Construction Of Multi-Band Orthogonal Multiwavelets With Multiplicity 2

The multi-band orthogonal multiwavelets with multiplicity 2 is a brand-new stage in the developing process of wavelet analysis. It has received people's attention for its many excellent properties. On one hand, it has multi-band wavelet's decomposing frequency characteristic on multi-channel, which also makes it to solve successfully and realize smoothly in the signal processing with a higher requirement of signal quality. On the other hand, it also has the same characteristic with multiwavelets, and it can simultaneously possess some desirable properties, such as compact support, symmetry, orthogonality, smoothness and so on. These properties play the indispensable role in processing aspects of image denoising.The previous construction idea on wavelet: firstly, find out the low-pass filter with corresponding to multi-band orthogonal multiwavelets with multiplicity 2. Secondly, find out high-pass filter of its corresponding multiwavelets by the relationship between low-pass filter and high-pass filter. And then find out the multi-band orthogonal multiwavelets function with multiplicity 2. However, matrix multiplication can not be interchangeable in nature, so in the actual calculation, it is very difficult to construct the multi-band orthogonal multiwavelets with multiplicity 2 acconding to this traditional way. Therefore, it is mainly studied on the construction methods of multi-band orthogonal multiwavelets with multiplicity 2 in this paper, and two methods are given as following.One kind, based on the idea of the multi-resolution analysis, first use the Riesz lemma to construct M -band complex-valued symmetrical orthogonal uni-scaling function of the Daubechies kind, then the M -band compactly supported orthogonal symmetric multi-scaling function with multiplicity 2 is constructed by using the real and imaginary parts of the above obtained complex-valued ones. Thus the M -band compactly supported orthogonal symmetric-antisymmetric multiwavelets function with multiplicity 2 is constructed, and its corresponding example is given.The other, under the situation that the low-pass filter is given, the correspondingα-band multiwavelets is obtained by introducing an unitary transformation of conjugate quadrature filter. Thus, one kind of method on constructingα-band compactly supported orthogonal multiwavelets is given, and the corresponding example is also given.Anyway, a new construction thought on multi-band orthogonal multiwave- lets is provided in this paper.