The aim of the work presented in this thesis is to develop a method of characterising the shape
of curves in the plane that is independent of the parameterisation of the curve. It is important
to remove the effect of a specific parameterisation of a curve because it is possible for two
curves to have the same shape while having different parameterisations. The characterisation is
accomplished by matching curves via deformations, and using the deformations to characterise
the difference between them. We specifically aim for a method that is able to characterise the
kind of complex curves found in cross sections of the human nasal cavity.
In order to match one curve to another, we derive the equations of motion for a geodesic flow, and seeking the
flow that deforms an embedded reference curve into the target curve we
wish to characterise. The geodesic
flow is itself characterised by a conjugate momentum on, and
normal to, the reference curve, giving a one dimensional descriptive signal of the deformation.
This descriptive signal contains all of the information required to generate the target curve
from the reference curve. We therefore say that this descriptive signal characterises the target
curve with respect to the reference curve.
The descriptive signal is found using a shooting approach, requiring a functional to measure
how closely overlaid are two curves. Formulating the problem as an optimisation problem,
we first present a parameterisation-independent functional based on geometric currents, but
show that we encounter problems in this matching functional due to local minima. We then
present a second approach in which we formulate the problem as a landmark matching problem.
Since we seek a characterisation that is independent of the choice of landmarks, and the
landmark matching functional is parameterisation dependent, we minimise the functional over
all reparameterisations of the reference curve. These two approaches solve equivalent problems.
We present the results of the reparameterisation-based matching, and show that they overcome
the problems observed in the currents-based method. In particular we demonstrate that the method is able to match complex nasal geometries, and show how the descriptive signal
can be used to interpolate between two dimensional slices of three dimensional objects to reconstruct
three dimensional surfaces representing the objects. Though here we implement the
geodesic
flow in two dimensions, we note that the
flow could be extended to three dimensional
space. Since the reparameterisation based matching functional is trivial to implement in
three dimensions, this would allow for the characterisation of both curves and surfaces in three
dimensional space.