A Class of Compactly Supported Orthonormal B-Spline Wavelets

Transcription

1 A Class of Compactl Supported Orthonormal B-Spline Wavelets Okkung Cho and Ming-Jun Lai Abstract. We continue the stud of constructing compactl supported orthonormal B-spline wavelets originated b T.N.T. Goodman. We simplif his constructive steps for compactl supported orthonormal scaling functions and provide an inductive method for constructing compactl supported orthonormal wavelets. Three eamples of compactl supported orthonormal B-spline wavelets are included for demonstrating our constructive procedure.. Introduction After the seminal construction of compactl supported orthonormal univariate wavelets (cf. []), there have been man attempts to construct compactl supported orthonormal wavelets using B-spline functions due to the facts that B-splines have nice refinement properties and eplicit representations. Three major research works along this direction are worth mentioning. In [] and [], two kinds of semi-orthonormal B-spline wavelets were constructed. One of them is compactl supported although the orthonormalit among the translates is lost. In [] the researchers initiated a fractal functional approach to construct compactl supported orthonormal wavelets from B-spline functions. Eamples of C 0 and C compactl supported B-spline wavelets were constructed. In [], the researchers used the interwining multiresolution analsis to show the eistence of compactl supported orthonormal B-spline wavelets using multi-wavelet technique. In [], the researchers used orthogonal polnomials to construct compactl supported smooth wavelets. An eample of C multiwavelets was shown. Furthermore, in [], the researchers etended the interwinding multiresoluton analsis to the bivariate setting. Eamples of compactl supported continuous piecewise linear spline wavelets are given. These approaches have an obvious difficulit that the number of wavelets is dependent on the Splines and Wavelets: Athens 00 Guanrong Chen and Ming-Jun Lai Editors pp.. Copright Oc 00 b Nashboro Press, Brentwood, TN. ISBN All rights of reproduction in an form reserved.

2 O. Cho. and M. J. Lai size of the support of the scaling functions. Recentl, another approach of multi-wavelets was given in [9] where Goodman showed how to construct compactl supported scaling functions using B-splines of an degree and indicated how to construct associated wavelets. One of the advantages of the approach in [9] is that the number of wavelets is alwas for B-splines of an degree. Although the construction of orthonormal scaling functions is clearl described, a constructive method of wavelets was given without an supporting eamples. One of reasons is that the construction is dependent on the factorization of positive definite matrices. The techniques in [] were used to factorize nonnegative Laurent polnomial matrices. The require a lot of manual computation. Followed the Goodman approach, we worked through his steps and found out that the computation of orthonormal scaling functions can be simplified b introducing a numerical approimation method of factorization of Laurent polnomial matrices and a new inductive method of constructing wavelets is given so that whole constructive procedure becomes much simpler. The purpose of this paper is to describe this new and simpler constructive procedure. One of our aims is to make these compactl supported orthonormal B-spline wavelets to become available to wavelet analsists as well as general wavelet practicianers. The paper is organized as follows. We first describe a general procedure in the preliminar section below. The procedure is similar to the one given in [9]. Then we eplain how to factorize Laurent polnomial matrices b using a smbol approimation method similar to the one in [] The convergence analsis of the method in the setting of Laurent polnomial matrices is given in [8]. This consists of Section. In Section, an inductive method for constructing compactl supported B-spline wavelets is introduced. In, we summarize the computational steps and present three eamples of compactl supported B-spline wavelets to illustrate the computation procedure.. Preliminaries Fi integers r > and d. Let φ,, φ r be compactl supported continuous real-valued functions in R d and Φ = (φ,, φ r ) T. We suppose that Φ is refinable. That is, there eist matrices A k s of size r r such that Φ() = k Z d A k Φ( k), R d.

3 Compactl Supported B-Spline Wavelets Also, we sa Φ is orthonormal if R d φ i ()φ j ( k)d = {, if i = j and k = 0, 0, otherwise for all i, j =,, r. Φ generates a space S if S consists of all finitel linear combination of integer translates of entries of Φ. Net we define a Grammian matri G = (G ij ) i,j=,,r of size r r associated with Φ b G ij (z) = k Z d z k R d φ i ()φ j ( k)d for all i, j =,, r with z C \ {0}. We note that Φ is orthonormal if and onl if its Grammian matri G is the identit. We suppose that Φ generates a space S. Then for an compactl supported functions ψ,, ψ s in S, there eists a finitel man nonzero matrices C k of size s r such that Ψ() = (ψ (),, ψ s ()) T = k Z d C k Φ( k). In terms of Fourier transform, we have Ψ(ω) = C(z) Φ(ω) where C(z) denotes the s r matri of Laurent polnomials, i.e., C(z) := C k z k. k Z d A square matri C(z) is said to be invertible if det(c(z)) is a monomial of z, e.g., αz m for a scalar α 0 and an integer m Z. It is clear that if C(z) is invertible, Ψ generates the same S. A proof of the following result can be found in literature (cf. e.g., [9]). Lemma. Fi d =. Suppose that Ψ is compactl supported and generates a space S. Let G(z) = (G ij (z)) i,j=,,r of size r r b G ij (z) = z k ψ i ()ψ j ( k)d k Z R for all i, j =,, r be the Grammian matri associated with Ψ. If the determinant of the Grammian matri G(z) is a nonzero constant, then there eists a Φ which is orthonormal and generates S. The converse is also true.

4 O. Cho. and M. J. Lai The above lemma reveals a ke for constructing orthonormal vector of scaling functions: find ψ,, ψ r which generate a space S such that its Grammian matri has a constant determinant. We now follow the steps in [9] to use B-splines for constructing an orthonormal vector of scaling functions with r =. Let N m be the normalized B-spline of order m, in terms of Fourier transform, ( e iω N m (ω) = iω Let V 0 = span{n m ( k), k Z} be the spline space. Since N m is a refinable function, for V being spanned b the integer translates of N m ( k), k Z, we have V 0 V. Thus, letting ψ () = N m () and ψ () = N m ( ), ψ and ψ generate V. On the other hand, b the dilation equation, there eist two finite sequences a k and a k+ such that N m () = k Z ) m a k ψ ( k) + k Z a k+ ψ ( k). Note that the Fourier transform of the above equation is N m (ω) = A(z) N m (ω) and N m (ω) =A 0 (z) ψ (ω) + A (z) ψ (ω) =A 0 (z) N m ( ω ) + A (z) z Nm ( ω ) where A 0 (z) = k Z a k z k and A (z) = k Z a k+ z k. It follows that A(z) = A 0 (z ) + za (z ) ( ) m + z It is known that A(z) =. Note that the proof of the following lemma is constructive. Lemma. There eist two Laurent polnomials B 0 (z) and B (z) of degree m such that A 0 (z)b 0 (z) + A (z)b (z) =.

5 Compactl Supported B-Spline Wavelets Proof: Recall ( + z = + z ) m m ( ) ( ) m j ( ) j n + z z = j + m ( ) ( m z j =:l(z)a(z) + l( z)a( z) ) m j ( ) j + z b using binomial epansion, where l(z) is a polnomial of degree m. Thus, we have =(A 0 (z ) + za (z ))l(z) + (A 0 (z ) za (z ))l( z) =A 0 (z )(l(z) + l( z)) + A (z )z(l(z) l( z)). That is, B 0 (z ) = l(z) + l( z) while B (z ) = z(l(z) l( z)). We now define a new spline function in terms of Fourier transform b Recall that M m (ω) = B (z) ψ (ω) + B 0 (z) ψ (ω). (.) N m (ω) = A 0 (z) ψ (ω) + A (z) ψ (ω) It follows that N m and M m generate V since the determinant of the following matri [ ] [ ] [ ] Nm (ω) A0 (z) A = (z) ψ (ω) (.) M m (ω) B (z) B 0 (z) ψ (ω) is constant. Furthermore, N m (), N m ( ), M m (), M m ( ) generate V. Define ψ = k Z α km m ( k) for some finitel man nonzero coefficients α k. We will show how to find such ( α k that the Grammian matri associated with {ψ, ψ, ψ }, G(z) = z k ψ i ()ψ j ( ) k)d has a constant determinant. Put k Z r(z) = k Z α k z k. R i,j=,, The computation in [8] shows det G(z ) = D(z)r(z)r(/z) + D( z)r( z)r( /z), (.)

6 8 O. Cho. and M. J. Lai where with D(z) = (a(z ) zb(z ))(a(z) zb(z) ), (.) a(z) = z k N m ()N m ( k)d, k Z b(z) = z k N m ()N m ( k )d. k Z Now we claim that there eists a polnomal p(z) 0 such that D(z)p(z) + D( z)p( z) =. (.) Once we have such a p(z), it follows from the Riesz-Féjer lemma that there eists a polnomial r(z) such that r(z)r(/z) = p(z). This r(z) is the polnomial we look for such that the determinant (.) of Grammian matri G(z) is a nonzero constant. To prove the claim (.), we need the following lemma (see a constructive proof in [0]). Lemma. Let p be a polnomal of degree n with all its zeros in [, ) having a positive leading coefficient. Then there eists a unique polnomial q with real coefficients of degree n such that p()q() + p( )q( ) =. for [0, ]. Moreover, ( ) n q() > 0 for (0, ). To use the lemma above, we need to eamine the zeros of D(z) = (a(z ) zb(z ))(a(z) zb(z) ). Let us simplif a(z ) zb(z ) a little bit more. a(z ) zb(z ) = z j N m ()N m ( j)d j Z R + z (j+) N m ()N m ( j )d j Z R = z j N m ()N m ( j)d j Z = z j j Z = z j j Z R R R N m ()N m ( + j)d N m ()N m (m j )d = N m (m j)z j = N m (j)z m j j Z j Z

7 Compactl Supported B-Spline Wavelets 9 where we have used the smmetric propert of B-spline functions, i.e., N m () = N m (m ). It is well-known that E m (z) := j Z N m(j)z j is an Euler-Frobinus polnomial which is never zero for e iω for an ω. The zeros of E m (z) are in (, 0) since all coefficients of E m (z) are positive. B the following Lemma, E m (z) can be written in terms of p() with = sin (ω/) and p() has onl zeros in [, + ). Net we consider a(z) zb(z). As above, a(z) = z j N m ()N m ( j)d = j Z R N m(m + j)z j j Z = [m/] (N m(m) + N m (m + j)(z j + /z j )) j= is a real polnomial in cos(ω) which can be converted to a polnomial in terms of = sin (ω/). So is a(z). Similarl, b(z) = z j N m ()N m ( j )d j Z j Z R = N m (m + j + )z j It follows that j Z = z / [m/] j= [m/] N m (m + j + )z (j+)/ = z /(N m(m + )z / + N m (m )z / + N m (m + )z / + N m (m )z / + ) = [m/] N z / m (m + j + )(z (j+)/ + z (j+)/ ). zb(z) = [m/] [m/] = i, N m (m + j + )(z (j+)/ + z (j+)/ ) N m (m + j + )N m (m + i + ) z i+j+ + z (i+j+) + z i j + z j i is again a real polnomial in cos(ω) which can be converted to a polnomial in terms of = sin (ω/) b Lemma below. The zeros of a(z) zb(z)

8 0 O. Cho. and M. J. Lai are contained in the zeros of a(z) which are located in [, + ). Therefore, Lemma implies that a polnomial p(z) eists such that D(z)p(z) + D( z)p( z) = and p(z) > 0. This completes the proof of our claim. Lemma. Let m c(z) = c j z j m be a polnomial which has onl zeros in (, 0) with real coefficients c j and c j = c j. Then there is a polnomal p of degree m such that and p has onl zeros in [, ). p() = c(e iω ), with = sin (ω/) Proof: Clearl, c(z) can be written as m m ( ) j z + /z c(z) = c 0 + c j cos(jω) = d j j= for some real coefficients d 0,, d m. Then we define p() = m d j ( ) j Then we can see that c(z) = p(/ (z + /z)/) = p(sin (ω/)). If p() = 0 with = / (z + /z)/ for z (, 0), then z + /z implies that. A major step in the computation of orthonormal scaling function vector is to factorize Grammian matri G(z) which will be discussed in the following section. This finishes the construction steps for compactl supported orthonormal scaling functions based on B-splines.. A Computational Method for Matri Factorization Let ψ, ψ, ψ be three compactl supported functions defined in the previous section. Since the determinant of the Grammian matri G(z) associated with Ψ = (ψ, ψ, ψ ) T is nonzero monomial, it can be factored into G(z) = B(z)B(z) with invertible polnomial matri B(z), where B(z) stands for the transpose and conjugate of B(z). In this section we discuss a computational method for the matri factorization. Although the method in [] is constructive, it requires a technique to factor a positive

9 Compactl Supported B-Spline Wavelets definite Hermitian matri into matrices with rational Laurent polnomials, a method to identif the location of poles, an epansion of the rational entries into a special format, and construction of unitar matrices to cancel these poles. It is reall not an eas task. To simplif the factorization, we describe a straightforward computational method to do such factorizations. The basic ideas are as follows. Let A be a bi-infinite matri with entries A ij = c i j. where {c j } is a finite sequence. Let be a bi-infinite sequence. Then = A is another bi-infinite sequence. Formall, the discrete Fourier transform of can be given b Y (ω) = j j e ijω = A(ω)X(ω) with X(ω) = k k e ikω and A(ω) = k c k e ikω. This is an identification of the bi-infinite matri A and Laurent polnomial A(ω). If A(ω) is smmetric, i.e., A( ω) = A(ω) and positive, we know that it can be factored into a polnomial B in e iω such that A(ω) = B(ω)B( ω) b Riesz-Féjer factorization. Then the matri A can be factored into a product of two matrices BB T, where B is a lower-trangular bi-infinite matri. This is indeed the case as discussed in []. For a positive definite matri M(ω) of size r r, we can write it as M(ω) = k m k e ikω with r r matrices m k. For simplicit, we assume that m k are matrices with real entries. Then we can identif M(ω) b a bi-infinite block matri M = (M ij ) <i,j< with M ij = m i j. When M(ω) can be factored into a product of two polnomial matrices N(ω) and N( ω), M can be factored into a product of two bi-infinite matrices N and N T. The converse is also true. Our computational method is to compute the Cholesk decomposition of a central section M l := [M ij ] i,j l of the bi-infinite matri M with l > being an integer. That is, let [M ij ] i,j l = N l N T l with lower triangular matri N l = [a ij ] i,j l of size rl rl. Let n 0 = [a ij ] r(l )+ i,j rl, n = [a ij ] r(l )+ i rl,r(l )+ j r(l ),... and define N l (ω) = k 0 n ke ikω, Then we can show that N l (ω) converges to N(ω) as l +. (See [8] for a proof.)

10 O. Cho. and M. J. Lai Eample. Let M(ω) = [ 8 + z + /z ] + z + /z be a Hermitian and positive definite matri. Then M(ω) = m 0 + m z + m /z with three matrices m, m 0, m of size. Let M be of size of 0 0 with m 0 on the diagonal blocks m on the upper diagonal blocks and m on the sub-diagonal blocks. The remaining entries are zero. Using computer software MATLAB, we find [ ].09 0 N l (ω) = [ It can be verified b using MAPLE to see that M(ω) = N l (ω)n l ( ω) T + o() with o() = 0 9. Our main result in this section is the following : Theorem. Let M be a bi-infinite matri associated with a positive definite Hermitian matri M(ω) and let N l be the Cholesk factorization of the central section M l. Then the block entr [a ij ] r(l )+ i,j rl converges to the m 0 eponentiall faster. Similar for the other block entries. We refer the reader to [8] for a proof and more numerical eamples. ] z. Construction of the Associated Wavelets For the Grammian matri G associated with ψ, ψ, ψ, let G(z) = B(z) B(z) be the Riesz-Fejér factorization as discussed in the previous section. Letting Φ(z) = B(z) Ψ(z), (.) we know that the Grammian matri of Φ is B(z) G(z)B (z) which is the identit matri and hence Φ = (φ, φ, φ ) T is an orthonormal refinable function vector. In this section we discuss how to compute the associated wavelets. We begin with Lemma. Φ is refinable. That is, letting Φ() = (φ (), φ (), φ (), φ ( ), φ ( ), φ ( )) T,

11 Compactl Supported B-Spline Wavelets there eists matri coefficients p i of size such that Φ() = k Z p k Φ( k) or Φ(ω) = P(z) Φ(ω), (.) where P(z) is a matri mask of size. Proof: Indeed, since Ψ is refinable, Ψ(ω) = C(z) Ψ(ω) (.) for a matri mask C(z) of size. If we denote b Ψ() = (ψ (), ψ (), ψ (), ψ ( ), ψ ( ), ψ ( )) T, the dilation relation of Ψ can be rewritten in terms of Ψ. That is, b (.), Ψ() = c k Ψ( k) k Z = c k Ψ( k) + c k+ Ψ( k ) c k Ψ( k). k Z k Z k Z In terms of Fourier transform, Ψ(z) = C(z) Ψ(z) with matri C(z) of size. B (.), Φ() = k Z b kψ( k) for matri coefficients b k of size. It follows that Φ() = k Z b k Ψ( k) Φ( ) = k Z b k Ψ( k ) and Φ() = k Z bk Ψ( k). In terms of Fourier transform, Φ(z) = B(z) Ψ(z). Note that B(z) is invertible because that both Φ and Ψ generates the same space S. We have Ψ(ω) = B(z) Φ(ω). Therefore, we have Φ(ω) =B(z) Ψ(ω) =B(z) C(z) Ψ(ω) =B(z) C(z) B(z) Φ(ω)

12 O. Cho. and M. J. Lai which completes the proof. Net using the dilation relation (.), the orthonormalit of φ i, i =,, implies I = Φ()Φ() T d R = p i Φ( i) Φ T ( j)dp T j (.) i,j Z = R i,j Z p i δ i,j I p T j = i Z p i p T i. Since Φ is of compact compact, we ma assume that onl m + terms p 0, p,, p m are nonzero matri coefficients. Furthermore, 0 = Φ()Φ( k) T d = = R m p i i,j= i,j= R Φ( i) Φ T ( j k)dp T j m m p i δ i,j+k I p T j = p i p T i k, for k =,, m. In particular, we have i=k (.) p m p T 0 = 0 (.) We now use induction on m to show how to construct three compactl supported orthonormal wavelets h, h, h S such that letting W := span{h ( i), h ( j), h ( k), i, j, k Z}, W is the orthogonal completement of S in S. That is, S = S W. More precisel, let H = (h, h, h ) T. (.) and the Fourier transform of H give Φ(ω) =P(z) Φ(ω) Ĥ(ω) =Q(z) Φ(ω) where Q(z) = i Z q iz i is a Laurent polnomial matri of size. The orthogonal completementness and the orthonormalit of h, h, h impl that the matri [ ] P(z) Q(z) is a unitar matri. That is, Q(z) is an unitar etension of P(z).

13 Compactl Supported B-Spline Wavelets It is trivial when m = 0. Indeed, in this case, P(z) = p 0 is a scalar matri. We simpl choose Q(z) to be a scalar matri which is an orthonormal m etension of p 0. Assume that for m, when P m (z) = p k z k is an orthonormal matri of, we can find Q m (z) such that [ ] Pm (z) Q m (z) is unitar. We now consider the case of m + : P m+ (z) = m+ p k z k satisfing orthonormal properties in (.) and (.). In particular, (.) implies that there eists a unitar matri U 0 of size such that p 0 U 0 = [0 p b 0 ] and p m+u 0 = [ p a m+ 0 ], where p b 0 is of size and the same for p a m+. Writing p k U 0 = [ p a k, pb k ] with pa k and pb k being of size. Then m+ m P m+ (z)u 0 = [ p a k zk, p b k zk ] Let Then it follows that U := k= [ z I ] 0. 0 I m m P m+ (z)u 0 U =[ p a k+z k, p b kz k ] = m [ p a k+, pb k ]zk. That is, P m (z) := P m+ (z)u 0 U has onl m + terms and is unitar. B induction, we can find an unitar etension Q m (z) such that [ ] Pm+ (z)u 0 U Q m (z) is unitar. Clearl, [ Pm+ (z)u 0 U Q m (z) ] U U 0 = [ Pm+ (z) Q m (z)u U 0 is also unitar. It follows that Q m+ (z) := Q m (z)u U 0 is an unitar etension of P m+ (z). This completes the induction procedure. Therefore, we conclude the following : ]

14 O. Cho. and M. J. Lai Theorem.. For the given refinable orthonormal functions φ, φ, φ, we can construct three associated wavelets h, h, h such that h i ( k) s are orthogonal to φ j ( m) for all j =,, and m Z, h i ( k) s are orthonormal among each other for all i =,, and k Z, and the linear span of h i ( k), i =,, and k Z forms a space W which is an orthogonal completement of S in S.. Eamples In this section we want to provide a few eamples based on the construction method in the previous sections. Recall that we have considered the B-spline of order m, N m (). Thus ψ () = N m (), ψ () = N m ( ). Now we organize our computation in the following four major steps: Step. Computation of M m (). First we find B 0 (z), B (z) satisfing the equation as in Lemma. And then we define M m () in terms of Fourier transform according to (.). Step. Computation of ψ (). We have to begin with the computation of the determinant of G(z), mainl we compute D(z) satisfing (.). Then b Lemma we find a polnomial p(z) 0 such that D(z)p(z) + D( z)p( z) =. A straightforward computation in [] gives r(z) = α k z k such that p(z) = r(z)r(/z) and so we get ψ () = α k M m ( k). Step. Computation of φ, φ, φ. We need to find the entries for the Grammian matri G(z) using ψ, ψ, ψ. Then b using the method in Section we factorize into G(z) = B(z)B(z) with the help from the computer software MAPLE. Therefore we can define the orthonormal scaling vector Φ = (φ, φ, φ ) in terms of Fourier transform : Φ(ω) = B(z) Ψ(ω). Step. Computation of the associated wavelets h, h, h. In this step, we follow Lemma. so that we have the dilation relation for φ, φ, φ : Φ(ω) = P(z) Φ(ω) where P(z) = B(z) C(z) B(z). Then b induction on m, i.e., steps in the proof of Theorem. we find the unitar etension Q(z) of P(z). Hence we define h, h, h in Fourier transform: Ĥ(ω) = Q(z) Φ(ω),

15 Compactl Supported B-Spline Wavelets where H = (h, h, h ) T. Following the above steps, we have the orthonormal scaling functions φ, φ, φ and the corresponding wavelet functions h, h, h for m =,, listed as follows :.. m= : Linear B-spline case We have the following scaling functions : φ () = N (), φ () = N () ( + ) N () + ( ) N ( ), φ () = α j N ( j) + β k N ( k) with α j s and β ks defined as follows: ( + ) ( + ) α 0 =, β 0 =, ( )( ) α =, β =, ( ) ( ) + α =, β =, ( ) ( ) + β = The wavelet functions associated with the above scaling functions are: h () = α j φ ( j)+ β k φ ( k)+ with α j s, β k s and γ ls defined as follows: γ l φ ( l) α 0 = , β 0 = , γ 0 = , α = , β = , γ = , α = , β = , γ = 0.0, α = , β = , γ = l=0. h () = α j φ ( j)+ β k φ ( k)+ γ l φ ( l) l=0

16 8 O. Cho. and M. J. Lai with α j s, β k s and γ ls defined as follows: α 0 = , β 0 = , γ 0 = 0.088, α = 0.09, β = 0.99, γ = , α = , β = 0.890, α = , β = γ = 0.908, h () = α j φ ( j)+ β k φ ( k)+ with α j s, β k s and γ ls defined as follows: γ l φ ( l) α 0 = , β 0 = , γ 0 = , α = , l=0 β = , γ = , α = 0.009, β = , γ = , α = 0.00, β = , γ = The graphs for three linear B-spline scaling functions and wavelets can be seen from Fig... And the masks associated with scaling functions φ, φ, φ and wavelet functions h, h, h are P(z) = p j z j and Q(z) = q j z j where the matri coefficients p 0, p, q 0, and q are listed as follows : p 0 = p = q 0 =

17 Compactl Supported B-Spline Wavelets Figure.. The scaling functions φ, φ, φ on the left colume and the asociated wavelet functions ψ, ψ, ψ on the right colume with the linear B-spline function N () on the top.

18 0 O. Cho. and M. J. Lai q = m= : Quadratic B-spline case Now b using the quadratic B-spline function N we have the following scaling functions : φ () = α j N ( j) + with α j s and β ks defined as follows: k= β k N ( k) α 0 =.98089, β = , β = , α = 0, β = , β = , α = , β = , β = , α = , β = 0.09, β = , α = , β 8 = , β = , α = , β 9 =.08, α = , β 0 =.00, φ () = α j N ( j) + with α j s and β ks defined as follows: k= β k N ( k) α 0 =.088, β =.9090, β = , α =.009, β =.8, β = , α =.0900, β = 0.09, β = , α = , β = 0.8, β = , α = 0.0, β 8 = 0.8, β = , α = , β 9 = , α = , β 0 = 0.0, φ () = α j N ( j) + β k N ( k)

19 Compactl Supported B-Spline Wavelets with α j s and β ks defined as follows: α 0 =.880, β 0 =.08, β 8 = , α =.9, β =.0908, β 9 = 0.008, α = 0.98, β = 0.098, β 0 = , α = , β = , β = , α = , β =.090, β = , α = , β = 0.800, β = , α = , β = 0.880, β = , β = , β = , The wavelet functions associated with the above scaling functions are: h () = α j φ ( j)+ β k φ ( k)+ with α j s, β k s and γ ls defined as follows: α 0 = , β 0 = , γ 0 = 0, γ l φ ( l) α = , β = , γ = , α = , β = 0.89, γ = , α = 0.088, l=0 β = 0., γ = , α = , β = , γ = 0.089, α = 0.89, β = 0.8, γ = , α = , β = 0.80, γ = 0.899, α = 0.080, β = 0.080, γ = 0.08, h () = α j φ ( j)+ β k φ ( k)+ with α j s, β k s and γ ls defined as follows: γ l φ ( l) l=0 α 0 = , β 0 = , γ 0 = 0, α = , β = , γ = , α = , β = , γ = , α = , β = , γ = , α = , β = 0.080, γ = , α = , β = , γ = 0.000, α = 0.00, β = 0.998, γ = 0.00, α = 0.889, β = 0.9, γ = 0.8,

20 O. Cho. and M. J. Lai h () = α j φ ( j)+ β k φ ( k)+ with α j s, β k s and γ ls defined as follows: α 0 = , β 0 = , γ 0 = 0, γ l φ ( l) α = , β = , γ = , α = , β = 0.000, γ = , α = 0.00, β = , γ = , α = 0.088, β = , l=0 γ = 0.008, α = , β = 0.080, γ = 0.09, α = , β = 0.8, α = , β = 0.998, γ = 0.080, γ = , The graphs for three quadratic B-spline scaling and wavelet functions can be seen from Fig... And the masks associated with scaling functions φ, φ, φ and wavelet functions h, h, h are P(z) = p j z j and Q(z) = q j z j where the matri coefficients p i s and q is are listed as follows : p 0 = p = p =

21 Compactl Supported B-Spline Wavelets p = q 0 = q = q = q = m= : Cubic B-spline case Finall, b using the cubic B-spline function, N we have the following scaling functions : 8 φ () = α j N ( j) + 0 k= β k N ( k) with α j s and β ks defined as follows

22 O. Cho. and M. J. Lai Figure.. The scaling functions φ, φ, φ on the left colume and the asociated wavelet functions ψ, ψ, ψ on the right colume with the quadratic B-spline function N () on the top.

23 Compactl Supported B-Spline Wavelets α 0 =.088, β = , β = , α = 0, β = , β = , α = 0.8, β = , β = , α = 0.89, β = 0.0, β = , α = , β 8 = 0.0, β = , α = 0.098, β 9 = 0.08, β 8 = , α = , β 0 = 0.0, β 9 = , α = , β = , β 0 = , α 8 = , β = 0.008, 9 φ () = α j N ( j) + 0 k= β k N ( k) with α j s and β ks defined as follows: α 0 =.0, β = , β = 0.00, α =.9, β = 0.08, β = , α =.09, β = , β = , α =.89909, β = 0.8, β = , α = 0.880, β 8 =.000, β = , α = , β 9 =.80, β 8 = , α = 0.009, β 0 = 0.090, β 9 = , α = , β = , β 0 = , α 8 = , β = 0.0, α 9 = , 8 φ () = α j N ( j) + 0 β k N ( k)

24 O. Cho. and M. J. Lai with α j s and β ks defined as follows: α 0 = 0.08, β 0 = , β = 0.08, α = , β = , β = 0.00, α =.8898, β =.980, β = 0.0, α =.9099, β =.09, β = , α = 0.99, β = 0.09, β = , α = 0.00, β =.8, β = , α = , β =.9098, β = , α = , β = 0.08, β 8 = , α 8 = , β 8 =.00, β 9 = , β 9 =.00888, β 0 = , β 0 = 0.9, The wavelet functions associated with the above scaling functions are: h () = α j φ ( j)+ β k φ ( k)+ γ l φ ( l) j= k= l= with α j s, β k s and γ ls defined as follows: α = , β = , γ = 0, α = , β = , γ = , α = 0.000, β = , γ = , α = 0.090, β = 0.090, γ = , α = 0.009, β = 0.00, γ = , α = , β = 0.99, γ = 0.0, α = 0.88, β = 0., γ = 0.998, α 8 = 0.09, β 8 = , γ 8 = 0.00, α 9 = , β 9 = , γ 9 = , h () = α j φ ( j)+ β k φ ( k)+ γ l φ ( l) j= k= l=

25 Compactl Supported B-Spline Wavelets with α j s, β k s and γ ls defined as follows: α = , β = , γ = 0, α = , β = , γ = 0, α = , β = , γ = , α = , β = , γ = , α = , β = , γ = , α = 0.09, β = , γ = 0.0, α = , β = 0.098, γ = , α 8 = 0.088, β 8 = 0.990, γ 8 = 0.80, α 9 = , β 9 = 0.000, γ 9 = 0.9, h () = α j φ ( j)+ β k φ ( k)+ γ l φ ( l) j= k= l= with α j s, β k s and γ ls defined as follows: α = , β = , γ = 0, α = , β = , γ = , α = 0.000, β = , γ = , α = , β = , γ = , α = , β = , γ = 0.088, α = , β = 0.09, γ = 0.009, α = 0.080, β = 0.0, γ = 0.00, α 8 = , β 8 = , γ 8 = 0.888, α 9 = 0.008, β 9 = , γ 9 = The graphs of the C cubic spline scaling functions and wavelets can be seen from Figure.. And the masks associated with scaling functions φ, φ, φ and wavelet functions h, h, h are P(z) = p j z j and Q(z) = q j z j

26 8 O. Cho. and M. J. Lai where the matri coefficients p j s and q j s are as follows : p 0 = p = p = p = p = q 0 = q = q =

27 Compactl Supported B-Spline Wavelets Figure.. The scaling functions φ, φ, φ on the left colume and the asociated wavelet functions ψ, ψ, ψ on the right colume with the cubic B-spline function N () on the top.

28 0 O. Cho. and M. J. Lai q = q = Acknowledgement: Results in this paper are based on the research supported b the National Science Foundation under the grant No. 0. References. C. K. Chui and J. Wang, A cardinal spline approach to wavelets, Proc. Amer. Math. Soc. (99), pp C. K. Chui and J. Wang, On compactl supported spline wavelets and a dualit principle, Trans. Amer. Math. Soc. 0(99), pp I. Daubechies, Orthonormal bases of compactl supported wavelets, Comm. Pure Appl. Math., (988), pp G. C. Donovan, J. S. Geronimo, D. P. Hardin, Interwining multiresolution analses and the construction of piecewise polnomial wavelets, SIAM J. Math. Anal., (99), pp G. C. Donovan, J. S. Geronimo, D. P. Hardin, and P. R. Massopust, Construction of orthogonal wavelets using fractal interpolation functions, SIAM J. Math. Anal. (99), pp G. C. Donovan, J. S. Geronimo, and D. P. Hardin, Orthogonal polnomials and the construction of piecewise polnomial smooth wavelets, SIAM J. Math. Anal., 0(999), pp G. C. Donovan, J. S. Geronimo, and D. P. Hardin, Compactl supported piecewise affine scaling functions on triangulation, Constr. Appro., (000), J. S. Geronimo and M. J. Lai, Factorization of multivariate positive Laurent polnomials, submitted, T. Goodman, A class of orthonormal refinable functions and wavelets, Constr. Appro. 9(00), pp.-0.

29 Compactl Supported B-Spline Wavelets 0. T. Goodman and C. A. Micchelli, Orthonormal cardinal functions, in Wavelets: Theor, Algorithms and Applications, C. K. Chui et. al. (eds.), Academic Press, San Diego, 99, 88.. D. Hardin, A. Hogan, and Q. Sun, The matri-valued Riesz lemma and local orthonormal bases in shift-invariant spaces, Adv. Comput. Math. 0(00), pp.-8.. M. J. Lai, On computation of Battle-Lemarie s wavelets, Math. Comp. (99), pp Okkung Cho Department of Mathematics Universit of Georgia Athens, GA 00 ocho@math.uga.edu and Ming-Jun Lai Department of Mathematics Universit of Georgia Athens, GA 00 mjlai@math.uga.edu