Digital Signal Processing Reference
In-Depth Information
3
Redundant Wavelet Transform
3.1 INTRODUCTION
Multiresolution transforms have two major uses: compression and what we may call
pattern or feature analysis. The latter includes signal restoration (denoising, decon-
volution, or deblurring) and object finding and measuring. The best filter bank or
multiresolution schema is likely to be different, depending on the application.
In this chapter, we begin with the general undecimated wavelet transform. We
then look at a transform that is partially decimated and partially not, with potential
storage economy.
The complex wavelet transform produces real and imaginary wavelet coeffi-
cients. This transform has a limited redundancy and provides approximate shift in-
variance and directional selectivity. Near-shift invariance, in turn, can lead to more
limited artifacts when carrying out image denoising.
The
starlet transform
is introduced and described as an undecimated isotropic
wavelet transform. This transform has been described in the past as the B
3
-spline a
trous wavelet transform.
We then discuss the principles of designing wavelet filters. In particular, because
we are dealing with redundant wavelet transforms, we design nonorthogonal filter
banks.
There follows a description of the pyramidal wavelet transform both in original
and Fourier domains. The wavelet transform is consolidated into a storage-efficient
pyramidal representation.
Before proceeding, we define a central notion used in the rest of the topic:
frames
. An operator
F
from a Hilbert space
H
to
K
is the frame synthesis operator
if its adjoint, that is, the analysis operator
F
∗
, satisfies
the generalized Parseval relation with lower and upper bounds
a
1
and
a
2
:
associated with a frame of
K
F
∗
u
2
2
2
≤
≤
<
≤
<
+∞
.
a
1
u
a
2
u
0
a
1
a
2
(3.1)
a
, in which case
FF
∗
=
The frame is tight when
a
1
=
a
2
=
a
I
, where
I
is the identity.
When
a
1
1, we have the Parseval relation. See Christensen (2002) for a
comprehensive account of frames.
=
a
2
=
a
=
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