Geometric modelling of the wavelet coefficients for image watermarking using optimum detector

In this study, a robust image watermarking method based on geometric modelling is presented. Eight samples of wavelet approximation coefficients on each image block are utilised to construct two line segments in the two-dimensional space. The authors change the angle formed between these line segments for data embedding. Geometrical tools are used to solve the tradeoff between the transparency and robustness of the watermark data. They analytically determine the probability density function of the embedding angle for Gaussian samples. Maximum-likelihood decoder is implemented in the receiver side. Owing to embedding in the angle between two line segments, the proposed scheme has high robustness against the gain attacks. In addition, using the low frequency components of the image blocks for data embedding, high robustness against noise and compression attacks has been achieved. Experimental results confirm the validity of the theoretical analyses given in this study and show the superiority of the proposed method against common attacks, such as Gaussian filtering, median filtering and scaling attacks.