Biomedical Engineering Reference
In-Depth Information
low, the image size is small, or the overlap of images is limited. In practice, the
multiresolution approach proves to be helpful. It can improve the optimization
speed, increase the capture range and the algorithm is relatively robust [42]. In
our implementation, the images were folded down to a 16
×
16
×
16 image as the
most coarse image. The resolution of the successive images was doubled until
the full image resolution or 64
×
64
×
64 was reached in all three dimensions.
We used fine resolutions beyond 64
×
64
×
64 when the image size permitted it.
But most cases did not exhibit any sizable improvements on the registration ac-
curacy, and there was almost no effect on the success rate. To obtain the coarse
images, the voxel values within a sampling volume are averaged. Although it
is a little slower than a simple subsampling approach, the averaging technique
results in a better registration [18].
10.3.6
Numerical Stability
Referring to the definitions of cross-entropy, reversed cross-entropy, and sym-
metric divergence, one may find that there may be a numerical problem under
some conditions. The instability is caused when the priori or joint probability
is zero. Ideally, one could sample a large dataset to get a better estimate of the
priori or use sophisticated sampling schemes to better estimate the joint proba-
bility. Since those are computationally demanding, we chose a simple, non-exact
approach that is described below.
For cross-entropy maximization, if the joint probability is zero, the contri-
bution to the cross-entropy is zero (0 log 0
=
0). If the marginal probability is
zero, the joint probability will be zero and will have no contribution to the cross-
entropy measure. In the cross-entropy calculation, therefore, the terms can be
ignored when the joint or marginal probabilities are zero.
For the cross-entropy minimization, the contribution to the cross-entropy
would be infinite when the priori probability is zero. We elected to ignore these
terms since the cross-entropy is being minimized. Alternatively, one could as-
sign a large value to the cross-entropy under this situation. When picking such
a large value, one should take into account the stop condition of the optimiza-
tion process. If the assigned value is too large, the optimization can prematurely
terminate. We chose a positive value, that resulted in the cross-entropy mini-
mization having a small capture range. Note that if the joint probability is zero,
the terms can also be ignored.
