Biomedical Engineering Reference
In-Depth Information
resolution later to get the final transformation. The advantage of solving the
problem on a course grid is that the algorithm requires fewer computations
per iteration than a finer grid. This results in reduced computation time at low
resolution. Each time the resolution of the grid is increased by a factor of two
in each dimension, the computation time increases by a factor of eight. The
drawback of solving the problem at low resolution is that there can be significant
registration errors due to the loss of detail in the down sampling procedure. The
trade-off between quicker execution times at low resolution and more accurate
registration at higher resolution can be exploited by solving the registration
problem at low spatial resolution during the initial iterations to approximate the
result and then increasing the spatial resolution to get a more accurate result at
the later iterations.
The spatial multiresolution approach works well with the frequency mul-
tiresolution approach provided by increasing the number of harmonics used
to represent the displacement fields. The number of harmonics used to rep-
resent the displacement fields is initially set small and then increased as the
number of iterations are increased. A low-frequency registration result is an
approximation of the desired high-frequency registration result. Computing the
gradient descent for a low-frequency basis coefficient at low spatial resolution
gives approximately the same answer as using high spatial resolution but the
computational burden is much less.
6.2.8
Tracking the Jacobian During the
Estimation Procedure
It is important to track both the minimum and maximum values of the Jacobian
during the estimation procedure. The Jacobian measures the differential vol-
ume change of a point being mapped through the transformation. At the start of
the estimation, the transformation is the identity mapping and therefore has a
Jacobian of one. If the minimum Jacobian goes negative, the transformation is
no longer a one-to-one mapping and as a result folds the domain inside out [31].
Conversely, the reciprocal of the maximum value of the Jacobian corresponds
to the minimum value of the Jacobian of the inverse mapping. Thus, as the max-
imum value of the Jacobian goes to infinity, the minimum value of the Jacobian
of the inverse mapping goes to zero.
