Biomedical Engineering Reference
In-Depth Information
the data is minimized. This threshold level depends on the noise and signal
relationships in the input data.
3.
stein unbiased estimated of risk
: Similar to minimax threshold but
T
n
is
determined by a different risk rule [42, 43].
4.
spatial adaptive threshold
:
T
=
σ
/σ
X
[44], where
σ
X
is the local variance
of the observation signal, which can be estimated using a local window
moving across the image data or, more accurately, by a context-based
clustering algorithm.
2
In many automatic denoising methods to determine the threshold value
T
,an
estimation of the noise variance
σ
is needed. Donoho
et al.
[45] proposed a
robust estimation of noise level
σ
based on the median absolute value of the
wavelet coefficients as:
median(
|
W
1
(
x
,
y
,
z
)
|
)
0
.
6745
σ
=
,
(6.41)
where
W
1
is the most detailed level of wavelet coefficients. Such estimator has
become very popular in practice and is widely used.
6.3.4 Summary
In general, multiscale denoising techniques involve a transformation process
and a thresholding operator in the transform domain. Research dedicated to
the improvement of such a technique has been explored along both directions.
Various multiscale expansions have been proposed, aimed at better adapta-
tion to signal and feature characteristics. Traditionally, an orthogonal base was
used for expansion [33], which leads to a spatial-variant transform. Various
artifacts, e.g. pseudo-Gibbs phenomena, were exhibited in the vicinity of dis-
continuities. Coifman
et al.
[40] proposed a translation-invariant thresholding
scheme, which averages several denoising results on different spatial shifts of
the input image. Laine
et al.
[38] prompted to an overcomplete representation
which allows redundancy in the transform coefficients domain and provides
a translation-invariant decomposition. Wavelet coefficients in an overcomplete
representation have the same size as the input image, when treated as a subband
image. Many denoising and enhancement techniques can be applied within a
