Biomedical Engineering Reference
In-Depth Information
3.4.2.1 The Head and Hat Algorithm
Pelizzari and colleagues
16, 17
proposed a surface fitting technique for registra-
tion of images of the head that became known as the “head and hat” algo-
rithm. Two equivalent surfaces are identified in the images. The first, from
the higher resolution modality, is represented as a stack of disks and is
referred to as the head. The second surface is represented as a list of uncon-
nected 3D points. The registration transformation is determined by itera-
tively transforming the (rigid) hat surface with respect to the head surface,
until the closest fit of the hat onto the head is found. The measure of closeness
of fit used is the square of the distance between a point on the hat and the
nearest point on the head, in the direction of the centroid of the head. The iter-
ative optimization technique used is the Powell method.
18
The Powell opti-
mization algorithm performs a succession of one dimensional optimizations,
finding in turn the best solution along each of the six degrees of freedom, and
then returning to the first degree of freedom. The algorithm stops when it is
unable to find a new solution with a significantly lower cost (as defined by a
tolerance factor) than the current best solution. This algorithm has been used
with considerable success for registering images of the head,
17
and has also
been applied to the heart.
19
As first described, the head surface was derived
from MR data, and the hat surface from PET data. The surfaces most com-
monly used are the skin surface (from MR and PET transmission images) or
the brain surface (from MR images and PET emission images).
3.4.2.2 Distance Transforms
The performance of the head and hat algorithm was improved by using a dis-
tance transform to preprocess the head images. A distance transform is
applied to a binary image in which object pixels (or voxels) have the value 1,
and other voxels have the value 0. All voxels in this image are labeled with
their distance from the surface of the object. By prelabeling all image vox-
els in this way, the computational cost per iteration can be substantially
reduced (potentially to a single address manipulation and accumulation
for each transformed hat surface point). A widely used distance transform is
the chamfer filter proposed by Borgefors.
20
This approach was used for rigid-
body medical image registration, e.g. by Jiang
21
and van Herk.
22
More recently,
exact Euclidean distance transforms have been used in place of the chamfer
transform.
23
Given estimates for the six degrees of freedom of the rigid-body transfor-
mation, the hat points are transformed and their distances from the head
surface are calculated from the values in the relevant voxels in the distance
transform. These values are squared and summed to calculate the cost asso-
ciated with the current transformation estimate. The risk of finding local
optima can be reduced by starting out registration at low resolution and
gradually increasing the resolution to refine the accuracy, combined with
outlier rejection to ignore erroneous points.
21
