Digital Signal Processing Reference
In-Depth Information
L
d
)
k
∈
U
j
,
K
β
2
2
j
,
fact, ((1
) can be interpreted as a local measure of signal-to-
noise ratio (SNR) in the block
U
j
,
K
. Such a block thresholding originates from the
James-Stein rule introduced by Stein (1981). As far as the choice of the threshold
parameter
/
j
,,
k
)
/
(
σ
λ
∗
is concerned, this is discussed next.
6.3.4 On the Choice of the Threshold
λ
∗
is crucial for good performance of the block thresh-
olding estimator. We here emphasize the main practical ideas. To get the optimal
behavior of block thresholding, it is enough to determine
The choice of the threshold
λ
∗
such that (A1) and
(A2) are satisfied. In the following, we first provide the explicit expression of
λ
∗
in the situation of an additive Gaussian noise sequence (
η
j
,,
k
)
j
,,
k
, not necessarily
white. This result is then refined in the case of white Gaussian noise.
6.3.4.1 Case of Frames
We suppose that, for any
j
∈{
,...,
−
}
∈
η
0
J
1
and any
B
j
,(
k
)
k
is a zero-mean
j
,,
Gaussian process with standard deviation
.Itwas
shown by Chesneau et al. (2009) that assumption (A2) can be reexpressed using
the covariance of the noise in the coefficient domain. Denote such a covariance
σ
j
,
at scale
j
and orientation
k
= E
(
η
η
k
); then (A2) is satisfied if
j
,,
k
,
j
,,
k
j
,,
Ca
||
k
−
k
||
,
where (
a
u
)
u
∈N
is a positive summable sequence, that is,
u
∈N
|
j
,,
k
,
k
|≤
Q
4
. In a nut-
shell, this means that assumption (A2) is fulfilled as soon as the noise in the co-
efficient domain is not too much correlated. For example, with an additive white
Gaussian noise in the original domain, with both wavelets and curvelets, this state-
ment holds true, and (A2) is verified (Chesneau et al. 2009). This result is useful as it
establishes that the block denoising procedure and its optimal performance apply to
the case of frames where a bounded zero-mean white Gaussian noise in the original
domain is transformed into a bounded zero-mean colored Gaussian process whose
covariance structure is given by the Gram matrix of the frame.
To apply the block thresholding procedure, it was also shown by Chesneau et al.
(2009) that the threshold
d
a
||
u
||
<
3
1
/
4
2
preserves the optimal proper-
ties of the estimator, where
C
and
Q
4
are the constants introduced previously, which
are independent of the number of samples
N
.
4
(2
CQ
4
)
1
/
2
λ
∗
=
+
6.3.4.2 Case of Orthobases
When (
η
j
,,
k
)
j
,,
k
reduces to an additive white Gaussian noise, as is the case when the
noise in the original domain is an additive white Gaussian noise and the transform is
orthonormal (e.g., orthogonal wavelet transform), it was proved by Chesneau et al.
(2009) that the best performance of block thresholding is achieved when
λ
∗
is the
root of
λ
−
log
λ
=
3, that is,
λ
∗
=
.
....
In fact, it was shown by Chesneau
et al. (2009) that this threshold works very well in practice, even with redundant
transforms that correspond to tight frames for which the threshold
4
50524
50524 is
not rigorously valid. Only a minor improvement can be gained by taking the higher
λ
∗
≈
4
.
