Biomedical Engineering Reference
In-Depth Information
treated as speed terms (motion in the normal direction), and the associated
gradient magnitude is computed using the up-wind scheme.
8.6.4.2 Sparse-Field Method
The computational burden associated with solving the 3D, second-order, non-
linear level set PDE is significant. For this reason several papers [34, 35] have
proposed narrow-band methods, which compute solutions only for a relatively
small set of pixels in the vicinity of
k
level set. The authors [36] have proposed a
sparse-field algorithm, which uses an approximation to the distance transform
and makes it feasible to recompute the neighborhood of the level set model at
each time step. It computes updates on a band of grid points, called the
active
set
, that is one point wide. Several layers around this active set are updated in
such a way as to maintain a neighborhood in order to calculate derivatives. The
position of the surface model is determined by the set of active points and their
values.
8.6.4.3 Incremental Projection Updates
The tomographic surface reconstruction problem entails an additional compu-
tational burden, because the measured data must be compared to the projected
model at
each iteration
. Specifically, computing
p
ij
can be a major bottleneck.
Computing this term requires recomputing the sinogram of the surface/object
model as it moves. In the worst case, we would reproject the entire model every
iteration.
To address this computational concern, we have developed the method of
incremental projection updates
(IPU). Rather than fully recompute
p
at every
iteration, we maintain a current running version of
p
and update it to reflect
the changes in the model as it deforms. Changes in the model are computed
only on a small set of grid points in the volume, and therefore the update time
is proportional to the area of the surface, rather than the size of the volume it
encloses.
The IPU strategy works with the the sparse-field algorithm as follows. At
each iteration, the sparse-field algorithm updates only the active layer (one voxel
wide) and modifies the set of active grid points as the surface moves. The incre-
mental projection update strategy takes advantage of this to selectively update
