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Title: Geodesic shooting for anatomical curve registration on the plane
Author: Clark, Allan
ISNI:       0000 0001 2424 9220
Awarding Body: Imperial College London
Current Institution: Imperial College London
Date of Award: 2012
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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.
Supervisor: Cotter, Colin ; Peiro, Joaquim Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral