Image registration under conformal diffeomorphisms : 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
Image registration is the process of finding an alignment between two or more images
so that their appearance matches. It has been widely studied and applied to several
fields, including medical imaging and biology (where it is related to morphometrics).
In biology, one motivation for image registration comes from the work of Sir D'Arcy
Thompson. In his book On Growth and Form he presented several examples where a
grid superimposed onto a two-dimensional image of one species was smoothly deformed
to suggest a transformation to an image of another species. His examples include
relationships between species of fish and comparison of human skulls with higher apes.
One of Thompson's points was that these deformations should be as `simple' as possible.
In several of his examples, he uses what he calls an isogonal transformation, which
would now be called conformal, i.e., angle-preserving. His claims of conformally-related
change between species were investigated further by Petukhov, who used Thompson's
grid method as well as computing the cross-ratio (which is an invariant of the Möbius
group, a finite-dimensional subgroup of the group of conformal diffeomorphisms) to
check whether sets of points in the images could be related by a Möbius transformation.
His results suggest that there are examples of growth and evolution where a
Möbius transformation cannot be ruled out. In this thesis, we investigate whether or
not this is true by using image registration, rather than a point-based invariant: we
develop algorithms to construct conformal transformations between images, and use
them to register images by minimising the sum-of-squares distance between the pixel
intensities. In this way we can see how close to conformal the image relationships are.
We develop and present two algorithms for constructing the conformal transformation,
one based on constrained optimisation of a set of control points, and one based
flow. For the first method we consider a set of different penalty terms that
aim to enforce conformality, based either on discretisations of the Cauchy-Riemann
equations, or geometric principles, while in the second the conformal transformation
is represented as a discrete Taylor series. The algorithms are tested on a variety of
datasets, including synthetic data (i.e., the target is generated from the source using a
known conformal transformation; the easiest possible case), and real images, including
some that are not actually conformally related. The two methods are compared on a
set of images that include Thompson's fish example, and a small dataset demonstrating
the growth of a human skull. The conformal growth model does appear to be validated
for the skulls, but interestingly, not for Thompson's fish.