Biomedical Engineering Reference
In-Depth Information
such as object boundaries and coarse textural detail, followed by registration of
fine detail. Sequential low-pass spatial filtering is used to achieve this goal. By
evolving the cut-off frequency of the spatial filter over computational time, the
influence of fine textural features in the image can be initially suppressed until
global registration is achieved. Fine structure can be registered subsequently by
gradually removing the spatial filter.
The spatial filter is applied by convolution of the image with a kernel
κ
(
X
).
For the template image field
T
,
T
∗
(
X
)
=
T
(
X
)
∗
κ
(
X
)
=
T
(
X
)
κ
(
X
−
Z
)
d
Z
,
(12.26)
B
where
T
(
X
) and
T
∗
(
X
) are the original image data and the filtered data respec-
tively in the spatial domain;
X
is a vector containing the material coordinates
and
Z
is the frequency representation of
X
. An efficient way to accomplish this
calculation is through the use of the discrete Fourier transform.
The convolution of the image data
T
(
X
) with the filter kernel
κ
(
X
)in
Eq. (12.26) becomes multiplication of
T
(
Z
) with K(
Z
) in the Fourier domain.
T
(
Z
) is the Fourier transform of
T
(
X
) and K(
Z
) is the Fourier transform of
κ
(
X
).
This multiplication is applied and then the transform is inverted to obtain the
convolved image in the spatial domain as shown below:
T
∗
(
X
)
=
−
1
{
T
(
Z
)K(
Z
)
}
.
(12.27)
Because of the very fast computational algorithms available for applying Fourier
transforms, this method is much faster than computing the convolution in image
space. In our implementation, a 3-D Gaussian kernel is used [38]:
κ
(
X
)
=
A
exp
X
·
X
2
σ
−
(12.28)
2
2
, the spatial variance is used to control the extent of blurring while
A
is a normalizing constant. Note that Eq. (12.28) is only valid for a 3-D vector
X
. The user specifies the evolution of the spatial filter over computational time
by controlling the mask and variance. In the specific results reported below,
the variance was set to a high value and evolved to remove the filtering as the
computation was completed (Fig. 12.2).
The practical application of spatial filtering is complicated by the fact that
the registration is nonlinear and is computed stepwise during the registration
process. At each step in the computational process, the spatial distribution of
Here,
σ
