Biomedical Engineering Reference
In-Depth Information
multiscale framework for spatial-frequency adaptation and solve certain noise
amplification problems. For a better separation of noise and signal components
in the transform domain, other multiscale representations have also been widely
investigated. Examples of such multiscale representations can be found in
Section 6.2.3.
The magnitude of the wavelet coefficients is related to the correlations be-
tween the signal and the wavelet basis function, which is the only criterion to
determine whether or not noise variation appears. Therefore, the selection of
the wavelet basis is a critical step in the design of the denoising and enhance-
ment procedure. Wavelet basis constructed from derivatives of spline functions
[46] were shown to have many advantages in denoising and enhancement. Such
wavelet functions, either symmetric or antisymmetric, are smooth with compact
support. Higher order spline function resembles Gaussian function, therefore
providing ideal spatial-frequency resolution for signal analysis. Moreover, mod-
uli of wavelet coefficients using first-derivative spline wavelets are proportional
to the magnitude of a gradient vector [47]. Analysis over such modulus therefore
provides extra information on directional correlations, and is especially impor-
tant for three or higher dimensional data analysis. Other wavelet basis func-
tions have also been developed to provide specific adaptation to different type
of signals. To name a few, slantlet [48], curvelet [49, 50], and ridgelet [51] were
designed to improve the correlations with edge information and were used for
edge-preserved denoising, while Fresnelets functions, based on B-spline func-
tions [52], were designed for processing of digital holography.
In a parallel direction, many research works on multiscale denoising focused
on improving thresholding operators. In the following discussion, “thresholding
operator” is a rather general concept that includes both denoising and enhance-
ment operators as described before. A determination of thresholding method
includes both selection of the thresholding operator and a decision or estima-
tion of the threshold parameters (threshold level, enhancement gain, etc.). Some
examples of thresholding operators designed to improve the basic thresholding
rules as shown in Eqs. (6.36)-(6.38) include the non-negative garrote threshold-
ing [53]:
⎧
⎨
0
,
if
|
x
|≤
T
G
T
(
x
)
=
(6.42)
ρ
|
x
|
>
T
,
T
2
x
,
⎩
x
−
if
