Image watermarking via fractional polar harmonic transforms

Invariant harmonic transforms based on the fractional Fourier transform are proposed in this paper. The so-called fractional polar harmonic transforms (FrPHTs) with the order parameter α are first defined, which are generalizations of the PHTs. Second, a watermarking scheme is presented and discussed in detail associated with the newly defined FrPHTs. Finally, the simulations are clearly performed to verify the well capabilities of the transforms on image watermarking, which show that the proposed transforms with suitable parameters outperform the traditional PHTs. In addition, the experimental results also demonstrate that the order parameter α has an effect on the performance of FrPHTs in the image watermarking robustness and can improve the watermarking safety.