A New Wavelet Watermarking Algorithm for Images using Edges Detection1

Olivier Le Cadet IMAG-LMC, BP 53, 38041 Grenoble Cedex 9, France Anne-Sophie Piquemal IMATI-CNR, Via Ferrata 1, 27100 Pavia, Italy

Abstract We present a new approach for watermarking images, in which the watermark embedding process employs multiresolution fusion techniques, and is based on edge detection. The original unmarked image is not required to extract the watermark.

Keywords: Wavelet method, Watermarking images, Edge detection

1

Introduction

The aim of watermarking images is to protect the ownership by including an invisible structure in the image, as a copyright information. To be effective, the watermark has to be imperceptible, discrete to prevent unauthorized removal, easily extracted by the owner, and robust to attacks, such as compression, erasing or filtering. These caracteristics immediatly induce a main contradiction. On one hand, it’s possible to include a lot of information in the less detailed parts of the image, without disturbing it, but then the watermarking method is not robust against compression. On the other hand, since we do not want to disturb the host image, it is possible to include only few information in the more detailed and robust parts of the image. To cure these problems, we introduce a new approach of watermaking, where the watermark is embedded using a multiscale data fusion approach [?] in which the original image and the watermark are both transform in some wavelet domains. At this point, the main drawback of the last data fusion methods lies in the fact that they mark all the most important wavelet coefficients at different scales. Nevertheless, a modification of wavelet coefficients 1

Work partially supported by the EC-IHP Network Breaking Complexity

1

corresponding to texture, modifies also the image. Then, the aim of our new approach is to mark only the wavelet coefficients corresponding to edges. By this way, we define a method which is robust against compression, since we mark only the most salient components of the image, and which does not modify the original image, since we only take care of the coefficients corresponding to edges.

2 2.1

The Watermarking Approach The Edge Detection

Following [?, ?], if θ is a gaussian kernel, we define two wavelets ψ 1(x, y) =

∂θ ∂x (x, y),

ψ 2(x, y) =

∂θ ∂y (x, y),

and the wavelet coefficients at the scale s, associated to a signal f (x, y) W p f (x, y, s) = f ∗ ψ¯sp(x, y), p = 1, 2, with ψsp(x, y) = 1s ψ p ( xs , ys ), and ψ¯sp(x, y) = ψsp(−x, −y), for p = 1, 2. In the following, we use an another representation of the image, given by the modulus and the angles at the scale s, M f (x, y, s) =

q

Af (x, y, s) = tan−1

|W 1f (x, y, s)|2 + |W 2 f (x, y, s)|2, 

W 2 f (x,y,s) W 1 f (x,y,s)

= π − tan−1





if W 1f (x, y, s) ≥ 0,

,

W 2 f (x,y,s) W 1 f (x,y,s)



, if W 1f (x, y, s) < 0.

Moreover, the decay of the wavelet transform amplitude across the scales is related to the uniform and pointwise Lipschitz regularity of the signal. Then a function f (x, y) is α−Lipschitz in ]a, b[×]c, d[ if and only if there exists a constant C > 0 sucht that, ∀x, y ∈]a, b[×]c, d[, M f (x, y, s) ≤ Csα . Finally, the edge detection algorithm used in the watermarking approach is divided in two main steps: – First of all, at each scale s, we search if a point (x, y) corresponds to a modulus maxima, by comparing M f (x, y, s) to his neighboors in the direction of the gradient. In this case we keep the values M f (x, y, s) in M f , and Af (x, y, s) in Af , otherwise we put M f (x, y, s) = Af (x, y, s) = 0. 2

– Secondly, a point (x0, y0 ) will be on an edge, if it corresponds to a modulus maxima through the scales. In this case we define the image of edges Cf , by Cf (x0 , y0) = M f (x0 , y0, 1) and zero otherwise. – Along the chains leading to a modulus maxima point, the lipschitz regularity of the edge point is computed.

2.2 2.2.1

The Watermaking Method Watermarking

The aim of this part is to embed a binary mark L defined as an array of N × N ones and zeros, in an image of size M × M , N < M . In this way, we need the image Cf , the modulus maxima M f and the corresponding angles Af , at scale s = 1, and finally the matrix of Lipschitz regularity αf , where for a point (x0 , y0) on an edge, we keep the value of α(x0 , y0), and put 50 otherwise. We note maxmod =

max

(x0 ,y0 )∈edges

Cf (x0 , y0),

(1)

and search the N 2 points {(xi, yi )}1≤i≤N 2 of lowest lipschitz-regularity α(xi , yi ). We define two square matrices, of size N × N , αmin for the N 2 values of the regularity, and index containing the positions of the corresponding point, ∀i, 1 ≤ i ≤ N 2, Index(i) = (xi , yi ) = position of the point whose lipschitz-regularity α(xi , yi) is αmin (i). The embedding of the mark is made by modifying the modulus maxima ˜ f , only at the points corresponding to the lowest regularity, and M f in M reading the mark L line per line. We define the following algorithm for i = 0, · · ·, N 2 (xi, yi ) = Index(i)   ˜ f (xi, yi ) = M f (xi , yi ) + L(i) · maxmod M end Table 1: Watermarking Algorithm ˜ 1 f and W ˜ 2 f using M ˜ f and Af , and then applying Finally, we compute W a deconvolution step, we obtain the watermarked image f˜.

3

2.2.2

Detection

Given the marked image f˜, we detect the watermark using as a reconstruc˜ , tion key the number maxmod (??). Following Section (??), we compute Cf ˜ f , Af ˜ , at scale s = 1, and finally the matrix of Lipschitz regularity αf M ˜ . Since the regularity is unchanged by the algorithm detailed in Table (??), we compute the N 2 points {(˜ xi, y˜i )}1≤i≤N 2 of lowest Lipschitz-regularity, whose positions should be unchanged with respect to those of the original image. Then, the algorithm of detection is given by for i = 1, · · ·, N 2 ˜ f (˜ if M xi , x ˜i) < maxmod, then this point is marked with 0 else this point is marked with 1 end for the other points, no mark Table 2: Detection Algorithm

3

Conclusion

We proposed the main steps of the definition of a new wavelet watermarking algorithm for images. Our poster session will show numerical results validating this method, showing its robustness against compression or filtering, and comparisons to already existing watermarking methods.

References [1] D. Kundur, D. Hatzinakos, A Robust Image Watermarking Method using Wavelet-Based Fusion, IEEE Sig. Proc. Society, 1997. [2] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 1998. [3] S. Mallat, W.L. Hwang, Singularity Detection and Processing with Wavelet, IEEE Trans. Info. Theory, 38(2), 617-643, 1992.

4