ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 9, Pages 32–39 (April 4, 2003) S 1079-6762(03)00109-4

A NOTE ON THE CONSTRUCTION OF NONSEPARABLE WAVELET BASES AND MULTIWAVELET MATRIX FILTERS OF

L^{2}(ℝ^{n}), WHERE n ≥ 2

ABDERRAZEK KAROUI

(Communicated by Guido Weiss)

Abstract. In this note, we announce a general method for the construction of nonseparable orthogonal wavelet bases of L^{2}(ℝ^{n}), where n ≥ 2. Hence, we prove the existence of such type of wavelet bases for any integer n ≥ 2. Moreover, we show that this construction method can be extended to the construction of n-D multiwavelet matrix filters.

1. Introduction

In this note, we are interested in the construction of dyadic nonseparable compactly supported wavelet bases of L^{2}(ℝ^{n}). This type of wavelet bases is defined as follows.

Definition 1. Consider a discrete set of functions

{Ψ_{j,k}^{i}(x) = 2^{−j/2}Ψ^{i}(2^{−j}x − k), j ∈ ℤ, k ∈ ℤ^{n}, i = 1, …, 2^{n} − 1},

obtained by translations and dilations of 2^{n}−1 mother wavelets Ψ^{i}, i = 1, …, 2^{n}−1. If the set {Ψ_{j,k}^{i}}_{i,j,k} satisfies the following three conditions:

(c_{1}) {Ψ_{j,k}^{i}}_{i,j,k} is orthogonal with respect to the usual L^{2} inner product

⟨f, g⟩ = ∫_{ℝ^{n}} f(x)\overline{g}(x) dx,

(c_{2}) ∀f ∈ L^{2}(ℝ^{n}), f(x) = ∑_{i=1}^{2^{n}−1} ∑_{j∈ℤ,k∈ℤ^{n}}⟨f, Ψ_{j,k}^{i}⟩Ψ_{j,k}^{i}(x), where the equal-

ity holds a.e., and (c_{3}) {Ψ_{j,k}^{i}}_{i,j,k} satisfies the stability condition, that is, ∃c_{2} > c_{1} > 0 such that

for any f ∈ L^{2}(ℝ^{n}), we have

c_{1}∥f∥_{2}^{2} ≤ ∑_{i=1}^{2^{n}−1} ∑_{j∈ℤ,k∈ℤ^{n}} |⟨f, Ψ_{j,k}^{i}⟩|^{2} ≤ c_{2}∥f∥_{2}^{2},

then {Ψ_{j,k}^{i}}_{i,j,k} is called a dyadic orthogonal wavelet basis of L^{2}(ℝ^{n}).

\begin{matrix}c \\ ◯ \end{matrix}2003 American Mathematical Society

Received by the editors December 14, 2001.

2000 Mathematics Subject Classification. Primary 39B42, 42C05; Secondary 42C15.

Key words and phrases. Multidimensional wavelet bases, multiwavelet bases, refinement equation, stability.

Note that in general, the set of mother wavelets Ψ^{i}, i = 1, …, 2^{n} − 1 is related to a mother scaling function Φ which is an appropriate solution of the following refinement equation:

(1)  Φ(x) = 2^{n} ∑_{k∈ℤ^{n}} α_{k}Φ(2x − k),

with ∫ Φ(x) dx = 1. Note that if in (1), (α_{k})_{k} is a finite sequence satisfying ∑_{k} α_{k} = 1 and if H_{0} denotes the n-variate trigonometric polynomial given by

H_{0}(ω_{1}, … ω_{n}) = ∑_{k} α_{k}e^{−ik·(ω_{1},...,ω_{n})},

then

Φ̂(ω_{1}, …, ω_{n}) = ∏_{j=1}^{∞} H_{0} (\frac{ω_{1}}{2^{j}} , …, \frac{ω_{n}}{2^{j}} ) .

In this case, the different mother wavelets Ψ^{i}, i = 1, …, 2^{n} − 1 are given by

Ψ̂^{i}(ω_{1}, …, ω_{n}) = H_{i} (\frac{ω_{1}}{2} , …, \frac{ω_{n}}{2} ) Φ̂ (\frac{ω_{1}}{2} , …, \frac{ω_{n}}{2} ) , ∀i = 1, …, 2^{n} − 1,

for some appropriate n-variate trigonometric polynomials, also called high-pass wavelet filters.

Finally, we should mention that unlike the 1-D case where an extensive work has been done, the construction of nonseparable orthogonal wavelet bases is still a challenging problem. To this date, only very few references have dealt with the construction of such multivariate orthogonal wavelet bases in some special cases; see [1]. Nonetheless, many references have dealt with the construction of multidimensional biorthogonal wavelet bases.

Finally, this note is organized as follows. In section 2, we construct the different orthogonal wavelet filters, candidates for generating nonseparable wavelet bases of L^{2}(ℝ^{n}), and study the stability of the associated wavelets. In section 3, we show how to extend the previous construction to the design of multidimensional multiwavelet filters.

Notation. In this note, we denote ω_{k} = (ω_{1}, …, ω_{k}) ∈ ℝ^{k}, η_{i} = (η_{i}^{1}, …, η_{i}^{n}) ∈ E_{n} = {0, π}^{n}, π_{k} = (π, …, π) ∈ ℝ^{k}.

2. Construction of n-D compactly supported and orthogonal wavelet bases

2.1. Construction of multivariate orthogonal wavelet filters. We first construct a family of 2-bands orthogonal and compactly supported low-pass wavelet filters (n-variate trigonometric polynomials) H_{n}(ω_{1}, …, ω_{n}). This family is given by the following proposition.

Proposition 1. Let H_{1}(ω) and G_{1}(ω) = e^{−iω}\overline{H}_{1}(ω + π) be any 1-D low-pass and the corresponding high-pass orthogonal wavelet filters of L^{2}(ℝ), respectively. Define an n-variate low-pass wavelet filter H_{n}(ω_{1}, …, ω_{n}) by the following iterative process:

∀ 2 ≤ k ≤ n  choose an integer 1 ≤ l_{k} ≤ k − 1,
P_{k−1}(ω_{1}, …, ω_{k}) = H_{k−1}(2ω_{1}, …, 2ω_{k−1}),
Q_{k−1}(ω_{1}, …, ω_{k}) = H_{k−1}(2ω_{1} + π, …, 2ω_{k−1} + π),
H_{k}(ω_{1}, …, ω_{k}) = P_{k−1}(ω_{1}, …, ω_{k−1})H_{l_{k}}(ω_{k−l_{k}+1}, …, ω_{k})
+ Q_{k−1}(ω_{1}, …, ω_{k−1})G_{l_{k}}(ω_{k−l_{k}+1}, …, ω_{k}),

where G_{l}(ω_{1}, …, ω_{l}) = e^{−iω_{l}} \overline{H_{l}}(ω_{1} + π, …, ω_{l} + π). Then H_{n}(0, …, 0) = 1. Moreover, H_{n} satisfies the following orthogonality condition:

(2) |H_{n}(ω_{1}, …, ω_{n})|^{2} + |H_{n}(ω_{1} + π, …, ω_{n} + π)|^{2} = 1, ∀(ω_{1}, …, ω_{n}) ∈ ℝ^{n}.

Proof. See [6]. □

It is well known that the construction of dyadic, orthogonal and compactly supported wavelet bases of L^{2}(ℝ^{n}) requires the construction of one low-pass filter H_{0} and 2^{n} − 1 high-pass filters H_{i}, i = 1, …, 2^{n} − 1. These different wavelet filters must satisfy the following matrix equation (see [4, 8]):

[ \begin{matrix}\overline{H_{0}(ω)} & \overline{H_{0}(ω + η_{1})} & ⋯ & \overline{H_{0}(ω + η_{2^{n}−1})} \\ \overline{H_{1}(ω)} & \overline{H_{1}(ω + η_{1})} & ⋯ & \overline{H_{1}(ω + η_{2^{n}−1})} \\ ⋮ & ⋮ & & ⋮ \\ \overline{H_{2^{n}−1}(ω)} & \overline{H_{2^{n}−1}(ω + η_{1})} & ⋯ & \overline{H_{2^{n}−1}(ω + η_{2^{n}−1})} \end{matrix} ]

× [ \begin{matrix}H_{0}(ω) & H_{1}(ω) & ⋯ & H_{2^{n}−1}(ω) \\ H_{0}(ω + η_{1}) & H_{1}(ω + η_{1}) & ⋯ & H_{2^{n}−1}(ω + η_{1}) \\ ⋮ & ⋮ & & ⋮ \\ H_{0}(ω + η_{2^{n}−1}) & H_{1}(ω + η_{2^{n}−1}) & ⋯ & H_{2^{n}−1}(ω + η_{2^{n}−1}) \end{matrix} ] = I_{2^{n}},

where the η_{i} are the different points of the set E_{n} = {0, π}^{n}. A solution to the above matrix equation is obtained by the use of a family of n − 1 matrices D_{i} ∈ ℤ^{n×n} satisfying the following three properties:

(P_{1}) ∀η_{j} ∈ E_{n}, D_{i}(η_{j}) = D_{i}(η_{j}^{s}) mod (2πℤ^{n}), where η_{j}^{s} = π_{n} − η_{j}.

(P_{2}) For all η_{j′} ≠ η_{j} and all η_{j′} ≠ η_{j}^{s}, we have D_{i}(η_{j}) ≠ D_{i}(η_{j′}) mod (2πℤ^{n}). (P_{3}) If F_{i} = D_{i}D_{i−1} ⋯ D_{1}(E_{n})/_{2πℤ^{n}}, then |F_{i}| = \frac{|E_{n}|}{2^{i}} and F_{i} is a symmetric

subset of E_{n}, i.e. ∀η ∈ F_{i}, η^{s} ∈ F_{i}.

Remark 1. If {e_{k}}_{k=1,...,n} denotes the usual basis of ℝ^{n}, and if ∀1 ≤ i ≤ n, we denote D_{i} the matrix defined by

D_{i}(e_{k}) = e_{k} if k ≠ i, i + 1, D_{i}(e_{i}) = e_{i} + e_{i+1}, D_{i}(e_{i+1}) = ∑_{k=1,k≠i}^{n} e_{k} − e_{i},

then the family {D_{i}; i = 1, …, n − 1} satisfies the above three properties.

The construction of H_{0} is then given by the following proposition.

Proposition 2. Let H_{0}(ω_{1}, …, ω_{n}) be the n-D filter of Proposition 1. Define H_{0} by

H_{0}(ω_{1}, …, ω_{n}) = ∏_{k=0}^{n−1} H_{0}(D_{k} ⋯ D_{0}(ω_{1}, …, ω_{n})).

Then H_{0}(0, …, 0) = 1. Moreover, H_{0} satisfies the following orthogonality condition:

(3)  ∑_{i=0}^{2^{n}−1} |H_{0}(ω_{n} + η_{i})|^{2} = 1, ∀ω_{n} ∈ ℝ^{n}.

Proof. Since H_{0}(0, …, 0) = 1, we have

H_{0}(0, …, 0) = ∏_{k=0}^{n−1} H_{0}(D_{k} ⋯ D_{0}(0, …, 0)) = 1.

Moreover, since

D_{1}(η_{i}) = D_{1}(η_{i}^{s}) mod (2πℤ^{n}), ∀η_{i} ∈ E_{n},

it follows that

∑_{i=0}^{2^{n}−1} |H_{0}(ω + η_{i})|^{2}

= ∑_{η_{i}∈F_{1}} [|H_{0}|^{2}(ω + η_{i}) + |H_{0}|^{2}(ω + η_{i}^{s})] · ∏_{k=1}^{n−1} |H_{0}|^{2}(D_{k} ⋯ (D_{1}(ω) + η_{i}))

= ∑_{η_{i}∈F_{1}} ∏_{k=1}^{n−1} |H_{0}|^{2}(D_{k} ⋯ (D_{1}(ω) + η_{i}))

= ∑_{η_{i}∈F_{2}}[|H_{0}|^{2}(D_{1}(ω) + η_{i}) + |H_{0}|^{2}(D_{1}(ω) + η_{i}^{s})]· ∏_{k=2}^{n−1} |H_{0}|^{2}(D_{k} ⋯ (D_{1}(ω) + η_{i}))

= ∑_{η_{i}∈F_{2}} ∏_{k=2}^{n−1} |H_{0}|^{2}(D_{k} ⋯ (D_{1}(ω) + η_{i}))

⋮

= ∑_{η_{i}∈F_{n−1}} |H_{0}|^{2}(D_{n−1} ⋯ D_{1}(ω) + η_{i})
= |H_{0}|^{2}(D_{n−1} ⋯ D_{1}(ω) + η_{0}) + |H_{0}|^{2}(D_{n−1} ⋯ D_{1}(ω) + η_{0}^{s}) = 1.
□

Once H_{0} is constructed, one is faced with the problem of finding the remaining 2^{n} − 1 high-pass wavelet filters H_{i}, i = 1, …, 2^{n} − 1, satisfying the following equations:

(4)  ∑_{η_{i}∈E_{n}} H_{j}(ω_{n} + η_{j})\overline{H_{j′}}(ω_{n} + η_{i}) = δ_{jj′}, ∀ 0 ≤ j, j′ ≤ 2^{n} − 1.

In general, a solution of (4) is hard to obtain. Nevertheless and thanks to the particular structure of H_{0}, a solution to the above equations is given by the following theorem. Note that the biorthogonal version of this theorem is given in [5].

Theorem 1. Let H_{0} be the n-D wavelet filter of Proposition 1. Let D_{1}, D_{2}, …, D_{n−1} be a set of matrices with integer coefficients and satisfying the three properties (P_{1}), (P_{2}) and (P_{3}). Define a filter H_{1} by

H_{1}(ω_{n}) = e^{−iω_{1}} \overline{H_{0}}(ω_{n} + π_{n}).

For i = 1, …, 2^{n} − 1, define H_{i} by

(5) H_{i}(ω_{n}) = ∏_{j=0}^{n−1}[ϵ_{j}^{i}H_{0}(D_{j}D_{j−1} ⋯ D_{0}ω_{n}) + (1 − ϵ_{j}^{i})H_{1}(D_{j}D_{j−1} ⋯ D_{0}ω_{n})],

where (ϵ_{0}^{i}, ϵ_{1}^{i}, …, ϵ_{n−1}^{i})_{i=1,2^{n}−1} are the different points of {0, 1}^{n}\(0, 0, …, 0). Then H_{i}, i = 0, 2^{n} − 1, is a solution of (4).

Proof. See [6]. □

2.2. Stability of the orthogonal wavelet bases. It is well known that a solution of (4) does not ensure the construction of wavelet bases of L^{2}(ℝ^{n}). In fact, the translates of the scaling function Φ derived from H_{0} must satisfy the stability condition, that is,

(6)  0 < c_{1} ≤ ∑_{k∈ℤ^{n}} |Φ̂(ω_{n} + 2πk)|^{2} ≤ c_{2} < +∞,

for some constants c_{1}, c_{2}.

It is known (see [3, 7, 9]) that if T_{H_{0}} denote the transition operator associated with H_{0} and defined by

T_{H_{0}}(f)(ω_{n}) = ∑_{η_{j}∈E_{n}} |H_{0}(\frac{ω_{n}}{2} + η_{j})|^{2}f(\frac{ω_{n}}{2} + η_{j})

and if V is a finite-dimensional subspace of n-variate trigonometric polynomials and V is invariant under the action of T_{H_{0}}, then the stability and the orthogonality of our wavelet basis can be deduced from the spectral radius of T_{H_{0}}/_{V}, the restriction of T_{H_{0}} to V. More precisely, if 1 is a simple eigenvalue of T_{H_{0}}/_{V} and if the other eigenvalues are inside the unit circle, then H_{0} gives rise to an orthogonal wavelet basis of L^{2}(ℝ^{n}). The following theorem provides us with an explicit invariant subspace V under T_{H_{0}} as well as the different coefficients of the matrix representing T_{H_{0}}/_{V} .

Theorem 2. Let H_{0} be the filter of Proposition 2 and

|H_{0}(ω_{1}, …, ω_{n})|^{2} = ∑_{m_{1},...,m_{n}=−M}^{M} β_{m_{1},...,m_{n}} cos [∑_{i=1}^{n} m_{i}ω_{i}] .

Let V = span Ω, where Ω = {U_{k} = cos [∑_{i=1}^{n} k_{i}ω_{i}] , k = (k_{1}, …, k_{n}) ∈ supp |H_{0}|^{2}}. Define a matrix B_{H_{0}} by

B_{H_{0}}(U_{k}) = 2^{n−1} ∑_{l,k,l+2k∈supp |H_{0}|^{2}} β_{k+2l}[U_{l+k} + U_{l}].

Then, B_{H_{0}} represents the restricted operator T_{H_{0}}/_{V} . In particular, if 1 is a simple eigenvalue of B_{H_{0}} and if the other eigenvalues are inside the unit circle, then H_{0} generates an orthogonal wavelet basis of L^{2}(ℝ^{n}).

Proof. See [6]. □

3. Construction of n-D multiwavelet filters

In this section we show that the techniques of the previous section can be used for the construction of orthogonal multiwavelet bases of L^{2}(ℝ^{n}). Note that a 1-D orthogonal multiwavelet basis of multiplicity r ≥ 2, r ∈ ℕ is an orthogonal and stable set generated by translations and dilations of a vector-valued mother multiwavelet function Ψ = (ψ^{1}, …, ψ^{r})^{T} . The associated vector-valued multiscaling function Φ = (ϕ^{1}, …, ϕ^{r})^{T} satisfies the functional equation

(7)  Φ(x) = ∑_{α∈ℤ} P_{α}Φ(2x − α),

where P_{α} is an r × r matrix with real coefficients. If we define the trigonometric matrix function P (ω) by

(8)  P (ω) = \frac{1}{2} ∑_{α∈ℤ} P_{α} exp(−iαω),

then by applying Fourier transform to (7), the latter can be written as

(9)  Φ̂(ω) = P (\frac{ω}{2} ) · Φ̂ (\frac{ω}{2} ) .

Unlike the one-dimensional case, where an extensive work has been done in the theory and the design of 1-D multiwavelet bases, only very few references have dealt with the theory and the design of multidimensional multiwavelet bases [2]. For the sake of simplicity, we will be concerned with the construction of n-D multiwavelet matrix filters in the special case r = 2. Moreover, we will use very often the following basic condition B on the low-pass matrix filter P (ω), which is a necessary condition for the existence of multiwavelet basis and it is due to [9]. This condition is given as follows:

Basic condition B. We say that the matrix mask P satisfies the basic condition B, if the following two conditions are satisfied:

(C_{1}) P (0) has the form [ \begin{matrix}1 & 0 \\ 0 & λ \end{matrix}], where λ is an (r − 1) × (r − 1) matrix such that

ρ(λ) < 1. Here ρ(·) denotes the spectral radius of a given matrix. (C_{2}) ∀μ ∈ {0, 1}^{n}, e_{1}^{T} P (μπ) = δ_{α}e_{1}^{T} , where e_{1}^{T} = (1, 0, …, 0) ∈ ℝ^{r}.

It is shown in [9] that if P (ω) satisfies condition B, then the infinite product ∏_{j=1}^{∞} P (\frac{ω}{2^{j}} ) converges to the r × r matrix [Φ̂(ω)A, 0, …, 0], where A is an eigenvector of P (0) corresponding to the eigenvalue 1. Also, it is well known that a necessary condition for the orthonormality of the r components of Φ is the following equality:

(10)  P (ω)P^{∗}(ω) + P (ω + π)P^{∗}(ω + π) = I_{r},

where P^{∗} = \overline{P}^{T} , the transpose of the conjugate of P. Also, it is well known that the construction of n-D dyadic multiwavelet matrix filters with multiplicity r = 2 requires the design of one low-pass 2 × 2 matrix filter P_{n}(ω) and 2^{n} − 1 high-pass 2 × 2 matrix filters Q_{n}^{i}(ω) such that the following matrix equation holds:

(11)  [ P_{n}(ω + μπ) ; Q_{n}^{1}(ω + μπ) ; ⋮ ; Q_{n}^{2^{n}−1}(ω + μπ) ]_{μ∈{0,1}^{n}} · [ P_{n}(ω + μπ) ; Q_{n}^{1}(ω + μπ) ; ⋮ ; Q_{n}^{2^{n}−1}(ω + μπ) ]_{μ∈{0,1}^{n}}^{∗} = I_{2^{2n}}.

One major difficulty in the design of multidimensional multiwavelet matrix filters is the noncommutativity of the matrix product. As we will see, this extra difficulty does not affect very much the construction method of the previous section. To solve (11), we need the result of the following proposition, whose proof is given in [6].

Proposition 3. Let P_{1}, Q_{1} be a 1-D matrix filters generating 1-D multiscaling function and multiwavelet with multiplicity 2. Let H_{k} be the p-variate wavelet filter of Proposition 1. For 2 ≤ k ≤ n, define two sequences of matrix filters P_{k}, Q_{k} by the following iterative process:

∀ 2 ≤ p ≤ k,  let
L_{p}(ω_{p}) = H_{p}(2ω_{p}), M_{p}(ω_{p}) = H_{p}(2ω_{p} + π_{p}),
P_{p}(ω_{p}) = L_{p}(ω_{p})P_{p−1}(ω_{p−1}) + \overline{M_{p}}(ω_{p})Q_{p−1}(ω_{p−1}),
Q_{p}(ω_{p}) = M_{p}(ω_{p})P_{p−1}(ω_{p−1}) − \overline{L_{p}}(ω_{p})Q_{p−1}(ω_{p−1}).

Then P_{n}(·), Q_{n}(·) is a solution of the following equation:

(12)  [ \begin{matrix}P_{n}(ω_{n}) & P_{n}(ω_{n} + π_{n}) \\ Q_{n}(ω_{n}) & Q_{n}(ω_{n} + π_{n}) \end{matrix} ] · [ \begin{matrix}P_{n}^{∗}(ω_{n}) & Q_{n}^{∗}(ω_{n}) \\ P_{n}^{∗}(ω_{n} + π_{n}) & Q_{n}^{∗}(ω_{n} + π_{n}) \end{matrix} ] = I_{4}.

Next, the following proposition gives us an n-variate matrix filter candidate for generating an n-D multiscaling function.

Proposition 4. Let P_{n} be as given by Proposition 3 and let D_{0} = I_{n} and D_{i}, i = 1, …, n − 1 be a set of matrices satisfying the three properties (P_{1}), (P_{2}) and (P_{3}). Define an n-variate matrix filter P_{n}(ω_{n}) by

(13)  P_{n}(ω_{n}) = ∏_{k=0}^{n−1} P_{n}(D_{n−k−1} ⋯ D_{0}ω_{n});

then P_{n} satisfies the basic condition B. Moreover, P_{n} is a solution of the following matrix equation:

(14)  ∑_{η_{k}∈E_{n}={0,π}^{n}} P_{n}(ω_{n} + η_{k})P_{n}^{∗}(ω_{n} + η_{k}) = I_{2}, ∀ω_{n} ∈ ℝ^{n}.

Proof. See [6]. □

Finally, the different 2^{n} − 1 matrix filters candidates for generating the different n-D mother multiwavelet functions are given by the following theorem, whose proof is given in [6].

Theorem 3. Let P_{n}, Q_{n} be as given by Proposition 3 and let P_{n} be as given by the previous proposition. Let D_{0} = I_{n} and let D_{1}, …, D_{n−1} be a set of matrices satisfying the three properties P_{1}, P_{2}, P_{3}. Define 2^{n} − 1 matrix filters Q_{n}^{i}, i = 1, …, 2^{n} − 1 by

(15) Q_{n}^{i}(ω) = ∏_{k=0}^{n−1} [ϵ_{k}^{i}P_{n}(D_{n−k−1} ⋯ D_{0}ω) + (1 − ϵ_{k}^{i})Q_{n}(D_{n−k−1} ⋯ D_{0}ω)] ,

where (ϵ_{0}^{i}, …, ϵ_{n−1}^{i})_{i=1,2^{n}−1} are the different points of {0, 1}^{n} \ {0_{n}}. The set

{P_{n}, Q_{n}^{1}, …, Q_{n}^{2^{n}−1}}

is a solution of (11).

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Université du 7 Novembre à Carthage, Institut Supérieur des Sciences Appliquées et de la Technologie de Mateur, 7030, Tunisia

E-mail address: abkaroui@yahoo.com