Biomedical Engineering Reference
In-Depth Information
method that approximates a digital shape by a parametric surface is described.
Since voxels belonging to a digital shape do not usually form a regular grid, we
choose the rational Gaussian (RaG) formulation [11, 12], which does not require
a regular grid of control points to represent a free-form shape. We will show
how to parametrize voxels in a digital shape and how to determine the control
points of a RaG surface that approximate the digital shape by the least-squares
method. The obtained RaG surface is then overlaid with the volumetric image
and the user is allowed to revise the surface by moving its control points.
7.2.1
Surface Approximation
Given a set of (control) points
{
V
i
:
i
=
1
,...,
n
}
, the RaG surface that approxi-
mates the points is given by [11, 12]
n
P
(
u
,v
)
=
V
i
g
i
(
u
,v
)
,
u
,v
∈
[0
,
1]
,
(7.1)
i
=
1
where
g
i
(
u
,v
)isthe
i
th blending function of the surface defined by
G
i
(
u
,v
)
j
=
1
G
j
(
u
,v
)
,
g
i
(
u
,v
)
=
(7.2)
and
G
i
(
u
,v
) is a 2D Gaussian of height 1 centered at (
u
i
,v
i
):
G
i
(
u
,v
)
=
exp
{−
[(
u
−
u
i
)
2
+
(
v
−
v
i
)
2
]
/
2
σ
2
}
.
(7.3)
{
(
u
i
,v
i
):
i
=
1
,...,
n
}
are the parameter coordinates associated with the points.
The parameter coordinates determine the adjacency relation between the points.
In the subsequent section, we will see how to estimate them. Formulas (7.1)-
(7.3) are for an open surface. If a surface is required to close from one side, like
a generalized cylinder, formula (7.3) should be replaced with
∞
exp
{−
[(
u
−
u
i
)
2
+
(
v
−
v
i
+
k
)
2
]
/
2
σ
2
G
i
(
u
,v
)
=
(7.4)
}
.
k
=−∞
If the opening at each end of a generalized cylinder converges to a point, a
closed surface will be obtained. In a cylindrical surface, a 2D Gaussian wraps
around the closed side of the surface infinitely. However, since a Gaussian ap-
proaches zero exponentially, its effect vanishes after a few cycles. Therefore,
in practice, the
∞
in formula (7.4) is replaced with a small number such as
1 or 2 [11]. An alternative method for obtaining a closed surface is to use the
