Applied and Computational Harmonic Analysis 11 305 31 (001 doi:10.1006/acha.001.0355 available online at http://www.idealibrary.com 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 3 1999; revised April 19 001; published online July 10 001 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-96-34833 and CCR-99-8889 and ARO Grant DAAD-19-00-1-051. Department of Mathematics and Computer Science University of Missouri St. Louis St. Louis Missouri 6311-4499. This author is also associated with the Department of Statistics Stanford University Stanford California 94305 (cchui@stat.stanford.edu. 3 Department of Mathematics and Computer Science University of Missouri St. Louis St. Louis Missouri 6311-4499. 305 1063-503/01 $35.00 Copyright 001 by Academic Press All rights of reproduction in any form reserved.
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
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
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. (.18 3. 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 1 + 1 (N t + 1 1. (3.
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 + 1...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.
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 = 1. 4. 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. 15 18]. 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 = 1...4 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 1... 7 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 = ].
LETTER TO THE EDITOR 311 FIG. 1. Box-spline M 111 and tight frame generators 1... 7. ( ( ( 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 1... 7 are either symmetric or antisymmetric. (See the pictures of these functions along with = M 111 in Fig. 1..
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 1993.. C. K. Chui An Introduction to Wavelets Academic Press Boston 199. 3. C. K. Chui Multivariate Splines CBMS-NSF Series in Applied Mathematics No. 54 SIAM Philadelphia 1988. 4. C. K. Chui and W. He Compactly supported tight frames associated with refinable functions Appl. Comput. Harmon. Anal. 8 (000 93 319. 5. I. Daubechies Ten Lectures on Wavelets CBMS-NSF Series in Applied Mathematics No. 61 SIAM Philadelphia 199. 6. A. Ron and Z. Shen Affine systems in L (R d : The analysis of the analysis operator J. Funct. Anal. 148 (1997 408 447.