Biomedical Engineering Reference
In-Depth Information
Because the wavelet basis
ψ
1
2
, and
ψ
3
are first derivatives of a cubic spline
,ψ
θ
, the three components of a wavelet coefficient
W
m
s
(
n
1
,
n
2
,
n
3
)
=
function
s
,ψ
m
,
n
1
,
n
2
,
n
3
,
k
=
1
,
2
,
3, are proportional to the coordinates of the gradient
vector of the input image
s
smoothed by a dilated version of
θ
. From these coor-
dinates, one can compute the angle of the gradient vector, which indicates the
direction in which the first derivative of the smoothed
s
has the largest ampli-
tude (or the direction in which
s
changes most rapidly). The amplitude of this
maximized first derivative is equal to the modulus of the gradient vector, and
therefore proportional to the wavelet modulus:
k
W
m
s
W
m
s
W
m
s
2
2
2
M
m
s
=
(6.44)
+
+
.
Thresholding this modulus value instead of the coefficient value consists of first
selecting a direction in which the partial derivative is maximum at each scale,
and then thresholding the amplitude of the partial derivative in this direction. The
modified wavelet coefficients are then computed from the thresholded modulus
and the angle of the gradient vector. Such paradigm applies an adaptive choice
of the spatial orientation in order to best correlate the signal features with the
wavelet coefficients. It can therefore provide a more robust and accurate se-
lection of correlated signals compared to traditional orientation selection along
three orthogonal Cartesian directions.
Figure 6.20 illustrates the performance of this approach at denoising a clinical
brain PET data set reconstructed by FBP with a ramp filter. The reconstructed
PET images, illustrated for one slice in Fig. 6.20(a), contain prominent noise in
high frequency but do not express strong edge features in the wavelet modulus
expansions at scale 1 through 5 as illustrated in Fig. 6.20(b)-(f).
Cross-Scale Regularization for Images with Low SNR.
As shown in
Fig. 6.20(b), very often in tomographic images, the first level of expansion (level
with more detailed information) is overwhelmed by noise in a random pattern.
Thresholding operators determined only by the information in this multiscale
level can hardly recover useful signal features from the noisy observation. On
the other hand, wavelet coefficients in the first level contain the most detailed
information in a spatial-frequency expansion, and therefore influence directly
the spatial resolution of the reconstructed image.
To have more signal-related coefficients recovered, additional information
or
a priori
knowledge is needed. Intuitively, an edge indication map could
