Canonical Multiwavelets As discussed in the previous chapters, wavelets have provided many beneficial features, including orthogonality, vanishing moments, regularity (continuity and smoothness), multiresolution analysis, among these features. Some wavelets are compactly supported in the time domain (Coifman, Daubechies) or in the frequency domain (Meyer), and some are symmetrical (Haar, Battle–Lemarie). On many occasions it would be very useful if the basis functions were symmetrical. For instance, it would be better to expand a symmetric object such as the human face using symmetric basis functions rather than asymmetric ones | Wavelets in Electromagnetics and Device Modeling. George W. Pan Copyright 2003 John Wiley Sons Inc. ISBN 0-471-41901-X CHAPTER SIX Canonical Multiwavelets As discussed in the previous chapters wavelets have provided many beneficial features including orthogonality vanishing moments regularity continuity and smoothness multiresolution analysis among these features. Some wavelets are compactly supported in the time domain Coifman Daubechies or in the frequency domain Meyer and some are symmetrical Haar Battle-Lemarie . On many occasions it would be very useful if the basis functions were symmetrical. For instance it would be better to expand a symmetric object such as the human face using symmetric basis functions rather than asymmetric ones. In regard to boundary conditions magnetic wall and electric wall are symmetric and antisymmetric boundaries respectively. It might be ideal to create a wavelet basis that is symmetric smooth orthogonal and compactly supported. Unfortunately the previous four properties cannot be simultaneously possessed by any wavelets as proved in 1 . To overcome the limitations of the regular . scalar wavelets mathematicians have proposed multiwavelets. There are two categories of multiwavelets and both of them are defined on finite intervals. The first class is that of the canonical multiwavelets that are based upon the vector-matrix dilation equation 2-4 this class will be studied in this chapter. The second class is based on the Lagrange or Legendre interpolating polynomials 5 which is similar in some respects to the pseudospectral domain method and as such facilitates MRA. VECTOR-MATRIX DILATION EQUATION Multiwavelets offer more flexibility than traditional wavelets by extending the scalar dilation equation p f hkv 2t - k 240 VECTOR-MATRIX DILATION EQUATION 241 into the matrix-vector version W t Ck W t - k k where Ck Ck rxr is a matrix of r x r W t Wo t Wr-1 t T is a column vector of r x 1 and r is the multiplicity of the .
