INFINITUDE OF MINIMALLY SUPPORTED TOTALLY INTERPOLATING BIORTHOGONAL MULTIWAVELET SYSTEMS WITH LOW APPROXIMATION ORDERS. Youngwoo Choi and Jaewon Jung

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1 Korean J. Math. (0) No. pp INFINITUDE OF MINIMALLY SUPPORTED TOTALLY INTERPOLATING BIORTHOGONAL MULTIWAVELET SYSTEMS WITH LOW APPROXIMATION ORDERS Youngwoo Choi and Jaewon Jung Abstract. By analyzing one-parameter families of totally interpolating multiwavelet systems of minimal total length with low approximation orders whose explicit formulas were obtained with the aid of well-known relations of filters we demonstrate the infinitude of such systems.. Introduction Wavelet theory has been a popular tool in signal/image processing computer graphics and many other applied mathematics. A wavelet is based on multiresolution analysis(mra) derived from one scaling function [6]. The Haar scaling function is the only orthogonal scaling function of compact support having symmetry and Shannon-like sampling properties [6]. Therefore multiwavelet theory or biorthogonal wavelets have been studied to achieve many desirable properties such as orthogonality symmetry short support and approximation order simultaneously. Totally interpolating multiwavelet systems have many advantages. The interpolating condition provides Shannon-like sampling property [ ] and perfect and fast reconstruction/decomposition algorithm in signal processing [7]. Moreover under the interpolating condition Received May 0. Revised July 0 0. Accepted August Mathematics Subject Classification: Primary C0; Secondary 6T60. Key words and phrases: biorthogonal multiwavelets stability totally interpolating. Corresponding author. c The Kangwon-Kyungki Mathematical Society 0. This is an Open Access article distributed under the terms of the Creative commons Attribution Non-Commercial License ( -nc/.0/) which permits unrestricted non-commercial use distribution and reproduction in any medium provided the original work is properly cited.

2 8 Youngwoo Choi and Jaewon Jung the equivalence of approximation and balancing orders was proved in terms of orthogonal/biorthogonal multiwavelet systems [ 7 9]. Therefore a prefiltering can be avoided with a totally interpolating orthogonal/biorthogonal multiwavelet system satisfying a suitable approximation order condition [ ]. It is well-known that both approximation order and short support conditions are important properties for applications such as denoising and compression. In this article we are mainly concerned with totally interpolating biorthogonal multiwavelet systems having minimal support and approximation order property. One can find sufficiently many refinable function vectors. For example a one-parameter family of interpolating refinable function vectors with a given approximation order is constructed in [7]. Nevertheless it is difficult to know how many interpolating multiwavelet systems there are that satisfy suitable regularity and stability. There are many totally interpolating orthogonal/biorthogonal multiwavelet systems of minimal total length for some given approximation orders [ 7 7]. Naturally we are concerned about a question how many such systems there are. In this article we investigate the cardinality of L -stable totally interpolating biorthogonal multiwavelet systems of minimal total length. Under FIR condition a degree of freedom provides the possibility of infinitely many totally interpolating biorthogonal multiwavelet systems from a suitable perturbation without lengthening their supports. This paper is organized as follows. The second section introduces elementary notions of biorthogonal multiwavelet systems interpolating condition approximation order and regularity. In third section we prove that such systems should have even total length and there are infinitely many L -stable systems with minimal total length for approximation orders and.. Preliminaries In this section we introduce basic notions on biorthogonal multiwavelet systems and interpolating properties. A vector function f L (R) is said to be L -stable if there are constants 0 < A B < such that A b kb k b kf ( k) B b kb k k= k= L k=

3 Minimally supported multiwavelet systems 9 holds for any vector sequence {b k } k Z l (Z) []. A vector function Φ = (φ φ ) T is said to be a multiscaling function of multiplicity if Φ satisfies a matrix refinement equation (.) Φ (t) = l Z P l Φ (t l) for some real matrices {P l }. The Fourier transform is defined by ( ˆΦ := ˆφ ˆφ ) T where ˆφj (ω) := φ j (t) e iωt dt with i = for j =. By taking Fourier transform (.) leads to ˆΦ (ω) = P ( ) ( ω ˆΦ ω ). Here P (ω) := k Z P ke iωk is called the two-scale matrix symbol or the refinement mask corresponding to Φ. { Let } Φ be a multiscaling function such that for some real matrices Pl Φ (t) = l Z P l Φ (t l). A pair of multiscaling functions Φ and Φ is said to be biorthogonal if Φ ( ) Φ ( k) = δ k0 I where f g := f (t) g (t) dt and δ kl denotes the Kronecker δ-symbol. Here and in what follows I and 0 denote the identity and zero matrices respectively. Consider multiwavelets Ψ and Ψ associated with {Φ Φ} given by Ψ (t) = Q l Φ (t l) and Ψ (t) = Q l Φ (t l) l Z l Z { } with real matrices {Q l } and Ql. A pair of multiwavelets Ψ and Ψ is said to be biorthogonal if for all k Z Ψ ( ) Ψ ( k) = δ k0 I and Φ ( ) Ψ ( k) = Φ ( ) Ψ ( k) = 0. For a given mask P (ω) the transition operator T : (L π) (L π) acting on matrix H (ω) with π-periodic square integrable entries in [] is defined by T H (ω) := P (ω) H (ω) P (ω) + P (ω + π) H (ω + π) P (ω + π).

4 0 Youngwoo Choi and Jaewon Jung Let H L be the space of matrices of trigonometric polynomials with degree at most L. A matrix is said to satisfy Condition E if it has a simple eigenvalue and the moduli of all other eigenvalues are less than. The Kronecker product X Y of X = (x jk ) jk= C and Y C [] is defined by ( ) x Y x X Y = Y. x Y x Y Since we assume that Φ is compactly supported we can write Φ (t) = L l=l P l Φ (t l) for some integer L < L. Let B n be the matrix given by B n := L l=0 P l n P l for L L L. Define the (L + ) (L + ) matrix (.) T := (B j k ) L jk= L. Then T : H L H L can be represented by T. Recall the following criterion for the L -stability. Theorem.. (Plonka and Strela []) The refinable function vector Φ is L -stable if and only if its symbol P (0) satisfies Condition E and the corresponding transition operator T (or T) restricted to H L satisfies Condition E where the eigenmatrix corresponding to the eigenvalue is positive definite for all ω R. A vector function f (t) = [f (t) f (t)] T is said to be interpolating if it satisfies the condition [ f (n) f ( n + )] = δ(n)i for n Z. If all of Φ Φ Ψ and Ψ in a biorthogonal multiwavelet system are interpolating then this system is said to be totally interpolating. Recall the following lemma: Lemma. (Zhang et al. [7]). Lemma.. If Φ is an interpolating multiscaling function then ( ) p P (ω) = (ω) (.) e iω p (ω) with p j (ω) := k Z cj k e iωk for some c j k R and j =. A filter is said to be a finite impulse response(fir) filter or have FIR property if it has finite duration. The following theorem for FIR property and the biorthogonality was shown in [ 7]: Theorem.. Let {Φ Φ} be a biorthogonal pair of interpolating multiscaling functions with two-scale matrix symbols P (ω) and P (ω)

5 Minimally supported multiwavelet systems as in (.). If both p (ω) and p (ω) are FIR filters then P (ω) is an FIR filter if and only if (.) c k+c l k c kc l k+ = C δ l+ m k Z k Z for some real constant C 0 and some odd integer m. Using biorthogonality and FIR property one can find P (ω) Q (ω) and Q (ω) from a given P (ω). { Theorem.. Let Φ Φ Ψ Ψ } be a totally interpolating biorthogonal multiwavelet system with FIR property. The refinement mask P (ω) dual to P (ω) can be obtained by p (ω) = C eim ω p ( ω + π) and p (ω) = C eim ω p ( ω + π). The corresponding high-pass filters are ( ) ( p Q (ω) = (ω) ) p and Q (ω) = (ω) e iω. p (ω) e iω p (ω) We say that a multiscaling function Φ provides approximation order M if there exist vectors yl m R such that l Z (ym l )T Φ(t l) = t m for all t R and m = 0... M. The following theorem of approximation property based on biorthogonality and FIR property is known in [ 0 7]. Theorem.. Let Φ(t) and Φ(t) be L -stable interpolating biorthogonal multiscaling functions. Assume that P(ω) and P(ω) are both FIR filters. Then both Φ(t) and Φ(t) provide approximation order M if and only if k n c k = ( )n ( C(m + ) n ) n+ k Z k Z k n c k+ = ( )n ( n + C(m + ) n ) n+ k n c k k Z k Z = + C( m ) n n+ and k n c k+ = ( )n ( C(m + ) n ) n+

6 Youngwoo Choi and Jaewon Jung for some nonzero real constant C some odd integer m and n = 0... M. The length of a refinement mask P (ω) = L l=l P l e iωl (or multiscaling function Φ (t) = L l=l P l Φ (t l) ) means the number L L + with L L and P L 0 P L 0. Similarly we say that a biorthogonal multiwavelet system {Φ Φ Ψ Ψ} has the total length T L T L + if T L is the minimum value of their first indexes and T L is the maximum value of their last indexes of all refinement masks in this system.. Existence of L -stable totally interpolating biorthogonal multiwavelet systems with minimal total length The primary concern of this section is how many L -stable totally interpolating biorthogonal multiwavelet systems with minimal total length exist. We begin with a simple property on the total length of a totally interpolating biorthogonal multiwavelet systems. Lemma.. The total length of a totally interpolating FIR biorthogonal multiwavelet system {Φ Φ Ψ Ψ} is even. Proof. Since Ψ and Ψ have the same support as Φ and Φ respectively we have only to consider the total length of {Φ Φ}. By Lemma. we can write P (ω) = L l=l P l e iωl with L 0 and L. Since p (ω) = C eim ω p ( ω + π) and p (ω) = C eim ω p ( ω + π) by Theorem. we have P (ω) = L m l= L m Pl e iωl for some odd integer m and real non-zero matrices P l. Therefore we have T L = min (L L m ) and T L = min (L L m ). The total length is either L + m + or L m + which finishes the proof.

7 Minimally supported multiwavelet systems There is a family of such systems of total length with approximation order whose the associated filters are follows: ( p (ω) = ) ( C + + ) C e iω and ( p (ω) = + ) ( C + ) C e iω. However one can check that each system is discontinuous. Theorem.. There are infinitely many L -stable totally interpolating biorthogonal multiwavelet systems with the minimal total length and the approximation order. Proof. We can find a totally interpolating multiwavelet system with the following filters: p (ω) = k= c k e iωk and p (ω) = k= c k e iωk where c = ( ) 6C + 8C 9 c 0 = 6 C 6 C c = ( ) + C c = 6 C 6 c = ( ) 8C + 8C + 9 c 0 = 8 C 8 + C c = ( ) + C c = 8 C 8 for some nonzero real constant C. When C = we have p (ω) = 7 eiω e iω + 6 e iω p (ω) = 6 eiω e iω 8 e iω p (ω) = 8 eiω e iω + 6 e iω and p (ω) = 6 eiω e iω 7 e iω. Let T ΦC be the set of transition matrix as in (.) and E ΦC be the set of absolute values of eigenvalues of T ΦC associated with Φ and some

8 Youngwoo Choi and Jaewon Jung nonzero real constant C. The sets E Φ E Φ and E Φ are given by { () 0.60() 0.() 0.69() E Φ } 6 () (0) { () 0.() () } 0.067() () (0) where λ(n) means that the multiplicity of λ is n. One can easily verify Condition E. If we let H C (ω) be suitably chosen eigenmatrix of T ΦC corresponding to simple eigenvalue then eigenmatrices H (ω) can be chosen as in Table where of T Φ and H (ω) of T Φ [H(ω)] jk := l Z hjk l check that the minima of [H det( H e iωl with some h jk l R for j k =. One can (ω)] and (ω))] det(h (ω)) [ H (ω)) are all positive. Furthermore one can choose eigenmatrices that depend smoothly on C with the aid of CAS packages such as Maple. Together with smooth dependence of eigenvalues on C we can conclude L -stability of the system for C sufficiently close to. Ma- trix representations of T ΦC H 0 and T L ΦC that depend smoothly on H 0 L C can be obtained. Sobolev exponents of Φ C and Φ C corresponding to C = are 0.68 and respectively. This implies the continuity of Φ C and Φ C for C sufficiently close to. Graphs of the system {Φ Φ Ψ Ψ } are Figures and. There are only two candidates of totally interpolating biorthogonal multiwavelet systems of approximation order with total length. Such systems are obtained with m = and C = 7 ± and they are dual to each other. The filters are given by p j (ω) = k=0 cj k e iωk for j = where c 0 = C c = C c = 6 8 C c = C c 0 = 6 8 C c = C c = C and c = 6 8 C.

9 Minimally supported multiwavelet systems Table. H (ω) and H (ω) for M = and m = in Theorem. l h l h l h l h l l h l h l h l h l (a) (b) (c) (d) Figure. φ φ φ and φ with M = and C = in Theorem. The associated filters p j q j and q j for j = are given by p (ω) = C e iω p ( ω + π) p (ω) = C e iω p ( ω + π) q j (ω) = p j (ω) and q j (ω) = p j (ω).

10 6 Youngwoo Choi and Jaewon Jung (a) (b) (c) (d) Figure. ψ ψ ψ and ψ with M = and C = in Theorem. Although the transition matrix T ΦC for C = 7 + satisfies condition E that of Φ C has spectral radius approximately 6.7. Therefore the system is not L -stable. In view of Lemma. one can deduce that the total length of a totally interpolating L -stable biorthogonal multiwavelet system is at least 6. On the other hand various examples of L -stable totally interpolating biorthogonal multiwavelet systems of approximation order with the minimal total length 6 have been known in [ 7] so far. Theorem.. There are infinitely many L -stable totally interpolating biorthogonal multiwavelet systems with the minimal total length 6 and the approximation order. Proof. For j = one can find a one-parameter family of totally interpolating biorthogonal multiwavelet systems based on filter components p j (ω) = c j k e iωk and p j (ω) = c j k e iωk k= k=

11 Minimally supported multiwavelet systems 7 where c = α ( γ c = 6 + ) ( α 8 C c 0 = 6γ C ( c = 6 + ) ( α 8 C c = γ ) 8 C c = β ( γ c = 6 + ) ( β 8 C c 0 = 6γ C ( c = 6 ) ( β 8 C c = γ 6 ) 8 C c l = C ( )l c l and c l = C ( )l c l with α := 8C + 6C + 8C + β := 8C + 8C + C + and γ := C + C for a nonzero real constant C and l =.... When C = we get p (ω) = 0 eiω + 80 eiω e iω e iω p (ω) = 0 eiω + 80 eiω e iω 9 e iω p (ω) = eiω + 6 e iω + 6 e iω 896 e ω p (ω) = eiω e iω 6 e iω e ω. The sets E Φ and E Φ of absolute values of eigenvalues of T Φ and respectively are given by T Φ E Φ E Φ { } 0.000() (6) ) ) { () () () } 0.006() 0(6) from which one can easily verify Condition E. One can check that the minima of [H (ω))] det(h (ω)) [ H (ω)] and det( H (ω)) are all positive. Furthermore one can choose eigenmatrices that depend smoothly on C. Together with smooth dependence of eigenvalues on

12 8 Youngwoo Choi and Jaewon Jung (a) (b) (c) (d) Figure. φ φ φ and φ with M = and C = in Theorem. C we can conclude L -stability of the system for C sufficiently close to. Matrix representations of T ΦC H 0 and T L ΦC that depend H 0 L smoothly on C can be obtained. Sobolev exponents of Φ and Φ are approximately are and respectively. This implies the continuity of Φ C and Φ C for C sufficiently close to. Graphs of the system {Φ Φ Ψ Ψ } are Figures and. Theorem.. There are infinitely many L -stable totally interpolating biorthogonal multiwavelet systems with the minimal total length 8 and the approximation order. Proof. A one-parameter family of totally interpolating biorthogonal multiwavelet systems with approximation order of total length 8 can be constructed with m =. If we let κ be a root of Z + (8C 6C 6C)Z +68C C 66C 7C +C + 968C = 0 and µ := C 8C + then the systems are determined

13 Minimally supported multiwavelet systems 9 (a) (b) (c) (d) Figure. ψ ψ ψ and ψ with M = and C = in Theorem. by the filter components p j (ω) = k= cj k e iωk for j = where c κ = 0µ c = C C 76C 6C 7 µ c = C κ 0µ c 0 = 6 C + (C 76C 6C 7) µ c = C + κ 0µ c = 8 6 C (C 76C 6C 7) µ c = 8 6 C κ 0µ c = (C 76C 6C 7) µ

14 60 Youngwoo Choi and Jaewon Jung and c = 6C + 8C + 8C µ c = 8 6 C 9C + C C + κ 0µ c = C (6C + 8C + 8C ) µ c 0 = C + (9C + C C + κ) 0µ c = 6 C + (6C + 8C + 8C ) µ c = C (9C + C C + κ) 0µ c = 8 6 C 6C + 8C + 8C µ c = 9C + C C + κ. 0µ Recall that the filter components p (ω) and p (ω) are determined by p (ω) = C e iω p ( ω + π) and p (ω) = C e iω p ( ω + π). We will analyze the filters corresponding to C = and α =. We get where c = p (ω) = c ke iωk and p (ω) = k= c ke iωk k= 7888 c = 7888 c = c 0 = c = c = c = c = c = 7888 c = c = c 0 = c = c = c = and c =

15 The sets E Φ T Φ E Φ E Φ Minimally supported multiwavelet systems 6 and E Φ respectively are given by of absolute values of eigenvalues of T Φ { () 0.068() () () () () () () and () () () } () 0() { () () () () () } 0.000() () () 0() which implies Condition E for T Φ H and T Φ. One can check that (ω) are positive definite. Sobolev exponents of Φ are.69 and.79 respectively. As in the proofs of previous (ω) and H and Φ theorems one can verify the L -stability. Graphs of the system {Φ Acknowledgements Φ Ψ Ψ } are Figures and 6. This paper was completed with Ajou University research fellowship of 0. References [] Y. Choi and J. Jung. Approximation and balancing orders for totally interpolating biorthogonal multiwavelet systems Bull. Korean Math. Soc. 8 (0) [] Y. Choi and J. Jung. Totally interpolating biorthogonal multiwavelet systems with approximation order Linear Multilinear Algebra 60 (0)

16 6 Youngwoo Choi and Jaewon Jung (a) (b) (c) (d) Figure. φ φ φ and φ with M = and C = in Theorem. (a) (b) (c) (d) Figure 6. ψ ψ ψ and ψ with M = and C = in Theorem.

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