ORTHOGONAL MULTI-WAVELETS FROM MATRIX FACTORIZATION

Transcription

1 J. Korean Math. Soc , No. 2, pp ORHOGONAL MLI-WAVELES FROM MARIX FACORIZAION Hongying Xiao Abstract. Accuracy of the scaing function is very crucia in waveet theory, or correspondingy, in the study of waveet fiter banks. We are mainy interested in vector-vaued fiter banks having matrix factorization and indicate how to choose bock centra symmetric matrices to construct muti-waveets with suitabe accuracy. 1. Introduction We consider the case of compacty supported muti-waveets. hat is, suppose Φ φ 1,..., φ r are scaing functions, and Ψ ψ 1,..., ψ r are the corresponding waveet functions, so that the foowing two-scae equations hod for a x R: M 1.1 Φx S n Φ2 x n, 1.2 Ψx n0 M n Φ2 x n. n0 Define the corresponding symbo functions as m 0 x : 1 S n x n, m 1 x : 1 n x n. 2 2 n Z As is we known, the orthonormaity of the muti-waveets impies the foowing PR condition 1.3 HxH 1/x I 2, with 1.4 Hx : m0 x m 0 x m 1 x m 1 x Received June 6, 2007; Revised June 3, Mathematics Subect Cassification. 42C40, Key words and phrases. muti-waveets, PR condition, accuracy, bock centra symmetric matrix. n Z. 281 c 2009 he Korean Mathematica Society

2 282 HONGYING XIAO p to now, it seems very difficut to find a soutions of the matrix equation 1.3 so peope hope to construct some specia soutions. A cass of PR waveet fiter banks was given in [2] for the genera case. In particuar, one soution for the muti-waveets case is as foows: 1.5 m k x 1 N 2 I r, xi r diag I r, x 2 I r V k, k 0, 1. Here N is a fixed positive integer, is any arbitrary 2r 2r rea orthogona matrix, and Ir I r 1.6 V : V 0, V 1. I r I r he foowing theorem coud be found in [2]. heorem 1.1. Suppose fiter banks are constructed as in hen the PR condition 1.3 is satisfied. It is we known that the inear phase of fiter banks corresponds to symmetry of the reated functions. It was aso pointed out in [2] that to ensure the uniform inear phase, i.e., to ensure that there exists a natura number s such that m k x x s m k 1/x, k 0, 1, we shoud choose to be r-bock centra symmetric matrices: P 0 Ir J 1.7 S S r, S, 0 Q J r I r where P, Q are r-th rea orthogona matrices and J r is the r-th reversa matrix. For ater convenience, et Gx 1 2 I r, xi r N diag I r, x 2 I r. 2. Accuracy conditions of mutipe scaing functions In the foowing sections, we concentrate on sufficient and necessary conditions so that the scaing functions have accuracy of order p. hat is, a poynomias with tota degree at most p 1 can be reproduced from inear combinations of the muti-integer transates of function Φ. Gibert Strang stated in [5] that to ensure accuracy, one must check the vaue of function m 0 and its derivatives at a aiasing frequencies which seems difficut to compute. By ony imposing some conditions on the functions m 0, m 1 at x 1, the author of this paper produced the foowing theorem, [9]. heorem 2.1. If Φ φ 1, φ 2,..., φ r are scaing functions, and the integer transates of φ 1,..., φ r are ineary independent, moreover, if the corresponding fiter bank satisfies the PR condition 1.3, then Φ have accuracy p if and ony

3 ORHOGONAL MLI-WAVELES FROM MARIX FACORIZAION 283 if there are p vectors ν 0,..., ν p 1, each ν being r 1 vector and ν 0 0, such that for a Z p {0, 1,..., p 1} : 2.1 2i m 0 1ν 2 ν, i m 1 1ν 0. By using this theorem, the next proposition is obtained for the fiter banks constructed as above. Proposition 2.2. If Φ φ 1, φ 2,...,φ r are scaing functions with the integer transates of φ 1,...,φ r being ineary independent, and, the corresponding fiter banks are constructed as in , then Φ have accuracy p if and ony if there are p vectors ν 0,..., ν p 1, each ν being r 1 vector and ν 0 0, such that for a Z p : 2.3 2i G 1V 0 ν 2 ν, i G 1V 1 ν 0. By this proposition, we must find p vectors ν 0,..., ν p 1 to meet the requirements of equations 2.3 and 2.4. he procedure is simpified as foows so that one must ony find a nonzero vector ν 0 which is the common eigen-vector corresponding to eigenvaue λ 0 of severa matrices. heorem 2.3. nder the assumptions of Proposition 2.2, the scaing functions have accuracy p if and ony if there exist a r 1 vector ν 0 0 such that ν 0 are common eigenvector corresponding to eigenvaue 0 of matrices B 1,...,B p 1, that is, 2.5 B n ν 0 0, n 1, 2,...,p 1. he matrices B are constructed iterativey as B 1 N 1, n n B n N n N n A, 2 n p 1 and 2.7 A 1 M 1, n 1 n A n M n M n A, 2 n p 1

4 284 HONGYING XIAO with 2.8 M : G 1V 0, N : G 1V 1. Furthermore, the soutions of equations 2.3, 2.4 are given as 2.9 ν n 2i n 2 n 1 A n ν 0, n 1, 2,..., p 1. Proof. If the assumptions of Proposition 2.2 are satisfied, one easiy checks that 2.10 G1V 0 I r, G1V 1 0 r, so that for any r 1 vector ν 0 0, equations 2.3 and 2.4 for p 1. hus, the corresponding scaing functions have at east accuracy of p 1. From Proposition 2.2, Φ have accuracy p 2 if and ony if there exists r 1 vector ν 0, ν 1 with ν 0 0 such that 2.11 G1V 0 ν 0 ν 0, G1V 1 ν 0 0, 2i 1 G 1 1V 0 ν 0 + G1V 0 ν 1 2ν 1, 2i 1 G 1 1V 1 ν 0 + G1V 1 ν 1 0. By using the reations 2.10 and the notations in , the ast equations are equivaent to B 1 ν 0 0, ν 1 2i 1 A 1 ν 0, this is ust what equations 2.5 and 2.9 states for p 2. Suppose this theorem hods for some p 2, next we wi prove by induction that it aso hods for p + 1. Proposition 2.2 states that Φ have accuracy p + 1 if and ony if there exists r 1 vector ν 0,..., ν p+1 with ν 0 0 such that 2.3 and 2.4 hods for a 0, 1,..., p. By induction, this is equivaent to B n ν 0 0, ν n 2i n 2 n 1 A n ν 0, n 1, 2,..., p 1; p p 2i p G p 1 V 0 ν 2 p ν p ; 0 p 0 p 2i p G p 1 V 1 ν 0. By using 2.10 and 2.12, the eft side of equation 2.14 equas to p p 2i p N p ν + 2i p N p ν 0 1 p p 2i p 2i N p 2 1 A ν 0 + 2i p N p ν 0 1 } p { p 2i p 2 1 N p A + N p 1 ν 0 2i p B p ν 0.

5 ORHOGONAL MLI-WAVELES FROM MARIX FACORIZAION 285 Simiary, by using 2.10 and 2.12, the eft side of equations 2.13 equas to p p 2i p M p ν + 2i p M p ν 0 + ν p p p 2i p 2i M p 2 1 A ν 0 + 2i p M p ν 0 + ν p } p 1 1 { p 2i p 2 1 M p A + M p 1 hus, equations are equivaent to ν 0 + ν p 2i p A p ν 0 + ν p. B n ν 0 0, ν n 2i n 2 n 1 A n ν 0, n 1, 2,..., p. So we have proved this theorem. 3. Computation of the derivatives G 1 heorem 2.3 propose a sufficient and necessary condition for the corresponding scaing functions to have accuracy of degree p. Note that this impies the necessity of computing the derivatives of Gx, that is, the derivatives of the product of severa functions. Next, we give the foowing resuts concerning the computation of derivatives Derivation of products of functions he first emma is cassica in mathematica anaysis which is caed Leibniz s formua. Lemma 3.1. Let fx f 1 xf 2 x. hen, for any natura number n, the n-th derivative of function f is n n 3.1 f n x f m 1 xf n m 2 x. m m0 Lemma 3.2. Let fx f 1 x f M x. hen, for any natura number n, the n-th derivative of f is 3.2 f 1 f M n x 1+ + Mn i 0 n! 1! 2! M! f1 1 x f M x. Proof. We wi prove this theorem by induction of M. For M 1, this theorem hods naturay. And, Lemma 3.1 states that 3.2 hods for M 2. M

6 286 HONGYING XIAO Suppose this theorem hods for some natura number M 2, then, by Lemma 3.1, f 1 f M+1 n x n n m m0 n n m m M+1n i 0 f 1 f M m xf n m M+1 x 1+ + Mm i 0 m! 1! 2! M! f1 1 x f M n + 1! 1! 2! M+1! f1 1 x f M+1 M+1 x. hus, this theorem aso hods for M Computation of the derivatives G 1 M xfn m M+1 x In this section we wi concentrate on the fiter banks which are constructed in Let f I r, xi r, and for 1,...,N, hen their derivatives are 3.3 f k I r, I r, k 0, r, I r, k 1, 0 r 2 r, k 2, Or equivaenty, f x diag I r, x 2 I r. f k 1 f k δ k I r, δ k + δ k 1 I r, I 2 r, k 0, 2 diag0 r, I r, k 1, 2, 0 2 r, k 3. f k 1 diagδ k I r, δ k + 2δ k 1 + 2δ k 2 I r. hus, by Lemma 3.2, the derivative of G at x 1 is given as in the next theorem. heorem 3.3. he derivatives of Gx is given as G k I r, I r r, I r k 1+ +k Nk, k 1+ +k Nk 1, k 1! k M! N diag δ k 0 r, δ k + 2δ k 1 + 2δ k 2I r k 1! N diag δ k 0 r, δ k + 2δ k 1 + 2δ k 2I r k 1! k M!.

7 ORHOGONAL MLI-WAVELES FROM MARIX FACORIZAION 287 Proof. By using Lemma 3.2 and reation 3.3, we have G k I r, I r r, I r k 1+ +k Nk k 1+ +k Nk 1 k 1! k M! fk1 1 1 f kn N 1 k 1! k 1! k M! fk1 1 1 f kn N 1. his combined with reations concudes the proof of this theorem. Athough this theorem gives a cose form of the derivative G 1, it is not very convenient for computationa purpose. It is impied from reations 3.3 and 3.6 that to compute G k 1, we shoud consider matrix mutipication of the foowing form M q1 q diag0 r, I r q where each q is characterized as in 1.7. Proposition 3.4. Let be characterized as in 1.7. hen for any 2 M N, we have 3.7 M Ir 1 0 r 0 r M 1 0 r Q Q +1 + J r P P +1 J r Proof. We wi prove this proposition by induction of M. 1 For M 2, by using the fact that 3.8 Ir P S S 0 r P Ir S we have Q Q S, M. 1 Ir 1 2 Ir 2 S P1 Q1 S 0 r P Ir S 1 P 2 S 0 r P Q 1 Q2 Ir S 2 S. Q 2 Note that for any r r matrices A, B, C, D, the foowing identities hod: A B r, Ir C D Ir 0 r D P S 1 P Q 1 Q S 2 Q 1 Q 2 + J r P1 P. 2J r hus, we have 1 Ir 1 2 Ir r Q 1 Q2+Jr P 1 P2 Jr 2. hat is, we have proved the reation 3.7 for M 2. 0 r

8 288 HONGYING XIAO 2 Suppose reation 3.7 hods for some M 2, then, by induction, 3.11 M+1 Ir 1 0 r 0 r Q M 1 Q Q+1+Jr P P+1 Jr M M+1 Ir M+1. Note that for any r r matrices A, B, C, D, F the foowing identities hod: 0 r A B 0 r, 0 r F C D Ir 0 r F D M P M+1 S M P M+1 Q M Q S M+1 Q M Q M+1 + J r PM P. M+1J r Consequenty, 0 M+1 r 0 r Ir 1 M 0 r Q Q +1 + J r P P +1 J r M+1. hat is, we have proved that this proposition hods for M + 1. Another question concerning the formua 3.6 is as foows: how to characterize the set { 1,..., N : N k, i {0, 1, 2}}. In fact we ony have to consider the foowing simper sets 3.12 S N,k : { 1,..., N : 1 2 N, N k, i {0, 1, 2}}. Proposition 3.5. Given a natura number N, suppose S N,k are defined in 3.12 for a nonnegative integers k. hen, we have 3.13 {0, 0,..., 0}, k 0, {1, 0,..., 0}, k 1, S N,k { {}}{ k 1,...,1, 0,...,0} S N,k 2, 2 k N, {2 N,..., 2 1 : 1,..., N S N,2N k }, N + 1 k 2N,, k 2N + 1, with S N,k : {2, 1,..., N 1 : 1,..., N S N,k }. Moreover, the cardinaities are [ k 2 ] k N, 3.14 #S N,k N [ k 1 2 ] N + 1 k 2N, 0 k 2N + 1.

9 ORHOGONAL MLI-WAVELES FROM MARIX FACORIZAION 289 Proof. o verify the equaities 3.13, we ony have to prove that the third equaity hods for a 2 k N, that is, S N,k { { 1,. }}..,1 { k, 0,...,0} S N,k 2. On the one hand, for any 1,..., N S N,k, we have either 1 2 or 1 1. If 1 2, then it is easy to verify that 2,..., N, 0 S N,k 2, thus, 1,..., N S N,k 2 ; in the case that 1 1, we have 1,..., N { 1,. }}..,1 { k, 0,...,0. So we have prove that { S N,k 1, {. }}..,1 { k, 0,...,0} SN,k 2. On the other hand, firsty we know that { 1,. }}.., 1 { k, 0,...,0 S N,k. For any 2, 1,..., N 1 S N,k 2 where 1,..., N S N,k 2, one can check easiy that N 0. Otherwise, if N 1, then it is impied that N N > k 2. hus, we have 2, 1,..., N 1 S N,k. Consequenty, we have proved that S N,k { 1, {. }}..,1 { k, 0,...,0} S N,k 2. Combining the two resut, the proof of 3.13 is finished. he equaity 3.14 can be easiy checked by using the resuts of Proposition 3.5 produces a recursive method to construct the sets S N,k. In what foows we wi show exacty what S N,k are. Proposition 3.6. Let the sets S N,k be defined as in hen for any 0 k N, we have [ k 2 ] k 2 N k+ S N,k {}}{{}}{{}}{ ,..., 2, 1,...,1, 0,...,0. 0 And, for those N + 1 k 2N, we have [N k 2 ] k+ N 2N k 2 S N,k {}}{{}}{{}}{ ,...,2, 1,...,1, 0,...,0. 0 Proof. One can easiy checks that equaity 3.16 is impied by 3.13 and We wi prove by induction of k that equaity 3.15 hods for a 0 k N. Firsty, it is easy to verify that this hods for k 0, 1. Furthermore, the fact that S N,2 {2, 0, 0,...,0, 1, 1, 0,..., 0} impies that this equaity aso hods for k 2. Suppose there are some 2 k N 1, such that equaity 3.15 hods for a 0 n k. By 3.13, we have 3.17 S N,k+1 { 1, {. }}..,1 { k+1, 0,...,0} S N,k 1, with SN,k 1 {2, 1,..., N 1 : 1,..., N S N,k 1 }.

10 290 HONGYING XIAO As was pointed in the proof of the ast proposition, for any 1,..., N S N,k 1, we have N 0. hus, by induction, [ k 1 2 ] +1 k 1 2 N k+ S N,k 1 {}}{{}}{{}}{ 2,...,2, 1,...,1, 0,...,0, so that S N,k { { 1,. }}..,1 { [ k+1 2 ] 0 0 k+1, 0,...,0 } [ k 1 2 ] 0 k+1 2 N k 1+ {}}{{}}{{}}{ 2,..., 2, 1,...,1, 0,...,0. +1 k 1 2 N k+ {}}{{}}{{}}{ 2,...,2, 1,...,1, 0,...,0, Note that to prove the equaity 3.18, we have used the equaity [ k+1 2 ] 1 [ k 1 2 ]. Combining the resuts in reation 3.6, and Propositions 3.4, 3.6, we propose the foowing theorem which seems more convenient to compute G 1. heorem 3.7. For 0 k N, we have k 1! k M! fk1 1 1 f kn N 1 k 1+ +k Nk [ k 2 ] k 0 1 p1< <pk N And, for N + 1 k 2N, we have k 1+ +k Nk [N k 2 ] 0 N k + N 0 r 0 r 2 k 2 k 1 k! p1 0 r Q Q + J P P J. pk p p+1 p p+1 k 1! k M! fk1 1 1 f kn N 1 1 p1< <pn N 1 0 r 0 r k! 2 k+ N p1 0 r N 1 1 Q p Q p+1 + J P p P p+1 J pn. Proof. For any n-tupes a 1,..., a n, denotes by Pa 1,..., a n the set of permutations of a 1,...,a n. First consider the case 0 k N, we have k 1+ +k Nk k 1! k M! fk1 1 1 f kn N 1 1,..., N S N,k k 1,...,k N P 1,..., N k 1! k M! fk1 1 1 f kn N 1

11 ORHOGONAL MLI-WAVELES FROM MARIX FACORIZAION 291 1,..., N S N,k k 1,...,k N P 1,..., N [ k 2 ] [ 2 fk1 0 k 1,...,k N Pe,k 1! M! fk1 1 1 f kn N f kn N 1] {}}{{}}{ where the N-tupes e,k are defined as e,k : 2,...,2, 1,...,1, { 0,. }}..,0, { note that the ast equaity hods due to Proposition 3.6. Next, we wi show ceary what the summation in the bracket is. For any given 0, 1,..., [ k 2 ], et C 2 : C 1 : 2 fk1 k 1,...,k N Pe,k k f kn N 1, 2 fq1 p 1 q 1,...,q k pẽ,k 1 p 1< <p k N k 2 1 f q k p k 1, {}}{{}}{ where ẽ,k : 2,..., 2, 1,...,1, we caim that C 1 C 2. On the one hand, it is straightforward to prove that the terms of summations are equa since N N k 2 k N k. On the other hand, we wi verify that any summation term of C 1 emerges aso in C 2. For fixed permutation of e,k, it is impied from reation 3.3 that ony k terms in the product 2 f k1 1 1 f kn N 1 counts. hus, there exists 1 p 1 < < p k N, and q 1,...,q k being permutation of ẽ,k so that 2 f k1 1 1 f kn N his concudes the proof of identity C 1 C 2. So we have k 1+ +k Nk k 1! k M! fk1 1 1 f kn N fq1 p 1 1 f q k p k 1. [ k 2 ] [ 2 fq1 p 1 0 q 1,...,q k Pẽ,k 1 p 1< <p k N 1 f q k p k 1] [ k 2 ] k 0 [ k 2 ] k 0 k! 2 fq1 p 1 1 p 1<p 2< <p k N 1 f q k p k 1 0 r 0 r 2 k 2 k 1 k! p1 0 r 1 p 1<p 2< <p k N 1 Q p Q1+p + J P P1+p J p pk.

12 292 HONGYING XIAO It shoud be noted that we have used Proposition 3.4 and the foowing facts: 1 for any 1,...,N, f 1 1 f Ir ; m n {}}{{}}{ 2 the number of permutations of 2,..., 2, 1,...,1 is m+n m. By the same trick we can prove the theorem for the case N+1 k 2N. 4. Numerica exampes Consider the case r 2. Assume that fiter banks are constructed as in where the 2 2 rea orthogona matrices P, Q are cosα sinα 4.19 P cosβ sin β, Q sinα cosα. sinβ cosβ Let γ α + β, then the matrices B 1, B 2 defined in heorem 2.3 are λ2 0 B 1 λ 3 I 2, B 2, 0 λ 3 with parameters λ 1 1 N 2 + cos γ, N N λ 2 N + sin γ N + 1 cos γ + N N λ 3 N sin γ N + 1 cos γ 1 k< N 1 k< N sin γ γ k, sin γ γ k. hus, by heorem 2.3, we have the foowing sufficient and necessary conditions for the corresponding scaing functions to have accuracy p 2, 3. heorem 4.1. When r 2, and waveet fiter banks are constructed as in and 4.19, then the corresponding scaing functions have at east accuracy of order p 2 if and ony if 4.20 N cos γ 1 2. Moreover, the corresponding scaing functions have at east accuracy of order p 3 if and ony if in addition to 4.20, the foowing equaity hods: 4.21 N sin γ + 1 k< N sin γ γ k ± 1 2.

13 ORHOGONAL MLI-WAVELES FROM MARIX FACORIZAION 293 Proof. By heorem 2.3, the scaing functions have at east second accuracy if and ony if there exists nonzero 2 1 vector ν 0 such that B 1 ν 0 λ 1 ν 0 0, this reduces to λ 1 0. Simiary, the scaing functions have at east third accuracy if and ony if there exists nonzero 2 1 vector ν 0 such that B 1 ν 0 λ 1 ν 0 0, B 2 ν 0 λ2 0 0 λ ν0 3 0, this reduces to either of the foowing two equaities: { { λ1 0, λ1 0, λ 2 0, λ 3 0. Figure 1. Waveet and scaing function with second accuracy By equation 4.20, to obtain second accuracy and the minima ength of the fiters, we shoud choose N 1, that is, cosγ In fact, in this case, ψ 2, φ 2 are the Daubechies s waveet function db2 and the corresponding scaing function. he graph of the above functions are potted in Figure 1. On the other hand, to obtain third accuracy and minima ength, we have to choose N 2. Here the equations 4.20 and 4.21 have four soutions: γ 1 π + arcsin0.5374, γ 2 arcsin0.9392; γ 1 π + arcsin0.9756, γ 2 π arcsin0.9600; γ 1 π arcsin0.5374, γ 2 arcsin0.9392; γ 1 π arcsin0.9756, γ 2 π + arcsin ake the first soution, and we present the graph of the above functions in Figure 2.

14 294 HONGYING XIAO Figure 2. Waveet and scaing function with third accuracy References [1] C. Cabrei, C. Hei, and. Moter, Accuracy of attice transates of severa mutidimensiona refinabe functions, J. Approx. heory , no. 1, [2] Q. H. Chen, C. A. Micchei, S. Peng, and Y. Xu, Mutivariate fiter banks having matrix factorizations, SIAM J. Matrix Ana. App , no. 2, [3] I. Daubechies, en Lectures on Waveets, CBMS-NSF Regiona Conference Series in Appied Mathematics, 61. Society for Industria and Appied Mathematics SIAM, Phiadephia, PA, [4] C. deboor, R. A. Devore, and A. Ron, Approximation orders of FSI spaces in L 2 R d, Constr. Approx , no. 4, [5] C. Hei, G. Strang, and V. Strea, Approximation by transates of refinabe functions, Numer. Math , no. 1, [6] R. Q. Jia, Approximation properties of mutivariate waveets, Math. Comp , no. 222, [7] Q. Lian, H. Xiao, and Q. Chen, Some properties on mutivariate fiter banks with a matrix factorization, Progr. Natur. Sci. Engish Ed , no. 2, [8] G. Ponka, Approximation order provided by refinabe function vectors, Constr. Approx , no. 2, [9] H. Xiao, Lattice structure for paraunitary inear-phase fiter banks with accuracy, Acta Math. Sin. Eng. Ser , no. 3, [10], Piecewise Linear Spectra Sequences and Waveet Fiter Banks, Doctora hesis, Graducate Schoo of CAS, May Coege of Science China hree Gorges niversity Yichang , P. R. China E-mai address: hongying x@ctgu.edu.cn