustness. The resultsshow that the proposed scheme can satisfy the requirements of both imperceptibility and robustness.

The original images and the watermarked images (Lena, Baboon and Peppers) obtained by the proposed algorithm areshown in Fig. 9(a1, b1, c1) and (a2, b2, c2), respectively. The differences between the original images and the watermarkedversions are magnified by a factor 10,000 and are shown in Fig. 9(a3, b3, c3). Based on (27), it is clear that the proposed

Watermarked image

Image partitioning

Partial calculation of radial features

Finding the phase of features

Finding the interval which include the phase

If the phase is greater than interval means?Watermark bit is 0

No

Watermark bit is 1

Yes

Fig. 8. Schematic diagram of watermark extraction.

a3 a2 a1

b1 b2 b3

c1 c2 c3 Fig. 9. Originl and watermarked images, (a1) original image of Lena, (a2) watermarked image of Lena with watermark of size 127 bits, (a3) difference imagebetween (a1) and (a2). (b1) original image of baboon, (b2) watermarked image of baboon with watermark of size 127 bits, (b3) difference imagebetween (b1) and (b2). (c1) original image of pepper, (c2) watermarked image of pepper with watermark of size 127 bits, (c3) difference image between (c1)and (c2).

102 S. Golabi et al. / Information Sciences 269 (2014) 94–105

method changes 127 pixel values of original image, less than 1. If intensities of 127 pixels of the original image are changedby 1, PSNR will be greater than 80 dB. So the PSNR of watermarked image will be greater than this.

After extracting the watermark, the bit error rate (BER) is computed using the original watermark and the extractedwatermark to evaluate the correctness of an extracted watermark. Bit Error Rate for the embedded and extracted watermarksequences of length N bits is listed as:

BERðW;W 0Þ ¼ 1N

XN

i¼1

jWðiÞ �W 0ðiÞj ð29Þ

where W and W 0 are the original watermark and the extracted watermark sequences.

S. Golabi et al. / Information Sciences 269 (2014) 94–105 103

In order to demonstrate, the robustness of the proposed watermarking algorithm, the watermarked image is attacked bysome geometric attacks. After these attacks on the watermarked image the extracted watermark sequence is compared withthe original one. In the following sub-sections, two attacks are examined: rotation and scaling. Our method shows good per-formance against these geometric attacks.

4.1. Rotation

Table 1 illustrates the robustness of the proposed method against the rotation attack for various degree rotations in termsof BER. As can be seen, this method has a good performance by various rotations especially in small angels.

4.2. Scaling

There are two types of image scaling, increasing and decreasing the size of image. For both types the scaling factor isexamined and the watermark is extracted as described in the extraction sub-section. Table 2 shows the results for variousscaling factors from 60% scaling down to 160% scaling up in terms of BER. As the results illustrate, the watermarked imagesshow strong resistance to the scaling attack.

4.3. Rotation with scaling

Fig. 10(a1 and a2), both are the examples of the rotation of the watermarked image in Fig. 9(a2) by 34�, one with croppingand the other without cropping. Experimental results show that the similarity measure for the cropping case is low. There-fore, as expected, for Fig. 10(a1) the proposed method does not have a good performance, but for Fig. 10(a2), it is more effi-cient. Fig. 10(a2) is rotation of the watermarked image by 34�, but to show all pixels, we apply image resizing by factor 1.39.Thus, rotation without cropping will use scaling itself. However, in general, another scaling can be applied to the rotatedwatermarked image. In this case, based on what has been mentioned in the extraction sub-section, the embedding locations

Table 1BER of the proposed method for various rotation angles for test images.

Rotation angle 0� 1� 2� 5� 8� 15� 30� 45� 80� 140� 170� 180�

Lena 0 0 0 0 0 0.05 0 0.37 0.007 0.16 0.32 0.31Baboon 0 0 0 0 0.007 0.039 0 0.38 0.33 0.025 0 0Pepper 0 0 0 0 0.007 0.039 0.078 0.36 0.047 0.071 0.023 0

Table 2BER of the proposed method for various scaling factors for test images.

Scaling factor 0.6 0.7 0.9 1.1 1.6

Lena 0.41 0.37 0.06 0.39 0.37Baboon 0.44 0.38 0.46 0.39 0.38Pepper 0.37 0.40 0.15 0.31 0.18

Fig. 10. (a1) Image of Lena after 34 degree rotation and with cropping, (a2) 34 degree rotation with scaling of Lena image without cropping.

Table 3BER of the proposed method for various geometric attacks on various images.

Image Attack

1� rotation 5� rotation 1� rotation and 2 scaling 1� rotation and 5 scaling and shift the origin to (2, 3)

Lena 0 0 0.11 0.13Baboon 0 0 0.09 0.12Pepper 0 0 0.07 0.1

Table 4Comparison between robustness of the proposed method and the method of [17] (BER).

Attack The proposed method Method of [16]

1� rotation with croppingLena 0.45 0.62Baboon 0.44 0.72Pepper 0.42 0.5

5� rotation with croppingLena 0.48 1Baboon 0.49 1Pepper 0.47 1

1.1 scalingLena 0.39 0.5Baboon 0.39 0.54Pepper 0.31 0.75

Table 5Comparison between robustness of the proposed method and the method of [6] (BER).

Attack The proposed method Method of [6]

30� rotationLena 0 0.60Baboon 0 0.73Pepper 0.078 0.69

5� rotationLena 0.18 0.60Baboon 0.35 0.66Pepper 0.16 0.61

1.5 scalingLena 0.39 0.40Baboon 0.39 0.53Pepper 0.31 0.44Lena 0.06 0.60

[0.7 0.9] scalingBaboon 0.46 0.86Pepper 0.15 0.73

104 S. Golabi et al. / Information Sciences 269 (2014) 94–105

are found and the watermark sequence is extracted. In Table 3, the robustness of the proposed method against various geo-metric attacks is evaluated. Note that, all of the practical implementations are performed by the nearest interpolation.

In each case, the bit error rate criterion (BER) has been used to evaluate the correction of the extracted bits. The resultsindicate the high performance of the proposed method.

Tables 4 and 5 show the robustness of the proposed method for various geometric attacks, as compared to other pub-lished results in [17,6], respectively. As papers [17,6] have used the attacks mentioned in Tables 4 and 5, these algorithmshave not been simulated and the comparison has been made based on these attacks. As can be seen, the proposed methodprovides better performance. Also, the comparison with the method described in [4] shows that the robustness results of theproposed method for some geometric attacks are close to the results of the method of [4], but the quality and the capacity ofthe proposed algorithm are higher than [4].

5. Conclusion

A blind and robust watermarking scheme was presented. This scheme is robust against geometric attacks and uses thefeatures of radial moments. At first, the invariant phase radial moments were computed, and the watermark was embedded

S. Golabi et al. / Information Sciences 269 (2014) 94–105 105

into the differential phase of various blocks of the original image by use of a special type of phase modulation called intervalphase modulation. In order to embed the watermark into the phase of moments, a particular method called partial calcula-tion of moments has been used. The implementation results show the improvement in the robustness and quality of water-marked images in comparison with other methods. This method has some advantages in comparison with other commonmethods: (1) This scheme does not use any kind of normalization, leading to the elimination of interpolation errors whichoccur during RST (Rotation, Scale, Transition) normalization. (2) The proposed method has the ability to detect and extractthe watermark and since the interval modulation is used, the watermarking scheme has high robustness. (3) This algorithmcan be adopted according to the user necessity. In order to achieve more robustness or quality larger or smaller intervals areused. (4) This method has the simple implementation and the ability to apply on various image formats. (5) The extractionprocess does not need any preprocessing and the watermark bits are extracted directly from the attacked image. This prop-erty increases the accuracy of the extraction process and decreases the computational errors.

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