Biomedical Engineering Reference
In-Depth Information
[
−
1
,
1]. Furthermore, any order of derivatives of
E
(
x
) exists and is continuous.
This property avoids creating any new discontinuities after enhancement.
6.3.3 Selection of Threshold Value
Given the basic framework of denoising using wavelet thresholding as discussed
in the previous sections, it is clear that the threshold level parameter
T
plays an
essential role. Values too small cannot effectively get rid of noise component,
while values too large will eliminate useful signal components. There are a
variety of ways to determine the threshold value
T
as will be discussed in this
section.
Depending on whether or not the threshold value
T
changes across wavelet
scales and spatial locations, the thresholding can be:
1.
global threshold
: a single value
T
is to be applied globally to all empirical
wavelet coefficients at different scales.
T
=
const
.
2.
level-dependent threshold
: a different threshold value
T
is selected for each
wavelet analysis level (scale).
T
=
T
(
j
)
,
j
=
1
,...,
J
,
J
being the coarsest
level for wavelet expansion to be processed.
3.
spatial adaptive threshold
: the threshold value
T
varies spatially depend-
ing on local properties of individual wavelet coefficients. Usually,
T
is also
level dependent.
T
=
T
j
(
x
,
y
,
z
).
While a simple way of determining
T
is as a percentage of coefficients maxima,
there are different adaptive ways of assigning the
T
value according to the noise
level (estimated via its variance
σ
):
1.
universal threshold
:
T
=
σ
√
2 log
n
[40], with
n
equal to the sample size.
This threshold was determined in an optimal context for soft thresholding
with random Gaussian noise. This scheme is very easy to implement, but
typically provides a threshold level larger than with other decision criteria,
therefore resulting in smoother reconstructed data. Also such estimation
does not take into account the content of the data, but only depends on
the data size.
2.
minimax threshold
:
T
=
σ
T
n
[41], where
T
n
is determined by a minimax
rule such that the maximum risk of estimation error across all locations of
