Digital Signal Processing Reference
In-Depth Information
capture the sparsity of the expansion. See the comprehensive overviews provided
by Antoniadis et al. (2001) and Fadili and Boubchir (2005).
6.2.2.4 On the Choice of the Threshold
There are a variety of methods for choosing the threshold value in equations (6.12)-
(6.14) in the case of additive Gaussian noise. These thresholds can be either global
or subband-dependent. The former means that the same threshold
t
is used for all
levels
j
and orientations
.
The hypothesis testing framework of Section 6.2.1 is a first way of setting the
threshold(s) by controlling the error rates either at individual or global levels. In
orthogonal wavelet denoising with white noise, Donoho and Johnstone (1994) pro-
posed the minimax threshold, which is derived to minimize the constant term in an
upper bound of the mean square error (MSE) risk involved in estimating a function
depending on an oracle.
For orthogonal transforms again with white noise, Donoho and Johnstone
(1994)
proposed the simple
formula
of the so-called universal threshold
t
2log
N
=
σ
(global) or
t
j
,
=
σ
2log
N
j
,
(subband-dependent), where
N
is the total number
of samples and
N
j
,
is the number of coefficients at scale
j
and orientation
.The
idea is to ensure that every coefficient, whose underlying true value is zero, will be
estimated as zero with high probability. Indeed, traditional arguments from the con-
centration of the maximum of
N
independent and identically distributed Gaussian
variables
ε
[
k
]
∼N
(0
,
1) tell us that as
N
→∞
,
√
2
N
b
/
2
−
1
b
Pr
max
1
b
log
N
N
ε
[
k
]
>
∼
log
N
.
π
≤
k
≤
=
/
π
For
b
log
N
. However, as this thresh-
old is based on a worst-case scenario, it can be pessimistic in practice, resulting in
smooth estimates. Refinements of this threshold can be found in the work of Anto-
niadis and Fan (2001).
The generalized cross-validation criterion (GCV) has been adopted for objec-
tively choosing the threshold parameter in wavelet-domain soft thresholding, for
both white (Jansen et al. 1997) and correlated Gaussian noise (Jansen and Bultheel
1998). GCV attempts to provide a data-driven estimate of the threshold that mini-
mizes the unobservable MSE. Despite its simplicity and good performance in prac-
tice, the GCV-based choice cannot work for any thresholding rule. Coifman and
Donoho (1995) introduced a scheme to choose a threshold value
t
j
,
by employing
Stein's unbiased risk estimator, SURE (Stein 1981), to get an unbiased estimate of
the MSE. However, this approach has serious drawbacks in low-sparsity regimes.
2, this probability vanishes at the rate 1
6.2.3 Orthogonal versus Overcomplete Transforms
The hard thresholding and soft thresholding introduced in Section 6.2.2 are the exact
solutions to equations (6.15) and (6.16) only when the transform is orthonormal.
But when it is redundant or overcomplete, this is no longer valid. We describe the
following two schemes, which use regularization to solve the denoising problemwith
redundant transforms.
