Multi-wavelet Construction Algorithm And Its Applications In Digital Watermarking

Multiwavelet is the new development of the wavelet analysis in recent years. Multiwavlelts are a natural generalization of traditional scalar wavelets. Multiwavelets have new features arising from the matrix structure of the two-scaling equation. For example, they can be simultaneously orthogonal, symmetric, short supported and of high vanishing moments, while scalar wavelets cannot.Firstly, the multi-scaling functions providing m approximation order requires the sufficient and necessary conditions in the time domain. On the bases of the time domain analysis, we construct the arbitrary order finite element multi-scaling function, using the properties of the finite element functions. In the time domain, the approximation usually requires the solution of large systems of nonlinear equations, but by switching to the frequency domain, the sufficient and necessary conditions is simple.The second part is firstly to introduce the two-scaling similarity transform. For the approximation and the refinement mask have the close relationship, the refinement mask corresponding the multi-scaling functions can be factorized by TST. Two factorization method are analysis in details: one is the special transformation matrix, the other is the general transformation matrix.The third part discuss the sufficiency of the construction of the multi-scaling functions. Then the other properties of the multi-scaling functions, such as regularity, symmetry, finite support and orthogonality, are analysis through two-scaling similarity transformation. Given a multi-scaling functions, the cofactor method is often used for the construction of dual scaling functions and biorthogonal multiwavelets. So, this method is modified in the article, and the finite element biorthogonal multiwavelets are constructed.The last, the standard (scalar) MRA make a step forward and, allow several scaling functions to generate a basis of the scaling space. On the base of the MRA, the transformation of multiwavelets can be complied. The multiwavelets is different with the scalar wavelets, so some problems must be analysis in the applications, such as the signal preprocessing/preprocessing, boundary and so on. In the application, we apply the multiwavelets to the digital watermarking technique, and comply a invisible image watermarking using CL multiwavelets system.