Digital Signal Processing Reference
In-Depth Information
On the basis of this morphological diversity concept, we present in the rest of
this chapter how we can derive fast algorithms for applications that involve such
complex dictionaries, that is, a union of several subdictionaries, where each has fast
analysis and synthesis operators. Applications tackled in this chapter are denoising,
deconvolution, component separation, and inpainting.
8.2 DICTIONARY AND FAST TRANSFORMATION
From a practical point of view, given a signal
x
, we will need to compute its forward
(or analysis) transform by multiplying it by
T
.
2
We also need to reconstruct (syn-
T
corresponding to each transform are never explicitly constructed in memory; rather,
they are implemented as fast implicit analysis and synthesis operators taking a signal
vector
x
and returning
thesize) any signal from its coefficients
α
. In fact, the matrix
and its transpose
T
x
=
T
x
(analysis side), or taking a coefficient vector
α
and
returning
(synthesis side). In the case of a simple orthogonal basis, the inverse of
the analysis transform is trivially
T
−
1
α
=
, whereas assuming that
is a tight frame
T
c
I
) implies that
T
+
=
c
−
1
(
is the Moore-Penrose pseudo-inverse transform
(corresponding to the minimal dual-synthesis frame). In other words, computing
=
α
is equivalent to applying
T
+
to
α
up to a constant. It turns out that
T
+
α
is the recon-
struction operation implemented by most implicit synthesis algorithms, including
some of those discussed in Chapters 2-5.
8.3 COMBINED DENOISING
8.3.1 Several Transforms Are Better Than One
Suppose that we observe
y
=
x
+
ε,
2
ε
where
) and
x
is a superposition of type (8.5). Our goal is to build an
estimator of
x
by exploiting the morphological diversity of its content.
Suppose that we are given
K
dictionaries
ε
∼N
(0
,σ
k
, each associated with a linear trans-
form (analysis) operator
T
k
.Let
=
[
,...,
K
] be the amalgamated dictionary
1
T
be the analysis operator of
and
T
.
According to Chapter 6, a sequence of estimates (
x
k
)
1
≤
k
≤
K
of
x
can be built by
hard thresholding each
=
α
k
:
T
k
HardThresh
λ
(
T
k
y
)
x
k
=
,
(8.6)
with a threshold
typically 3 to 4 times the noise standard deviation. Given the
K
individual estimates
x
k
, a naive but simple aggregated estimator is given by averag-
ing them, that is,
λ
K
1
K
x
=
x
k
.
(8.7)
k
=
1
This is, for instance, the idea underlying translation-invariant wavelet denoising
by cycle spinning (Coifman and Donoho 1995), where the averaging estimator is
T
is to be replaced by the adjoint
∗
.
2
For a complex dictionary,
