The approximation of Cartesian coordinate data by parametric orthogonal distance regression.
This thesis is concerned with the approximation of Cartesian coordinate data by
parametric curves and surfaces, with an emphasis upon a technique known as parametric
orthogonal distance regression (parametric ODR). The technique has become
increasingly popular in the literature over the past decade and has applications in a
wide range of fields, including metrology-the science of measurement, and computer
aided design (CAD) modelling. Typically, the data are obtained by recording points
measured in the surface of some physical artefact, such as a manufactured part.
Parametric ODR involves minimizing the shortest distances from the data to
the curve or surface in some norm. Under moderate assumptions, these shortest
distances are orthogonal projections from the data onto the approximant, hence the
nomenclature ODR. The motivation behind this type of approximation is that, by
using a distance-based measure, the resulting best fit curve or surface is independent
of the position or orientation of the physical artefact from which the data is obtained.
The thesis predominately concerns itself with parametric ODR in a least squares
setting, although it is indicated how the techniques described can be extended to
other error measures in a fairly straightforward manner. The parametric ODR problem
is formulated mathematically, and a detailed survey of the existing algorithms
for solving it is given. These algorithms are then used as the basis for developing
new techniques, with an emphasis placed upon their efficiency and reliability. The
algorithms (old and new) detailed in this thesis are illustrated by problems involving
well-known geometric elements such as lines, circles, ellipse and ellipsoids, as well
as spline curves and surfaces. Numerical considerations specific to these individual
elements, including ones not previously reported in the literature, are addressed. We
also consider a sub-problem of parametric ODR known as template matching, which
involves mapping in an optimal way a set of data into the same frame of reference as
a fixed curve or surface.