All the experiments were performed on sets of known arbitrary-transformed data obtained by the following procedures.
Data type A: The images selected for the experimental computations consisted o f a set of 2-D adjacent T2-weighted MR slices. To simulate the registering data for this assessment, the set of original grey-scale MR data were misregistered arbitrarily by different known transformation parameters (e.g. 5, 8 and 10 voxels, shift, and 5, 8,10 degrees, rotation, with respect to x, y, and z direction). These data sets were not truncated after misregistration. The surface of these data sets were formed and stored for surface fitting process and thus for the evaluation o f various aspects of the registration algorithm.
Data type B: The original 2-D MR slices were degraded before applying the arbitrary misregistration, in order to simulate PET and SPECT data in a known geometrical environment. This degradation was performed by applying a Gaussian filter to the original MR data and then adding Gaussian noise to them.
Data type C: To provide a representative set o f data for the real images taken by an imaging system and thus to simulate a more practical set of misregistered images, an alternative set of data was generated. The MR data were arbitrarily misregistered by setting different physical parameters (e.g. rates o f gradients) of the imaging system itself (i.e. setting different combinations o f x-y-z gradient amplitude o f the MR slice-selection parameters which generate the slices in different directions).
with these markers were registered by checking the marker position in both studies. Knowing the geometric information between the two original data sets, MR and SPECT, the multi resolution process was then applied and verified.
In the first set of experiments, an attempt was made to register the generated (well- defined) images with the original MR slices using the sequential multi-resolution algorithm. Different parameters (e.g. resolution and threshold) were set and tested in these experiments. In order to verify the fidelity and the performance characteristics o f the sequential and multi resolution methods, alternative sets of experiments were designed, in which all the mismatch measures {MSD or MDE values) were examined in a search window around the optimal transformation (true known location). These transformations were applied following the registration o f the two arbitrary-misregistered data sets. For obtaining the performance of the process at other low resolution levels, the original high resolution siufaces were degraded (i.e. geometrically scaled) before the registration. Then the corresponding geometrical transformation (e.g. a shift o f 1 voxel on image size of 32^ in respect of 8 voxels shift on image size o f 256^) was applied at each level to register the two surfaces, originally, before assessing the search window.
When checking all the transformations in a search window, the minimization process recognizes some locations whose MDE values are less than a pre-defined threshold (or their cumulative errors are less than the variable threshold). Thus if only the known mismatch locations are checked, then any match location recognized by the process must be a false match. The probability that such an observed occurrence o f the distance error being less than ilncshold T is called False Alarm Probability (FAP). The window search designed for FAP constitutes all the transformations defined in the proposed search window (i.e. n\y^°=(8x24-l)^ locations corresponding to -8 to 8 voxels, shift, and -8 to +8 degrees, rotation, in x,y and z directions; see section 6.3) except the actual true match location itself and except those neighbouring locations which can be assumed as match locations (e.g. W^°=( 1x2+1)® locations corresponding to a shift of -1 to +1 voxel, and a rotation o f -1 to +1 degree, around the true match location; see region k l in figure 6.6). In order to evaluate the most uncertain false match locations, a smaller window size was considered for evaluating FAP in most experiments. This was done by constituting only the locations which need a minor extra transformation (e.g. ±2 to ±3 voxels, shift, or ±2 to ±3 degrees, rotation; see region k2 and k3 in figure 6.6) in order to be counted as match location.
This helps to established the response of the registration process to various unacceptable transformations which are uncertain. An equivalent transformation range was applied in other resolution levels.
Similar sets o f experiments were designed using the above simulated data sets, but at a number of locations which can be assumed as being acceptable matches (defined as true match locations). These locations can be defined as the neighbouring transformations o f the true match, where they just pass the test for being as a true match location (i.e. they are very close to the true match location; e.g. region k l as shown in figure 6.6). In these experiments, the images were also registered in advance using the known transformation parameters. Then the distance errors {MDE values) at true match location and its neighbourhood (see figure 6.6 for a schematic diagram of these neighbours) were examined and compared to a preset threshold. The probability of the occurrence of the error being less than the threshold is, then, counted as True Match Probability {TMP). Ideally, the suggested match and mismatch locations need a small perturbation to reverse their behaviour to being as mismatch or match location, respectively.
In order to be able to use the central limit theorem, and thereby to have a normal error distribution, the cumulative error (being the sum o f a number o f individual distance errors; e.g. 6 to 10 individual errors as confirmed empirically) was used. The errors were then compared to the corresponding variable threshold (i.e. the threshold corresponding to the number sample points which contribute to the error). The results o f applying these experiments are outlined in the next section.