| As the research of wavelet is deepened and the range of its application is extended extensively, more demands are added on the construction of wavelet base. In image processing, the wavelet of L2(R) will cause distortion on the boundary by truncation error. In order to exterminate this phenomenon, one must modify the wavelet on the boundary. This gives rise to the wavelet on the interval. At the same time, it is desirable for wavelet possess orthogonality, symmetry, short support and higher vanishing moments simultaneously, this leads to the advent of the multiwavelets. The dissertation discussed the construction of these two kinds of wavelet and their applications in the image denoising and mainly included following aspects:The general theory about periodic wavelet was discussed. The pre-wavelet on [0,1] was constructed with the splines wavelet and fold function. And thecorresponding Mallat's algorithm was presented. The construction of Daubechies orthogonal wavelet on the interval [0,1] was analyzed. And the biorthogonal wavelet on the interval [0,1] was constructed similarly. As an application, the orthogonal and biorthogonal wavelet on the interval [0,1] with vanishing moments 2 and 3 respectively was used to removel the noise of image. Experiments show that these wavelets on the interval not only have a good denoising result, but also need no extension on the boundary.The approximation order of multi-scaling vector was studied. The approximation order is increased by adding the factor number in the factorization of mask of multi-scaling vector with TST, and symmetry, short support of multi-scaling vector are maintained simultaneously. A simple method for selecting the transform matrix which preserve the above property of multi-scaling vector was presented. With the dilation property of the fractal interpolation function, one multi-scaling vector and it's refinement equation was presented. The multi-scaling vector possess symmetry and approximation order 1 at least by selecting a suitable parameter. Then choosing suitable transform matrix, and applying TST to it, the satisfactory approximation order of the multi-scaling vector was achieved.The biorthogonal multi-scaling vector with higher approximation order was constructed using TST. Although the TST increases the multi-scaling vector's approximation order, maintains the symmetry and short support, but destroy the orthogonality. It is known that the dual multi-scaling vector has some approximation order, the multi-wavelet functions have same vanishing moment. The TST was appliedto multi-scaling vector to increase it's approximation order, and with PR condition, the dual of transformed multi-scaling vector was obtained. Then the special TST was taken for this dual, the dual of original multi-scaling vector with higher approximation order obtained, finally the multi-wavelet function with higher vanishing moment maybeconstructed.Design of prefilter of multiwavelets was discussed. For the prefilter designed by Xia with interpolating property of GHM multiwavelet, a modified interpolating prefilter was presented. It is impotent to design the prefilter which maintain the orthogonality and approximation order of central filter bank. The orthogonal quasi-interpolation filter preserving approximation order designed by Hardin and Roach maintain the orthogonality and approximation order of central filter bank and give a proper direction in the design of prefilter.The balanced multiwavelet was studied. For low-pass matrix filter P( ), the orthogonal matrix R was selected to ensure the constant signal as a characteristic signal of balanced low-pass matrix filter RTP( )R, the corresponding balanced high-pass matrix filter is Q( )R or RTQ( ))R which maintain the orthogonality and symmetry or orthogonality only respectively. As an application, the OPTFR-multiwavelet constructed by Jiang was balanced and applied to image denoising and fusion. Experiments show that the result is satisfactory.Construction of biorthogonal matrix filter with GMP order...
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