Registration of functional and anatomical brain volumes provide more accurate localisation of regions of interest and therefore greatly enhance diagnosis. Awareness of the associated errors incurred by the registration is necessary to provide a basis for the determination of the significance of conclusions based on the registered images. The ability to measure error or the deviation from a correct or ``true'' value presupposes the knowledge of that true result or value - the correct registration. If the true value is not known, it may be at best approximated by calculation of the mean. A true or nominal registration results in zero translation and rotation errors. Illegitimate errors (what Bevington calls ``blunders'' [Bev69]), systematic errors, and random errors all may contribute to produce deviations from the true value when a measurement is repeated. Illegitimate errors are often obvious and thus easily remedied and systematic errors may usually be corrected retrospectively upon comparison of data with expected results. Random errors usually result from physical statistical fluctuations on a microscopic scale being measured on a macroscopic scale.
When measuring room temperature with a mercury thermometer for example, the variability in time of the kinetic energies of the atoms and molecules which constitute the air and that impinge on the probe results in random variation in the measured expansion of the mercury. The net result is a sample of variable temperature readings even at constant room temperature. For the case of image registration, random variation in the specification of the location of an anatomical point results from the statistical nature of the spatial map produced by the low resolution radionuclide based imaging modality and the finite response of the measuring instrument - the electronic and mechanical imaging system and the imperfect human observer. From the development of the previous chapter, Procrustes analysis produces approximations to the true registrations which are limited by the amount of perturbation which exists between the homologous configurations. A theoretical difficulty arises when it is considered that exact homology between configuartions is insufficient for the provision of a true registration when matching clinical scans. Configurational and anatomical homology is necessary.
It is easy to imagine 4 points situated uniformly on an outline of a
circle. If 4 exactly homologous points which were chosen as landmarks for
registration were in fact off by some angle , registration with the ILM
technique will still produce a Procrustes statistic of zero but the
rotation error will be exactly
(see figure 4.1). The corollary
is also true. A true registration does not necessarily yield a residual of
zero [Mac]. The angle
may only be measured if the true
registration, accounting for configurational and anatomical homology, were
known. If there is an imperfect homology envelope between the two
configurations but the true registration were known, multiple
registrations will produce deviations from the true registration such that
a distribution of translation and rotation errors will result. The mean
of the errors will be at zero, since a true registration has zero error.
If the true registration had in fact included some systematic error, the
mean of the distribution of the errors will be shifted from zero. For a
fixed number of points in a configuration, the distribution of deviations
from the exactly correct registration (configurationally and anatomically)
will only be dependent on the amount of perturbation (homology error)
between the configurations. This is the basis of the methods for the
studies in the next chapter.
The presupposition of the knowledge of the true registration may be afforded by point simulations or external fiducials on real scan subjects and thus allow for the quantification of error. Point simulations are advantageous in that the exactly correct registration is known configurationally and anatomically. They are clinically unrealistic because real scans do not have homogeneous perturbation distributions. This is why validations based on the real scans with external fiducials are necessary. The disadvantage of an external fiducial based measurement is the finite possibility of both configurational and anatomical homology being non-zero. Both methods are thus presented in the next chapter to doubly glean insight from the point simulation's results and validate the real scan results against the point simulations.
Previous validations of the ILM technique have incorrectly employed the Procrustes statistic as a measure of the registration error [MPCP89][EMT+91]. Many papers which introduce methods for registration or attempt to validate existing ones unacceptably specify registration errors as simplistic mean Cartesian distance measures between specific point pairs after registration [PCS+89][SHC+92][WMC93][SBG+90][PEN+91][SHS+92][RTL+93][HZJ+90], subtraction images [HMS93], or as residual measurements [WQC+93][OTP91][SHC+92][PEN+91][MPCP89][EMT+91][SAX+93]. While different methods do provide measures of an error, the utility of an error measurement should be based on its ability to tenably describe bounds for decisions based on registrations. Quantitatively, the uncertainty in the anatomical specification of a point in a registered functional image is difficult if one has to deal with subtraction images and the interpretation of contrast differences as a measure of the of error. In the same vein, although measurements of error based on the mean of Cartesian distances does specify the net error at specific points, it is a dimensionally limited quotation and therefore inadequate if one wishes to investigate any dimensional biases in the errors.