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For the point simulations, m points were randomly generated in 3-D
space within spheres of radius r or shells with outer radius 
and thickness of 5 mm to produce the target configuration, A. A is
randomly translated and rotated (uniform distribution)
about the center of the specified
spherically symmetric boundary within 
mm and
degrees, respectively, to produce an exactly corresponding but
un-registered configuration, B. The un-registered configuration
with homogeneous homology error which is to be transformed for
registration, 
, is produced by perturbing the 3-D position of
each point of B within homogeneous Gaussian error envelopes with
ranging from 2.5 mm to 20 mm. For different radii for
the shells and spheres, r and 
respectively,
(ranging from 35 mm to 90 mm), each m (ranging from 4 to 35 point
pairs) simulated for a radius, 100 simulations, (where 
for 
),
are produced and registered for each 
. More briefly,
the simulation encompasses iterations which are five loops deep:
- FOR a shell and sphere configuration.
- FOR different radii of the above configurations, r.
- FOR different numbers of homologous point pairs, m, for each r
and configuration.
- FOR different homology errors envelopes for each m, r,
and configuration.
- DO the creation of the data set to be registered and register
them N = 100 times.
- Upon termination of each DO loop, calculate the mean,
standard deviation, and
statistic of
the translation and rotation errors over the sample
population of N = 100.
Six test points, two per axis, were
symmetrically situated about the
center of either the sphere or shell to encode the correct
registration in the simulation for each A
.
No homology error was
introduced to the test points so that the test points in each
were exactly homologous to their pair in A. The correct translations,
, and rotations, 
, about the same center of
rotation were calculated for each using the test points. The
simulated registrations of the N 's to the same A and their
associated shifts, 
, and rotations, 
, were calculated and
the uncertainty of the distributions (from their differences from
and 
) were found to ultimately yield
and 
through each dimension and
about each axis for each combination of factors. These simulations
were written as batch files for Matlab
and
run on an SGI
(Silicon Graphics Interface) workstation
with twin 200 MHz Challenge processors.
Next: 5.3.2 Real Scans
Up: 5.2 Methods
Previous: 5.2 Methods