Biomedical Engineering Reference
In-Depth Information
Figure 2.
The point marked by a cross has a relatively distinctive GMI-based attribute
vector. The color-coded image on the right shows the degree of similarity between the
attribute vector of the marked (by crosses) point and the attribute vector of every other point
in the brain. 1 is maximum and 0 minimum similarity. See attached CD for color version.
the template and the individual. This is due, in part, to the ambiguity in finding
point correspondences. For example, if many candidate points in an individual
image have similar attribute vectors to that of a particular template voxel, then
this introduces an ambiguity that results in local minima of the corresponding en-
ergy function. In contrast, consider the situation in which there are a few anchor
points for which correspondence (the value of the shape transformation) can be
determined rather unambiguously, perhaps because each anchor point's attribute
vector is very different for all but its corresponding anchor point. In that case,
the shape transformation on all other (non-anchor) points could be determined
via some sort of interpolation from the anchor points. This problem would not
have local minima. Of course, the cost function being minimized would be only
a lower-dimensional approximation, compared to a cost function involving every
single voxel in the image. HAMMER is based on this fact, and forms successive
lower-dimensional cost functions, based initially only on key anchor points, and
gradually involving more and more points. More points are considered as a bet-
ter estimate of the shape transformation is obtained, and potential local minima
are avoided. Anchor points are defined based on how distinctive their attribute
vectors are.
A third feature of HAMMER is that it is inverse consistent, in terms of the
driving correspondences. This means that if the individual is deformed to the tem-
plate, instead of the converse, the mapping between any two driving points during
this procedure would be identical. This feature is a computationally fast approxi-
mation to the problem of finding fully 3D inverse consistent shape transformations
originally proposed by Christensen [65]. Representative results elucidating HAM-
MER's performance are shown in Figures 3 and 4.
