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    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
    (Massey University, 2017) Tufail, Muhammad Yousuf
    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 on gradient 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.
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    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
    (Massey University, 2016) Zarredooghabadi, Raziyeh
    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 specific 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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    Least-squares optimal interpolation for direct image super-resolution : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Engineering at Massey University, Palmerston North, New Zealand
    (Massey University, 2009) Gilman, Andrew
    Image super-resolution aims to produce a higher resolution representation of a scene from an ensemble of low-resolution images that may be warped, aliased, blurred and degraded by noise. There are a variety of methods for performing super-resolution described in the literature, and in general they consist of three major steps: image registration, fusion and deblurring. This thesis proposes a novel method of performing the first two of these steps. The ultimate aim of image super-resolution is to produce a higher-quality image that is visually clearer, sharper and contains more detail than the individual input images. Machine algorithms can not assess images qualitatively and typically use a quantitative error criterion, often least-squares. This thesis aims to optimise leastsquares directly using a fast method, in particular one that can be implemented using linear filters; hence, a closed-form solution is required. The concepts of optimal interpolation and resampling are derived and demonstrated in practice. Optimal filters optimised on one image are shown to perform nearoptimally on other images, suggesting that common image features, such as stepedges, can be used to optimise a near-optimal filter without requiring the knowledge of the ground-truth output. This leads to the construction of a pulse model, which is used to derive filters for resampling non-uniformly sampled images that result from the fusion of registered input images. An experimental comparison shows that a 10th order pulse model-based filter outperforms a number of methods common in the literature. The use of optimal interpolation for image registration linearises an otherwise nonlinear problem, resulting in a direct solution. Experimental analysis is used to show that optimal interpolation-based registration outperforms a number of existing methods, both iterative and direct, at a range of noise levels and for both heavily aliased images and images with a limited degree of aliasing. The proposed method offers flexibility in terms of the size of the region of support, offering a good trade-off in terms of computational complexity and accuracy of registration. Together, optimal interpolation-based registration and fusion are shown to perform fast, direct and effective super-resolution.