Harmonic Transform Theory Based On Matrix Lie Groups And Its Application In Image Watermarking

In this thesis, some kinds of harmonic transforms based on matrix Lie group-s are proposed to improve the information safety in the image watermarking. The experiments verify that the proposed methods can influence on performances of water-marking, and improve the information safety.The emphases in this thesis are summarized as follows.1. Three invariant harmonic transforms based on the special linear group SL(2, R) are proposed, which include the first polar linear canonical transform (PLCT1), the sec-ond polar linear canonical transform (PLCT2) and the two-dimensional linear canonical transform series (2D LCT series). We analyze the capabilities of the PLCT1, PLCT2 and the 2D LCT series on image representation. The experimental results show that the 2D LCT series has much stronger capability on the image representation with re-spect to characters, but not better than the other transforms corresponding to the Lena image. Moreover, due to the varieties of parameters, the performance of the image representation is going bad when parameters used in the reconstructing process are inconsistent with those in the decomposing process. We assert that the proposed transforms can be used for the protection of the image safety because they have free parameters than the traditional methods.2. Novel invariant harmonic transforms based on the fractional Fourier transform and the orthogonal group SO(2) are proposed. The so called fractional polar harmonic transforms with the order parameter a are firstly defined, which are generalizations of the polar harmonic transforms (PHTs). Secondly, a watermarking scheme is presented and discussed in detail associated with the newly defined fractional polar harmonic transforms. Finally, the simulations are clearly performed to verify the well capabil-ities of the novel 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 a has an effect on the performance of FrPHTs in the image watermarking robustness and can improve the watermarking safety.3. The polar linear canonical transform with parameters in SL(2,R) are applied in image watermarking. Moreover, a watermarking scheme associated with the PLCT is presented, and the simulations are performed to verify the importance of parameters of the PLCT and improve the capabilities of the existing results on the watermarking information’s security. The experimental results show that due to the varieties of parameters, if parameters chosen in the extraction process are inconsistent with those used in the embedding process, then the bit error rate is bigger which means it is worse than that of consistent parameters in the two processes. This confirms the fact that parameters of the PLCT have superior security for watermark information.4. Quaternion harmonic transforms based on the quaternion theory and the har-monic transforms with matrix parameters are proposed, which include the quaternion polar linear canonical transform, the two dimensional quaternion linear canonical trans-form series and the quaternion polar complex exponential transform. We verify that the quaternion harmonic transforms can be applied in color image representation by the experiments.