INTRODUCTION
In recent years, the common users have become extremely easy to produce good quality forgery images with the development of the powerful digital photoediting tools. While these editing tools and technologies have achieved many exciting advances in art and science but also led to some serious social, legal and scientific issues. There are many image forgery issues had been ever happened in the world. The first image forgery issue happened in 1860, forgers placed Lincoln’s head in Senator John Calhoun’s body in a image aim to obtain desired political benefits. Original image is shown in Fig. 1a, forged image is shown in Fig. 1b. If these issues used in scientific discovery, court evidence, news reports and other occasions will have the potential negative impact on society. So, proceed with digital image forgery authentication research will be of great significance.
There are many methods for image forgery. Farid (2004) classified image forgery authentication schemes into six representative categories: Enhanced, composited, morphed, retouched, painted, computer generated. The image composited operation is also called as the copymove forgery.

Fig. 1(ab): 
(a) Original image and (b) Forged image 
It is the most common digital image forgery (Nguyen and Katzenbeisser, 2012; Hsu and Chang, 2007; Kumar et al., 2011) in which a part of a digital image is copied and pasted to another part in the same image in order to cover an important image feature (Bayram et al., 2009). In the actual application, it is likely to be subjected to rotation, scaling, or additive noise for better covering the forgery marks and achieving the blended purposes. In the past decades, many approaches about copymove blind forgery are proposed. Fridrich et al. (2003) extracted Discrete Cosine Transform (DCT) coefficients which are robust to lowpass filtering and compression. Popescu and Farid (2004) proposed to apply Principal Components Analysis (PCA) to obtain compact representations for the blocks. And then they (Popescu and Farid, 2005) employed ExpectationMaximization approach to expose resample traces. Zhao (2012) proposed an approach based on PCA of projection feature. As tampered image is divided into many overlapped small image blocks. The horizontal projection and vertical projection of each image block regained to build up a matrix. And the experiment shows that the computation of this approach is more rapid than Posucue approach. Li et al. (2007) proposed a blind forensics approach based on Discrete Wavelet Transform (DWT) and Singular Value Decomposition (SVD) to detect the specific artifact. Since the robustness of these previously blind authentication approaches against noise and geometric deformation especial antirotation and antiscaling is poor. Qian et al. (2013) proposed a rotation angle estimation approach. It based on pixel variance periodicity introduced by interpolation and established a mapping between image rotation angle and the periodicity of pixel variance. But the approach is robust for rotation angles between 5° and 45°. Hence, it must be consider a novel approach to detect a good accuracy for a large angle rotation and scaling.
Hu (1962) proposed a theory of TwoDimensional (2D) moment invariants for planar geometric figures. His fundamental theorem method is wellknown as HU moments. Hu (1962) used geometric figures linear transforms to construct classic invariant moments containing translation, rotation and scale etc. It is laid the foundation for the image moment invariant theory. Hu (1962) moment invariants are no scaling moment invariants. Later, people found that many improved moment invariants. The representative approaches have the Zernike moments (Liao and Pawlak, 1998), Legendre moments (Yap and Paramesran, 2005), Tchebichef moments (Mukundan et al., 2001), Krawtchouk moments (Yap et al., 2003) and orthogonal FourierMellin moments (Derrode and Ghorbel, 2001). Orthogonal moment invariant is superiority to nonorthogonal moment invariant. But the problem of the orthogonal moment invariant for image geometric moment invariant analysis is lack of essentially scaling invariance (Teh and Chin, 1988). So some improved approaches are proposed. Huang et al. (2008) proposed an approach to detect copymove forgery in digital images by extracting Scale Invariant Feature Transform (SIFT) descriptors of an image which are invariant to changes in illumination, rotation, scaling etc. Owing to the similarity between copied region and pasted region, descriptors are matched between each other to seek for any possible forgery in images. Chihaoui et al. (2014) proposed an approach uses the Scale Invariant Feature Transform (SIFT) and uses the Singular Value Decomposition (SVD) by matching between identical features. He and Wang (2010) proposed a approach use the radon transform to construct moment invariants. Lei et al. (2011) proposed a novel robust hashing approach for image authentication based on Radon and Discrete Fourier Transform (DFT). Kakar and Sudha, (2012) proposed an authentication approach based on transforminvariant features. But the authentication approach is limited by modifying the MPEG7 image signature tools descriptors. Yap et al. (2010) proposed an approach which is called Polar Harmonic Transforms (PHTs), namely, Polar Complex Exponential Transform (PCET), Polar Cosine Transform (PCT) and Polar Sine Transform (PST). These a set of transforms can be used to generate rotation moment invariant. The experiments conducted shown that unlike the wellknown Zernike and pseudoZernike moments, the kernel computation of PHTs is extremely simple and has no numerical stability issue. This implies that PHTs encompass the orthogonality and invariance advantages of Zernike and pseudoZernike moments but are free from their inherent limitations. The problem of this approach is also lack of scaling invariants. Wang et al. (2009) proposed rotational invariance based on Fourier analysis in polar and spherical coordinates. Zhang et al. (2010) proposed a general approach to construct a complete set of Orthogonal FourierMellin Moment (OFMM) invariants. By establishing a relationship between the OFMMs of the original image and those of the image having the same shape but distinct orientation and scale, a complete set of scaling and rotation invariants is derived. Sheng and Shen (1994) proved that antirotation, antiscaling and orthogonality performance of the Fouriermellinmomentinvariants are better than Zemike and Hu moments. And the feature description ability, especially the description of the small image performance is also better. Papakostas et al. (2007) pointed out OFMMS can be widely applied in many fields such as image blind authentication. since the radial polynomials used in OFMMs have much more zeros than those of the other orthogonal moments, the OFMMs have the capability to describe high spatial frequency components of an image.
In general, one of the challenge here is the geometric moment invariants applied to detect the image forgery, especially applied to the copymove blind image authentication are less. Another challenge is to find the robust representations for the approach, that the copymove blocks can be detected more accuracy under modifications. In view of this, we focus on these two aspects and propose an approach to extract RadonPseudoFourierMellinMomentInvariants (RPFMMIs) features from the image blocks by using RadonPseudoFourierMellinTransform (RPFMT). These features would be not only applied to robust to Gaussian noise addition but also applied to rotation and scaling moment invariants.
AUTHENTICATION APPROACH BASED ON RADONPSEUDOFOURIERMELLINTRANSFORM
Radon and FourierMellin transforms are the fundamental tools used in the proposed approach. In this section we first briefly review these transforms and their properties in order to establish the proper ties of the proposed approach.
Radontransform: Mathematician Radon (1917) proposed Linear Radon transform. The basic idea of radon transform is the duality of point and line. It has a good antinoise performance, can detect the image translation, rotation, scaling and other geometric transform.
In mathematics, the classical Radon transform in TwoDimensions (2D) is the integral transform consisting of the integral of a function over straight lines (Galigekere et al., 2000; Beylkin, 1987; JafariKhouzani and SoltanianZadeh, 2005a) Radon transform reflects the relationship between the image and its projection. Now supposed a 2D function f(x, y)εL^{2} (D), its Radon transform R(r, θ) is a function defined on the space of straight lines L in 2D plane by the line integral along each such line. The Radon transform of a 2D function f_{D}(x, y) is defined as:
where, δ is a Dirac Delta function. The value of the r on a given straight line is the integral of f_{D}(x, y) along this line and θ is the angle between the line and the yaxis (Beylkin, 1987; JafariKhouzani and SoltanianZadeh, 2005a, b). Radon transform is given in Fig. 2. Due to the inherent properties of the Radon transform, it is a useful tool to detect linear trends in images and capture the directional information of the images.

Fig. 2:  Radon transform of f_{D}(x, y) 
From Eq. 1, the discrete Radon transform equation is defined as:
where, M and N are the dimensions of rows and columns in the image matrix.
From Eq. 1, R (r, θ) is the integral of f_{D}(x, y) along this line L:
Suppose r_{1}, r_{2},…,r_{n} is the perpendicular distance of a line from the origin and projection angle θ_{1}, θ_{2},…,θ_{m} is the angle formed corresponds to r_{1}, r_{2},…,r_{n} by the distance vector. The R(r_{i}, θ_{k}) represents the Radon transform of point (r_{i}, θ_{k}). Radon transform matrix (Gan and He, 2011) is defined in Eq. 4:
The normal form of the affine transform is defined as:
where, is a Det.
The translation, scaling and rotation expressions of the Radon transform (Hu, 1962) is given as:
where, λ is scaling factor, i = 1,2,3…., x_{i}^{;} and y^{;}_{i} are new coordinates after the Radon transform.
Rotation feature of Radon transform: From Eq. 6, it can be write that:
where, g_{D}(x’, y’) is f_{D}(x, y) rotate to a new coordinate.
Suppose:
x’ = x cosφ+y sinn, y’ = x sinn+y cosφ
where, dx’ = cosφ dx, dy’ = cosφ d.
In both Eq. 1, 6 and 7, rotation feature of Radon transform is defined as:
Scaling feature of Radon transform: From Eq. 6, the scaling factor is simplified as:
Suppose λ>0 then:
where, g_{D} (x, y) is a new scaling function of fD (x’, y’).
From Eq. 1 and 10, scaling feature of Radon transform is defined as:
The Radon transform of g_{D}(x, y) by rotation and scaling is defined as:
Radonpseudofouriermellintransform: The radial moments of order r with repetition of image intensity function are defined as D(s, k). For detecting the forgery image, Wang et al. (2008), Jing et al. (2010), Liu et al. (2011) and Derrode and Ghorbel (2001) introduced a set of FourierMellinTransform moments. In TwoDimension (2D) polar coordinates, D(s, k) is the RadonFourierMellinTransform of a 2D function g(s, k) with order s^{1} and it can be defined as:
where, s is complex number, e^{ikθ} is complex exponential, k = 0, ±1, ±2 is order of circular harmonic function. To simplify the Eq. 13, further improvement are suggested in Eq. 14 by using real number u to display complex numbers. Equation 14 is defined as:
where, k = 0, ±1, ±2,…, u = 1, 2, 3… are real numbers and the priors k and u are supposed to be equal to 2 in this study.
Construct RadonPseudoFourierMellinTransform moment invariants: From Eq. 12 and 14, it can be supposed that:
The rotation and scaling of RadonPseudoFourierMellinTransform moment invariant is defined as:
where, D_{f}(u, k) can be transformed from R_{f}(u, k) by using RadonPseudoFouriermellinTransform (RPFMT). D_{g}(u, k) is RadonPseudoFourierMellinTransform of R_{f}(u, k). From Eq. 15, a function R_{f}(u, k) can be transformed from a scaling factor to magnitude factor and transformed from a rotate factor to a phase factor by using RPFMT approach. From Eq. 15, we also can construct scaling and rotation moment invariants of Z_{f}(u, k) by using RPFMT approach:
Given a RPFMT moment invariant function Z(u, k), we can construct its mean, standard deviation, skewness, kurtosis. Equation 1722 are defined as following forms:
where, Eq. 17 is Z(u, k) mean value, Eq. 18 is magnitude, Eq. 19 is absolute value of magnitude, Eq. 20 is standard deviation, Eq. 21 is skewness and Eq. 22 is kurtosis.
These six eigenvalues with scaling and rotation moment invariants can be detected scaling and rotation tampered for copymove image. Hence, we can detect the overlapping square blocks of image and extract six eigenvalues of each pixel from these overlapping blocks to construct feature matrix. ThreeDimension (3D) feature matrix of image can be defined as:
The size of overlappingblock image matrix is dxd. In order to simplify the feature matrix and apply lexicographic sorting to detect forgery image, Eq. 23 can be divided to six categories 2D feature matrix and improved the authentication accuracy by multiply authentication. Equation 24 is defined as:
where, H_{f1}[d, d] is a 2D matrix for function f_{1} in Eq. 17.
MATERIALS AND METHODS
Proposed technique approach steps: In this study, our approach is applied to detect copymove forgery image. If the target image is RGB image, we use standard color space conversion to converted it to grayscale image firstly. Then the grayscale image is divided into overlapping blocks. The overlapping blocks is projected to construct planar projective transform matrixes from projection space by Radon transform, extract the overlapping blocks scaling and rotation feature to construct moment invariants for the next step. Based on the above scaling and rotation feature, a RadonPseudoFourierMellinTransform approach was proposed to construct moment invariants feature matrixes and then eigenvalues are extracted. Lexicographic Sorting is applied to matrices and finally it is judged, detected and located forgery region based on the confidence distance whether the wellordered adjacent images are copymove image blocks.
The experimental results show that the proposed approach has a good robustness and it can detect and authenticate types of a large angle rotation and scaling transform to copymove images. The approach has more accuracy compared above approaches.
The detailed steps of the proposed approach are described as follows:
Step 1: 
Using standard color space conversion transform original RGB image into grayscale image S which image size is MxN 
Step 2: 
Divide S into fixedsized overlapping square blocks which block size is dxd. The number of overlapping blocks is L= [Md+1] x[Nd+1] 
Step 3: 
Using RadonPseudoFourierMellinTransforms approach to construct 3D moment invariants matrix from Eq. 23 or six 2D moment invariants matrixes from Eq. 24 for each overlapping block and extract its eigenvalues. Using 3D feature matrix of each overlappingblock to construct 4D feature matrix of S and extract its eigenvalues 
Step 4: 
Each 3D matrix of block is also consisted of six such 2D matrixes from Eq. 24 which is suitable for Lexicographic matrix. Form a row vector matrix K using the feature vectors, where i is the row vector number of a overlapping block. The total number of row vectors is L. The obtained matrix K has a dimension of LxH, where, H is the number of elements in each row vector 
Step 5: 
Lexicographic matrix K in a dictionary sequence. Denote the sorted matrix as K1 which is the same dimension as K. K1i is the i th row vector of K1. Record the overlapping block in K and K1i with (Xi, Yi) as its top left corner coordinates 
Step 6: 
Conduct the matching authentication of similar blocks and identify the forgery position. During the window movement, when the matching function value is the maximum or over a level, the center corresponding to the window will be the position of matching similar point 
Step 7: 
Repeat the 46 steps six times by using Eq. 24 and detect, locate, identify the forgery region more accuracy 
RESULTS AND DISCUSSION
In the present experiments, all experiments are operated on the computers architecture with Intel (R) Core™2 Duo CPU T8300 at 2.4 GHz, 2 GB RAM and Windows XP system. Image overlapping blocks are square overlapping block of 16x16. MATLAB 7.0 processing software is applied to realize the approach in this study and other relevant approaches.
In this experiment, the performance of the proposed approach was analyzed in the presence of forgery images which have scaled and rotated copymove region. To further demonstrate the benefit of RadonpseudoFourierMellinTransform approach in this study, an image named “HelloKitycake” was forged whose size is 1024x768 as an example. This image region was tampered with the translation, ±10% scaling, 30°, 60°, 90° clockwise rotation and additive Gaussian noise by using PhotoShop software. The detection results were presented in antitranslation, antiscaling, antirotation and antiGaussian Noise and then the accuracy rates (percents) for the authentication of copymove forgery image by using RPFMT were also presented. In the last experiment, the performance of the approaches in study were also compared (Popescu and Farid, 2004; Qian et al., 2013; Yap et al., 2010) with the present proposed approach.
Authentication in translation tampered testing: The authentication results for tampered “HelloKityCake” images are shown in Fig. 3. Figure 3a is the original image without tampered operation, the copied region translated in Fig. 3b and 3c is the authentication with the present approach.
Authentication in scaling tampered testing: The authentication results for tampered “HelloKityCake” images are shown in Fig. 4. Figure 4a is the copied region scaled 10% and Fig. 4c is the authentication with the proposed approach.
The authentication results for tampered “HelloKityCake” images are shown in Fig. 5. Figure 5a is copied region scaled 10% and Fig. 5c is the authentication with the present proposed approach.

Fig. 3(ac): 
Authentication for translation tampered (a) Original image, (b) Copymove
forgery image and (c) Detected image 

Fig. 4(ab): 
Authentication for 10% scaling tampered, (a) Copymove forgery image and (b)
Detected image 

Fig. 5(ab): 
Authentication for 10% scaling tampered, (a) Copymove forgery image and
(b) Detected image 
Authentication in rotation tampered testing: The authentication results for tampered “HelloKityCake”images are shown in Fig. 6. Figure 6a is the copied region rotated 30° clockwise and Fig. 6b is the authentication with the proposed approach.
The authentication results for tampered “HelloKityCake” images are shown in Fig. 7. Figure 7a is the copied region rotated 60° clockwise and Fig. 7b is the authentication with the proposed approach.
The authentication results for tampered “HelloKityCake” images are shown in Fig. 8. Figure 8a is the copied region rotated 90° clockwise and Fig. 8b is the authentication with the proposed approach.
Authentication in added Gaussian noise tampered testing: The authentication results for tampered “HelloKityCake” images are shown in Fig. 9. Figure 9a is the copied region added Gaussian noise (μ = 0.01, δ^{2} = 0.01) and Fig. 9b is the authentication with the present proposed approach.
The authentication results for tampered “HelloKityCake” images are shown in Fig. 10. Figure 10a is the copied region added Gaussian noise (μ = 0.05, δ_{2} = 0.1) and Fig. 10b is the authentication with the present proposed approach.
Performances of authentication approach: Hundred natural images were downloaded from the Internet used for experiment. Then images were tampered with the abovesaid manner.

Fig. 6(ab): 
Authentication for 30° clockwise rotation tampered, (a) Copymove forgery image and (b) Detected image 

Fig. 7(ab): 
Authentication for 60° clockwise rotation tampered, (a) Copymove forgery image and (b) Detected image 

Fig. 8(ab): 
Authentication for 90° clockwise rotation tampered, (a) Copymove forgery image and (b) Detected image 

Fig. 9(ab): 
Authentication for additive Gaussian noise tampered, (a) Copymove forgery image and (b) Detected image 

Fig. 10(ab): 
Authentication for additive Gaussian noise tampered, (a) Copymove forgery image and (b) Detected image 
We tampered the image region with the translation, ±10% scaling, 30, 60 and 90° clockwise rotation and additive Gaussian noise by using Ps. We used d = 16 as a block size and slide the blocks one pixel each time to detect image. It was supposed that the smallest size of duplication would be at least 32x32. In this case, if a image size is there would be (3216+1)x(3216+1) = 289 connected duplicated blocks. The above tampered image samples are tested with the approach in this paper and the testing results are showed in Table 1.
Table 1:  Authentication rates of the present proposed approach on 100 forgery images with copymove regions undergone different types of distortion 

Table 2:  Comparison for copymove forgery image authentication approaches 

Table 1 shows that the present proposed approach has relatively high authentication. The authentication rates is at least 80%, even if the copied region of tampered images is undergone translation, scaling, rotation and added Gaussian noise tampered.
Comparison for copymove authentication approaches: The approaches, described by Popescu and Farid (2004), Qian et al. (2013) and Yap et al. (2010) were implemented and the robustness of these approaches was also compared in Table 2.
From Table 2, the approach of Popescu and Farid (2004) has the poorer robustness for the copymove forgery detection. This approach only can detect translated forgery. The approach of study (Qian et al., 2013) is an improved approach and it has the fairly good robustness, can detect such as translation, some certain extent scaling, some small degrees rotated forgery region but can not detect a large degree rotated forgery region which rotated degrees is larger than 45°. The approach based on the RadonPseudoFourierMellinTransform in this study has more superiority than the compared approaches.
CONCLUSION
Copymove forgery is a very common type of image forgery approach and this approach belongs to blind authentication which is applied so widely. In this study, we study the problem of copymove blind forgery authentication, the image geometric structure and focus on its scaling and rotation invariant moments. To detect the forgeries under the modifications, we proposed the RadonPseudoFourierMellinTransform (RPFMT) approach which is invariant to scaling and rotation. The present experimental results show that we have achieved good results and we can detect copymove forgery very accurately even if the forged image is scaled ±10%, large angle rotated. And we also compared the robustness of our approach with the previously proposed approaches which use PCA and Periodicity within pixel variances, PHTs and we showed that our approach is more robust to various types of processing and the approach can be used in regions like copymove image forgery authentication. Our experiments for all schemes are going on for additive Gaussian noise. But our approach in paper uses the Lexicographic Sorting that make the authentication efficient low. For example, if the image is large size, such as size of MxN, The numbers of subblock are L = (Md+1) x(Nd+1), the numbers of image pixels are W = (Md+1)x(Nd+1 xdxd). The eigenvalues of 3D in each pixel may be calculated in Lexicographic Sorting approach that will result in computationally very expensive.