Image Processing Reference
In-Depth Information
f
x
=(
f
i
+1
,j,k
−
f
i−
1
,j,k
)
/
2
∆i
(3.30)
f
xx
=(
f
i
+1
,j,k
−
2
f
ijk
+
f
i−
1
,j,k
)
/
4
∆i
(3.31)
f
xy
=(
f
i
+1
,j
+1
,k
−
f
i−
1
,j
+1
,k
−
f
i
+1
,j−
1
,k
+
f
i−
1
,j−
1
,k
)
/
4
∆i∆j,
(3.32)
where
∆i
and
∆j
are sampling intervals (= the length of voxel edges) in the
i
and the
j
directions, respectively. They are unit length usually. The value
of
∆k
may be larger when the interval between slices is larger than the voxel
size within a slice as is often seen in CT images of the human body. Various
other methods and equations to numerically evaluate differentials are found
in topics about numerical analysis.
Remark 3.8.
Values of derivatives can be estimated by executing a curve
fitting in the neighborhood of each voxel. The outline of the procedure is as
follows in the case of 3D image processing:
Given an input image
F
=
{
f
ijk
}
, we consider to fit a suitable function
φ
(
x, y, z
;
)atavoxel(
i, j, k
) and its neighborhood, where
x
,
y
,and
z
repre-
sent variables in the coordinate axes
i
,
j
,and
k
,and
a
a
is a parameter vector
specific to the function
φ
. Then we obtain values
a
=(
a
1
,a
2
,...,a
M
)which
minimizes the following estimation error
e
ijk
e
ijk
=
(
p,q,r
)
[
f
pqr
−
φ
(
p, q, r
;
a
)]
,
(3.33)
where
means the sum over the neighborhood
(
i, j, k
)ofavoxel(
i, j, k
).
Next, we estimate the density
f
ijk
at a voxel (
i, j, k
)by
φ
(
i, j, k
;
N
). Values
of derivatives at the same voxel are also approximated by
∂φ/∂x,∂
2
φ/∂x
2
,
etc.
Process of calculation is obvious. The value of
a
a
is obtained by solving the
equation.
2
(
p, q, r
)[
f
pqr
−
∂e
ijk
/∂a
k
=
)](
∂φ/∂a
k
)=
0
,k
=
1
,
2
,...,M
(3.34)
The solution is obtained from values of (
i, j, k
) and density values of an input
image in the neighborhood
−
φ
(
p, q, r
;
a
((
i, j, k
)). The calculation process is simplified
by putting the origin at the center of the neighborhood
N
N
((
i, j, k
)), because it
is enough to calculate values of
φ
x
(0
,
0
,
0;
a
),
φ
xx
(0
,
0
,
0;
a
), etc., without using
values of
φ
x
(0
,
0
,
0;
) etc. at a general position of (
i, j, k
). The underlying idea
here is the same as the processing of a 2D image as is seen in [Haralick81,
Haralick83]. Examples of applications to 3D images are found in [Brejl00,
Hirano03].
a
3.3.4 Basic characteristics of difference filter
Derivatives are approximated by differences in digital image process. A local
processing to perform this is called a
difference operation
(
difference operator
,
Search WWH ::
Custom Search