Biomedical Engineering Reference
In-Depth Information
Since the three components of the surface are independently defined, to
minimize
E
2
, we minimize
E
x
,
E
y
, and
E
z
, separately. To minimize
N
E
x
=
[
x
(
u
j
,v
j
)
−
X
j
]
2
(7.8)
,
j
=
1
since
n
x
(
u
j
,v
j
)
=
x
i
g
i
(
u
j
,v
j
)
,
(7.9)
i
=
1
we minimize
x
i
g
i
(
u
j
,v
j
)
−
X
j
2
N
n
E
x
=
(7.10)
.
j
=
1
i
=
1
This involves determining the partial derivatives of
E
x
with respect to the
x
i
's,
setting the partial derivatives to zero and solving the obtained system of equa-
tions. This results in
N
n
g
k
(
u
j
,v
j
)
[
x
i
g
i
(
u
j
,v
j
)
−
X
j
]
=
0;
k
=
1
,...,
n
.
(7.11)
j
=
1
i
=
1
This represents a system of
n
linear equations, which can be solved for
{
x
i
:
i
=
1
,...,
n
}
. Since RaG basis functions monotonically decrease from a center
point, if
σ
is not very large, Eq. (7.11) will have a diagonally dominant matrix
of coefficients, ensuring a solution. In the same manner,
{
y
i
:
i
=
1
,...,
n
}
and
{
z
i
:
i
=
1
,...,
n
}
can be determined by minimizing
E
y
and
E
z
, respectively. Note
that the above process positions the
n
control points of a RaG surface so that
the surface will approximate the
N
image voxels with the least sum of squared
errors.
n
depends on the size and complexity of the shape being approximated.
n
is typically a few hundred.
Since shape voxels are mapped to a sphere, spherical parameters are ob-
tained for them. Assuming the approximating surface is represented by
P
(
u
,v
),
the distance of voxel
V
i
=
(
x
i
,
y
i
,
z
i
) to the surface is estimated from
E
(
u
i
,v
i
)
=
||
V
i
−
P
(
u
i
,v
i
)
||
. The adjacency information between the control points is pro-
vided in the
u
and
v
parameter coordinates. Therefore, index
i
is arbitrary and
the control points with their associated nodes can be rearranged in Eq. (7.1)
without having any effect in the obtained surface.
When the standard deviation in a RaG surface is very small, the surface fol-
lows individual voxels. The selected standard deviation should be large enough
