A WATERMARKING METHOD RESISTANT TO GEOMETRIC ATTACKS

Transcription

1 A WATERMARKING METHOD RESISTANT TO GEOMETRIC ATTACKS D. Simitopoulos, D. Koutsonanos and M.G. Strintzis Informatics and Telematics Institute Thermi-Thessaloniki, Greece. Abstract In this paper, a novel watermarking scheme able to resist geometric attacks is analyzed. The proposed method performs watermarking of images in the raw domain. A perceptual model is used in order to define the strength of the embedded watermark. For resistance to scaling and rotation attacks, two generalized Radon transformations of the image are introduced, while resistance to translation is accomplished through a localization of the watermarking method based on feature points of the image. Experimental evaluation demonstrates that the proposed scheme is able to withstand a variety of attacks including common geometric attacks. INTRODUCTION Watermarking of images is a technology that has attracted a lot of attention the recent years. In order to verify the robustness of the proposed watermarking schemes, a variety of attacks may be imposed to the image []. Among them, geometric attacks such as scaling, rotation and translation are easy to apply and may lead many watermark detectors to total failure due to loss of synchronization between the embedded and the correlating watermark. Lately, a lot of watermarking methods resistant to geometric attacks were presented in the literature. These may be divided in three categories. In some approaches, the watermark embedding is performed in a domain invariant to geometric attacks [2], while in others an additional pattern is embedded in the image in order to be able to revert the geometric attack [3]. Yet another approach for resisting geometric attacks is based on geometrically transforming a reference watermark both in the embedding and detection (correlation) according to a characteristic of the image content. A similar approach was presented in [4] where characteristics of the image content were extracted using Principal Component Analysis. The watermarking method presented in this paper, is also based on the latter approach to geometric resistant watermarking. 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 two onedimensional generalized Radon transformations that are applied to the image. According to characteristic values extracted from the two transformations during the embedding and detection of the watermark, a reference watermark is scaled and rotated before embedding or correlation-based detection respectively. This way, synchronization between the embedding and the correlating watermark is achieved. The experimental results demonstrate the resistance of the proposed scheme to geometric attacks as well as other common attacks. One- Dimensional Generalized Radon Transformations One-Dimensional Generalized Radon Transformations Two one-dimensional generalized Radon transformations [5] for resistance to scaling and rotation attacks respectively are introduced. The Radial Integration Transform (RIT) is used to 36

2 deal with rotation attacks, while the Circular Integration Transform (CIT) is used to cope with scaling attacks. Radial Integration Transform The RIT of a function f(x,y) is defined as the integral along a straight line that begins from the origin (x o,y o ) and has angle θ with respect to the horizontal axis (see Fig.a) The RIT is given by the following equation: + ( θ) = ( + cosθ, + sin ) where u is the distance from the origin (x o,y o ). Rf f xo u yo u θ du () 0 x y If f(x,y) is an image and g( x, y) = f, is the image scaled by s (s>0) in both s s directions, then the RIT of image g(x,y) is given by: R ( θ ) = sr ( θ ) g f (2) Therefore, the RIT amplitude of the scaled image is only multiplied by the factor s. If f (, r ϕ ) is an image written in polar form and gr (, ϕ) = f( r, ϕ ϕa ) is the image rotated byφ a around the ( r, φ ) system origin, then the RIT of image grϕ (, ) is given by: R ( θ ) = R ( θ φ ) g f a (3) Therefore, the RIT of the rotated image is translated byφ a. Circular Integration Transform The CIT of a function f(x,y) is defined as the integral of f(x,y) along a circle curve ( ) h ρ with center (x o,y o ) and radius ρ (see Fig. b). The CIT is given by the following equation: 2 ( ) π ( ) ( ρ ) 0 C ( ) cos, sin cos, sin f ρ = f xo + ρ θ yo + ρ θ du = xo + ρ θ yo + ρ θ ρdθ (4) h where du is the elementary arc length over the integration path and dθ is the corresponding elementary angle. x y If f(x,y) is an image and g( x, y) = f, is the image scaled by s (s>0) in both s s directions, then the CIT of image g(x,y) is given by: Cg( ρ) = scf( ρ) (5) s Therefore, the CIT of the scaled image is scaled with the same scaling factor and its amplitude is also multiplied by the factor s. 37

3 If f ( r, ϕ ) is an image written in polar form and gr (, ϕ) = f( r, ϕ ϕa ) is the image rotated byφ a around the ( r, φ ) system origin, then the CIT of image grϕ (, ) is given by: C ( ρ) = C ( ρ) g f (6) Therefore, the CIT of an image is independent of rotation. y (x o,y o ) Integration path θ du dx dy y (x o,y o ) ρ dθ θ Integration path ds x x (a) (b) Discrete RIT and CIT Fig. (a) RIT integration path, (b) CIT integration path. A discrete form of the RIT equation () is given by J R( t θ) = I( x0 + j ucos ( t θ), y0 + j usin( t θ) ), t =,..., T (7) J j= where u and θ are the constant step sizes of the corresponding variables, J is the number of points that lie on the line with orientation θ and are located between the origin and the end of 360 the image I(x,y) in that direction, and T =. θ Similarly, a discrete form of the CIT equation (4) is given by Τ Ck ( ρ) = I( x0 + k ρcos ( t θ), y0 + k ρsin( t θ) ), k=,..., K Τ t= where ρ and θ are the constant step sizes of the corresponding variables, K ρ is the radius 360 of the smallest circle that encircles the image I(x,y), and Τ=. θ Resisting geometric transformations using RIT and CIT The RIT and CIT properties given in equations (2), (3), (5) and (6) are very desirable in watermarking applications in which resisting the geometric attacks performed on the watermarked image is required. This can be accomplished by the watermark embedding and detection scheme described in the following. First, a two-dimensional watermark is created, to be used as a reference watermark. The orientation of the reference watermark corresponds to a reference angle θ r and one of its dimensions (its width for example) corresponds to a reference size ρ r. Then, the values of the (8) 38

4 parameters θ max and ρ max which maximize the RIT and CIT coefficients respectively, define the geometric transformation that the reference watermark should undergo before additive embedding in the original image. Specifically, the reference watermark is first scaled by the scaling factor ρmax sw = (9) ρr and rotated by the angle r (0) w = θmax θr and then embedded in the image. Let us assume a scaling and a rotation attack to the watermarked image by s α and θ α respectively. Based on the RIT and CIT properties, it is obvious that after the attack, the location of the new maximum will be θ max + θ α for RIT and s α ρ max for CIT. Subsequently, in ρmax the detection process, if the reference watermark is scaled by the scaling factor sw = s α ρ and rotated by the angle rw = θmax θr + θa, the embedding and the correlating watermark will be synchronized (same scale and orientation), which is vital for a successful detection. This concept for resisting geometric attacks can only be applied if a specific location of the image content can be accurately traced both in the embedding and in the detection (possibly after a geometrical attack that would change its position). This location will serve as an origin for RIT and CIT in embedding and detection and also for positioning the embedding and the correlating watermark. A robust feature point extracted from the image content would be appropriate for this cause. For this reason, a corner point is extracted from the image. Watermark embedding The proposed watermarking scheme, which is based on the concept described in the previous section, performs the watermark embedding in the following steps (see Fig. 2a): A random two-dimensional sequence of + and - is created based on a secret key. Each value of the sequence is spread in blocks sized B B (B=2). This block-based watermark will be used as the reference watermark. The Harris corner detector [6] is applied in the image and the most robust among the detected corners is extracted. Using the location of the most robust corner as the origin, the RIT and CIT are applied to the image. Then, θ max and ρ max are found and the rotation parameter r w and the scaling parameter s w are calculated. The reference watermark is geometrically transformed using r w and s w. A perceptual analysis for each image pixel X(i,j) is performed as in [7], in order to determine the strength of the watermark. Additive embedding of the watermark in the spatial domain is performed. After experimenting with a set of 500 photographic images where the watermark was embedded, no noticeable degradation of the image quality was observed and the minimum and average PSNR observed were 39.97db and 4.76db respectively. Watermark detection For the detection of the watermark, a correlation-based detection scheme is applied. The detection process can be divided in the following steps (see Fig. 2b): r 39

5 . The watermark W is created using the owner's secret key and is then transformed to a block-based watermark W B as in the embedding process. 2. The Harris corner detector is applied to the image. 3. The RIT and CIT of the image are calculated using the location of the corner detector response maximum as the origin. Then, the values of the parameters ρ max and θ max that maximize the RIT and CIT coefficients respectively are found. 4. The scaling and rotation parameters s w and r w, that will be used to create the correlating watermark W C, are estimated using the detected corner as the origin for RIT and CIT. 5. The correlating watermark W C is created by first geometrically transforming the watermark W B and then by applying the sign function to the values of the geometrically transformed watermark as in the embedding process. 6. The local mean Y lm (i,j) for each pixel Y(i,j) of the test image is calculated. The local mean is calculated over a 7 x 7 window centered at the pixel location. 7. The correlation value c between the watermarked and possibly distorted N image pixel values Y(i,j) (after the local mean Y lm (i,j) is subtracted) and the watermark values W C (i,j) is calculated: c = () ( Y( i, j) Ylm ( i, j) ) WC ( i, j) N i j 8. The correlation value c is compared to the threshold T, which is defined according to the allowed false alarm probability P FA of the detection scheme. Reference Watermark Reference Watermark Original Image Corner Detection Perceptual Analysis CIT RIT ρ max θ max Geometrical Transformation of Watermark Rotated and Scaled Watermark Attacked Watermarked Image Harris Corner Detector CIT RIT ρ max θ max Correlation Geometrical Transformation of Watermark Synchronized Watermark Additive Embedding Decision Watermarked Image (a) (b) Fig. 2 (a) Watermark embedding process, (b) Watermark detection process. Experimental Results The experiments presented in the following were all performed using the image Lena. The corner detected at (x 0 =80, y 0 =435) was used as the origin for the RIT and CIT transforms. The parameters of these transforms were set to the following values: θ=0.0, s=0.5, θ r =0 o for the RIT transform and ρ=0., θ = , ρ r =250 for the CIT transform. Linear interpolation was used for the geometrical transformations of the watermark. Various attacks were directed on the watermarked image. The corresponding errors of the estimated geometrical transformations of the image, and the corresponding correlator values are presented in Table. Finally, the correlator value for the watermarked image Lena using 5000 different correlating watermarks is given in Fig. 3. The 000th watermark is the valid correlating 40

6 watermark. The experimental mean and variance of the correlator value derived from this test were very close to the corresponding theoretical values which were used to determine the threshold T=0.2 for a given false alarm probability P FA =0-3. Scaling Attack Rotation parameters JPEG Q=70 - Estimation Scaling error Rotation Correlation value c Threshold T Table Correlator output results for various attacks on the watermarked image Lena Fig. 3 Correlator values for the watermarked image Lena using 5000 different correlating watermarks. References [] F. A. P. Petitcolas, Watermarking schemes evaluation, IEEE Signal Processing, vol. 7, no.5, pp , [2] J. O. Ruanaidh and T. Pun, Rotation, scale and translation invariant spread spectrum digital image watermarking, Signal Processing, vol. 66, no. 3, pp , May 998. [3] S. Pereira and T. Pun, Robust template matching for affine resistant image watermarks, IEEE Trans. Image Processing, vol. 9, no. 6, pp , June [4] P. Bas and B. Macq, A new video-object watermarking scheme robust to object manipulation, in ICIP, Thessaloniki, Greece, Oct. 200, vol. 2, pp [5] C.H. Chapman, Generalized Radon transforms and slant stacks, Geophys. J. R. ast. Soc., vol 66, pp , 98. [6] C. Harris and M. Stephen, A combined corner and edge detector, in Alvey Vision Conference, 988, pp [7] D. Simitopoulos, D. Koutsonanos, and M. G. Strintzis, Image watermarking resistant to geometric attacks using Generalized Radon Transformations, DSP 2002, Santorini, Greece, June 2002, pp