Image registration using finite dimensional lie groups : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand
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Date
2016
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Massey University
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Abstract
D'Arcy Thompson was a biologist and mathematician who, in his 1917 book `On
Growth and Form', posited a `Theory of Transformations', which is based on the observation
that a smooth, global transformation of space may be applied to the shape
of an organism so that its transformed shape corresponds closely to that of a related
organism. Image registration is the computational task of finding such transformations
between pairs of images.
In modern applications in areas such as medical imaging, the transformations are often
chosen from the infinite-dimensional diffieomorphism group. However, this differs from
Thompson's approach where the groups are chosen to be as simple as possible, and
are generally finite-dimensional. The main exception to this is the similarity group
of translation, rotation, and scaling, which is used to pre-align images. In this thesis
the set of planar Lie groups are investigated and applied to image registration of the
types of images that Thompson considered. As these groups are smaller, successful
registration in these groups provides more specifc information about the relationship
between the images than diffeomorphic registration does, as well as providing faster
implementations. We build a lattice of the Lie groups showing which are subgroups of
each other, and the groups are used to perform image registration by minimizing the
L2-norm of the difference between the group-transformed source image and the target
image. A robust, practical, and efficient algorithm for registration in Lie groups is
developed and tested on a variety of image types.
Each successful registration returns a point in a Lie group. Given several related images
(such as the hooves of several animals) it is possible to find smooth curves that pass
through the Lie group elements used to relate the various images. These curves can
then be employed to interpolate points between the set of images or to extrapolate to
new images that have not been seen before. We discuss the mathematics behind this
and demonstrate it on the images that Thompson used, as well as other datasets of
interest.
Finally, we consider using a sequence of the planar Lie groups to perform registration,
with the output from one group being used as the input to the next. We call this multiregistration,
and have identified two types: where the smallest group is a subgroup
of the next smallest, and so on up a chain, and where the groups are not directly
related, i.e., separated on the lattice. We demonstrate experimentally that multiregistration
can provide more information about the relationship between images than
simple registration. In addition, we show that transformations that cannot be obtained
by a single registration in any of the groups considered can be successfully reached.
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Lie groups, Image registration, Mathematical models, Research Subject Categories::MATHEMATICS::Applied mathematics