Compactly Supported Orthogonal Symmetric Scaling Functions

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1 Applied and Computational Harmonic Analysis 7, ) Article ID acha , available online at on Compactly Supported Orthogonal Symmetric Scaling Functions Eugene Belogay and Yang Wang School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia Communicated by Pierre G. Lemarié-Rieusset Received January 17, 1997; revised February 9, 1998 Daubechies 1988, Comm. Pure Appl. Math. 41, ) showed that, except for the Haar function, there exist no compactly supported orthogonal symmetric scaling functions for the dilation =. Nevertheless, such scaling functions do exist for dilations > as evidenced by Chui and Lian s construction 1995, Appl. Comput. Harmon. Anal., 68 84) for = 3); these functions are the main object of this paper. We construct new symmetric scaling functions and introduce the Batman family of continuous symmetric scaling functions with very small supports. We establish the exact smoothness of the Batman scaling functions using the joint spectral radius techniue Academic Press Key Words: wavelets; orthogonal scaling function; symmetric scaling function 1. INTRODUCTION Compactly supported wavelets are typically constructed from a compactly supported single scaling function that generates a multiresolution analysis [5, 14].It is important and nontrivial) to construct scaling functions and hence wavelets) with desirable properties, such as orthogonality, high regularity, symmetry, and small support. Recall that a multiresolution analysis MRA) with dilation factor, where Z and >1, is a seuence of nested subspaces of L R) V V 1 V 0 V 1 V such that V j = span { f j x k): k Z } for some fx) L R),and V j = L R), j Z 137 V j ={0}. j Z /99 $30.00 Copyright 1999 by Academic Press All rights of reproduction in any form reserved.

2 138 BELOGAY AND WANG The function fx) is called the scaling function of the MRA. We shall consider only multiresolution analyses generated by a compactly supported scaling function fx) with orthogonal integer translates. Such an orthogonal scaling function generates an orthogonal MRA which leads to an orthonormal wavelet basis for L R) [14]. Let fx) L R) be a compactly supported orthogonal scaling function of a MRA with dilation.then R fx)dx 0, and fx)satisfies a dilation euation fx)= k Z c k fx k), c k =, where c k are real, and c k 0 for only finitely many k Z [6]. Since the set {fx k): k Z} is orthogonal in L R), the coefficients {c k } must satisfy k Z c k c k+j = δ 0,j, j Z, 1.1) k Z where δ 0,j is the Kronecker symbol. The converse is not true, though; in order for the integer translates of fx) to be orthogonal, certain conditions often overlooked in the study of wavelets) in addition to 1.1)mustbemet[10]. First, Daubechies [5] constructed a family of minimally supported orthogonal scaling functions for dilation = and studied their asymptotically increasing smoothness using Fourier analytic methods. Then, Heller [11], Steffen et al. [15], and Welland and Lundberg [17] constructed compactly supported orthogonal scaling functions for dilations > we describe Heller s construction briefly in Section 3). In applications, such as digital imaging, it is often desirable to use scaling functions that are symmetric. Daubechies [5] showed that if =, then the only symmetric orthogonal scaling function is the Haar function χ [0,1). In order to construct symmetric orthogonal scaling functions, one has to consider dilations > a construction for = 3 is due to Chui and Lian [3]). An alternative approach for = is to give up orthogonality and consider nearly orthogonal symmetric scaling functions [1, 1]. Construction of orthogonal symmetric scaling functions for arbitrary dilations > is the main object of this paper. In Section we introduce definitions and basic results on scaling functions and scaling seuences. In Section 3 we restate Heller s explicit general formula for orthogonal scaling seuences [11]. We use this formula to construct symmetric orthogonal scaling functions for an arbitrary dilation 3 in Section 4. We establish necessary and sufficient conditions for scaling functions to be symmetric, based on the modulus of their symbols. Finally, in Section 5 we introduce a new family of symmetric orthogonal scaling functions with short support the Batman family) and compute their smoothness using the joint spectral radius of matrices.. PRELIMINARY RESULTS Fix an integer. Let S R) denote the set of all real seuences c ={c k : k Z}, such that k Z c k = and c k = 0 for all but finitely many k Z. It is known [7] that for each c S R) there exists a uniue compactly supported c x) in the sense of tempered

3 ORTHOGONAL SYMMETRIC SCALING FUNCTIONS 139 distribution) satisfying c x) = k Z c k c x k) for almost all x R, and ˆ c 0) = 1..1) We call c x) the scaling function corresponding to c. DEFINITION.1. The symbol of c S R) is the trigonometric polynomial M c ω) = 1/) k Z c ke ikω. A seuence c S R) is -orthogonal if k Z { if j = 0, c k c k+j = 0 if j 0..) We recall some well-known properties of -orthogonal seuences. A proof of i) can be found in Gröchenig [10];ChuiandLian[3] proved ii). PROPOSITION.. i) c S R) is a -orthogonal seuence if and only if 1 M c ω + πk ) = 1, for all ω R. k=0 ii) If the seuence c S R) is -orthogonal, then j Z c k+j = 1 for all k Z. We define two transformations on S R), thetranslation τ n for a given n Z and the reflection γ,by τ n {c k }) := {c k n } and γ{c k }) := {c k }. The corresponding scaling functions satisfy γc) x) = c x), τn c)x) = c x + n ), n Z. 1 We also define the convolution of b ={b k } and c ={c k } in S R) by { 1 b c := i } b i c k i : k Z. Please note the extra factor 1/. It follows that b c S R), and M b c ω) = M b ω)m c ω). DEFINITION.3. We say that b and c in S R) are euivalent and denote it by b c, if c = τ n b),orc = τ n γb),forsomen Z. THEOREM.4. Let b, c S R). Then M b ω) = M c ω) if and only if there exist a, e, e S R), such that b = a e, c = a e, and e e..3)

4 140 BELOGAY AND WANG Proof. Suppose.3) holds. Then M e ω) = e inω M e ω) if e = τ n e), andm e ω) = e inω M e ω) if e = τ n γe). In either case, M b ω) = M a ω) = M c ω). Conversely, suppose that M b ω) = M c ω). Without loss of generality we shall assume that b 0 0, and b i = 0foralli<0, for otherwise we can consider an euivalent shifted) seuencewith this property;we shall assume the same for c. Make the substitution z = e iω,letbz) = M b ω) and Cz) = M c ω), and define Bz) = z n B1/z), where n = degb). Now the assumption reads Bz) Bz) = Cz) Cz). Note that Bz) and Cz) must have the same degree and hence the same number of zeros counted with their multiplicity). Let Az) = gcdbz), Cz)). ThenBz) = Az)Ez) and Cz) = Az)E z) for some Ez),E z) R[z]. By assumption, Az)Ez)Ãz)Ẽz) = Az)E z)ãz)ẽ z). Since gcdez), E z)) = 1, we obtain that E z) = Ẽz). Now.3) follows immediately by letting M a ω) = Ae iω ), M e ω) = Ee iω ), M e ω) = E e iω ), and observing that E z) = Ẽz) implies e e. Theorem.4 suggests that the -orthogonal seuences can be classified by the suare of the modulus of their symbols. Although Mallat [14] proved the following theorem for =, the proof generalizes easily to all >1. THEOREM.5. Let c S R) be a -orthogonal seuence. Then c x) L R). 3. SCALING SEQUENCES OF ARBITRARILY HIGH ACCURACY In this short section we recall Heller s classification [11] ofall-orthogonal seuences for any given accuracy r 1seealso[15, 17]). Let be a fixed integer and. The simplest -orthogonal seuence is the Haar seuence h ={h k },whereh k = 1for 0 k<and h k = 0 otherwise. We denote its symbol by Hω):= 1/) 1 k=0 eikω. A seuence c S R) is r-accurate, orhaving accuracy r, ifm c ω) = H r ω)gω) for some trigonometric polynomial Gω). Proposition. states that every -orthogonal seuence is at least 1-accurate and that a seuence c is -orthogonal and has accuracy r if and only if there exists a real trigonometric polynomial Gω) such that 1 H ω + πk ) r G ω + πk ) = ) k=0 There are infinitely many such polynomials G, which becomes evident from the following lemma by Heller [11, Theorem 3.3]. LEMMA 3.1 Heller). if and only if 1 H ω + πk ) r G ω + πk ) = 0 k=0 Gω) = 1 cosω) r n kc n e inω, where c n = 0 for all but finitely many n.

5 ORTHOGONAL SYMMETRIC SCALING FUNCTIONS 141 Because of Lemma 3.1, we only need a particular solution G r to 3.1). Heller [11] derived the following formula for such a solution, namely, the minimal degree solution to 3.1); where the coefficients p n are given by r 1 G r ω) := p n 1 cosω) n, 3.) n=0 p n = r r 1) k 1 + +k 1 =n j=1 1 kj + r 1 r 1 ) 1 cos πk j ) kj r 3.3) for = 1 + 1, and p n = r r 1) k 0 +k 1 + +k 1 =n k0 + r 1 r 1 ) 1 ) kj + r 1 1 cos πk j r 1 j=1 ) kj r 3.4) for = 1 +. The following theorem summarizes these results and characterizes all -orthogonal seuences with given accuracy. THEOREM 3. Heller [11]). Let Pω) be a trigonometric polynomial. Then Pω)= M c ω) for some -orthogonal seuence c of accuracy at least r if and only if Pω) 0 for all ω R and Pω)= Hω) r Gω) for some Gω) of the form Gω) = G r ω) + 1 cosω) r n kc n cos nω, where G r ω) is defined by 3.) 3.4). We shall use Theorem 3. to construct symmetric scaling functions for any given accuracy r. 4. SYMMETRIC SCALING FUNCTIONS A seuence c S R) is symmetric if c = τ n γc) for some n Z. A function fx)is symmetric if fx)= fa x) for some a R. LEMMA 4.1. Let c S R).Thenc is symmetric if and only if c x) is. Proof. Suppose that c is symmetric. Then c = τ n γc) for some n Z. Therefore ) n c x) = τn γc)x) = c 1 x, and so c x) is symmetric. Conversely, suppose that c x) is symmetric and c x) = c a x). Without loss of generality, assume that c ={c k : k Z} so that c 0 c n 0andc k = 0forallk/ [0,n]. Then c x) is supported exactly on [0,n/ 1)], forcing a = n/ 1). Hence,

6 14 BELOGAY AND WANG c x) = τn γc)x). We claim that c = τ n γc). Indeed, note that ˆ c ω) = M c ω ) ) ω ˆ c, and ˆ τn γc)ω) = M τn γc) Therefore, ) ) ω ω M c = M τn γc) for all ω R, which yields c = τ n γc). ω ) ˆ τn γc) ω THEOREM 4.. i) Suppose that c S R) is symmetric. Let M c ω) = Pcos ω), where Pt) R[t].Then Pt)= G t) or ) 1 + t Pt)= G t), 4.1) for some Gt) R[t], G1) = 1. ii) Conversely, suppose that Pt)= G t), orpt)= 1 + t)/)g t), wheregt) R[t] and G1) = 1. Then there exists a uniue up to euivalence) symmetric c S R), such that M c ω) = Pcos ω). Proof. i) If c S R) is symmetric, then c = τ n γc) for some n Z,andM c ω) = e inω M c ω). Hence, M c ω) = M c ω) M c ω) = e imω M c ω) = e inω/ M c ω) ). Since M c ω) 0 is real, the imaginary part of e inω/ M c ω) must be 0. Now, if n = k, then e inω/ M c ω) = Re e ikω M c ω) ) = Gcosω) for some Gt) R[t]. Hence, M c ω) = G cos ω).ifn = k + 1, then e inω/ M c ω) = Re e ik+1)ω/ M c ω )) = G cos ω ) ). for some Gt) R[t]. Hence, M c ω) = G cos ω ). 4.) But cos ω/) = cosω),so G cos ω ) = G 1 cos ω) + cos ) ω G cos ω) for some G 1 t), G t) R[t]. On the other hand, by 4.), M c ω) = G cosω/)) = Pcos ω). It follows that either G 1 t) = 0orG t) = 0. Since G t) 0, it follows that G 1 t) = 0. Thus, ) ω M c ω) = cos G cos ω) = cosω) G cos ω), which proves i).

7 ORTHOGONAL SYMMETRIC SCALING FUNCTIONS 143 ii) First, we prove the existence. Suppose that Pt)= G t). ThenM c ω) = Gcosω) defines a symmetric c S R), because M c ω) = M c ω). Now suppose that Pt) = 1 + t)/)g t). Then M c ω) = e iω/ cos ω Gcosω) = eiω + 1 e iω G + e iω ) 4.3) defines a symmetric c S R), because M c ω) = e iω M c ω). Next, we show by contradiction that the symmetric c S R) is uniue, up to euivalence. Assume that there is another symmetric c S R), such that M c ω) = Pcos ω). By Theorem.4, thereexista, e, e S R), such that c = a e, c = a e, and e e. Therefore there exists some k Z, such that M c ω) = e ikω M a ω)m e ω) or M c ω) = e ikω M a ω)m e ω), depending on the type of euivalence between e and e. In the first case, we must have c = τ k c), andsoc and c are euivalent. In the second case, since both c and c are symmetric, we have M a ω)m e ω) = e in 1ω M a ω)m e ω) M a ω)m e ω) = e in ω M a ω)m e ω). Hence, M a ω) = ein 1+n )ω M a ω). ThisimpliesthatM aω) =±e inω M a ω), where n = n 1 + n )/ is clearly an integer. Since M a 0) = 1, we have M a ω) = e inω M a ω). The euivalence of c and c now follows from M c ω) = e ikω M a ω)m e ω) = e ik+n)ω M c ω). Remark. It is possible for a nonsymmetric c S R) to satisfy 4.1). For example, let c 0 = 4, c 1 = 4, c =, and all other c k = 0. This c is obviously nonsymmetric; nevertheless, M c ω) = 5 4cosω). EXAMPLE 4.1. For accuracy r = 1 and arbitrary >3, Theorem 3. implies that M c ω) = Hω) Gω), where Gω) = cosω) n kc n cos nω. Choosing Gω) = cosω)c 1 cosω + c cosω) and applying 4.1) and4.3), we obtain two scaling seuences c 1 and c,givenby M c1 ω) = 1 Hω) α + 1 α)e iω + 1 α)e iω + αe i3ω) M c ω) = 1 Hω) β + 1 β)e iω + 1 β)e iω + βe i3ω),

8 144 BELOGAY AND WANG FIG. 1. The Batman scaling function c1 for = 3Example4.1). where α = 1 6 and β = The scaling seuence c 1 corresponds to the continuous Batman scaling function Fig. 1), while c corresponds to a discontinuous scaling function Fig. ). For = 3, the corresponding two wavelets symmetric and antisymmetric) are shown in Fig. 3. In Section 5, we study the Batman function in more detail. EXAMPLE 4.. Consider -orthogonal seuences c for dilation = 5 and accuracy r =. Choose Gω) = cosω) + 1 cosω) a 1 cosω + a cos ω ) where t := 1 cosω. = 1 + 8t + t + a 1 + a )t a 1 + 4a )t 3 + a 1 t 4, FIG.. The discontinuous) scaling function c in Example 4.1.

9 ORTHOGONAL SYMMETRIC SCALING FUNCTIONS 145 FIG. 3. Two Batman wavelets long and short) for dilation = 3. By Theorem 3., the symbol M c ω) of c satisfies M c ω) = Hω) 4 Gω). Solving for a 1, a to complete the suare for Gω), we obtain two solutions: G 1) ω) = 1 + 4t 4t ) and G ) ω) = 1 + 4t 8 3 t). The two symmetric scaling seuences are: c 1 = 1 { 1, 0, 0,, 3, 6, 5, 6, 3,, 0, 0, 1} 4.4) 5 c = 1 {,, 1, 6, 9, 16, 19, 16, 9, 6, 1,, }. 4.5) 15 Figures 4 and 5 depict the corresponding continuous scaling functions. As we shall see in Section 5, only c x) is differentiable. EXAMPLE 4.3. As r grows, it becomes increasingly harder to find symmetric scaling seuences by hand. Fortunately, Theorem 4. can be aided by standard software tools, such as Mathematica. Figure 6 shows two minimal support symmetric scaling functions in the case = 4andr = 3; the polynomial Pt), defined in Theorem 4., hastheform Pt)= t)g t).

10 146 BELOGAY AND WANG FIG. 4. Continuous scaling function c1, = 5, r = Example4.). 5. THE BATMAN SCALING FUNCTION Let 3. In Example 4.1 Fig. 1) we introduced the Batman scaling function, which corresponds to the -orthogonal seuence { c = α, 1, 1 α, 1,...,1, 1 α, 1 } }{{},α, where α = ) 3 The refinement euation has the form fx)= αf x) + 1 fx 1) + 1 α)f x ) + fx 3) + + fx + 1) + 1 α)f x )+ 1 fx 1) + αf x ). FIG. 5. Smooth scaling function c for = 5, r = Example4.).

11 ORTHOGONAL SYMMETRIC SCALING FUNCTIONS 147 FIG. 6. Two scaling functions for = 4, r = 3Example4.3). Let φ x) denote the Batman scaling function corresponding to the scaling seuence givenby5.1). The support of φ x) is precisely [0,+ )/ 1)], which yields [0,.5] for = 3, and [0, ] for = 4. In what follows, we show that φ x) is continuous for every 3. We use the joint spectral radius to compute the Hölder exponent of φ x). The joint spectral radius techniue is presented in more detail by Daubechies and Lagarias [9], Berger and Wang [], and Lagarias and Wang [13]. Consider the general two-scale dilation euation fx)= N c n fx n), 5.) n=0 where c 0 c N 0and. If fx) is a compactly supported solution to 5.), then suppf =[0,N/ 1)]. LetL =[N/ 1)]. DefinetheL-dimensional vector vx) by vx) =[fx),fx+ 1),..., f x + L 1)] T, 0 x 1.

12 148 BELOGAY AND WANG Denote the space of real L L matrices by M L R), and define k matrices P k M L R), 0 k 1, by Now the dilation euation 5.) can be rewritten in the form where x [0, 1] has the base expansion and σx is the fractional part of x, Iterating 5.4), we obtain P k ) ij = c i 1) j 1)+k. 5.3) vx) = P d1 vσ x), 5.4) x = 0.d 1 d d 3..., 0 d j 1, j= 1,,..., σx = 0.d d 3 d 4... vx) = P d1 P d P dn vσ n x). By Proposition., the matrices P k are column stochastic; i.e., the entries in each column add up to one. Hence, the vector [1, 1,...,1] is a common left 1-eigenvector of all P k. Therefore, all P k can be simultaneously block-triangularized by a real nonsingular L L matrix Q whose first row is [1, 1,...,1]: QP k Q 1 = [ 1 0 A k ], 0 k 1. The following statement by Collela and Heil [4, Theorem 3] and Wang [16, Theorem.5] relates the Hölder exponentof the scaling function f to the spectral properties of the matrices A k ; we omit the proof. PROPOSITION 5.1. Denote the joint spectral radius ˆρA 0,A 1,...,A 1 ) by ˆρ, and let s = log 1/ ˆρ). Suppose that ˆρ <1.Then i) the scaling function fx)satisfying 5.) is continuous; ii) fx) C s ɛ,butfx) / C s+ɛ, for any 0 <ɛ<s; iii) fx) C s if and only if the semigroup of matrices generated by {A k / ˆρ : 0 k 1} is bounded. Now we can compute the exact smoothness of the Batman scaling function as a direct application of Proposition 5.1. THEOREM 5.. The Batman scaling function φ x) is continuous for each 3, and its Hölder exponent is log 4/ 6). Proof. We consider the case 4 first. In this case, L =[ + )/ 1)]=, and the matrices P 0,...,P 1 are matrices. Let [ ] 1 1 Q =. 0 1

13 ORTHOGONAL SYMMETRIC SCALING FUNCTIONS 149 Then [ ] QP k Q =, A k where all A k are scalars. By the definition 5.3), and a straightforward computation yields [ ] ck c P k = k 1, c +k c +k 1 ˆρA 0,A 1,...,A 1 ) = max A 0, A 1,..., A 1 ) = 6 4. By Proposition 5.1, the theorem is proved for 4. In the case = 3, the support of φ 3 x) is [0,.5], sol = 3 and the matrices P k are 3 3. We have [ ] QP k Q 1 =, by taking Q = A k Here, A 0 = [ α 0 α 1 α and α = 1/ c 6/4. Note that ] [ ] [ 1 α α 1 α, A 1 =, A = 0 α A 1 := max { a 11 + a 1, a 1 + a } 1 α 0 0 defines a matrix norm on M R) actually, the operator norm induced by the l 1 -norm in R ). Since 6 α = 1 < 1 6 α = 4, it follows that A k 1 = 1 α = 6/4, for k = 0, 1,. As a result, ˆρA 0,A 1,A ) 6/4, and the semigroup generated by {4/ 6A k : k = 0, 1, } is bounded [, Lemma II]. On the other hand, ˆρA 0,A 1,A ) 6/4, because 6/4 is an eigenvalue of A 0. Thus, which proves the theorem for = 3. ˆρ =ˆρA 0,A 1,A ) = 6 4, Using the same techniue, we can show that the smooth hat scaling function Fig. 5), which corresponds to the 5-orthogonal scaling seuence defined in 4.5), is differentiable. THEOREM 5.3. Let = 5, and c = 15 1 {,, 1, 6, 9, 16, 19, 16, 9, 6, 1,, }. The symmetric scaling function c x) is differentiable. ],

14 150 BELOGAY AND WANG Sketch of a proof. To prove the theorem, we assemble the five matrices P k,0 k 4. 1 A straightforward computation reveals that they have a common left 5 -eigenvector [1,, 3], in addition to the common left 1-eigenvector [1, 1, 1]. The rest of the proof involves simultaneously triangularizing P k and applying results by Daubechies and Lagarias [8]; we omit the details. ACKNOWLEDGEMENT We are greatly in debt to Chris Heil for his valuable advice and generous assistance with regard to the references. We are thankful to Jeff Geronimo and George Donovan for helpful discussions. We express our gratitude to Jian Ao Lian for the conversation we had in Pittsburgh, which motivated our study. Finally, we express our appreciation for the extremely constructive and helpful comments by the anonymous referees. REFERENCES 1. E. Adelson, E. Simoncelli, and R. Hingorani, Orthogonal pyramid transforms for image coding, in Proc. SPIE Visual Comm. and Image Processing II, Cambridge, MA, M. A. Berger and Y. Wang, Bounded semigroup of matrices, Linear Algebra Appl ), C. Chui and J. A. Lian, Construction of compactly supported symmetric and antisymmetric orthonormal wavelets with scale = 3, Appl. Comput. Harmon. Anal. 1995), D. Collela and C. Heil, Characterizations of scaling functions: Continuous solutions, SIAM J. Matrix Anal. Appl ), I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math ), I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematic, Vol. 61, SIAM, Philadelphia, I. Daubechies and J. C. Lagarias, Two-scale difference euations I: existence and global regularity of solutions, SIAM J. Appl. Math. 1991), I. Daubechies and J. C. Lagarias, Two-scale difference euations II: local regularity, infinite products of matrices and fractals, SIAM J. Appl. Math ), I. Daubechies and J. C. Lagarias, Sets of matrices all infinite products of which converge, Linear Algebra Appl ), K. Gröchenig, Orthogonality criteria for compactly supported scaling functions, Appl. Comput. Harmon. Anal ), P. Heller, Rank M wavelets with N vanishing moments, SIAM J. Matrix Anal. Appl ), P. Heller, J. Shapiro, and R. O. Wells, Jr., Optimally smooth symmetric uadrature mirror filters for image coding, in Proc. SPIE 491, Wavelet Applications for Dual Use, Orlando, FL, April J. C. Lagarias and Y. Wang, The finiteness conjecture for the generalized spectral radius of a set of matrices, Linear Algebra Appl ), S. Mallat, Multiresolution analysis and wavelets, Trans. Amer. Math. Soc ), P. Steffen, P. Heller, R. Gopinath, and C. Burrus, Theory of regular M-band wavelet bases, IEEE Trans. Signal Processing ), Y. Wang, Two-scale dilation euations and the mean spectral radius, Random Comput. Dynam ), G. Welland and M. Lundberg, Construction of compact p-wavelets, Constr. Approx ),