Dyadic-based factorizations for regular paraunitary filterbanks and M-band orthogonal wavelets with structural vanishing moments

Abstract

Paraunitary filterbanks (PUFBs) can be designed and implemented using either degree-one or order-one dyadic-based factorization. This work discusses how regularity of a desired degree is structurally imposed on such factorizations for any number of channels M ≥ 2, without necessarily constraining the phase responses. The regular linear-phase PUFBs become a special case under the proposed framework. We show that the regularity conditions are conveniently expressed in terms of recently reported M-channel lifting structures, which allow for fast, reversible, and possibly multiplierless implementations in addition to improved design efficiency, as suggested by numerical experience. M-band orthonormal wavelets with structural vanishing moments are obtained by iterating the resulting regular PUFBs on the lowpass channel. Design examples are presented and evaluated using a transform-based image coder, and they are found to outperform previously reported designs.

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 1, JANUARY 2005193
Dyadic-Based Factorizations for Regular Paraunitary
Filterbanks and ?-Band Orthogonal Wavelets with
Structural Vanishing Moments
Ying-Jui Chen, Member, IEEE, Soontorn Oraintara, Senior Member, IEEE, and Kevin S. Amaratunga, Member, IEEE
Abstract—Paraunitaryfilterbanks (PUFBs) can be designed and
implemented using either degree-one or order-one dyadic-based
factorization. This paper discusses how regularity of a desired de-
gree is structurally imposed on such factorizations for any number
of channels
?, without necessarily constraining the phase
responses. The regular linear-phase PUFBs become a special case
under the proposed framework. We show that the regularity con-
ditions are conveniently expressed in terms of recently reported
-channel lifting structures, which allow for fast, reversible, and
possibly multiplierless implementations in addition to improved
design efficiency, as suggested by numerical experience.
orthonormal wavelets with structural vanishing moments are
obtained by iterating the resulting regular PUFBs on the lowpass
channel. Design examples are presented and evaluated using a
transform-based image coder, and they are found to outperform
previously reported designs.
-band
Index Terms—Regular paraunitary filterbank, dyadic-based
factorization, vanishing moment, wavelet.
I. INTRODUCTION
A
operation stands for conjugate transpose
. Namely,
If
is both PU and FIR, it is automatically lossless, and the
synthesisfilterscanbefounddirectlyfromtheanalysisfiltersby
inspection (in fact, by time reversal and complex conjugation)
[1]. In this paper, we will consider exclusively causal and FIR
paraunitary filterbanks (PUFBs).
The McMillan degree and the order of an
matrix
are two distinct but important concepts. The
(McMillan) degree of
refers to the minimum number
of delay elements required for its implementation. A minimal
structure of
is one that uses this minimum number of
delay elements in it; as a contrast, the order of
the highest power of
in
less than the order.
N
said to be paraunitary (PU) if
-channel filterbank with polyphase matrixis
, where the
and time-reversal
is unitary on the unit circle,.
polyphase
refers to
. As a result, the degree is no
Manuscript received October 1, 2003; revised December 27, 2003. The as-
sociate editor coordinating the review of this manuscript and approving it for
publication was Dr. Xiang-Gen Xia.
Y.-J. Chen was with the Intelligent Engineering Systems Laboratory, Massa-
chusetts Institute of Technology, Cambridge, MA 02139 USA. He is now with
MediaTek, Inc. Hsinchu, Taiwan, R.O.C. (e-mail: yrchen@alum.mit.edu).
S. Oraintara is with the Electrical Engineering Department, University of
Texas at Arlington, Arlington, TX 76011 USA (e-mail: oraintar@uta.edu).
K. S. Amaratunga is with Intelligent Engineering Systems Labora-
tory, Massachusetts Institute of Technology, Cambridge, MA 02139 USA
(kevina@mit.edu).
Digital Object Identifier 10.1109/TSP.2004.838962
Any PUFB
torization
of degreealways assumes the fac-
, where
is the degree-one
is unitary [1]. Each
with
paraunitary building block, and
can be implemented using only one delay element [1]. It is
named the dyadic-based structure as it involves the dyadic form
. In [2], the use of dyadic-based structure for filterbank
design was studied and was shown to outperform the Givens
rotation-based parameterization. Generalizing the above de-
gree-constrained structure, Gao et al. have recently proposed a
factorization given the order of the PUFB [3].
Regularity of a filterbank is equivalent to the number of van-
ishing moments of the
-band wavelets [4], which are suitable
for approximating the Sobolev space as they are orthogonal to
polynomials up to a certain order [4], [5]. As such, the decay of
wavelet coefficients [5]–[7] and the accuracy of approximation
are both determined by the degree of regularity, which is further
related to the smoothness of the scaling function. One smooth-
ness index is the Sobolev regularity, which measures the
finite-energy differentiability of the scaling function [5], [8]. As
onecanexpect,themoreregularthefilterbank,thesmootherthe
scaling function, and the more derivatives it has. In many appli-
cations such as smooth signal interpolation, approximation, and
data compression [5], [7], [9]–[11], regular filterbanks are very
desirable.
In [4], [12], and [13], a closed-form expression for
scaling filter
was derived, and a technique for con-
structing a family of PUFBs or
by further assuming a given unitary matrix
chosen in an ad hoc fashion—the issue of how to choose
was not fully addressed. Consequently, the resulting PUFB may
not be optimal given certain design criteria, and faces the same
problem of being (McMillan) degree-constrained as pointed
out in [2].
For the class of
-channel linear-phase PUFBs (a.k.a.
GenLOT [14]) with
even, the imposition of up to two
degrees of regularity on the lattice structure was discussed in
[15]. The regularity conditions were expressed in terms of the
Givens rotation angles of the lattice components. On the other
hand, for the most general class of
without the linear-phase constraint, the imposition of structural
regularity has not been reported, except when
which regularity of degree one is guaranteed if all the rotation
angles of the lattice structure sum up to
solve this problem in its most general form by considering a
higher degree of regularity and an arbitrary number of channels
or
-regular
fromwas proposed
, which was
-channel regular PUFBs
for
[5]. We aim to
1053-587X/$20.00 © 2005 IEEE

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194 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 1, JANUARY 2005
without necessarily constraining the phase responses.
The resulting design outperforms and spans a larger class than
the regular GenLOT [15]. Preliminary results can be found in
[16].
In the following, we will first present the preliminaries nec-
essary for complete parameterizations of a PUFB, including the
degree-one and order-one factorizations (Section II). We will
thenfocusontheimpositionofregularityonthesetwoclassesof
dyadic-based structures (Section III), with important properties
and conditionsderivedand geometric interpretations given.The
special class of linear-phase PUFBs is revisited within the pro-
posed framework, and the corresponding regularity conditions
are shown to simplify in this case. All the regularity conditions
on the dyadic-based structures are shown to be conveniently ex-
pressed in terms of recently reported
ization (Section IV), which allows for efficient and reversible
implementations of the filterbank, and results in faster conver-
gence than the Givens rotation-based parameterization in the
design phase. Regular lifting structures are proposed. Finally,
based on the derived regularity conditions, design examples are
presented(Section V)and evaluatedina transform-basedimage
coder (Section VI)—the resulting regular PUFBs outperform
existingonesintermsofbothsubjectiveandobjectivemeasures.
Concluding remarks are found in Section VII.
The following notations will be used. Bold-faced characters
denote eithera columnvectoror a matrix,e.g.,
column of an
-indexed matrix
. When references are made to the th
element of an
-vector, we use
.For
basisvectorfor
isthe-vector
with the 1 in the th position.
all zeros and all ones, respectively.
identity and reverse identity matrices, respectively. The dimen-
sion subscript
will be omitted if the dimension is clear from
the context. The rank of a matrix
constant matrixis said to be unitary if
-channel lifting factor-
and.The th
, withis denoted as
or, equivalently,
,the thstandard
andare the
and
-vectors of
are the
is denoted as. An
.
II. REGULARITY AND DYADIC-BASED FACTORIZATION FOR
PARAUNITARY FILTERBANKS—PRELIMINARIES
In this section, the factorization of a paraunitary matrix into
building blocks involving the dyadic form
viewed, where
is either a unit-norm vector or a unitary ma-
trix(notnecessarilysquare).Somepropertiesofthedyadicform
canbefoundin[2].Thedefinitionandimplicationsofregularity
will also be covered in this section.
is briefly re-
A. Householder Transformation [17]
The
-dimensional Householder transformation
a given vectorin
a plane
with unit normal
geometry, it can be derived that
maps
to a mirror image
, i.e.,
with respect to
. By simple
(1)
Apparently, this is an invertible, length-preserving and thus or-
thogonal transformation. Since
is also a mirror image of
with respect to the plane, the inverse ofis simply itself
.
Given a nonzero vector
with
, one can choose a unit vector
aligns with the desired coordinate axis
and a desired coordinate axis
such that the transformation
Inthiscase,onecanshowthat
forany
that
andaproperchoiceof
[1].
such
B. Householder Factorization of Unitary Matrices
A unitary matrix can be factored as a product of Householder
matrices. Let
beunitary. There exists a Householder
transformation
that aligns the 0th column of
namely
with,
for some unit-norm vector
. Such a process can be repeated on
and some square unitary matrix
, and so on, to arrive at
(2)
where
Householder factorization of
diag. Hence, the
is given by
(3)
Note that the vectors
take the following form:
... ...
...
......
(4)
where
denotes possibly nonzero values.
C. Complete Dyadic-Based Factorizations for Paraunitary
Filterbanks
Two types of dyadic-based factorizations provide a complete
parameterization of a PUFB with or without length constraint;
they are the order-one factorization [3] and the degree-one fac-
torization [1], [3], respectively.
1) Degree-OneParaunitary
(McMillan) degree of a multi-input–multi-output causal system
refers to the minimum number of delay elements required to
implement it. The following degree-one paraunitary building
blocks are cascaded to increase the degree of an
PUFB.
Lemma1 (Degree-One
[1]): The dyadic-based structure with parameter vector
BuildingBlock: The
-band
ParaunitaryBuildingBlock
(5)

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CHEN et al.: DYADIC-BASED FACTORIZATIONS FOR REGULAR PARAUNITARY FILTERBANKS195
is thedegree-one paraunitary building block: Anydegree-
raunitary polyphase matrix
pa-
can be factored as
(6)
where
as the degree-one factorization and is complete for any given
degree
.
Remark: This structure covers a larger class than the
GenLOT [14] as the linear-phase constraint is not assumed
by (6). Each
increases the filter length by
cascaded and, therefore, is of order one. However, it is not the
most general building block of order one. In fact, an
PU building block of order one can have degree up to
2) Order-One Paraunitary Building Block: To generalize
the degree-one factorization, complete and minimal structures
for paraunitary filterbanks of a given order were proposed in
[3]. Such PUFBs are constructed by cascading an appropriate
number of the following order-one PU building blocks.
Lemma 2 (Order-One Paraunitary Building Block [3]): The
dyadic-based structure with parameter matrix
is unitary:. This structure is referred to
upon
.
(7)
is the order-one paraunitary building block for some integer
with. Any order-
can be factored as
paraunitary polyphase matrix
(8)
forsome
structure is referred to as the order-one factorization of
is complete for any given order
monotonically ordered
unitaryandsomeintegers.This
. It
, and the integers can be
(9)
or
(10)
without affecting the completeness of the structure.
Remarks: In (7), the parameter matrix
orthonormal columns,
consists of
(11)
with
wehave
is thus
cascade of
.Sincethematrix isunitary,
,andthedegreeof
in (7) can be decomposed into a[1]. In fact,
degree-one PU building blocks:
(12)
where for each
PU building block (5) with parameter vector
from (11); on the other hand, given an orthonormal basis
some of any nonzero subspace of
the corresponding degree-one PU building blocks
product(12)isalwaysoforderone.Inviewof
is the minimum-degree order-one building block. In practice,
order-one factorization is preferred as one cares more about
the filter length than the McMillan degree of the PUFB. It is
also necessary because when the order or filter length is spec-
ified, it allows for more design flexibility than the degree-one
factorization.
is the degree-one
coming
and
, the
D. Regularity of PUFBs and Sobolev Smoothness
An
-channel finite impulse response (FIR) PUFB is said
to be
-regular or have
degrees of regularity if its scaling
filter
has a zero of multiplicity
unity
for
be expressed as
at the
. Namely,
th roots of
can
(13)
where
tion can be shown to be equivalent to the number of zeros at
DC of the bandpass and highpass filters (vanishing moments)
[4], [18]. In particular, a PUFB is
bandpass filters
have
, namely
is FIR, satisfying. This condi-
-regular if and only if the
multiple zeros at, for
... ...(14)
where
polyphase matrix of the PUFB.
Closelyrelatedtoregularityorvanishingmomentsisthecon-
cept of smoothness of the
-band wavelet basis. The Sobolev
regularity of a filterbank measures the
the corresponding scaling function
functions
) and is completely determined by the scaling
filter
. In particular, assume for the moment that
in (13) has been appropriately normalized such that
, and letbe the associated convolution matrix. Then, the
Sobolev regularity or Sobolev smoothness
for, andis the Type-I
differentiability of
(and, thus, the wavelet
as
is given by
where
eigenvalue of its argument.
or transfer operator associated with
tion operator of a filter
, anddenotes the largest
is referred to as the transition
[5], [8]. The transi-
captures how the cascade algorithm
converges in
underiteration[5].Asrevealed
, or
equivalently,thestabilityof
by the above relation, the smaller the spectral radius of
, the

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196IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 1, JANUARY 2005
smoother the scaling function associated with (13). The max-
imumsmoothnessfor
-regularityisachievedbyB-splines,for
which
notebelowaMATLABfunctiontoconstructthetransitionmatrix.
, and. We
function T trans matrix(h, M)
%TRANS MATRIX Computes the M-band transition
% matrix
%
? ???? ? ? ??
% associated with a filter h.
? ? ??? ?? ?????? ??;
??????? ? ? ????? ;
?
????????? ? ? ;
?? ?? ?????????? ;
???
? ??????;
? ???
????????? ??? ???????? ;
III. DYADIC-BASED FACTORIZATION FOR PARAUNITARY
FILTERBANKS WITH STRUCTURAL REGULARITY
A. One-Regular PUFBs
We begin our discussion with the construction of one-regular
PUFBs. The following Lemmas will help establish the one-reg-
ular results for both the degree-one and order-one factorizations
(6) and (8).
Lemma 3: In the Householder factorization of a unitary ma-
trix
as in (3), the 0th column of
by the unit-norm
and the entry
Proof: Consider the special form (4) the
Lemma 4: A degree-0 (and thus order-0)
with Type-I analysis polyphase matrix
ular if and only if the 0th row of
particular, these elements are equal to
nitudes and equal phases) for arbitrary
Proof: Consider (13) with
Lemma4isequivalenttotheType-IIsynthesispolyphasema-
trix
having identical elements
column.
Lemma 5: A degree-0 (and thus order-0)
with Type-I analysis polyphase matrix
ular if and only if the Householder factorization of
is completely determined
of.
take on.
-channel PUFB
is one-reg-
has identical elements. In
(equal mag-
.
and.
of its 0th
-channel PUFB
is one-reg-
(15)
where
diagis such that
with
(16)
(17)
where
case, we have
can be either 1 or , andis any real number. In this
(18)
where
(19)
Proof: By Lemma 4,
if each element of the 0th row equals
Lemma 3, the 0th row of
is one-regular if and only
. By
in (15) is
. This gives
or
for some sign parameter
One can then obtain
, as.
and
, and hence
(16) and (17). Now, since the PUFB is one-regular, (14) implies
(18), which in turn implies
tary. In fact,
Theorem 1: A degree-
PUFB (6) is one-regular if and only
if
is one-regular, as in Lemma 5.
Proof: Since
asis uni-
.
Using (8) and (12), one can establish the following one-reg-
ular result for the order-one factorization.
Corollary1: Anorder- PUFB(8)isone-regularifandonly
if
is one-regular as in Lemma 5.
Remarks: This also establishes that regularity of degree one
iscompletelydeterminedbytheunitarymatrix
irrespective of the filter length and the McMillan degree. Fur-
thermore,theHouseholdermatrix
trolling factor for one degree of regularity. An order-
can have degree ranging from
in(6)and(8),
in(15)istheonlycon-
PUFB
to.
B. Two-Regular PUFBs
Having developed the conditions for one-regularity, we are
nowreadytoderivethetwo-regularityconditionsonthedyadic-
based structures.
1) Two-Regular Dyadic-Based Structures with or Without
Length Constraint:
Theorem 2 (Two-Regular Dyadic-Based Structure): A
degree-
PUFB (6) is two-regular if and only if we have the
following.
1)
is one-regular as in Lemma 5.
2) The unit-norm parameter vectors
are such that
ofas in (5)
(20)
where
and
Proof: Setting
ization (6) for
in (14) with the degree-one factor-
, we have

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CHEN et al.: DYADIC-BASED FACTORIZATIONS FOR REGULAR PARAUNITARY FILTERBANKS197
from (19), and
for some
the equation, we arrive at (20).
Similarly, we have the following two-regular result for the
order-one factorization (8).
Corollary 2 (Two-Regularity With Length Constraint): An
order-
PUFB (8) is two-regular if and only if we have the
following.
1)
is one-regular as in Lemma 5.
2) Theunitaryparameter
. By deleting the 0th rows from both sides of
matrices
of the order-one PU
are such that
building blocks
(21)
where
Proof: Using the order-one factorization for
, one again obtains
in (14)
and
with
and thus (21).
Having obtained the two-regular dyadic-based structures, we
now present the following two Lemmas, which are useful for
further analysis.
Lemma 6 (Norm of
): For each of the terms
the norm is upper-bounded by
,
(22)
The equality holds when
This norm bound will be used to prove the triangle inequality
in the subsequent sections.
Lemma7(ConstantNorm): IfthePUFB
regular, the norm of
, as appears in (20) and (21), will be
constant, irrespective of
.
.
isatleastone-
Proof: Since
, we have
is obtained by deleting the 0th entry
of
(23)
independent of
Based on the above, the first result is the minimum McMillan
degree required for two-regularity.
Theorem 3 (Minimum Degree for Two-Regularity): For a
PUFB to be two-regular, its McMillan degree has to be at least
one.
Proof: Taking the norm of (20) and using (22) and (23)
gives
.
(24)
from which it can easily be seen that
the PUFB, has to be at least one for the inequality to hold.
Note that this result is consistent with the fact that, for a two-
regular PUFB, the minimum order is one, and the filter length is
thus
[4], which is a stronger requirement. One should also
note that if the linear-phase property is imposed, this minimum
length is increased to
[15].
2) Existence of Two-Regular Solutions: Not all choices of
unit-norm vectors
satisfy (20) for the degree-one factoriza-
tion and similarly for the order-one case (21). In particular, the
parameter vectors
and
equalities imposed by (20) and (21), respectively. Take (20) for
example. Dividing it by
thesummation to the otherside of theequation, we have thefol-
lowing inequalities:
, which is the degree of
have to satisfy the triangle in-
and moving the first terms of
(25)
for
appealing to triangle inequality and Lemma 6. The following
theorem summarizes the results obtained from this idea.
Theorem4(Two-Regular
two-regular PUFB with the degree-one factorization (6). In
the corresponding two-regular condition (20), suppose that
has been given. Then, there always exist
unit-norm vectors
in, satisfy (20), regardless of the choice of
, where it is understood that
if.
. The last inequality is obtained by
Feasibility—I): Considera
, which, together with
, for any