REGULARITY AND CONSTRUCTION OF BOUNDARY MULTIWAVELETS FRITZ KEINERT Abstract. The conventional way of constructing boundary functions for wavelets on a finite interval is to form linear combinations of boundary-crossing scaling functions. In this article we focus instead on boundary functions defined by recursion relations. We show that the number of boundary functions at each end is uniquely determined, and derive conditions for determining regularity from the boundary recursion coefficients. We then develop an algorithm based on linear algebra which can be used to construct boundary functions with maximal regularity.. Introduction The Discrete Wavelet Transform (DWT) is designed to act on infinitely long signals. For finite signals, the algorithm breaks down near the boundaries. This can be dealt with by extending the data by zero padding, extrapolation, symmetry, or other methods [4, 9,, ], or by constructing special boundary functions [, 6, 7]. Boundary functions can be constructed as linear combinations of boundary-crossing interior functions, or from boundary recursion relations. The standard examples in the literature have both properties, but the recursion relation approach can produce new boundary functions which cannot be derived using linear combinations. There are also linear combinations which are not refinable, but these are of little practical value, since the DWT algorithm requires recursion coefficients. One way to find suitable boundary recursion coefficients is based on linear algebra. The infinite banded Toeplitz matrix representing the DWT is replaced by a finite matrix, by suitable end-point modifications. A particular such method for scalar wavelets is presented in Madych [9]. The Madych approach can be generalized to multiwavelets under an additional assumption, which may or may not be satisfied for a given multiwavelet. We present a modified method which does not require this extra assumption. Linear algebra completions are not unique; they all include multiplication by an arbitrary orthogonal matrix. A random choice of matrix does not in general produce coefficients that correspond to any actual boundary function. Random choice also does not provide any approximation orders at the boundary. Thus, we investigate the question of continuity and approximation order at the boundary in some detail. We then show how to impose these constraints in the algorithm, and in the process remove much or all of the non-uniqueness. Mathematics Subject Classification. 4C4. Key words and phrases. wavelets on an interval, multiwavelets, discrete wavelet transform, boundary handling. Department of Mathematics, Iowa State University, Ames, IA 5, keinert@iastate.edu.
Regularity and Construction of Boundary Wavelets The results given below are based on the papers [] and []. For more details, and most of the proofs, we refer to the original papers.. Wavelets and Multiwavelets We assume that the reader is familiar with the basics of wavelet theory, so we only review a few basic concepts that we need in this paper. A multiresolution approximation (MRA) of L prq is a chain of closed subspaces tv n u, n P Z, Ă V Ă V Ă V Ă Ă L prq satisfying (i) V n Ă V n` for all n P Z; (ii) fpxq P V n ðñ fpxq P V n` for all n P Z; (iii) fpxq P V n ùñ fpx n kq P V n for all n, k P Z; (iv) Ş npz V n tu; (v) Ť npz V n L prq; (vi) there exists a function vector φ pxq φpxq. φ r pxq, φ i P L prq, such that tφ j px kq : ď j ď r, k P Zu is an orthonormal basis for V []. The function φ is called the multiscaling function of the given MRA. r is called the multiplicity. φ satisfies a recursion relation φpxq? ÿ k H k φpx kq. (.) The recursion coefficients H k are r ˆ r matrices. The orthogonal projection P n of a function s P L into V n is given by P n s ÿ kpzxs, φ nk yφ nk, where φ nk : n{ φp n x kq. P n s is interpreted as an approximation to s at scale n. Q n P n` P n is also an orthogonal projection onto a closed subspace W n which is the orthogonal complement of V n in V n` : V n` V n W n. Q n s is interpreted as the fine detail in s at resolution n. An orthonormal basis of W is generated from the integer translates of a single function vector ψ P L prq, called a multiwavelet function. Since W Ă V, ψ can be represented as ψpxq? ÿ n G n φpx nq (.) for some coefficient matrices G n.
Regularity and Construction of Boundary Wavelets The Discrete Wavelet Transform (DWT) takes a function s P V n for some n, and decomposes it into a coarser approximation at level m ă n, plus the fine detail at the intermediate levels. s P n s P m s ` It suffices to describe the step from level n to level n. Since the signal s P V n V n W n, we can represent it by its coefficients s nk xs, φ nky, d nk xs, ψ nky as s ÿ s nkφ nk ÿ s n,jφ n,j ` ÿ d n,jψ n,j. k j j n ÿ k m Q n s. Here s denotes the transpose of s. We find that s n,j ÿ H k j s nk, d n,j ÿ k k If we interleave the coefficients at level n. s n, d psdq n n,, s n, d n, the DWT can be written as where psdq n s n, T T T T T T T T T, T k. G k j s nk. ˆHk The matrix is orthogonal. Signal reconstruction corresponds to s n psdq n. G k H G k`. (.) A multiscaling function φ has approximation order p if all polynomials of degree less than p can be expressed locally as linear combinations of integer shifts of φ. That is, there exist row vectors c jk, j,..., p, k P Z, so that x j ÿ k c jkφpx kq. (.4) For orthogonal wavelets c jk where µ j is the jth continuous moment of φ, ż µ j x j φpxq dx. jÿ l ˆj k l j l µ l, (.5)
4 Regularity and Construction of Boundary Wavelets A high approximation order is desirable in applications. A minimum approximation order of is a required condition in many theorems.. Boundary Functions The original DWT is defined for infinitely long signals. In practice, we can only transform a signal of finite length. What do we do near the boundaries? A frequently used approach is data extension, for example by zeros, by symmetry, or by linear or higher order extrapolation [4, 9,, ]. We will be using instead special boundary wavelets. We make the following assumptions: The original functions φ, ψ are orthogonal, continuous, provide approximation order p ě, and are supported on r, Ns. The interval r, Ms on which the signal and the wavelets are defined is long enough so that the left and right boundary functions do not overlap. We use all of the interior functions, that is, all φpx nq, ψpx nq whose support lies in r, Ms. There are L (scalar) left boundary scaling functions, R right boundary functions, collected into function vectors φ L, φ R. Likewise, we have L, R left and right boundary wavelet functions ψ L, ψ R. The left boundary functions have support on r, N s, the right boundary functions have support on rm N `, Ms. The boundary functions are regular, which means they are orthogonal, continuous, and provide approximation order ě. This implies that φ L, φ R are not identically zero at the endpoints. The numbers L and R of boundary functions are the same at every level. Note that L and R are generally not equal, and they are not in general multiples of r = multiplicity of interior functions. Lemma.. [] Under our assumptions, L and R are uniquely determined. We sketch the idea of the proof, since some of the results are needed later. Assume at first that there are only four recursion coefficients, so only two matrices T, T. Orthogonality of means T T ` T T I, T T. From this one can show that ρ j rankpt j q satisfy ρ ` ρ r, and that T, T have a common SVD ˆIρ ˆ T U V, T U V I ρ. Here I k denotes the k ˆ k identity matrix. With an argument due to Strang [] one can show L ρ, R ρ. If there are more than two matrices T j, we form block matrices. For example if we have T,..., T, we use T T ˆT T T T, ˆT T T T T T T T There are two systematic ways to construct boundary functions. We show the details for the left endpoint only, since the notation is easier there.
Regularity and Construction of Boundary Wavelets 5 The first method uses linear combinations of boundary-crossing functions (see e. g. [7]). φ L pxq ÿ k N` C k φpx kq for x ą. Each C k is of size L ˆ r. The second method uses recursion relations. For the interior functions, these are given by (.) and (.). For the boundary functions at the left end, they are φ L pxq? Aφ L pxq `? N ÿ B k φpx kq, k ψ L pxq? Eφ L pxq `? N ÿ F k φpx kq. A and E are of size L ˆ L, each B k and F k is of size L ˆ r. Under our assumptions, the decomposition matrix on the interval r, M s takes the the form L L L K T T T K.. T M T.. TK.................. (.).... T T T K R R R K Here ˆ ˆ Z ^ A L, L E k Bk B k N, k,..., K, K, F k F k and similarly at the right end. Theorem.. [] Assume that the boundary functions are both refinable and linear combinations. Then A, B k, C k are related by CV AC, (.) CW B, where B `B B B N, k C `C N` C N` C, H N H N H V N H N H N H N............. H, H H. W............. H 4 H N. H H N H N H N
6 Regularity and Construction of Boundary Wavelets Some observations about the two approaches are as follows. Eq. (.) is a form of eigenvalue problem. There are only a small number of choices for C, A, B. In fact, if there is a single boundary scaling function (i. e. L ), (.) is a standard eigenvalue problem. Not every refinable boundary function is a linear combination of boundary-crossing functions. Conversely, not every linear combination of boundary-crossing functions is refinable. Refinability is required for applications. Without the existence of A, B k there is no DWT algorithm. Linear combinations are not required. If boundary functions as constructed as linear combinations, then continuity is automatic, and approximation order is easy to enforce. If the approximation order is greater than or equal to the number of boundary scaling functions, refinability is automatic. Orthogonality is hard, and this is where most of the effort lies. If boundary functions are constructed from recursion relations, then orthogonality is easy. Continuity and approximation order are harder to impose, but a suitable algorithm is detailed in this paper. 4. Construction of Boundary Functions Using Linear Algebra In [9], Madych proposed an algorithm for constructing a decomposition matrix M of the form (.). His algorithm was for scalar wavelets. It turns out that it generalizes to multiwavelets only if L R. Our algorithm does not require this assumption. Assume again initially that there are only four recursion coefficients. The general case is handled as before, by forming block matrices. It is sufficient to demonstrate the idea on the ˆ orthogonal block matrix T T T T. T T We use pre- and postmultiplication by orthogonal matrices to make T in the lower left disappear. Let U and V be the orthogonal matrices from the joint SVD of T and T. Let U U U, V V V, U V then U V I ρ I ρ I ρ I ρ I ρ I ρ (4.) By inspection, a technique that produces the correct structure is to move the first ρ columns to the end, and then interchange the first ρ rows with the last ρ rows. That amounts to multiplying from the right and left with appropriate permutation matrices.
Then Regularity and Construction of Boundary Wavelets 7 P L U V P R I ρ I ρ I ρ I ρ I ρ I ρ Now let Q L, Q R be arbitrary orthogonal matrices of size ρ ˆ ρ, ρ ˆ ρ, respectively. We multiply from the left with U L QL U Q R and from the right with Let U R I ρ V V. I ρ Q L pq L, Q L, q, Q R pq R, Q R, q, V pv V q (4.) with Q L,, Q L, of size ρ ˆ ρ, Q R,, Q R, of size ρ ˆ ρ, and V, V of size r ˆ ρ, r ˆ ρ, respectively. Then U L P L U L V P R U R L T T R R where L Q L,, L Q L, V, R Q R, V, R Q R,. If we consider Madych s approach in our notation, we find that it amounts to moving the second set of columns in (4.) to the end, so that we end up with I ρ I ρ I ρ I ρ I ρ I ρ This only yields matrices of the correct size if ρ ρ. Madych only considered scalar wavelets, where this condition is automatic. The drawback of the algorithm is that Q L, Q R are arbitrary, and the recursion coefficients produced by most random choices do not correspond to actual boundary functions, nor do they provide any approximation orders. In the following, we first investigate how to determine continuity and approximation orders from the boundary recursion coefficients. After that, we will show how to find suitable matrices Q L, Q R to impose these conditions.
8 Regularity and Construction of Boundary Wavelets 5. Regularity Properties of Refinable Boundary Functions First we consider continuity. The cascade algorithm at the left boundary is given by φ L,n` pxq? Aφ L,n pxq `? N ÿ B k φpx kq k Starting with an initial function φ L, with support r, N s, we see that φ L, is given on rpn q{, N s by a linear combination of translates of φpxq. This part of φ L, does not change in future steps. After step, φ L, is given on rpn q{4, pn q{s by a linear combination of translates of φp4xq, and so on. Thus, the cascade algorithm converges to a continuous limit on p, N s. The only question is what happens at x. It is easy to show that if the spectral radius ρp? Aq ă, the cascade algorithm will converge, but φ L pq. If ρp? Aq ą, the cascade algorithm will diverge for most starting guesses. It is conceivable but unlikely that there will be continuous solutions. We will not pursue this further. We say that? A satisfies Condition Epmq if it is diagonalizable, has an m-fold eigenvalue, and all other eigenvalues are less than in magnitude. In this case,? A n Ñ Γ `r,..., r m l. l m where r j, l j are the right and left eigenvectors to eigenvalue, normalized to l j r j. Theorem 5.. [] If? A satisfies condition E(m), then φ L p n x q Ñ Γ φ L px q ` p? B qpi?h ı q φpx q as n Ñ 8. In other words: Starting with any point x ą, the sequence of point values φ L p n x q converges. Thus, continuity of φ L at is equivalent to the condition that the limit is independent of x. This appears hard to verify, but we will see below that this is actually a consequence of approximation order. It can be shown that our regularity assumptions imply that pi?h q exists. The following graph shows an example where the limit of φp n x q exists for every x, but the resulting function is not continuous at. It is based on the Daubechies D wavelet, with A? { and B p? 6{4, q.
Regularity and Construction of Boundary Wavelets 9 6 D, discontinuous example, iteration 6 D, discontinuous example, iteration 5 5 4 4.5.5.5.5 6 D, discontinuous example, iteration.5.5.5.5 6 D, discontinuous example, iteration 5 5 4 4.5.5.5.5.5.5.5.5 We next consider approximation order. The approximation order conditions for interior functions are stated in (.4). For left boundary functions, approximation order p means x j l j φ L pxq ` ÿ c jkφpx kq, j,..., p, for suitable vectors l j. kě Theorem 5.. [] The left boundary functions have approximation order p if the interior functions do, and if there exist vectors l j so that for j,..., p? l j A j l j,? (5.) Bm γ jm, m,..., N, where the γ jm are known vectors: l j γ jm j c jm? tm{u ÿ k c jkh m k. Corollary 5.. If? A satisfies condition Epq, then approximation order implies continuity.
Regularity and Construction of Boundary Wavelets Proof. Approximation order at x means ÿ l φ L pxq ` µ φpx kq, x ě, which for x P r, s reduces to l φ L pxq ` µ φpxq. The approximation order conditions (5.) for j begin with? A The second of these implies l l l µ l kě l,? B µ µ? H,? B µ µ? H,...? B I?H, so that Γ φ L pxq ` p? B qpi?h ı q φpxq? r l φ L pxq ` l B I?H j r l φ L pxq ` µ φpxq r. It is not immediately clear whether continuity and φ L pq implies approximation order. We conjecture that this is not the case. 6. Imposing Regularity We now describe how to construct suitable orthogonal matrices Q L, Q R for use in the algorithm described in section 4, to impose continuity and approximation orders on the boundary functions. This will also remove some or all of the arbitrariness in the construction. Let Λ be the matrix with columns l j (see theorem 5.). It is easy to see that if we replace the boundary function vector φ L by Mφ L for an invertible matrix M, the new boundary functions still span the same space, and remain refinable. They also remain orthogonal if M is orthogonal. The effect of M on the coefficients A, B, C and the matrix Λ is A Ñ Ã MAM, B Ñ B MB, C Ñ C MC, Λ Ñ Λ ΛM. The key observation is that by using a suitable M, we can assume that Λ is upper triangular. We don t actually need to find this M.
Regularity and Construction of Boundary Wavelets One can show that it is possible to construct vectors l e, l P spanpe, e q,..., and a matrix Q L which enforce the approximation order conditions one by one. The details are rather lengthy, and can be found in []. This algorithm will build one boundary function per approximation order, up to the approximation order provided by the interior functions. If the number of boundary functions is higher than the interior approximation order, we can fill up the remaining rows of recursion coefficients arbitrarily. 7. Examples Example 7.. For the Daubechies scaling function with two vanishing moments we get one boundary function at each end. We can impose approximation order at each end..5 D, left end, regular.5 D, right end, regular.5.5.5.5.5.5.5.5.5.5.5.5.5.5 Example 7.. The Chui-Lian multiwavelet CL() [5] has approximation order. We need boundary functions at each end, so we can impose approximation order at the ends. This multiwavelet is symmetric, and the boundary functions are mirror images of each other..5 CL() left end, approximation order.5 CL() right end, approximation order.5.5.5.5.5.5.5.5.5.5 Example 7.. The DGHM multiwavelet [8] has approximation order. We find ρ, ρ, so we have only a single boundary function at the left end, but three at the right end. At the left end we can only enforce approximation order. At the right end we can enforce approximation order. This determines two of the boundary functions. The third function (determined by the third row of Q R ) is arbitrary, except
Regularity and Construction of Boundary Wavelets that the diagonal entry in Z should be smaller than? { to ensure that the completion is regular. As an example, we set the third column of Z to zero, and let the QR-factorization built into Matlab choose an appropriate third row of Y. The result was DGHM left end, approximation order DGHM right end, approximation order.5.5.5.5.5.5.5.5...4.6.8..5.5.5.5.5 References [] A. Altürk and F. Keinert, Regularity of boundary wavelets, Appl. Comput. Harmon. Anal., (), pp. 65 85. [], Construction of multiwavelets on an interval wavelets, Axioms, (), pp. 4. [] L. Andersson, N. Hall, B. Jawerth, and G. Peters, Wavelets on closed subsets of the real line, in Recent advances in wavelet analysis, vol. of Wavelet Anal. Appl., Academic Press, Boston, MA, 994, pp. 6. [4] C. Brislawn, Classification of nonexpansive symmetric extension transforms for multirate filter banks, Appl. Comput. Harmon. Anal., (996), pp. 7 57. [5] C. K. Chui and J.-a. Lian, A study of orthonormal multi-wavelets, Appl. Numer. Math., (996), pp. 7 98. Selected keynote papers presented at 4th IMACS World Congress (Atlanta, GA, 994). [6] C. K. Chui and E. Quak, Wavelets on a bounded interval, in Numerical methods in approximation theory, Vol. 9 (Oberwolfach, 99), vol. 5 of Internat. Ser. Numer. Math., Birkhäuser, Basel, 99, pp. 5 75. [7] A. Cohen, I. Daubechies, and P. Vial, Wavelets on the interval and fast wavelet transforms, Appl. Comput. Harmon. Anal., (99), pp. 54 8. [8] G. C. Donovan, J. S. Geronimo, D. P. Hardin, and P. R. Massopust, Construction of orthogonal wavelets using fractal interpolation functions, SIAM J. Math. Anal., 7 (996), pp. 58 9. [9] W. R. Madych, Finite orthogonal transforms and multiresolution analyses on intervals, J. Fourier Anal. Appl., (997), pp. 57 94. [] S. G. Mallat, Multiresolution approximations and wavelet orthonormal bases of L prq, Trans. Amer. Math. Soc., 5 (989), pp. 69 87. [] G. Strang and T. Nguyen, Wavelets and filter banks, Wellesley-Cambridge Press, Wellesley, MA, 996. [] J. R. Williams and K. Amaratunga, A discrete wavelet transform without edge effects using wavelet extrapolation, J. Fourier Anal. Appl., (997), pp. 45 449.