Biomedical Engineering Reference
In-Depth Information
with a single expansion coefficient for a particular pair of frequency and orien-
tation.
6.3 Noise Reduction and Image Enhancement
Using Wavelet Transforms
Denoising can be viewed as an estimation problem trying to recover a true
signal component
X
from an observation
Y
where the signal component has
been degraded by a noise component
N
:
Y
=
X
+
N
.
(6.34)
The estimation is computed with a thresholding estimator in an orthonormal
basis
B
={
g
m
}
0
≤
m
<
N
as [32]:
m
=
0
ρ
m
(
X
,
g
m
)
g
m
,
N
−
1
X
=
(6.35)
where
ρ
m
is a thresholding function that aims at eliminating noise components
(via attenuating or decreasing some coefficient sets) in the transform domain
while preserving the true signal coefficients. If the function
ρ
m
is modified to
rather preserve or increase coefficient values in the transform domain, it is
possible to enhance some features of interest in the true signal component with
the framework of Eq. (6.35).
Figure 6.9 illustrates a multiscale enhancement and denoising framework
using wavelet transforms. An overcomplete dyadic wavelet transform using
biorthogonal basis is used. Notice that since the DC cap contains the overall
energy distribution, it is usually not thresholded during the procedure. As shown
in this figure, thresholding and enhancement functions can be implemented in-
dependently from the wavelet filters and easily incorporated into the filter bank
framework.
6.3.1 Thresholding Operators for Denoising
As a general rule, wavelet coefficients with larger magnitude are correlated with
salient features in the image data. In that context, denoising can be achieved by
applying a thresholding operator to the wavelet coefficients (in the transform
