Biomedical Engineering Reference
In-Depth Information
image are defined at specific grid points, to compute the joint pdf by binning
voxel value pairs, one needs to interpolate the voxel values in the transformed
space. There are different interpolation methods in three-dimensional space.
Tsao [16] compared eight such methods. Interpolation artifacts are reported
which introduce spurious fluctuations in the similarity measures and impact the
optimization behavior. For simplicity, the trilinear interpolation was used in our
study.
10.3.4
Optimization
Powell's multidimensional direction set optimization is used to minimize the
three similarity measures when favorable priori estimates are used using Brent's
method in one-dimensional search [39]. The same technique is also used to min-
imize the negated similarity measures when unfavorable priori estimates are
applied. The direction matrix is initialized to a unitary matrix. The vector is
(
θ
x
,θ
y
,θ
z
,
t
x
,
t
y
,
t
z
), as explained before. A reordering of these registration pa-
rameters is possible which may improve the optimization speed as in Maes
et al.
[15]. We did not try to optimize the parameter order since the order may be image
content dependent and an exhaustive trial seems impractical (there are 6!
=
720
different combinations although one may try a subset of them). Furthermore,
Powell's optimization may use six independent directions which do not nec-
essarily correspond to the six desired directions as the search proceeds (see
[15, 39]).
The stop criterion in the Powell's algorithm was set to
ftol
=
0
.
001.
10.3.5
Multiresolution
The Powell's optimization method cannot guarantee that a global optimal value
will be found since it can be easily trapped to a local optimum. To find a true
global optimal value, Ritter
et al.
[40] successfully applied the simulated an-
nealing [39] algorithm to two-dimensional image registration. It is a stochastic
method and is slow, which limits its application to three-dimensional image reg-
istration. Pluim
et al.
used genetic algorithms [17] or others used multistarter
methods to pinpoint a global optimum. However, the desired registration is
frequently a local rather than global extremum of the similarity measure, as dis-
cussed in Fitzpatrick
et al.
[41]. This can occur when the image resolution is
