Construction of orthonormal multi-wavelets with additional vanishing moments.

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Construction of Orthonormal Multi-Wavelets
with Additional Vanishing Moments
Charles K. Chuia,∗
aDepartment of Mathematics & Computer Science, University of Missouri–St.
Louis, St. Louis, MO 63121, USA, and
Department of Statistics, Stanford University, Stanford, CA 94305, USA
Jian-ao Lianb,†
bDepartment of Mathematics, Prairie View A&M University, Prairie View, TX
77446, USA
Abstract. An iterative scheme for constructing compactly supported orthonormal
(o.n.) multi-wavelets with vanishing moments of arbitrarily high order is established.
Precisely, let φ = [φ1,···,φr]?be an r-dimensional o.n. scaling function vector with
polynomial preservation of order (p.p.o.) m, and ψ = [ψ1,···,ψr]?an o.n. multi-
wavelet corresponding to φ, with two-scale symbols P and Q, respectively. Then a
new (r + 1)-dimensional o.n. scaling function vector φ?:= [φ?,φr+1]?and some
corresponding o.n. multi-wavelet ψ?are constructed in such a way that φ?has
p.p.o. = n > m and their two-scale symbols P?and Q?are lower and upper triangular
block matrices, respectively, without increasing the size of the supports. For instance,
for r = 1, if we consider the mthorder Daubechies o.n. scaling function φD
φ?:= [φD
for r = 2, if we use the symmetric o.n. scaling function vector φ in our earlier work
[3], then we obtain a new pair of scaling function vector φ?= [φ?,φ3]?and multi-
wavelet ψ?that not only increase the order of vanishing moments but also preserve
symmetry.
m, then
m,φ2]?is a scaling function vector with p.p.o. > m. As another example,
Keywords: scaling function vectors, multi-wavelets, two-scale symbols, two-scale
equations, orthonormality, compactly supported functions
Dedicated to Charles A. Micchelli in Honor of His Sixtieth Birthday
1. Introduction
A vector-valued function φ := [φ1,···,φr]?is called a refinable func-
tion vector of dimension r, or r-refinable function vector, if φ ∈ (L2)r,
L2:= L2(I R), and satisfies φ(x) =?M
j=0Pjφ(2x − j), P0, PM ?= Or,
for some finite matrix sequence {Pj} ⊂ I Rr×r, where Or denotes the
∗Supported in part by NSF grants CCR-9988289 and CCR-0098331 and Army
Research Office under grant DAAD 19-00-1-0512.
†Supported in part by Army Research Office under grant DAAD 19-01-1-0739.
c ? 2003 Kluwer Academic Publishers. Printed in the Netherlands.
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C. Chui & J.-A. Lian
zero square matrix of order r. The r-refinable function vector φ is
called an r-scaling function vector, if φ is L2-stable, meaning that
{φ?(·− k) : 1 ≤ ? ≤ r; k ∈ Z Z} is a Riesz basis of V0, where, for j ∈ Z Z,
Vj:= closL2 span2j/2φ?(2j·−k) : 1 ≤ ? ≤ r; k ∈ Z Z
??
,
which is also called a multiresolution analysis (MRA) of L2, provided
L2= closL2 ∪j∈Z ZVj. Corresponding to an r-scaling function vector
φ, a vector-valued function ψ(x) = [ψ1,···,ψr]?=
k), Q0, QN ?= Or, for some finite matrix sequence {Qj} ⊂ I Rr×r, is
called a (semi-orthogonal) multi-wavelet, if {ψ?(· − k) : 1 ≤ ? ≤ r;k ∈
Z Z} is a Riesz basis of the L2-orthogonalcomplementary subspace W0⊂
V1relativetoV0, i.e.,V1= V0⊕⊥W0. A function vector η = [η1,...,ηr]?
is said to be orthonormal (o.n.) if it satisfies ?η?(· − k),η??(· − k?)? =
δ?,?? δk,k?, ?, ??= 1,···,r; k, k?∈ Z Z. Hence, an o.n. multi-waveletψ cor-
responding to an o.n. r-scaling function vector φ generates an o.n. basis
{2−j/2ψ?(2j· −k) : ? = 1,...,r; j,k ∈ Z Z} of L2.
The above time-domain definitions have frequency-domain formula-
tions
?N
k=0Qkφ(2x −
?φ(ω) = P
?ψ(ω) = Q
?
?
e−iω/2??φ
e−iω/2??φ
?ω
?ω
2
?
?
,P(z) =1
2
M
?
N
?
j=0
Pjzj,
(1.1)
2
,Q(z) =1
2
j=0
Qjzj,
where the polynomial matrices P,Q ∈ πr×r, π := π(I R), are called
two-scale symbols of φ and ψ, respectively. To study the smoothness
of o.n. scaling function vectors, we consider the usual H¨ older spaces
Cγ:= Cγ(I R), where γ > 0 and 0 < γ − ?γ? < 1, defined by
Cγ(I R) =
f :
?
|f(?γ?)(·+ h) − f(?γ?)(·)| ≤ C |h|γ−?γ??
.
(1.2)
Finally, the supports of φ and ψ are defined by suppφ = ∪r
and suppψ = ∪r
The first main objective of this paper is to introduce an iterative
scheme for constructing compactly supported o.n. scaling function vec-
tors and multi-wavelets in terms of any given o.n. scaling function (or
scaling function vector) of lower dimension in order to achieve addi-
tional vanishing moments for the multi-wavelets. Precisely, let φ be
an o.n. scaling function (vector) with polynomial preservation of order
(p.p.o.) m, and ψ an o.n. (multi-)waveletcorresponding to φ, with two-
scale symbols P and Q, respectively. Then a new o.n. (r + 1)-scaling
?=1suppφ?
?=1suppψ?, respectively.
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Orthonormal Multi-Wavelets with Vanishing Moments
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function vector φ?:= [φ?,φr+1]?and some corresponding o.n. multi-
wavelet ψ?can be constructed in such a way that φ?has p.p.o. = n > m
and that their two-scale symbols P?, Q?are lower and upper triangular
block matrices, without increasing the support. Furthermore, if φ is
symmetric and/or anti-symmetric, φ?and ψ?can be constructed to
preserve symmetry and/or anti-symmetry with only slight increase
in support. For instance, for r = 1, if we start with the mthorder
Daubechies o.n. scaling function φD
function vector with p.p.o. > m. As another example, for r = 2, if we
use the symmetric o.n. scaling function vectors φ = [φ1,φ2]?in our
earlier work [3], then we obtain new pairs of scaling function vectors
φ?= [φ?,φ3]?and multi-waveletsψ?that not only increase the order of
vanishing moments but also preserve symmetry. An important feature
of our method is that the corresponding multi-wavelets are easy to
construct.
The main results of this paper will be presented in §2 and proved in
§3. An interesting example of o.n. φ and ψ with support= [0,1] will be
considered in §4. This is a natural vector-valued extention of the Haar
wavelet. For r = 1 and with φ1= φD
multi-wavelets will be constructed in §5. The construction of o.n. sym-
metric scaling function vectors and multi-wavelets will constitute §6,
where r = 2 and φ1being scaling function vectors of those in [3] and
GHM [5].
m, then φ?:= [φD
m,φ2]?is a scaling
m, o.n. scaling function vectors and
2. Main Results
A scaling function vector φ has p.p.o. = m, if m is the largest integer for
which there is a set of row vectors {a0,?}m−1
that satisfy, for ? = 0,...,m− 1,
?
k
?
k
j∈Z Z
?=0⊂ I R1×r, with a0,0?= O1×r
?
?
k=0
(−1)k1
2k
?
?
?
a0,?−k?
j∈Z Z
(2j)kP2j =
1
2?a0,?;(2.1)
?
?
k=0
(−1)k1
2k
?
a0,?−k?
(2j + 1)kP2j+1 =
1
2?a0,?,
(2.2)
(see [8] for details). As shown in [11], if a scaling function vector φ has
p.p.o. = m, then det(P(z)) must be divisible by (1 + z)m. However,
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C. Chui & J.-A. Lian
this condition is not sufficient. As a simple counterexample, consider


which satisfies (1+z)3|det(P(z)). If P is the two-scale symbol of some
scaling function vector φ, then, with φ1being normalized as the sec-
ond order cardinal B-spline, the corresponding autocorrelation symbol
Φ(z) :=?

2(1 − β)
Hence, since Φ must be positive definite for |z| = 1, we have |β| < 1
(see, e.g., [4] and [6]), which implies that φ has p.p.o. = 2 when β ?= 1/4,
and p.p.o. = 3 when β = 1/4. In general, by applying (2.1)–(2.2), we
can establish the following.
P(z) =


?1 + z
2
α
?2
0
1 + z
2
β



,αβ ?= 0,
n∈Z Z(?φj(·),φk(· − n)?)1≤j,k≤2zn, is given by


Φ(z) =

1 + 4z + z2
6z
α
α
2(1− β)(1+ z)
2α2(2+ β)
3(1+ β)(1− β)2
?
1 +1
z
?



.
Theorem 2.1. Let φ be an r-scaling function vector with two-scale
symbol P and p.p.o. = m, and consider the polynomial matrix
?P(z) Or×1
where
a(z) =1
2
k∈Z Z
B(z) = [B1(z),...,Br(z)] =:1
P?=
B(z) a(z)
?
,
(2.3)
?
akzk;(2.4)
2
?
k∈Z Z
bkzk,
(2.5)
with row vectors bk, k ∈ Z Z. Then a necessary condition for P?to
be the two-scale symbol of some (r + 1)-scaling function vector φ?=
[φ?,φr+1]?that provides p.p.o. = n > m is that
?1 + z
s(1) = 1.
a(z) =
1
2m
2
?n−m
s(z);(2.6)
(2.7)
By applying Theorem 2.1 to any pair of o.n. scaling function vectors
and multi-wavelets,not only do we obtain a new scaling function vector
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Orthonormal Multi-Wavelets with Vanishing Moments
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with higher p.p.o., but also some corresponding o.n. multi-wavelet can
be easily constructed. Precisely, we have the following.
Theorem 2.2. Let φ be an o.n. r-scaling function vector with two-
scale symbol P and p.p.o = m, and ψ be its corresponding o.n. multi-
wavelet with two-scale symbol Q(z). Then there is φr+1such that φ?=
[φ?,φr+1]?is an o.n. (r+1)-scaling function vector with p.p.o. ≥ m+1
and two-scale symbol P?given by (2.3)–(2.5) with
B(z) =
r
?
j=1
αjQj(z),
(2.8)
a(−1) = 0,a(1) =
1
2m,
(2.9)
|a(z)|2+ |a(−z)|2= 1 −
r
?
j=1
|αj|2=
1
22m,
|z| = 1,
(2.10)
for some constants α1,...,αr. In addition, ψ?is an o.n. multi-wavelet
corresponding to φ?, with two-scale symbol Q?being upper triangular
block matrix, given by
?
O1×r
Q?(z) =
AQ(z) 2ma(z)[ζ1,...,ζr]?
2mz2L−1a(−z)?
?
,
(2.11)
where A is a constant matrix, deg(a) = 2L − 1, and [ζ1,...,ζr]?is a
constant vector, such that
[α1,...,αr]A?+
1
2m[ζ1,...,ζr] = 0,
(2.12)
AA?+ [ζ1,...,ζr]?[ζ1,...,ζr] = Ir,
(2.13)
where Iris the identity matrix of order r.
One of the important features of the construction procedure de-
scribed here is that it can be applied repeatedly without increasing the
support (or filter length). However, if the support of φ?needs to be
increased, B in (2.8) can be chosen as
B(z) =
r
?
j=1
αj(z2)Qj(z),
(2.14)
for some Laurent polynomials α1,...,αr, while A in (2.11) is chosen
to be a Laurent polynomial matrix in z2. By doing so, the governing
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