Robust Watermarking Using n-Diagonalization Based on Householder Transform

Digital image watermarking based on singular value decomposition (SVD) is highly robust against misuse, but lacks the ability to distinguish whether watermarks are correct due to the importance of singular values being lower than two orthogonal matrices. To achieve highly accurate watermark extraction while maintaining high robustness, we propose robust watermarking based on discrete wavelet transform (DWT) and n-diagonalization formalized by Householder transformation. We propose that DWT be used to ensure visibility and that n-diagonalization be used to control information quantity related to watermark extraction accuracy. Experimental results confirm the robustness of our proposed method and that the extraction accuracy of the proposed method is approximately 2 times better than that of SVD.

FINDING AND DISENTANGLING COMPLICATED UNITARY OPERATORS BY VIRTUE OF THE INTEGRATION OVER DIRAC'S KET–BRA OPERATORS

Usually complicated unitary operators in exponential form are hard to handle and the transformation stated by them are not physically clear until these operators are disentangled. By virtue of the technique of integration within an ordered product (IWOP) of operators, we can disentangle some complicated unitary operators and then reveal their physical role. Many new unitary operators can be found after new quantum mechanical representations are constructed by virtue of the IWOP technique. The unitary operators for permutation, Hilbert transform, Householder transform, and Hardmad transform can also be introduced. The IWOP technique thus bridges this mathematical gap between classical mechanics and quantum mechanics and provides a new route connecting classical transformations to quantum mechanical unitary operators. In this way, the symbolic method exhibits further beauty and elegance and can be increasingly used for developing various fields of quantum physics.