Invisible Digital watermarks have been proposed as a method for discouraging illicit copying and distribution of copyright material. In recent years it has been recognized that embedding information in a transform domain leads to mo re robust watermarks. A major difficulty inwatermarking in a transform domain lies in the fact that constraints on the allowable distortion at any pixel may be speci ed in the spatial domain. The central contribution of the paper is the proposal of an approach which takes into account spatial domain constraints in an optimal fashion. The main idea is to structure the watermark embedding as a linear programming problem in which we wish to maximize the strength of the watermark subject to a set of lin- ear constraints on the pixel distortions as determined by a masking function. We consider the special cases of embedding in the DCT domain and wavelet domain using the Haar wavelet and Daubechies 4-tap lter in conjunction with a masking function based on a non-stationary Gaussian model, but the algorithm is applicable to any combination of transform and masking functions. Our results indicate that the proposed approach performs well against lossy compression such as JPEG and other types of ltering which do not change the geometry of the image.
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