The solution to the problem of registration first requires a description of the current orientation and position of the brain volumes to be matched. A rigid body may be described as a set of at least two points whose distance or distances between the points remains fixed [Gol50]. Given two identical rigid bodies described with respect to the same frame of reference and representing the same object, differing only in spatial position and orientation, a matching or registration of those two rigid bodies may be obtained with a linear transformation wherein one of the bodies is held fixed and the other is first translated and then rotated into the position and orientation of the static body. The assumption that the two objects to be registered represent the same rigid body removes the necessity of the inclusion of shears or warping of the body in the transformation required for the matching.
In the simple one dimensional case of two vectors, registration may be
obtained by shifts along one dimension to match general spatial position
and a single reflection (a rotation of only for one dimension)
to match vector direction. The three-dimensional case for registration is
equally simple. Given two volumetric data sets of the same brain obtained
from MR and SPECT, the assumption of a rigid body for three-dimensional
registration may be preserved since the brain doesn't suffer from gross
deformations within the skull between scans [TS76]. The
representation of the brain as a rigid body in the MR and SPECT images may
thus be sufficiently described by a set of points in 3-D space. If a set
of points describing the MR volume, A, and the SPECT volume, B, are chosen
so that the corresponding elements of point sets A and B describe the
same physical location in the brain exactly, then the transformation which
makes points sets A and B coincident is the same transformation which
registers the SPECT and MR brain volumes from the same patient. Again, in
addition to the assumption of a rigid body, the points sets must describe
the same rigid body - the scans are from the same patient. Points
sets A and B do not exactly describe the same physical location in the
brain in practice because of the finite spatial resolutions of the imaging
modalities and the difference in the nature of information that is
actually being imaged. A homology error is said to exist in the selection
of anatomically corresponding point pairs. SPECT imaging of the brain
with
Tc-HMPAO produces a spatial map of tracer perfusion in its attempt
to reflect cerebral blood flow while MR produces a spatial map based partly
on proton densities, the rotational freedom of hydrogen molecules,
and the hydrogen
content of different tissues. The registration problem is therefore less
trivial in fact because of the existence of a homology or correspondence
error between the points sets which makes the use of a direct analytical
inversion procedure impossible. Clearly, some form of minimization or
maximisation
of a quality factor which describes the rigid matching is required. This
is the basic procedure of all registration techniques [Che93]. The
Procrustes algorithm
[GL83][Sib78] produces
a rigid fitting where an analytical best fit registration is achieved by
minimizing a quality factor given by the squared distances between the two
points sets to produce the transformation for the optimal matching of the
two configurations in a least squares sense by three translations, three
rotations, and an optional global scaling. Since a least squares fit or
transformation is produced, there are still finite errors in the
registration. The residual sum of squares between the points after
matching is the Procrustes statistic. The selection of points sets A and
B and their use for the registration of the cerebral data volumes they
describe using the Procrustes algorithm is the basic procedure of the ILM
technique, first applied to the registration of brain volumes by Evans et
al [EBM+88][EMT+91]. The 3-D positions of the points describing the
brain as a rigid body may be given in real physical units if the voxel
dimensions are known so that the scaling may be simply found by taking
the ratios of appropriate voxel dimensions after regsitration.
The points used to describe the rigid body
may not be arbitrarily selected because they must additionally describe
points which are deemed homologous in the SPECT and MR images for proper
use in Procrustes analysis. Without the restriction of homology, the
definition of a rigid body may be trivially found by thresholding or
segmentation. It is the simple additional requirement of data homology
which edifies the computational elegance of the ILM technique when
compared to other retrospective methods of anatomical/functional brain
registration such as the surface-fitting technique [PCS+89],
principal axes technique [ABKC90] and their variants
[SHS+92][OTP91][MBL+91][GAFC86]. In those techniques, the
definition of rigid bodies are found from surfaces or volumes with no
preassigned element to element correspondence. These methods
suffer when brains are either grossly abnormal or extremely symmetric,
when images are from different modalities and when the entire brain volume
definition is unavailable from both modalities. Unless the registration
method is an analytical technique (i.e. based on the Procrustes statistic
or on the calculation of the orientation of the principal axis), then
laborious and computationally intensive global searches must usually be
made for the proper registration. This opens the possibility for the
divergence of the algorithms used and the possibility of multiple
solutions.
In the self-calibrating techniques which use stereotactic frames or external fiducials, the rigid bodies used are not the brains themselves which require the registration but rather the external markers and stereotactic marker systems. The rigid bodies are the discrete systems of particles which constitute the external landmark system. In contrast to the other retrospective techniques, they are assumed to be exactly homologous and fixed with respect to the brain, and are thus used to yield transformations for registration by solving simple matrix equations. Problems obviously arise if the external marker systems are not fixed with respect to the brain or if they are not available in the brain volumes to be registered. Additionally, the literature is greatly lacking in their explanations of the methods used for marker location, which in practice tends to compromise the homology of marker pair locations so that the residuals obtained are also non-zero. As a result, iterative methods must also necessarily be employed to produce reliable registrations. Comparatively, the ILM technique assumes an inherent homology error and uses an analytical procedure without the usual computational intensity. The advantage of the simplicity of the ILM technique is its use of a limited number of discrete homologous points which sufficiently describe the orientation of the brain as an existing and viable internal rigid body. Thorough reviews of practically implemented registration methods may be found in the literature [Che93][Cor90].