Biomedical Engineering Reference
In-Depth Information
ward specific features in the data. One must choose those properties of the
input data to which the model will be attracted and what role the shape of the
model will have in the deformation process. Typically, the deformation process
combines a data term with a smoothing term, which prevents the solution from
fitting too closely to noise-corrupted data. There are a variety of surface-motion
terms that can be used in succession or simultaneously, in a linear combination
to form
F
(
x
) in Eq. (8.4).
Curvature
: This is the smoothing term. For the work presented here we use the
mean curvature of the isosurface
H
to produce
∇·
∇
φ
|∇
φ
|
F
curv
(
x
)
=
H
=
(8.6)
.
The mean curvature is also the normal variation of the surface area (i.e., min-
imal surface area). There are a variety of options for second-order smoothing
terms [41], and the question of efficient, effective higher-order smoothing
terms is the subject of ongoing research [7, 31, 42]. For the work in this
chapter, we combine mean curvature with one of the following three terms,
weighting it by a factor
β
, which is tuned to each specific application.
Edges
: Conventional edge detectors from the image processing literature pro-
duce sets of “edge” voxels that are associated with areas of high contrast. For
this work we use a gradient magnitude threshold combined with nonmaxi-
mal suppression, which is a 3D generalization of the method of Canny [16].
The edge operator typically requires a scale parameter and a gradient thresh-
old. For the scale, we use small, Gaussian kernels with standard deviation
σ
=
[0
.
5
,
1
.
0] voxel units. The threshold depends on the contrast of the vol-
ume. The distance transform on this edge map produces a volume that has
minima at those edges. The gradient of this volume produces a field that
attracts the model to these edges. The edges are limited to voxel resolution
because of the mechanism by which they are detected. Although this fitting
is not sub-voxel accurate, it has the advantage that it can pull models toward
edges from significant distances, and thus inaccurate initial estimates can
be brought into close alignment with high-contrast regions, i.e. edges, in the
input data. If
E
is the set of edges, and
D
E
(
x
) is the distance transform to
those edges, then the movement of the surface model is given by
F
edge
(
x
)
=
n
·∇
D
E
(
x
)
.
(8.7)
