Orthogonal M-band compactly supported interpolating wavelet theory

Recently, 2-band interpolating wavelet transform has attracted much attenti on. It has the following several features: (i) The wavelet series transform coefficients of a signal in the multiresolution subspace are exactly consi stent with its discrete wavelet transform coefficients; (ii) good approxima tion performance; (iii) efficiency in computation. However orthogonal 2-ban d compactly supported interpolating wavelet transform is only the first ord er. In order to overcome this shortcoming, the orthogonal M-band compactly supported interpolating wavelet basis is established. First, the unitary in terpolating scaling filters of the length L = MK are characterized. Second, a scheme is given to design high-order unitary interpolating scaling filte rs. Third, a parameterization of the unitary interpolating scaling filters of the length L = 4M is made. Fourth, the orthogonal 2-order and 3-order th ree-band compactly supported interpolating scaling functions are constructe d. Finally, the properties of the orthogonal M-band compactly supported int erpolating wavelets and the approximation performance of the Mallat project ion 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 th e band-limited signal, the quantitative estimate of its upper bounds is giv en. The results show that the Mallet projection has the same approximation order as the orthogonal projection, and particularly for the orthogonal eve n number-order M-band compactly supported interpolating scaling function, t hey have the same approximation performance. The quantitative result also s hows 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 determi ne the initial scale and at the same time estimate the initial scaling coef ficients without prefiltering according to the error requirement.