Image Processing Reference
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2M thod
Anisotropic diffusion filtering as proposed by Weickert [
10
,
11
] uses the diffusion
tensor to steer the filtering process, which allows for directional, anisotropic
smoothing. The diffusion tensor is based on the structure tensor that uses first-
order derivative information to describe structures in an image. The principal
directions used for smoothing are thus based on the structure tensor [
10
]:
∇u
˃
∇u
˃
)
J
ˁ
(
∇u
˃
)=
K
ˁ
∗
(
(1)
where
K
is the Gaussian kernel with standard deviation
ˁ
(integration scale),
over which the orientation information is averaged, and
∇u
˃
is the gradient of the
image
u
at scale
˃
. Conventionally the structure tensor is based on the intensity
information in a 2D or 3D image, with either two or three spatial dimensions.
Pixels or voxels with similar intensity values located in close proximity to one
another (low first-order derivative) are assumed to belong to the same structure,
whereas high intensity variations (high first-order derivative), are considered
as transitions from one structure to another. Four dimensional CT perfusion
scans, however, consist of three spatial dimensions and one additional temporal
dimension. Therefore, the fourth dimension [
9
] in CT perfusion scans can be used
to distinguish between different types of tissues. Thus, instead of one intensity
value per voxel, a range of intensity values (the time-intensity profile [
9
]) is
available for each voxel in the 3D spatial domain. This is especially beneficial
to differentiate between structures with contrast enhancement (e.g. arteries and
veins, see Figure
1
), due to the injection of contrast material prior to the CT
perfusion scan acquisition. Similar time-intensity profiles are likely to belong to
the same structure. In [
9
], the sum of squared differences (SSD) is used as a
similarity measure:
T −
1
)=
1
T
− I
(
ʾ
(
x, y, z, t
)))
2
ʶ
(
ʾ,
x
(
I
(
x
(
x, y, z, t
))
(2)
t
=0
where T is the size of the temporal dimension,
I
(
x
(
x, y, z, t
)) is the intensity
value of voxel
(
x, y, z
) at time point
t
and
I
(
ʾ
(
x, y, z, t
)) is the intensity value
of a neighboring voxel
ʾ
(
x, y, z
) at time point
t
. We propose to use the SSD simi-
larity measure (
2
) between the time-intensity profiles to determine the first-order
derivatives that are used to determine the structure tensor (
1
). The first-order
derivatives can be approximated by using a finite differences scheme [
11
]. Instead
of using the intensity values to determine the finite differences, the SSD values
are used. In this way, the structure tensor is based on the structures described by
the 4th dimension of the CT perfusion data. Principle axis transformation of the
matrix with 4th dimension first-order derivative information provides the eigen-
vectors and eigenvalues of the 3D structure tensor
J
ˁ
(
x
∇u
˃
). The eigenvectors of
this structure tensor are then used as the eigenvectors of the diffusion tensor.
Consequently, the orientation of the diffusion tensor is based on the structures
that are defined by the fourth dimension of the CT perfusion data.
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