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 fre- quency 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 antisym- metric boundaries, respectively. It might be ideal to create a wavelet basis that is symmetric, smooth, orthogonal, and compactly supported. Unfortunately, the previ- ous four properties cannot be simultaneously possessed by any wavelets, as proved in [1]. To overcome the limitations of the regular (i.e., 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 multi- wavelets 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 in- terpolating polynomials [5], which is similar in some respects to the pseudospectral domain method and as such facilitates MRA. 6.1 VECTOR-MATRIX DILATION EQUATION Multiwavelets offer more flexibility than traditional wavelets by extending the scalar dilation equation ϕ(t) =  h k ϕ(2t − k) 240 Wavelets in Electromagnetics and Device Modeling. George W. Pan Copyright ¶ 2003 John Wiley & Sons, Inc. ISBN: 0-471-41901-X VECTOR-MATRIX DILATION EQUATION 241 into the matrix-vector version |φ(t)=  k C k |φ(2t − k), where C k =[C k ] r×r is a matrix of r ×r, |φ(t)=(φ 0 (t) ···φ r−1 (t)) T is a column vector of r × 1, and r is the multiplicity of the multiwavelets. By taking the jth derivative, we have |φ ( j) (t)=  k C k 2 j |φ ( j) (2t − k). Let us denote a matrix (t) =        φ 0 (t)φ  0 (t) ··· φ (r−1) 0 (t) φ 1 (t)φ  1 (t) . . . . . . φ r−1 (t) . . .φ (r−1) r−1 (t)        . (6.1.1) Then (t) =  |φ(t)|φ  (t) ··· |φ (r−1) (t)  =   C k |φ(2t − k) 2  C k |φ  (2t − k) ··· 2 r−1  C k |φ (r−1) (2t − k)  =  k C k |φ(2t − k)      1 2 . . . 2 r−1      r×r , or (t) =  C k (2t − k) −1 , (6.1.2) where  −1 = diag{1, 2, ,2 r−1 }. (6.1.3) Equation (6.1.2) can be verified as follows: Show. LHS = (t) =    φ 0 (t)φ (r−1) 0 (t) ··· ··· ··· φ r−1 (t)φ (r−1) r−1 (t)    242 CANONICAL MULTIWAVELETS =  |φ(t)|φ (1) (t) ··· |φ (r−1) (t)  =   C k |φ(2t −k) 2  C k |φ (1) 2t−k ···2 (r−1)  C (r−1) k |φ(2t −k)  . RHS =  k C k (2t − k) −1 =  (C k ) r×r  |φ(2t − 2)|φ (1) (2t − 2)···|φ (r−1) (2t − k)  ·       1 ··· ··· ···0 020 ···0 002 2 ···0 ··· ··· ··· ··· 000···2 r−1       r×r =  k (C k ) r×r      φ 0 (2t − k)φ  0 (2t − k) ··· φ (r−1) 0 (2t − k) φ 1 (2t − k)φ  1 (2t − k) ··· φ (r−1) 1 (2t − k) ··· ··· ··· ··· φ r−1 (2t − k)φ  r−1 (2t − k) ··· φ (r−1) r−1 (2t − k)      ·     1 2 ··· 2 r−1     =  k (C k ) r×r      φ 0 (2t − k) 2φ  0 (2t − k) ··· 2 (r−1) φ (r−1) 0 (2t − k) φ 1 (2t − k) 2φ  1 (2t − k) ··· ··· ··· ··· φ r−1 (2t − k) 2φ  r−1 (2t − k) ··· 2 (r−1) φ (r−1) r−1 (2t − k)      . Hence (t) =  C k (2t − k) −1 . (6.1.4) In the construction of the multiwavelets, we may use either the frequency domain approach or the time domain approach. The frequency approach is more elegant but requires more extensive mathematical background. We select the latter approach, which seems to be easier to follow despite being more cumbersome. 6.2 TIME DOMAIN APPROACH We begin with the vector dilation equation |φ(t)=  k C k |φ(2t − k), (6.2.1) TIME DOMAIN APPROACH 243 which has an explicit form of         φ 0 (t) φ 1 (t) · · · φ r−1 (t)         = n−1  k=0 [C k ] r×r         φ 0 (2t − k) φ 1 (2t − k) · · · φ r−1 (2t − k)         , where n is the order of approximation (see Eq. (6.2.6)). Let us denote an infinite-dimensional matrix L =       ··· ··· ··· C 3 C 2 C 1 C 0 ··· ··· C 3 C 2 C 1 C 0 C 3 C 2 C 1 C 0 ··· ···       . Then (6.2.1) becomes |(t)=L |(2t), (6.2.2) where |(t)=[···φ(t − 1) |φ(t)|φ(t +1) |···] T . The explicit form of (6.2.2) is       ··· |φ(t − 1) |φ(t) |φ(t + 1) ···       =       ··· C 3 C 2 C 1 C 0 ··· C 3 C 2 C 1 C 0 ··· C 3 C 2 C 1 C 0 ···             ··· |φ(2t − 1) |φ(2t) |φ(2t + 1) ···       , or       ··· C 3 C 2 C 1 C 0 ··· C 3 C 2 C 1 C 0 ··· C 3 C 2 C 1 C 0 ···             ··· |φ(2t − 1) |φ(2t) |φ(2t + 1) ···       =       ··· |φ(t − 1) |φ(t) |φ(t + 1) ···       . (6.2.3) Let us pick out the row that represents |φ(2t) and |φ(t) in (6.2.3), namely ···+C 2 |φ(2t − 2)+C 1 |φ(2t − 1)+C 0 |φ(2t)=|φ(t). If we replace t by t − 1, then (6.2.1) becomes  k C k |φ(2t − k − 2)=|φ(t − 1). 244 CANONICAL MULTIWAVELETS Explicitly, the equation above is ···+C 1 |φ(2t − 3)+C 0 |φ(2t − 2)=|φ(t − 1), which is one row above in (6.2.3). Notice the two-unit shift in the row of the matrix L that corresponds to the equation above. Now consider the monomials t j , j = 0, 1, ,r − 1, which span the scaling subspace. The φ(·) are the basis function in V r . Therefore t j := G j (t) = ∞  k=−∞ y [j] k |φ(t − k)=y [j] |(t), (6.2.4) where y [j] |=  ···y [j] 0 |y [j] 1 |y [j] 2 |···  and each piece y [j] k | is a row vector with r components that matches the vectors |φ(t − k). Substituting (6.2.2) into (6.2.4), we obtain G j (t) =y [j] |(t)=y [j] | L |(2t). On the other hand, we may rewrite this as G j (t) = t j = 2 −j (2t) j = 2 −j y [j] |(2t) . Hence y [j] | L |(2t)=2 −j y [j] |(2t), and therefore y [j] | L = 2 −j y [j] |. (6.2.5) The previous equation implies that L has eigenvalue 2 −j for the left eigenvector y [j] |.Thatistosay,ifL has eigenvalues 1, 2 −1 , 2 −2 , ,2 −( p−1) with left eigen- vectors y [j] |, then G j (t) = ∞  k=−∞ y [j] k |φ(t − k). A special and important case is j = 0, in which case  k y [0] k |φ(t − k)=1 = t 0 . In the remainder of this section, we will list definitions, lemmas and theorems that will form a solid foundation of multiwavelets in the time domain. Definition. A multiscalet |φ(t) has approximation order n if each monomial t j , j = 0, ,n − 1 is a linear summation of integer translations |φ(t − k) CONSTRUCTION OF MULTISCALETS 245 such that t j = ∞  k=−∞ y [j] k |φ(t − k), j = 0, 1, ,n − 1, (6.2.6) almost everywhere. Lemma 1. Suppose that φ j (t) ∈ L 1 for j = 0, ,r − 1 and the translates φ j (t − k), k ∈ Z, are linearly independent. Then |φ(t) provides an approximation of order n if and only if L has eigenvalues 2 −j corresponding to the left eigenvectors y [j] |=  ···y [j] 0 |y [j] 1 y [j] 2 |···  with a component y [j] k |= j  =0  j l  (−k) j− u [] |, j = 0, 1, ,n − 1, (6.2.7) where u [] | are constant vectors that will be given in (6.12.12). Lemma 2. Suppose that y [j] | is given by (6.2.7) and that L corresponds to a multiscalet with an approximation order n.Then y [j] |L = 2 −j y [j] |, j = 0, ,n − 1, if and only if the following finite equations are held:  k y [j] k |C 2k+1 = 2 −j u [j] | (6.2.8)  k y [j] k |C 2k = 2 −j y [j] 1 |=2 −j j  =0 (−1) j−  j   u [] | for j = 0, 1, ,n − 1. (6.2.9) Equations (6.2.8) and (6.2.9) are referred to as the approximation conditions. The proofs of Lemma 1 and Lemma 2 are provided in the Appendix to this chapter. 6.3 CONSTRUCTION OF MULTISCALETS We begin with the approximation conditions (6.2.8) and (6.2.9):  k y [j] k |C 2k+1 = 2 −j u [] |, (6.3.1)  k y [j] k |C 2k = 2 −j y [j] 1 | = 2 −j j  =0 (−1) j−  j   u [] |, (6.3.2) 246 CANONICAL MULTIWAVELETS which are a system of nonlinear equations in terms of matrix components and the starting vectors u [j] |. These equations can be solved effectively only for low approximation orders with a small number of dilation coefficients. Fortunately, in electromagnetics, the order is usually ≤ 4. An intervallic function of order r is a multiscalet |φ(t)=(φ 0 (t) φ r−1 (t)) T (6.3.3) consisting of intervallic φ j , which are piecewise polynomials of degree 2r − 1 with r − 1 continuous derivatives. For all r, φ j (t) = 0 only on two intervals [0, 1] and [1, 2]. The function value and its r −1 derivatives are specified at each integer node. If the intervallic functions are defined on [0, 2], then they are alternatively symmetric and antisymmetric about t = 1. The translations of these functions span V 0 . The dilation equation may be written as |φ(t)=  k C k |φ(2t − k) = C 0 |φ(2t)+C 1 |φ(2t − 1) + C 2 |φ(2t − 2). (6.3.4) Since the support is [0, 2], the only nonzero coefficients are C 0 , C 1 ,andC 2 .There are r basis functions at each node, and C i are matrices of r × r (i = 0, 1, 2).The polynomials of degree 2r − 1on[0, 1] and [1, 2] can be determined by  d dt  k φ j (1) = δ k, j , k, j = 0, ,r − 1, (6.3.5)  d dt  k φ j (0) = 0 =  d dt  k φ j (2), k, j = 0, ,r − 1, (6.3.6) where δ k, j is the Kronecker delta. The symmetry and antisymmetry about t = 1aregivenby φ j (2 − t) = (−1) j φ j (t), j = 0, ,r − 1. (6.3.7) Notice that C 0 |φ(2)=0 = C 2 |φ(0) by (6.3.6). Equations (6.3.5) and (6.3.6) may be expressed compactly as (n) = δ 1,n I, where δ 1,n is the Kronecker delta, I is the identity matrix of r × r,and(t) was defined in (6.1.4) as (t) = (|φ(t)|φ  (t)···|φ (r−1) (t)) =    φ 0 (t)φ (r−1) 0 (t) ··· ··· ··· φ r−1 (t)φ (r−1) r−1 (t)    with φ ( j) i (t) :=(d/dt) j φ i (t), i, j = 0, ,r − 1. CONSTRUCTION OF MULTISCALETS 247 Example 1 The multiscalets for multiplicity r = 2are φ 0 (t) = (3t 2 − 2t 3 ), φ 1 (t) = t 3 − t 2 for t ∈[0, 1], (6.3.8) φ 0 (t) = φ 0 (2 − t), φ 1 (t) =−φ 1 (2 −t) for t ∈[1, 2]. (6.3.9) We can verify that (t)| t=1 =  φ 0 (t)φ  0 (t) φ 1 (t)φ  1 (t)  | t=1 =  10 01  . It is easy to find that φ 0 (1) = 1, φ 1 (1) = 0, φ  0 (t)| t=1 =[6t − 6t 2 ] t=1 = 0, φ  1 (t)| t=1 =[3t 2 − 2t]| t=1 = 1. The curves of φ 0 (t) and φ 1 (t) with explicit expressions are plotted in Fig. 6.1. Recall from (6.1.2) and (6.1.3) that (t) =  C k (2t − k) −1  −1 = diag{1, 2, ,2 r−1 }. Let us evaluate the dilation coefficients by taking t = m/2, m ∈ Z in (6.1.4),   m 2  =  k C k (m − k) −1 =  k C k δ 1,m−k  −1 = C m−1  −1 . (6.3.10) Since  has a support of [0, 2],allC k = 0fork ≥ 3. For the three nonzero coeffi- cients, we have from (6.3.10) that C 0 =   1 2  , C 1 = (1) = I =  = diag  1, 1 2 , ,  1 2  r−1  , (6.3.11) C 2 =   3 2  . While C 1 was given in (6.3.11) for any multiplicity r, C 0 and C 2 can be obtained for the case of r = 2 in the next example. For arbitrary r, the general expressions of C 0 and C 2 will be derived later in this section. 248 CANONICAL MULTIWAVELETS 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 t φ 0 (t), φ 1 (t) φ 0 (t) φ 1 (t) FIGURE 6.1 Multiscalets of r = 2 from analytic expression. Example 2 Evaluate C 0 and C 2 for r = 2. Solution (t) =  φ 0 (t)φ  0 (t) φ 1 (t)φ  1 (t)  =  (3t 2 − 2t 3 ) 6(t − t 2 ) t 3 − t 2 (3t 2 − 2t)  for t ≤ 1. (6.3.12) Hence   1 2  =  1 2 3 2 − 1 8 − 1 4  , C 0 =   1 2   =  1 2 3 2 − 1 8 − 1 4  10 0 1 2  =  1 2 3 4 − 1 8 − 1 8 .  . (6.3.13) To evaluate C 2 , we need ( 3 2 ). However, we cannot set t = 3 2 in (6.3.12). In- stead, (3/2) may be found from ( 1 2 ) by symmetry/antisymmetry about t = 1 (see Fig. 6.2), yielding   3 2  =  1 2 − 3 2 1 8 − 1 4  . CONSTRUCTION OF MULTISCALETS 249 Therefore C 2 =   3 2   =  1 2 − 3 2 1 8 − 1 4  10 0 1 2  =  1 2 − 3 4 1 8 − 1 8  . (6.3.14) Next let us derive C 0 and C 2 for arbitrary r . The property (6.3.7) may be written in amatrixformas |φ((2 − t)=S|φ((t), (6.3.15) where S =     1 −1 ··· (−1) r−1     = S −1 . Applying the dilation equation (6.3.4) to (6.3.15), we obtain LHS =|φ(2 −t) = C 0 |φ(4 − 2t) +C 1 |φ(3 − 2t) +C 2 |φ(2 − 2t)  = C 0 |φ[2 −(2t − 2)] + C 1 |φ[2 −(2t − 1)] + C 2 |φ(2 − 2t)  = C 0 S|φ(2t − 2)+C 1 S|φ(2t − 1)+C 2 S|φ(2t), where the last equality was arrived at by using the symmetry–antisymmetry property of (6.3.15). Applying the dilation equation (6.3.4) to the right-hand side of (6.3.15), we have RHS = S[C 0 |φ(2t)+C 1 |φ(2t − 1)+C 2 |φ(2t − 2)]. Equating both sides, we have C 0 |φ(2t)+C 1 |φ(2t − 1)|C 2 |φ(2t − 2)=S −1 C 2 S|φ(2t) + S −1 C 1 S |φ(2t − 1) + S −1 C 0 S |φ(2t − 2). By linear independence of translations φ(2t − k), we claim that C 0 = S −1 C 2 S = SC 2 S −1 . (6.3.16) The component expression of (6.3.16) is [C 0 ] ij = (−1) i+j [C 2 ] ij . [...]... p δ p,q δ1,k δk,m+1 k∈Z 1 2 · 2k − m ˘ ORTHOGONAL MULTIWAVELETS ψ(t ) 255 Hence φ(2t), φ(2t − m) 1 = δ0,m −2 (6.3.23) Equation (6.3.23) will be used in Section 6.5 ˘ 6.4 ORTHOGONAL MULTIWAVELETS ψ(t ) In the previous section the orthogonal multiscalets were constructed Naturally, one ˘ ˘ expects to build the corresponding multiwavelets Multiwavelets ψ0 (t), , ψr −1 (t) are orthogonal to multiscalets... − 0.1 − 0.15 0 0.5 1 1.5 2 2.5 − 0.2 3 0 0.5 1 1.5 2 2.5 3 ˘ ˘ FIGURE 6.4 Intervallic multiwavelets ψ0 (t) and ψ1 (t) of r = 2 6.5 INTERVALLIC MULTIWAVELETS ψ(t ) The orthogonal multiwavelets constructed in the previous section are orthogonal to the multiscalets in the standard L 2 sense However, these multiwavelets are oscillatory and have relatively wide supports Most inconveniently, they are not... the multiwavelet, Walter introduced the orthogonal finite element multiwavelets [4], which we referred to as the intervallic multiwavelets to avoid confusion with the finite element method (FEM) in electromagnetics This multiwavelet family is comprised of the intervallic multiwavelet and its dual, namely the intervallic dual multiwavelets As usual, we denote the closed linear span V p in L 2 (R) of {φ0... 0.2 0 − 0.2 1 1.1 1.2 1.3 1.4 1.5 t 1.6 1.7 1.8 1.9 FIGURE 6.7 Intervallic dual multiwavelets of r = 2 2 WORKING EXAMPLES 273 and ˜ ψ0 (t) φ0 (2t − 2) = ˜ φ1 (2t − 2) ψ1 (t) ˜ ˜ The intervallic dual multiwavelets ψ0 and ψ1 are plotted in Fig 6.7 Example 4 Derive explicit expressions of the multiwavelets and dual multiwavelets for r = 2 Solution D1 = −1 2 1 − 16 3 2 1 8 D2 = , 1 0 0 1 2 , D3 = −1 2 −3... in the Sobolev sampling sense Hence we introduce the dual multiwavelets The dual ˜ multiwavelets ψ are related to φ by    φ0 (2t − 2) 1   ···     = ···      ··· φr −1 (2t − 2)  ˜ ψ0 (t)   ···  → φ j (2t − 2) = 2 j ψ j (t),  ˜   ··· ˜ 2(r −1) ψr −1 (t)  2 ··· j = 0, 1, , r − 1 Detailed study of the dual multiwavelets is deferred to Section 6.7 6.6 MULTIWAVELET EXPANSION... orthogonal wavelets are constructed according to Eq (6.4.1) They are plotted in Fig 6.4 By construction, the scalets are orthogonal to the wavelets and their integer translations Hence V1 = V0 ⊕ W0 , V2 = V0 ⊕ W0 ⊕ W1 , Wi ⊥ W j for i = j Unfortunately, a wavelet is not orthogonal to its translations, nor a scalet to its translations 258 1 0.8 0.6 0.4 0.2 0 − 0.2 − 0.4 − 0.6 − 0.8 −1 CANONICAL MULTIWAVELETS... 260 CANONICAL MULTIWAVELETS In general, −1 Dm = (−1)m T C3−m , m ∈ Z (6.5.2) Verification One can verify from (6.5.2) that D2 = −1 C T 1 D0 = −1 C T 3 , =0 because C3 = 0 Since only C0 , C1 , C2 = 0, (m = 3, 2, 1 in (6.5.2)), we obtain ψ(t) = D1 φ(2t − 1) + D2 φ(2t − 2) + D3 φ(2t − 3) T = − −1 C2 φ(2t − 1) + φ(2t − 2) − −1 C T 0 φ(2t − 3) (6.5.3) Figure 6.5 illustrates the two multiwavelets, ψ0 and ψ1... is also held for the wavelets The property C0 = SC2 S may be extended to the G as G 0 = SG4 S G 1 = SG3 S, where S = diag{1, −1, , (−1)r −1 } As a result the first two equations in (6.4.5) become identical to the remaining two Employing this pattern of the G and also X = SXS Y = SY T S, we obtain [X ]i j = [X ] ji = = φi (t)φ j (t) dt φi (2 − t)φ j (2 − t) dt ˘ ORTHOGONAL MULTIWAVELETS ψ(t ) 257 φi... 1)φ p (t − k), (6.6.10) 264 CANONICAL MULTIWAVELETS and let f ∈ V1 with expansion (6.6.6) Hence the difference f (t) − f 0 (t) = r −1 f ( j) k + 1 2 2− j φ j (2t − 2k) k j=0 f ( j) (k + 1)[2− j φ(2t − 2k − 1) − φ j (t − k)] + k The first summation on the RHS of the previous equation is related to the intervallic dual multiwavelet ˜ 6.7 INTERVALLIC DUAL MULTIWAVELETS ψ(t ) The intervallic dual wavelet... By using the dilation equations of both scalets and wavelets, we integrate φi (2t − k)φ j (2t − m) A change of variable converts these inner product integrals into φi (2t − k)φ j (2t − m) dt = 1 2 φi (t)φ j (t − m + k) dt, (6.4.3) where φi and φ j are supported on [0, 2] Hence the only inner products needed are the two matrices 256 CANONICAL MULTIWAVELETS X= (t) T (t), Y = (t) T (t − 1) = (t + 1) T . possessed by any wavelets, as proved in [1]. To overcome the limitations of the regular (i.e., scalar) wavelets, mathematicians have proposed multiwavelets. There are two categories of multiwavelets,. VECTOR-MATRIX DILATION EQUATION Multiwavelets offer more flexibility than traditional wavelets by extending the scalar dilation equation ϕ(t) =  h k ϕ(2t − k) 240 Wavelets in Electromagnetics and Device. INTERVALLIC MULTIWAVELETS ψ( t ) The orthogonal multiwavelets constructed in the previous section are orthogonal to the multiscalets in the standard L 2 sense. However, these multiwavelets are oscilla- tory