Biomedical Engineering Reference
In-Depth Information
Grayscale features—gradient magnitude
: Surface models can also be attracted
to certain grayscale features in the input data. For instance, the gradient
magnitude indicates areas of high contrast in volumes. By following the
gradient of such grayscale features, surface models are drawn to minimum
or maximum values of that feature. Typically, grayscale features, such as the
gradient magnitude, are computed with a scale operator, e.g., a derivative-of-
Gaussian kernel. If models are properly initialized, they can move according
to the
gradient of the gradient magnitude
and settle onto the edges of an
object at a resolution that is finer than the original volume.
If
G
(
x
) is some grayscale feature, for instance
G
(
x
)
=|∇
I
(
x
)
|
, where
I
(
x
) is the input data (appropriately filtered—we use Gaussian kernels with
σ
≈
0
.
5), then
F
grad
(
x
)
=
n
·
(
±∇
G
(
x
))
,
(8.8)
where a positive sign moves surface toward maxima and the negative sign
toward minima.
Isosurface
: Surface models can also expand or contract to conform to isosur-
faces in the input data. To a first order approximation, the distance from a
point
x
∈
U
to the
k
-level surface of
I
is gi
ven by
(
I
(
x
)
−
k
)
/
|∇
I
|
.Ifwelet
g
(
α
) be a fuzzy threshold, e.g.,
g
(
α
)
=
α/
√
1
+
α
2
, then
F
iso
(
x
)
=
g
I
(
x
)
−
k
|∇
I
|
(8.9)
causes the surfaces of
φ
to expand or contract to match the
k
isosurface
of
I
. This term combined with curvature or one of the other fitting terms
can create “quasi-isosurfaces” that also include other considerations, such
as smoothness or edge strength.
8.3.3 Framework Results
Figure 8.4 presents one slice from an MRI scan of a mouse embryo, and an
isosurface model of its liver extracted from the unprocessed dataset. Figure 8.5
presents 3D renderings of the sequence of steps performed on the mouse MRI
data to segment the liver. The first step is the initialization, which includes
smoothing the input data, thresholding followed by a a flood fill to remove
isolated holes, and finally applying morphological operators to remove small
gaps and protrusions on the surface. The second (surface deformation) step
