IEEE SIGNAL PROCESSING MAGAZINE [127] NOVEMBER 2005
when
|d
c
( j, n)| > 0
. As with the Fourier transform, complex
wavelets can be used to analyze and represent both real-
valued signals (resulting in symmetries in the coefficients)
and complex-valued signals. In either case, the
C
WT enables
new coherent multiscale signal processing algorithms that
exploit the complex magnitude and phase. In particular, as we
will see, a large magnitude indicates the presence of a singu-
larity while the phase indicates its position within the support
of the wavelet [81], [83], [113], [117].
The theory and practice of discrete complex wavelets can be
broadly classed into two schools. The first seeks a
ψ
c
(t )
that
forms an orthonormal or biorthogonal basis [9], [11], [37], [64],
[108], [114]. As we show below, this strong constraint prevents
the resulting
C
WT from overcoming most of the four DWT
shortcomings outlined above. The second school seeks a redun-
dant representation, with both
ψ
r
(t )
and
ψ
i
(t )
individually
forming orthonormal or biorthogonal bases. The resulting
C
WT
is a
2×
redundant tight frame [26] in 1-D, with the power to
overcome the four shortcomings.
In this article, we will focus on a particularly natural
approach to the second, redundant type of
C
WT, the dual-
tree approach, which is based on two FB trees and thus two
bases [55], [57]. As we will see, any
C
WT based on wavelets
of compact support cannot exactly possess the Hilbert
transform/analytic signal properties, and this means that
any such
C
WT will not perfectly overcome the four DWT
shortcomings. The key challenge in dual-tree wavelet design
is thus the joint design of its two FBs to yield a complex
wavelet and scaling function that are as close as possible to
analytic. From Figure 3, we see that we can reach quite close
to the ideal even with quite short filters.
As a result, the dual-tree
C
WT comes very close to mirror-
ing the attractive properties of the Fourier transform, includ-
ing a smooth, nonoscillating magnitude (see Figure 1); a
nearly shift-invariant magnitude with a simple near-linear
phase encoding of signal shifts; substantially reduced aliasing;
and directional wavelets in higher dimensions. The only cost
for all of this is a moderate redundancy:
2×
redundancy in 1-D
(
2
d
for
d
-dimensional signals, in general). This is much less
than the
log
2
N×
redundancy of a perfectly shift-invariant
DWT [22], [63], which, moreover, will not offer the desirable
magnitude/phase interpretation of the
C
WT nor the good
directional properties in higher dimensions.
COMPLEX WAVELET COMPLEXITIES
The design of complex analytic wavelets raises several unique
and nontrivial challenges that do not arise with the real DWT. In
this section, we overview them and discuss a straightforward but
limited approach to the
C
WT that provides a jumping off point
for the dual-tree.
ANALYTICITY VERSUS FINITE SUPPORT
It is often desired in wavelet-based signal processing that the
wavelet be well localized in time. (In many applications, the
wavelet
ψ(t)
will actually have finite support.) Finitely sup-
ported wavelets are of special interest because, in this case,
the DWT can be easily implemented with finite impulse
response (FIR) filters. However, a finitely supported function
can never be exactly analytic, because the Fourier transform
of a finitely supported function can never be exactly zero on
an interval
[A, B]
with
B > A
(on any set of positive measure
to be exact) let alone on the entire positive or negative fre-
quency axis [77]. Thus, any exactly analytic wavelet must have
infinite support (and slow decay, in fact).
Thus, if we want finitely supported wavelets, then we must
accept wavelets that are only approximately analytic and a
C
WT
that is only approximately magnitude/phase, shift invariant, and
free from aliasing. We can relax the finite support condition, but
the resulting infinitely supported wavelets are beyond the scope
of this article. The design challenge will be to see how close we
can get to analyticity. Unfortunately, the standard approach to
designing and implementing wavelet transforms (with FIR or
infinite impulse response (IIR) filters) has basic limitations even
for approximately analytic wavelets, as we now illustrate.
ANALYTICITY VERSUS PERFECT RECONSTRUCTION
The question of how to design filters
h
0
(n)
and
h
1
(n)
satisfying
the perfect reconstruction (PR) conditions so that the wavelet
ψ(t)
has short support and vanishing moments was answered
by Daubechies [25]. Note, however, that Daubechies’ wavelets
are not analytic. Can we design the filters
h
i
(n)
in Figure 24
such that the corresponding scaling function and wavelet given
by (60) and (59) are complex and (approximately) analytic?
While complex filters satisfying the PR conditions have been
developed [11], [42], [64], [123], those solutions do not give ana-
lytic wavelets and do not have the desirable properties of analyt-
ic wavelets described previously. (They do, however, have
desirable symmetry properties.) It turns out that the design of a
complex (approximately) analytic wavelet basis is more difficult
than the design of a real wavelet basis. If we follow the standard
approach for wavelet design, then problems arise when we
require the wavelet to be analytic.
So that the dyadic dilations and translations of a single func-
tion
ψ(t)
(the wavelet) constitute a basis for signal expansion,
ψ(t)
must satisfy certain constraints. Unfortunately, these con-
straints make it difficult to design a wavelet
ψ(t)
that is also
analytic. Specifically, analytic solutions are not possible because
the PR conditions (see “Real-Valued Discrete Transform and
Filter Banks”) require that
H
0
e
j ω
H
0
e
j ω
+ H
1
e
j ω
H
1
e
j ω
= 2
for
−π ≤ ω ≤ π.
Suppose that
h
1
(n)
is (approximately) ana-
lytic. Then
H
1
(e
j ω
) ≈ 0
for
−π<ω<0,
which in turn
implies that
H
0
(e
j ω
)
H
0
(e
j ω
) ≈ 2
for
−π<ω<0.
That is, nei-
ther
H
0
(z)
nor
H
0
(z)
is a reasonable low-pass filter and, conse-
quently, the dilation equation does not have a well-defined
solution. Therefore, the wavelet corresponding to the usual
DWT cannot be approximately analytic.
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