PARAMETRIZATION OF MULTIWAVELET ATOMS FRITZ KEINERT Abstract. It is known that the polyphase matrix of any biorthog- onal multiwavelet of compact support can be factored into a con- stant matrix, zero or more projection factors, and an atom. The projection factors are of the form F (z) = (I −S)+Sz with S2 = S. a The atom is of the form A(z) = I + A1z + ··· + Aaz , with det(A(z)) = 1. We give an explicit algorithm for generating all possible atoms of given type. This can be used to numerically gen- erate biorthogonal multiwavelets that are optimal by some user- defined criterion. 1. Introduction A multiwavelet is given by a multiscaling function φ and a multi- wavelet function which satisfy matrix recursion relations p X φ(x) = 2 Hkφ(2x − k); k p X (x) = 2 Gkφ(2x − k): k A convenient way to represent the recursion coefficients is through the polyphase matrix H (z) H (z) P (z) = 0 1 ; G0(z) G1(z) where Hi(z), Gi(z) are the polyphase symbols X k X k Hi(z) = H2k+iz ;Gi(z) = G2k+iz : k k we assume all functions have compact support, so P is a matrix Laurent polynomial. 2000 Mathematics Subject Classification. 42C40. Key words and phrases. wavelets, multiwavelets, wavelet atoms, polyphase factorization. This work was partially supported by ARC Large Grant 390 1404 1099 14 1 at Flinders University, Adelaide, South Australia. 1 2 Fritz Keinert March 2, 2009 If P and P~ represent a biorthogonal multiwavelet pair, the biorthog- onality conditions are equivalent to P (z)P~(z)∗ = I: (1.1) If P , P~ and Q, Q~ are two polyphase matrix pairs which satisfy (1.1), so does the pair PQ, Q~P~. Conversely, it is possible to factor polyphase matrices into more elementary building blocks. This has in fact been done in a number of different ways. Some possible building blocks are • Projection factors ([8], [12], [17]), which have the form F (z) = (I − S) + Sz; where S is a projection, that is S2 = S, and I is the identity matrix. • Lifting factors ([3], [4], [9]), which have the form I 0 IB(z) L(z) = or U(z) = : A(z) I 0 I • Diagonal matrices with monomial terms on the diagonal. • Constant matrices. It is also possible to combine elementary factors with some free pa- rameters to produce polyphase matrices from scratch. Numerical op- timization can then be used to select the wavelets which are optimal with respect to some user-defined criterion. Such approaches have been used for example in [1], [11], and [14]. In [8], [12] it was shown that the polyphase matrix of any biorthog- onal multiwavelet of compact support can be factored in the form P (z) = MF1(z)F2(z) ··· Fn(z)A(z); (1.2) where M is a constant nonsingular matrix, Fj are projection factors, and A(z) is an atom. An (a; b)-atom has the form a A(z) = I + A1z + ··· + Aaz ;Aa 6= 0; b B(z) = I + B1z + ··· + Bbz ;Bb 6= 0; with A(z)B(z) = I. This implies det(A(z)) = det(B(z)) = 1, so in linear algebra terms, an atom is a pair of co-monic unimodular matrix polynomials. The name \atom" is taken from [8]; in [12], atoms are called pseudo-identity matrix pairs. The goal of this paper is to give an explicit parametrization of all atoms of given type, so that they can be numerically generated by a computer program. The atoms can then be combined with other types of elementary factors to produce multiwavelets. As an intermediate March 2, 2009 Multiwavelet Atoms 3 result and special case, we also obtain a parametrization of all nilpotent matrices of given size. The factorization (1.2) has the advantage that if we want to construct multiwavelets with a given number of recursion coefficients, there are only a few possibilities for the type of atom and the number of pro- jection factors necessary. For example, all biorthogonal pairs with P , P~ both linear can be constructed from one constant matrix M, one projection factor, and a (1; 1)-atom. Other factorization approaches often contain redundant parameters, or don't produce all possible multiwavelets of given type. A construction of biorthogonal multiwavelets based on nilpotent ma- trices is given in [13], [14]. These papers construct wavelets by multi- plying together several terms of the form I + Az, A nilpotent. In the notation of this paper, these are (1; b)-atoms. This approach is different from the one taken here. The outline of the paper is as follows. In x2, we give a brief review of multiwavelets, polyphase matrices and atoms, and introduce some basic notation. In x3, we define the companion matrix C of a given co-monic matrix polynomial A of degree a. We show that A and B = A−1 form an (a; b)-atom if and only if C is nilpotent of degree d, with a + b = d: (1.3) In x4, we investigate Jordan bases of a general nilpotent matrix Z, and show how they can be partially orthonormalized. In x5, we apply the results of x4 to parameterize nilpotent matrices of size n × n. Each possible eigenstructure (number and length of Jordan chains) leads to a separate parametrization. In x6, we specialize the results of x5 to the case where Z = C is a companion matrix. The special structure of C simplifies the algorithm and reduces the number of types, as well as the number of parameters for each type. This provides the desired parametrization of atoms. Examples are given at the end of x4, 5 and 6, as well as in the final x7. 2. Multiwavelets and Atoms In this section we introduce some basic notation, review multiwavelets and polyphase matrices, motivate the definition of an atom, and show how atoms are related to the problem of creating biorthogonal multi- wavelets. 4 Fritz Keinert March 2, 2009 Wavelet theory is based on the concept of refinable functions. For multiwavelets, the starting point is a refinable function vector of mul- tiplicity r. This is an r-vector of real- or complex-valued functions 0φ1(x)1 . φ(x) = @ . A ; x 2 R; φr(x) which satisfies a two-scale recursion relation p X φ(x) = 2 Hkφ(2x − k): (2.1) k The coefficients Hk are r × r matrices. The function vector φ is called a multiscaling function if it generates a multiresolution approximation (MRA) of L2(R). An MRA is a sequence 2 of nested subspaces · · · ⊂ V−1 ⊂ V0 ⊂ V1 ⊂ · · · of L (R) with properties T (i) j Vj = f0g, S 2 (ii) j Vj = L (R), (iii) f(x) 2 Vj =) f(2x) 2 Vj+1; j 2 Z, −j (iv) f(x) 2 Vj =) f(x + 2 k) 2 Vj; j; k 2 Z, (v) The functions fφi(x − k): i = 1; : : : ; r; k 2 Zg form a Riesz basis of V0. Another function vector is called a multiwavelet function if its integer translates f i(x − k): i = 1; : : : ; rg form a Riesz basis of a space W0 so that V0 ⊕ W0 = V1: Since W0 ⊂ V1, satisfies a two-scale recursion p X φ(x) = 2 Gkφ(2x − k): (2.2) k Two multiwavelets φ, and φ~, ~ are called biorthogonal if ~ ~ hφi; φki = h i; ki = δik; (2.3) ~ ~ hφi; ki = h i; φki = 0: If φ~ = φ, ~ = , the multiwavelet is called orthogonal. A convenient way to represent the recursion coefficients Hk, Gk is through the polyphase matrix H (z) H (z) P (z) = 0 1 ; G0(z) G1(z) March 2, 2009 Multiwavelet Atoms 5 where Hi(z), Gi(z) are the polyphase symbols X k X k Hi(z) = H2k+iz ;Gi(z) = G2k+iz : k k We restrict attention to the case where all functions have compact support. In this case, P , P~ are matrix Laurent polynomials. The biorthogonality conditions (2.3) are equivalent to P (z)P~(z)∗ = I; (2.4) where I is the identity matrix. For a constant matrix P , P ∗ denotes the P j ∗ Hermitian transpose; for a matrix polynomial P (z) = Pjz , P (z) is defined by ∗ X ∗ −j P (z) = Pj z : j A matrix polynomial P (z) which satisfies P (z)P (z)∗ = I is called para- unitary. Orthogonal multiwavelets correspond to paraunitary poly- phase matrices. Matrix polynomials P , P~ which satisfy (2.4) do not necessarily gen- erate multiwavelets; further conditions can be imposed which insure that the two-scale recursion relations (2.1) and (2.2) produce multi- scaling and multiwavelet functions and MRAs (see [10]). However, relation (2.4) is sufficient to characterize perfect reconstruction filter- banks for signal processing applications. If P , P~ and Q, Q~ are two biorthogonal pairs, then PQ and P~Q~ form another biorthogonal pair. This opens the possibility of factoring a given P into elementary building blocks, or conversely constructing polyphase matrices from simpler factors. Two of these building blocks are projection factors and atoms. A projection factor has the form F (z) = (I − S) + Sz; where S is a projection, that is S2 = S. It is easy to verify that its dual is given by F~(z) = (I − S∗) + S∗z: If S is an orthogonal projection, so that S∗ = S, then F is paraunitary. Such an F is called an orthogonal projection factor. An (a; b)-atom is a pair of matrix polynomials a A(z) = I + A1z + ··· + Aaz ;Ab 6= 0; b (2.5) B(z) = I + B1z + ··· + Bbz ;Bb 6= 0 with A(z)B(z) = I: (2.6) 6 Fritz Keinert March 2, 2009 One possible factorization approach, which was rediscovered inde- pendently a number of times, is described in the following two theo- rems.