Orthogonal M-band compactly supported interpolating wavelet theory

Recently, 2-band interpolating wavelet transform has attracted much attention. It has the following several features: (i) The wavelet series transform coefficients of a signal in the multiresolution subspace are exactly consistent with its discrete wavelet transform coefficients; (ii) good approximation performance; (iii) efficiency in computation. However orthogonal 2-band compactly supported interpolating wavelet transform is only the first order. In order to overcome this shortcoming, the orthogonal M-band compactly supported interpolating wavelet basis is established. First, the unitary interpolating scaling filters of the length L = MK are characterized. Second, a scheme is given to design high-order unitary interpolating scaling filters. Third, a parameterization of the unitary interpolating scaling filters of the length L = 4M is made. Fourth, the orthogonal 2-order and 3-order three-band compactly supported interpolating scaling functions are constructed. Finally, the properties of the orthogonal M-band compactly supported interpolating wavelets and the approximation performance of the Mallat projection are discussed. For the smooth signal in L-2(R), the asymptotic formula of the approximation error of the Mallat projection is obtained, and for the band-limited signal, the quantitative estimate of its upper bounds is given. The results show that the Mallet projection has the same approximation order as the orthogonal projection, and particularly for the orthogonal even number-order M-band compactly supported interpolating scaling function, they have the same approximation performance. The quantitative result also shows that the selection of the initial scale depends on the distribution of the signal frequency and the regularity order of the scaling function. For the given scaling function and signal, using these results one can determine the initial scale and at the same time estimate the initial scaling coefficients without prefiltering according to the error requirement.