ROTATION, TRANSLATION AND SCALING INVARIANT WATERMARKING USING A GENERALIZED RADON TRANSFORMATION

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1 ROTATION, TRANSLATION AND SCALING INVARIANT WATERMARKING USING A GENERALIZED RADON TRANSFORMATION D. SIMITOPOULOS, A. OIKONOMOPOULOS, AND M. G. STRINTZIS Aristotle University of Thessaloniki Electrical and Computer Engineering Dept. Thessaloniki, Greece A watermarking scheme able to resist geometric attacks is presented in this paper. The watermark embedding and detection are performed in a domain which is invariant to geometric attacks such as rotation, scaling and translation. The invariant domain is derived by applying a new generalized Radon transform to the image. The ability of the proposed method to withstand geometric attacks is evaluated experimentally. 1. Introduction Digital image watermarking is a technology which has attracted many researchers in the past few years. A lot of watermarking techniques are laying emphasis on robustness against common digital image processing operations, such as compression or filtering. However it is clear that even the smallest geometric distortions can irreparably harm the detection process of a watermarking scheme due to loss of synchronization between the embedded and the correlating watermark. These geometric distortions usually include translation, rotation and scaling (RST) operations which can be easily applied on an image. Recently, many watermarking techniques have been proposed which are oriented towards robustness against geometric attacks. In some approaches an additional pattern is embedded in the image along with the watermark in order to be able to revert the geometric attack 1. There are also watermarking methods 2, where the embedding and detection processes are taking place in a domain which is invariant to geometric transformations. The watermarking technique presented in this paper is based on the latter approach for resisting geometric attacks. First, a corner detection scheme detects corners in the image content and finds the most robust among them. This corner is used as an origin for a new generalized Radon transform (GRT) 3 followed by a two dimensional Fourier Transform. Through this sequence of transformations 1

2 2 an invariant to RST attacks domain is created. The watermark is embedded in this domain, and then, by applying the inverse sequence of transformations we obtain the watermarked image. A correlation-based detection scheme is applied in order to detect the existence of the watermark. The detection is blind, i.e. the original image is not required. The detection is performed in the invariant domain in order to ensure that the embedded and the correlating watermark will be synchronized even in the case of geometric attacks. The resistance of the proposed scheme is demonstrated through experimental results. 2. RST invariance using the Spiral Integration Transform In order to achieve robustness to translation a Harris corner detector 4 is first applied to the image. In this way a set of corners in the image content is detected and the most robust among them is selected. This corner is used as an origin for the application of a new generalized Radon transform, the Spiral Integration Transform (SIT), which is applied in order to achieve robustness against rotation and scaling. If f(r, φ) is the image in a polar coordinate system, the SIT of f(r, φ) is the integral of f(r, φ) along the line (in a polar coordinate system) given by r = e a (φ ω), r [, + ). This line (see Fig.1a) in the polar coordinate system, corresponds to a spire (see Fig.1b) in a cartesian coordinate system. The SIT is given by: Figure 1. system. r S f (a, ω) = ω Integration path 2π e a = tan(z) (a) z φ f(r, φ)δ(r e a (φ ω))rdrdφ (1) (b) Most robust corner The integration path in (a) the polar coordinate system and (b) the cartesian coordinate Consider now the rotated by φ and scaled by b image g(r, φ) =f ( r b,φ φ ) written in a polar coordinate system. The SIT of this image will be: S g (a, ω) =b S f (a ln(b),ω φ ) (2)

3 3 Therefore, the SIT of the scaled and rotated image is translated by log(b) along the a axis and cyclically shifted by φ along the ω axis, and its magnitude is scaled by b. Since the DFT magnitude is translation invariant 2, using equation (2) it is easy to show that A g = b A f, where A f and A g are the magnitudes of the spectrum of S g and S f respectively. As can be seen, the magnitude of the SIT spectrum of the scaled and rotated image is only scaled by b. However, we need not be concerned with the magnitude scaling of the spectrum, since we intend to use a correlation-based detector that is invariant to scaling. Therefore, it is clear that the magnitude of the SIT spectrum is invariant to rotation and scaling. In addition, due to the use of the corner detection scheme for selecting the origin of SIT, invariance to image translation is achieved. 3. Embedding As shown in section 2 the magnitude of the SIT spectrum of an image f(x, y) is invariant to RST operations. For this reason, the watermark embedding and detection should take place in this domain. However, if the watermark is added to the magnitude of the SIT spectrum, then the inverse transformations (IDFT, inverse SIT and inverse polar mapping) should be applied to the watermarked spectrum in order to obtain the watermarked image in the spatial domain. In such a case the original image may be degraded by the transformations. For this reason, after the watermark is embedded in the magnitude of the SIT spectrum and IDFT is applied, we subtract the SIT of the original image from the SIT of the watermarked image. In this way a representation of the watermark in the SIT domain is obtained and after inverse SIT and inverse polar mapping are applied, the spatial representation of the watermark is additively embedded in the original image. This is depicted in Fig. 2 and is described in detail in the following. The watermark W is the product of a zero-mean random sequence of the values +1 and -1 and the constant embedding strength factor h. A secret key is used in order to produce a seed for the watermark sequence generation. In order to ensure that the IDFT of the watermark is real, the watermark values should comply to the following rule: W (k, n) =W ( k, n). For this reason, first the watermark is generated in the first and the fourth quadrant of the spectrum, and then the generated watermark is duplicated in the third and the second quadrant respectively. In addition, the area of the spectrum in which the watermark is embedded, is a ring-shaped area that covers the middle frequencies: {, if ρ < R1 and ρ > R W (ρ, θ) = 2 (3) ±1 if R 1 <ρ<r 2 where ρ = k 2 + n 2 and θ = arctan(n/k). This area was selected for two

4 4 User ID Watermark Magnitude + DFT Phase X IDFT Watermarked DFT - SIT SIT of the watermark Polar Mapping Inverse SIT Harris Corner Detector Original Image Inverse Polar Mapping + Watermarked Image Figure 2. Block diagram of the embedding process. reasons. Firstly, it was found that very high valued coefficients appear in the low frequencies of the SIT spectrum. After experimenting with various images, it was found that these outliers deteriorate the detector performance and for this reason embedding and detection in this area was avoided. In addition, the magnitude of the SIT spectrum in the high frequencies is generally small. In a significant number of these coefficients it is smaller than the embedding strength h and these coefficients may be zeroed after embedding. Therefore, the high frequencies were also not used for embedding and detection. After the watermark generation process is completed, the watermark coefficients W (k, n) are added to the SIT spectrum magnitude values A(k, n). Then IDFT is applied and the resulting SIT of the watermarked image is subtracted from the original SIT. The SIT of the watermark is obtained, which must be transferred to the spatial domain so that it can be added to the original image. As mentioned in section 2, each element of the SIT domain is the result of an integration along a line of the form r = e a (φ ω) in a polar coordinate system. In

5 5 order to invert the SIT of the watermark, each SIT coefficient is distributed along the pixels belonging to the corresponding line. This means that the value of each SIT coefficient is divided by the number of pixels that exist in each line and the resulting value is added to every pixel that belongs to the corresponding line. This leads to a representation of the watermark in the polar coordinate system. The final stage of the embedding process includes transformation of the watermark from the polar representation to the spatial domain and addition to the original image. 4. Detection For the detection of the watermark a correlation-based scheme is applied. The detection process can be divided into the following steps: First, the most robust corner of the test image is detected. Then, the SIT is applied to the test image. In the following, the DFT is applied to the SIT of the image and its magnitude is obtained. The symmetrical watermark W is created using the owner s secret ID as described in section 3. Finally, the correlation metric c between the values A(k, n) of the magnitude of the SIT spectrum, and the correlating watermark values W (k, n) is calculated: c = N 1 N 2 k= n= N 1 A(k, n) W (k, n) N 2 k= n= A(k, n) 2 (4) where N 1 and N 2 are the dimensions of the 2D-DFT image. The correlation metric c is compared to the threshold T, which is defined according to the allowed false alarm probability P FA of the detection scheme, as described in Experimental results The experiments presented in the following were performed using standard test images such as Lena, Baboon and Fishing boat. Linear interpolation was used for the SIT, the polar mapping and their inverses. Table 1. Correlation metric values under various attacks. Attacks Lena Baboon Fishing boat No attack Rotation by 45 degrees Downscaling by JPEG Q= Median filtering 3x Gaussian Noise N(,1)

6 6 Table 1 presents the correlation metric value for both the original and the attacked images. As can be seen, apart from RST attacks, filtering, noise addition and compression attacks were also tested. Furthermore, the values of the correlation metric for the watermarked image Lena (see Fig. 3a) using 5 different correlating watermarks were computed and are given in Fig. 3b. The 1th watermark is the valid correlating watermark. As can be seen, the value of c for the 1th watermark is significantly larger than other values corresponding to different watermarks. In addition, all these values are below the threshold T = 4.75 for false alarm probability P FA = Correlation metric c Threshold T (a) Watermark Figure 3. (a) Watermarked image Lena, (b) Correlator values for the watermarked image Lena using 5 different correlating watermarks. (b) 6. Conclusions In this paper a watermarking method that uses a new generalized Radon transform, the SIT transform, is presented. The proposed method offers resilience to RST attacks without using the original image in the detection. Furthermore, experimental results demonstrate that the embedded watermarks resist filtering, noise addition and compression. References 1. S. Pereira and T. Pun, Robust template matching for affine resistant image watermarks, IEEE Trans. Im. Proc., vol. 9, no. 6, pp , June J. O Ruanaidh and T. Pun, Rotation, scale and translation invariant spread spectrum digital image watermarking, Signal Processing, vol. 66, no. 3, pp , May C. H. Chapman, Generalized Radon transforms and slant stacks, Geophys. J. R. ast. Soc., vol. 66, pp , C. Harris and M. Stephen, A combined corner and edge detector, in Alvey Vision Conference, 1988, pp I. J. Cox, J. Kilian, T. Leighton, and T. Shamoon, Secure spread spectrum watermarking for multimedia, IEEE Trans. Im. Proc., vol. 6, no. 12, pp , Dec 1997.