SIAM
J.
MATH.
ANAL.
Vol.
24,
No.
2,
pp.
499-519,
March
1993
1993
Society
for
Industrial
and
Applied
Mathematics
014
ORTHONORMAL
BASES
OF
COMPACTLY
SUPPORTED
WAVELETS
II.
VARIATIONS
ON
A
THEME*
INGRID
DAUBECHIES
Abstract.
Several
variations
are
given
on
the
construction
of
orthonormal
bases
of
wavelets
with
compact
support.
They
have,
respectively,
more
symmetry,
more
regularity,
or
more
vanishing
moments
for
the
scaling
function
than
the
examples
constructed
in
Daubechies
[Comm.
Pure
Appl.
Math.,
41
(1988),
pp.
909-996].
Key
words,
wavelets,
orthonormal
bases,
regularity,
symmetry
AMS(MOS)
subject
classifications.
26A16, 26A18,
26A27,
39B12
1.
Introduction.
This
paper
concerns
the
construction
of
orthonormal
bases
of
wavelets,
i.e.,
orthonormal
bases
{$jk;
j,
kZ}
for
L2(R),
where
(1.1)
q%(x)
2-;/2q(2-;x-
k)
for
some
(very
particular!)
L2(E).
The
functions
(1.1)
are
wavelets
because
they
are
all
generated
from
one
single
function
by
dilations
and
translations.
Note
that
wavelets
need
not
be
orthogonal
or
even
linearly
independent.
In
fact,
the
"first"
wavelets
were
neither
[1],
[2].
See
[3],
[4]
for
discussions
of
wavelet
expansions
using
nonindependent
wavelets,
with
continuous
[3]
or
discrete
[4]
dilation
and
translation
labels.
Even
the
special
case
of
orthonormal
wavelets
need
not
always
be
of
the
form
(1.1).
Basic
dilation
factors
different
from
2
are
possible:
there
exist
orthonormal
bases
in
which
this
factor
is
any
rational
p/q
>
1
[5];
in
more
than
one
dimension
we
may
even
choose
a
dilation
matrix
instead
of
an
isotropic
dilation
factor.
In
these
more
general
cases,
it
may
be
necessary
to
introduce
more
than
one
(but
always
a
finite
number).
We
shall
restrict
ourselves
to
one
dimension
here,
and
to
the
dilation
factor
2,
as
in
(1.1).
Bases
with
factor
2
are
by
far
the
easiest
to
implement
for
numerical
computations.
All
interesting
examples
of
orthonormal
wavelet
bases
can
be
constructed
via
multiresolution
analysis.
This
is
a
framework
developed
by
Mallat
[6]
and
Meyer
[7],
in
which
the
wavelet
coefficients
(f,
Ojk)
for
fixed
j
describe
the
difference
between
two
approximations
of
f,
one
with
resolution
2
j-,
and
one
with
the
coarser
resolution
2
.
The
following
succinct
review
of
multiresolution
analysis
suffices
for
the
understanding
of
this
paper;
for
more
details,
examples,
and
proofs
we
refer
the
reader
to
[6]
and
[7].
The
successive
approximation
spaces
V
in
a
multiresolution
analysis
can
be
characterized
by
means
of
a
scaling
function
ok.
More
precisely,
we
assume
that
the
integer
translates
of
b
are
an
orthonormal
basis
for
the
space
Vo,
which
we
define
to
be
the
approximation
space
with
resolution
1.
The
approximation
spaces
V
with
resolution
2
are
then
defined
as
the
closed
linear
spans
of
the
bk
(k
7/),
where
(1.2)
dpjk
2-J/adp(2-Jx-
k).
To
ensure
that
projections
on
the
V
describe
successive
approximations,
we
require
Vo
c
V_l,
which
implies
(1.3)
*
Received
by
the
editors
May
29,
1990;
accepted
for
publication
(in
revised
form)
May
23,
1992.
?
Mathematics
Department,
Rutgers
University,
New
Brunswick,
New
Jersey
08903
and
AT&T
Bell
Laboratories,
600
Mountain
Avenue,
Murray
Hill,
New
Jersey
07974.
499
500
INGRID
DAUBECHIES
This
imposes
a
restriction
on
b:
since
b
Vo
c
V_l=Span{b_lk;
k7/},
there
must
exist
c.
such
that
(1.4)
(x)
c,,
(2x
n).
In
order
to
have
a
complete
description
of
L2(),
we
also
impose
(1.5)
fq
V
{0},
U
L().
jZ jZ
For
every
multiresolution
analysis
as
described
above,
there
exists
a
corresponding
ohonormal
basis
of
wavelets
defined
by
(1.6)
(x)
Z
(-1)"c_,+6(2x-
n),
where
c,
are
the
coefficients
in
(1.4).
We
can
prove
[6],
[7]
(see
also
below)
that
the
4o,
are
then
an
orthonormal
basis
for
the
orthogonal
complement
Wo
of
Vo
in
V_I.
This
phenomenon
repeats
itself
at
every
resolution
level
j.
It
follows
that,
for
every
j,
the
(f,
qgk)
determine
the
difference
in
information
between
the
approximations
Pf
P-lf
at
resolutions
2
j,
2
j-,
respectively:
Pj-lf--
Pf+
E
(f,
q’jk)qgk.
Consequently,
by
(1.3)
and
(1.5),
the
(jk’
j,
k
7/)
constitute
an
orthonormal
basis
for
().
One
advantage
of
the
"nested"
structure
of
a
multiresolution
analysis
is
that
it
leads
to
an
efficient
tree-structured
algorithm
for
the
decomposition
and
reconstruction
of
functions
(given
either
in
continuous
or
sampled
form).
Instead
of
computing
all
the
inner
products
(f,
ltjk
directly,
we
proceed
in
a
hierarchic
way:
mcompute
(f,
(jk)
for
the
finest
resolution
level
j
wanted
(if
the
data
are
given
in
a
discrete
fashion,
then
these
discrete
data
can
just
be
taken
to
be
(f
--then
compute
(f
q-k)
and
(f
b-k)
at
the
next
finest
resolution
level
by
applying
(1.4)
and
(1.7),
1
(f,
qg-,k)
,
(--
1)"C-,,+2k+l(f
6j,,),
--iterate
until
the
coarsest
desired
resolution
level
is
attained.
The
total
complexity
of
this
calculation
is
lower,
despite
the
computation
of
the
seemingly
unnecessary
(f,
b2k),
than
if
the
(f,
q%)
were
computed
directly.
This
brief
review
shows
how
to
construct
an
orthonormal
basis
of
wavelets
from
any
"decent"
function
b
satisfying
an
equation
of
type
(1.4).
An
example
of
such
a
construction
is
given
by
the
Battle-Lemari6
wavelets,
consisting
of
spline
functions
[8],
[9],
[10].
In
general,
constructions
starting
from
a
choice
of
4
lead
to
4,
q,
which
are
not
compactly
supported
(see,
e.g.,
[15],
[25]
for
a
more
detailed
discussion).
The
construction
can,
however,
also
be
viewed
differently.
The
Fourier
transform
of
(1.4)
is
which
implies
(1.7)
(s:)
[=
mo(2-Jsc)]
(0),
ORTHONORMAL
BASES
OF
COMPACTLY
SUPPORTED
WAVELETS
II
501
with
mo()=
1/2
.
c,,
e
i",
so
that,
up
to
normalization,
b
is
completely
determined
by
the
c..
Fixing
the
c.,
therefore,
also
defines
a
multiresolution
analysis.
The
c.
have
to
satisfy
certain
conditions.
Combining
(bok,
4o)=
6k
with
(1.4)
immediately
leads
to
(1.8)
C,,C.-2k
26k0,
where
we
have
assumed,
as
we
shall
do
in
the
sequel,
that
the
c.
are
real.
In
terms
of
too(sO),
(1.8)
can
be
rewritten
as
(1.9)
Imo()l+
Imo(:+
r)l
2=
1.
To
ensure
that
b
is
well
defined,
the
infinite
product
in
(1.7)
must
converge,
which
implies
too(0)--
1
or
(1.10)
c=2.
It
follows
that
4
is
uniquely
determined
by
(1.4),
up
to
normalization,
which
we
fix
by
requiring
dx
4(x)=
1.
One
can
show
(see,
e.g.,
[12])
that
(1.9)
implies
that
b
is
in
L-(),
but
unfortunately
(1.8)
is
not
sufficient
to
guarantee
orthonormality
of
the
bo,.
A
counterexample
is
Co=C3
1,
all
other
c,-0,
which
leads
to
b(x)=]
for
0
<-
x
<
3,
4(x)
0
otherwise.
Such
counterexamples
are
rare,
however.
If
N
N
3,
then
the
example
above,
o
3
1,
is
the
only
one.
For
a
detailed
discussion,
see
12],
13],
[22].
If
we
exclude
these
thin
sets
of
"bad"
choices
for
the
c
(which
can
be
done
by
various
means
[6], [7],
[12]
[13],
[15]),
then
we
can
build
orthonormal
bases
of
wavelets
starting
from
the
c,.
Once
orthonormality
of
the
bOk
is
established,
all
the
rest
follows
easily.
Formula
(1.6)
for
q
leads
immediately
to
orthogonality
of
the
qOl
and
4Ok,
1
(_l).c
_.++,c,._(_,.
1
2
(-1)
c_++c,_
0.
The
last
equality
follows
from
the
substitution
n
m
+
2(k
+
l)
+
1
for
the
summation
index
n.
Similar
manipulations
prove
and
(1.11)
k
It
follows
that
both
{b-1,;
n
;7}
and
{(0k,
0k;
k
Z}
are
orthonormal
bases
for
V_I.
(In
other
words,
(1.8)
ensures
that
(1.4)
and
(1.6)
describe
an
orthonormal
basis
transformation.)
It
follows
that
Wo
Span
(qOk)
is
the
orthogonal
complement
of
V0
in
V_,
and
hence
that
the
{qgk;
J,
k
7/}
constitute
an
orthonormal
basis
for
L2(R).
Constructing
q
from
the
c,
rather
than
from
b
has
the
advantage
of
allowing
better
control
over
the
supports
of
b
and
q.
If
c,
0
for
n
<
N1,
n
>
N2,
then
support
(b)c
[N,
N2]
(see
[lla],
[14]).
In
[15]
this
method
was
used
to
construct
orthonormal
bases
of
wavelets
with
compact
support,
and
arbitrarily
high
preassigned
regularity
(the
size
of
the
support
increases
linearly
with
the
number
of
continuous
derivatives).
These
orthonormal
basis
functions
and
the
associated
multiresolution
analysis
have
502
INGRID
DAUBECHIES
been
tried
out
for
several
applications,
ranging
from
image
processing
to
numerical
analysis
[16].
For
some
of
these
applications,
variations
on
the
scheme
of
[15]
were
requested,
emphasizing
other
properties.
The
goal
of
this
and
the
next
paper
is
to
present
a
number
of
these
variations.
The
construction
in
[15]
relied
on
the
identity
s
(N--l+J)[(cosoz)2rV(sina)2J+(sina)2(cosa)2]
1.
(1.12)
j=O
j
Since
(1.12)
suggests
the
choice
(1
13)
mo()=(l+ei)
1
2
Q(e’)’
where
Q
is
a
trigonometric
polynomial
with
real
coefficients
such
that
(1.14)
IQ(e’)l
---
j=o
j
2
By
(1.12),
any
such
mo
will
satisfy
(1.9).
To
determine
0,
we
have
to
extract
the
"square
root"
of
the
right-hand
side
of
(1.5).
This
can
be
done
by
using
a
lemma
of
Riesz
[17].
Denote
the
right-hand
side
of
(1.14)
by
Pc(ei),
and
extend
PN
to
all
of
C.
We
have
PN(Z)--PN()
and
Pc(z-1)
PN(Z).
Consequently,
the
zeros
of
Pn
come
either
in
real
duplets,
rk
and
r{
or
in
complex
quadruplets,
Zl
l,
z-f
and
-
P(z)
=4-
\
N-
1
]z-
(z-
rk)(Z--
r;
1)
[I
(Z--
Zl)(Z- l)(Z--
Z;1)(Z-
;1)
=4-
\N-1]
.U
(Z
ZI)(Z
l)(Z,
Z-1)(/--
Z
-1)
It
follows
that
PN(e’)
[Q(e’t)[
,
with
(1.15)
Q(z)=2-N+I(
2N-211/2
N-l/
(z-r,)
(zZ+lz,
lZ-2Zlz,
Re
z,)
This
gives
a
recipe
for
the
construction
of
mo:
(1)
For
given
N,
determine
the
zeros
of
PN;
(2)
Choose
one
zero
out
of
every
pair
of
real
zeros
r,
r[
of
PN,
and
one
conjugated
pair
out
of
every
quadruplet
Zk,
Z-
(3)
Compute
the
product
Q,
and
substitute
into
(1.12).
The
result
is
a
polynomial
in
e
of
degree
2N
1,
corresponding
to
an
orthonormal
basis
of
wavelets
in
which
the
basic
wavelet
has
support
width
2N-1.
Since
(1.6)
can
be
rewritten
as
0(l)
ei((/-)+)m
o
+
"rr
ORTHONORMAL
BASES
OF
COMPACTLY
SUPPORTED
WAVELETS
II
503
and
since
(1.13)
has
a
zero
of
order
N
at
7r,
it
follows
that
qN
has
N
vanishing
moments,
dxxld/N(X)
=0,
=0,
1,...,
N-
1,
which
is
useful
for
quantum
field
theory
[18]
and
numerical
analysis
applications
[19].
The
regularity
of
the
PN
constructed
in
15]
increases
linearly
with
their
support
width,
qN
C
a(N),
with
limu_
N-la(N)
.2075
[23],
[24],
[25].
Plots
of
and
q
for
various
values
of
N
can
be
found
in
[15],
[25].
Depending
on
the
application
they
had
in
mind,
several
scientists
(mathematicians
or
engineers)
have
requested
possible
variations
on
the
construction
in
[15].
The
following
are
the
most
recurrent
wish
items.
(1)
More
symmetry:
the
functions
,
0
in
[15]
are
very
asymmetric.
Complete
symmetry
is
incompatible
with
the
orthonormal
basis
condition
(see
[15,
p.
971],
or
2
below),
but
is
less
asymmetry
possible?
(2)
Better
frequency
resolution"
orthonormal
bases
with
basic
multiplication
factor
2
correspond
to
frequency
intervals
of
1
octave.
Is
better
possible
(e.g.,
1/2
octave),
without
giving
up
compact
support?
(3)
More
regularity:
is
better
regularity
than
in
[15]
achievable
for
the
same
support
width
?
(4)
More
vanishing
moments:
for
a
fixed
support
width
2N-1,
the
PN
of
[15]
have
the
maximum
number
of
vanishing
moments.
The
functions
eu
do
not
satisfy
any
moment
condition,
except
dx
eN(X)=
1.
For
numerical
analysis
applications,
it
may
be
useful
to
give
up
some
zero
moments
of
0
in
order
to
obtain
zero
moments
for
,
i.e.,
to
have
dx&(x)
1,
(1.16)
I
dx
xlch(x)
O,
1,...,
L,
dxxl(x)
=0,
l=0,...,
L.
How
can
such
,
be
constructed?
They
would
have
the
advantage
that
inner
products
with
smooth
functions
are
particularly
appealing:
f
dx
b-jk(x)f(x)--
2J/2
f
dx
qb(2J(x-2-Jk))f(x)
2-J/f(2-Yk)
+
correction
terms
in
f+l
(use
the
Taylor
expansion
off
around
2-2k;
the
second
through
(L+
1)th
terms
vanish
because
of
(1.16)).
Moreover,
if
the
(L+
1)th
derivative
of
f
is
uniformly
bounded,
then
the
correction
terms
in
this
formula
are
of
order
2
-(/’+l/2)j.
The
purpose
of
this
and
the
next
paper
is
to
show
how
such
variations
can
be,
constructed.
In
2
we
handle
symmetry,
in
3
regularity,
and
in
4
vanishing
moments
for
.
The
next
paper
shows
how
to
obtain
better
frequency
localization.
2.
More
symmetry.
If
we
restrict
our
attention
to
orthonormal
bases
of
compactly
supported
wavelets
only,
then
it
is
impossible
to
obtain
which
is
either
symmetric
or
antisymmetric,
except
for
the
trivial
Haar
case
(Co
1,
Cl
=-l,
all
other
c,
=0).
This
is
the
content
of
the
following
theorem.