Applied and Computational Harmonic Analysis 11, (2001) doi: /acha , available online at

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1 Applied and Computational Harmonic Analysis (001 doi: /acha available online at on LETTER TO THE EDITOR Construction of Multivariate Tight Frames via Kronecker Products 1 Charles K. Chui and Wenjie He 3 Communicated by Amos Ron Received November ; revised April ; published online July Abstract Integer-translates of compactly supported univariate refinable functions φ i such as cardinal B-splines have been used extensively in computational mathematics. Using certain appropriate direction vectors the notion of (multivariate box splines can be generalized to (non-tensor-product compactly supported multivariate refinable functions from the φ i s. The objective of this paper is to introduce a Kronecker-product approach to build compactly supported tight frames associated with using the two-scale symbols of the univariate tight frame generators associated with the φ i s. 001 Academic Press 1. INTRODUCTION Extension from univariate function spaces to the multivariate setting can be trivially accomplished by considering the space generated by tensor-products of the univariate basis functions. When these univariate basis functions are compactly supported refinable functions in terms of dilation and translation not only the procedure for introducing (multivariate box-splines [1 3] can be used to create the generating basis functions of the multivariate function spaces but these multivariate generating functions also remain to be refinable with multivariate Laurent polynomial two-scale symbols. The objective of this paper is to introduce an algorithmic approach for constructing tight frames associated with compactly supported multivariate refinable generating functions. This approach features a recipe for selecting the columns of the Kronecker product of the orthonormal matrix extensions of the univariate two-scale symbols in order to formulate the multivariate orthonormal matrix extension of the multivariate two-scale symbol which in turn gives the symbols for writing down the desired multivariate tight frame generators. This recipe will be derived in Section 3 where its consequences will be stated as Theorem 1. 1 Partially supported by NSF Grants DMI and CCR and ARO Grant DAAD Department of Mathematics and Computer Science University of Missouri St. Louis St. Louis Missouri This author is also associated with the Department of Statistics Stanford University Stanford California (cchui@stat.stanford.edu. 3 Department of Mathematics and Computer Science University of Missouri St. Louis St. Louis Missouri /01 $35.00 Copyright 001 by Academic Press All rights of reproduction in any form reserved.

2 306 LETTER TO THE EDITOR Preliminaries on orthonormal matrix extension due to Ron and Shen [6] a criterion for the validity of such extensions observed in our earlier work [4] and a key formula for Kronecker products will be discussed in Section. In the final section we consider the construction of tight frames of box-splines in the three- and four-directional meshes in R.. PRELIMINARIES This section is devoted to the introduction of necessary notations and preliminary results..1. Multivariate Refinable Functions Let be a compactly supported refinable function in L (R s with two-scale relation (x = k Z s p k (x k (.1 for some finite sequence {p k } or equivalently where ˆ (ω = P(e iω/ ˆ (ω/ ω R s P(z := 1 s p k z k (. k Z s is a Laurent polynomial in C s. Note that refinable functions are less restrictive than scaling functions (cf. [ 5] in that { ( k: k Z s } is not required to constitute a Riesz basis of the L (R s closure of its algebraic span. To construct from t s compactly supported univariate refinable functions φ i in L (R with finite two-scale sequences {p ik } or equivalently univariate Laurent polynomials P k (z := 1 p kj z j k= 1...t (.3 j Z let A be any nonsingular s-dimensional square matrix with row vectors y 1...y s and introduce additional vectors s y v = η vk y k η vk Z (.4 s<v t.then defined by its Fourier transform ˆ (ω := ˆφ i (ω y i ω R s (.5 i=1 is a refinable function with two-scale s-variate Laurent polynomial symbol s P(z = P(z 1...z s := P k (z k. (.6 v=s+1 P v( s l=1 z η vl l

3 LETTER TO THE EDITOR 307 Here z k := e iω y k/ k = 1...sandfors<v t z v = e i s η vk ω y k / = Indeed by using (.5 and (.6 we have ˆ (ω = ˆφ k (ω y k = s z η vk k. (.7 P k (e iω yk/ ˆφ k (ω y k / = P(e iω/ ˆ (ω/... Orthonormal Matrix Extensions Let P(z be an s-variate Laurent polynomial. For every lɛ{0... s 1} consider its (unique binary representation or equivalently l = s ɛk l k 1 ɛk l {0 1} (.8 and adopt the standard multivariate notation l σ(l:= (ɛ1 l...ɛl s (.9 ( 1 σ(l z = (( 1 ɛl 1 z1...( 1 ɛl s z s (.10 where z := (z 1...z s. The problem of orthonormal matrix extension is to find s-variate Laurent polynomials Q 1 (z...q N (z such that the matrix satisfies P(( 1 σ(0 z P(( 1 σ(s 1 z Q 1 (( 1 σ(0 z Q 1 (( 1 σ(s 1 z R N (z := Q N (( 1 σ(0 z Q N (( 1 σ(s 1 z (.11 R N (zr N (z = I s z T s. (.1 Here T denotes the unit circle in C and the subscript N which denotes the number of Laurent polynomials Q i (z to be constructed is called the order of the matrix extension R N (z. The importance of this orthonormal matrix extension problem is that under very mild conditions on the refinable function with two-scale Laurent symbol P(z Ron and Shen [6] proved that the compactly supported s-variate functions l l = 1...N with ˆ l (ω = Q l (e iω/ ˆ (ω/ l = 1...N (.13

4 308 LETTER TO THE EDITOR generate a tight frame { sj/ l ( j x k : j Z k Z s and l = 1...N} (.14 of L (R s. For the univariate setting it was shown in our earlier work [4] that a Laurent polynomial P(z admits orthonormal matrix extension if and only if it satisfies P(z + P( z 1 z T. (.15 Moreover if (.15 holds two procedures were given in [4] one for constructing orthonormal matrix extension of order and the other for constructing symmetric/antisymmetric orthonormal matrix extension of order 3 for symmetric P(z..3. Kronecker Products of Rectangular Matrices The objective of this paper is to introduce a Kronecker-product approach to solving the orthonormal matrix extension problem (.1 for constructing tight frames of L (R s. Recall that the Kronecker product of an m n matrix A =[a ij ] with a p q matrix B =[b ij ] is an mp nq matrix defined by A B =[a ij B] (.16 where block matrix formulation is used. As an immediate consequence of the property (A B (A B = (A A (B B (.17 of Kronecker products for rectangular matrices of arbitrary dimensions where the asterisk indicates a complex conjugate of the transpose matrix we have the following. LEMMA 1. Let A and B be rectangular matrices of dimensions m n and p q with m n and p q respectively such that A A = I n and B B = I q.then (A B (A B = I nq. ( COMPACTLY SUPPORTED MULTIVARIATE TIGHT FRAMES Let t s and P 1 (z... P t (z be univariate two-scale symbols as in (.3 that admit orthonormal matrix extensions R N1 (z... R Nt (z of orders N 1...N t respectively. By using the notation in (.7 and observing that the Kronecker product operation is associative we introduce the (N + 1 t matrix where z = (z 1...z s is extended from T s to C s and K N (z = R N1 (z 1 R Nt (z t (3.1 N := (N (N t (3.

5 LETTER TO THE EDITOR 309 Since each P i (z admits an orthonormal matrix extension R Ni (z of order N i i = 1...t it follows from Lemma 1 that K N (zk N (z = I t z T s. (3.3 An important property of the matrix K N (z is that its first row contains all of the entries of the first row of the matrix R N (z in (.11 for the mapping σ(ldefined in (.9. To see this recall that for each l = 0... s 1 σ(l {0 1} s is uniquely determined by the binary representation of l in (.8 so that by introducing the notation we have s ɛv l := ɛk l η vk (mod v = s t (3.4 P(( 1 σ(l z = P i (( 1 ɛl i zi l = 0... s 1 (3.5 i=1 where the notation in (.7 is used. But from (.6 and the definition of K N (z we see that the t components of the first row vector of K N (z are given by P 1 (±z 1 P t (±z t with all possible t combinations of ± signs so that P(( 1 σ(l z in (3.5 is one of these entries. To be more precise we observe by (.16 and (3.1 that the first row block matrices of K N (z have the form [P 1 (z 1 R N (z R Nt (z t P 1 ( z 1 R N (z R Nt (z t ]. (3.6 Consequently for each l = 0... s 1 the (ɛ l 1 t 1 + ɛ l t + +ɛ l t 0 + 1th entry of the first row of (3.6 is P(( 1 σ(l z for l = 0... s 1. Thus by using the notation d l := t ɛi l t i l= 0... s 1 (3.7 i=1 we may construct an (N + 1 s matrix R N (z with exactly the same formulation as (.11 by using the (d l + 1th column of K N (z as the (l + 1th column of R N (z l = 0... s 1. By (3.3 we see that this R N (z satisfies (.1. As a consequence of the above discussion and by applying the results stated in Section. we have established the following result. THEOREM 1. Let t s and P 1 (z... P t (z be univariate Laurent polynomials that admit orthonormal matrix extensions of orders N 1...N t respectively. Then the s-variate Laurent polynomial P(z defined in (.6 by using (.4 also admits an orthonormal matrix extension R N (z given by (.11 with N given by (3.forsomes-variate Laurent polynomials Q l (z l = 1...N. Furthermore if φ 1...φ t are univariate refinable functions with two-scale symbols P 1 (z... P t (z respectively such that ˆφ 1 (0... ˆφ t (0 0 and P 1 ( 1 = = P t ( 1 = 0 then the Laurent polynomials Q l (z are the two-scale symbols of the compactly supported L (R s functions l l = 1...Nin(.13 that generate a tight frame (.14 of L (R s.

6 310 LETTER TO THE EDITOR We remark that if the univariate Laurent polynomials in the above theorem are symmetric then symmetric or antisymmetric frame generators 1... N can be constructed in view of the fact that the Kronecker product operation preserves symmetry. In addition since the largest value of N 1...N t is the number of frame generators l s is between t 1and3 t 1; but if symmetry or antisymmetry is desired this number of frame generators may increase up to 4 t 1. The reason for the lower bound is that if φ l is an orthonormal scaling function in L (R then we may choose N l = BOX-SPLINE TIGHT FRAMES Let M n (x denote the cardinal B-spline of order n [3 pp. 1 and ]. We choose y 1 = e 1 y = e y 3 = e 1 + e y 4 = e 1 e in (.4 and φ 1 := M l φ := M m φ 3 := M n φ 4 := M p in (.5 to define ˆ.Forlmnp>0 this bivariate refinable function is the box-spline M lmnp on a four-directional mesh with Fourier transform given by ˆ (ω = ˆM lmnp (ω = ˆM l (ω e 1 ˆM m (ω e ˆM n (ω (e 1 + e ˆM p (ω (e 1 e = ˆM l (ω 1 ˆM m (ω ˆM n (ω 1 + ω ˆM p (ω 1 ω ( 1 e iω 1 l ( 1 e iω m ( 1 e i(ω 1 +ω n ( 1 e i(ω 1 ω p = iω 1 iω i(ω 1 + ω i(ω 1 ω and two-scale symbol ( 1 + l ( z1 1 + m ( z 1 + z1 z n ( 1 + z1 /z p p(z = P(z 1 z =. For details see [1] and [3 pp ]. We remark that the Z -translates of M lmnp do not constitute a Riesz basis of the L (R -closure of their algebraic span. For p = 0 = M lmn0 =: M lmn are box-splines on a three-directional mesh. The simplest such example is the Courant element M 111 whose two-scale symbol has a seventh-order orthonormal matrix extension R 7 (z formulated via the Kronecker product K 7 (z = K 7 (z 1 z = [ 1+z1 1 z 1 1 z 1 1+z 1 ] [ 1+z 1 z 1 z 1+z ] [ 1+z1 z 1 z 1 z 1 z 1 z 1+z 1 z Recall that the nth column of R 7 (z is the (d n 1 + 1th column of K 7 (zforn = where d l is given by (3.7. That is R 7 (z is an 8 4 matrix whose column vectors are given in consecutive order by the first sixth fourth and seventh columns of K 7 (z. From this we can write the two-scale symbols of the tight frame generators as follows. ( ( ( 1 + z1 1 + z 1 z1 z Q 1 (z = ( ( ( 1 + z1 1 z 1 + z1 z Q (z = ( ( ( 1 + z1 1 z 1 z1 z Q 3 (z = ].

7 LETTER TO THE EDITOR 311 FIG. 1. Box-spline M 111 and tight frame generators ( ( ( 1 z1 1 + z 1 + z1 z Q 4 (z = ( ( ( 1 z1 1 + z 1 z1 z Q 5 (z = ( ( ( 1 z1 1 z 1 + z1 z Q 6 (z = ( ( ( 1 z1 1 z 1 z1 z Q 7 (z = Observe that are either symmetric or antisymmetric. (See the pictures of these functions along with = M 111 in Fig. 1..

8 31 LETTER TO THE EDITOR REFERENCES 1. C. de Boor K. Höllig and S. Riemenschneider Box Splines Applied Mathematical Sciences No. 98 Springer-Verlag New York C. K. Chui An Introduction to Wavelets Academic Press Boston C. K. Chui Multivariate Splines CBMS-NSF Series in Applied Mathematics No. 54 SIAM Philadelphia C. K. Chui and W. He Compactly supported tight frames associated with refinable functions Appl. Comput. Harmon. Anal. 8 ( I. Daubechies Ten Lectures on Wavelets CBMS-NSF Series in Applied Mathematics No. 61 SIAM Philadelphia A. Ron and Z. Shen Affine systems in L (R d : The analysis of the analysis operator J. Funct. Anal. 148 (