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 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

How round is a protein? Exploring protein structures for globularity using conformal mapping.

We present a new algorithm that automatically computes a measure of the geometric difference between the surface of a protein and a round sphere. The algorithm takes as input two triangulated genus zero surfaces representing the protein and the round sphere, respectively, and constructs a discrete conformal map f between these surfaces. The conformal map is chosen to minimize a symmetric elastic energy E S (f) that measures the distance of f from an isometry. We illustrate our approach on a set of basic sample problems and then on a dataset of diverse protein structures. We show first that E S (f) is able to quantify the roundness of the Platonic solids and that for these surfaces it replicates well traditional measures of roundness such as the sphericity. We then demonstrate that the symmetric elastic energy E S (f) captures both global and local differences between two surfaces, showing that our method identifies the presence of protruding regions in protein structures and quantifies how these regions make the shape of a protein deviate from globularity. Based on these results, we show that E S (f) serves as a probe of the limits of the application of conformal mapping to parametrize protein shapes. We identify limitations of the method and discuss its extension to achieving automatic registration of protein structures based on their surface geometry

Computing Teichm\"{u}ller Maps between Polygons

By the Riemann-mapping theorem, one can bijectively map the interior of an $n$-gon $P$ to that of another $n$-gon $Q$ conformally. However, (the boundary extension of) this mapping need not necessarily map the vertices of $P$ to those $Q$. In this case, one wants to find the ``best" mapping between these polygons, i.e., one that minimizes the maximum angle distortion (the dilatation) over \textit{all} points in $P$. From complex analysis such maps are known to exist and are unique. They are called extremal quasiconformal maps, or Teichm\"{u}ller maps. Although there are many efficient ways to compute or approximate conformal maps, there is currently no such algorithm for extremal quasiconformal maps. This paper studies the problem of computing extremal quasiconformal maps both in the continuous and discrete settings. We provide the first constructive method to obtain the extremal quasiconformal map in the continuous setting. Our construction is via an iterative procedure that is proven to converge quickly to the unique extremal map. To get to within $\epsilon$ of the dilatation of the extremal map, our method uses $O(1/\epsilon^{4})$ iterations. Every step of the iteration involves convex optimization and solving differential equations, and guarantees a decrease in the dilatation. Our method uses a reduction of the polygon mapping problem to that of the punctured sphere problem, thus solving a more general problem. We also discretize our procedure. We provide evidence for the fact that the discrete procedure closely follows the continuous construction and is therefore expected to converge quickly to a good approximation of the extremal quasiconformal map.Comment: 28 pages, 6 figure

Learning shape correspondence with anisotropic convolutional neural networks

Establishing correspondence between shapes is a fundamental problem in geometry processing, arising in a wide variety of applications. The problem is especially difficult in the setting of non-isometric deformations, as well as in the presence of topological noise and missing parts, mainly due to the limited capability to model such deformations axiomatically. Several recent works showed that invariance to complex shape transformations can be learned from examples. In this paper, we introduce an intrinsic convolutional neural network architecture based on anisotropic diffusion kernels, which we term Anisotropic Convolutional Neural Network (ACNN). In our construction, we generalize convolutions to non-Euclidean domains by constructing a set of oriented anisotropic diffusion kernels, creating in this way a local intrinsic polar representation of the data (`patch'), which is then correlated with a filter. Several cascades of such filters, linear, and non-linear operators are stacked to form a deep neural network whose parameters are learned by minimizing a task-specific cost. We use ACNNs to effectively learn intrinsic dense correspondences between deformable shapes in very challenging settings, achieving state-of-the-art results on some of the most difficult recent correspondence benchmarks

Exposição à teratógenos e anormalidades oculares congênitas em pacientes brasileiros portadores da sequência de Möbius

Purpose: To assess the sociodemographic profiles, teratogen exposures, and ocular congenital abnormalities in Brazilian patients with Möbius sequence. Method: Forty-four patients were recruited from the Brazilian Möbius Sequence Society. This cross-section comprised 41 patients (age, mean ± standard deviation, 9.0 ± 5.5 years) who fulfilled the inclusion criteria. The parent or caregiver answered a questionnaire regarding sociodemographic data and pregnancy history. Patients underwent ophthalmological assessments. They were subdivided into groups according to misoprostol exposure during pregnancy, and the two groups were compared. Results: Mothers/caregivers reported unplanned pregnancies in 36 (88%) cases. Of these, 19 (53%) used misoprostol during their first trimesters. A stable marital status tended to be more frequent in the unexposed group (P=0.051). Incomplete elementary school education was reported by two (11%) mothers in the exposed group and by three (14%) mothers in the unexposed group (P=0.538). The mothers' gestational exposures to cocaine, marijuana, alcohol, and cigarettes were similar in both groups (P=0.297, P=0.297, P=0.428, and P=0.444, respectively). One (5%) case of Rubella infection during pregnancy was found in the unexposed group. The main malformations in the exposed and unexposed groups were the following: strabismus (72% and 77%, respectively), lack of emotional tearing (47% and 36%, respectively), and lagophthalmos (32% and 41%, respectively). Conclusion: Stable marital statuses tended to be more frequent among mothers that did not take misoprostol during pregnancy. Exposures to other teratogens and the main ocular abnormalities were similar in both groups.Objetivo: Descrever o perfil sóciodemográfico, exposição à teratógenos e anormalidades oculares congênitas em pacientes brasileiros portadores da sequência de Möbiu Método: Quarenta e quatro pacientes recrutados da Sociedade Brasileira de Sequência de Möbius foram examinados. Este estudo transversal incluiu 41 pacientes que preencheram os critérios de inclusão do estudo (média das idades: 9,0 ± 5,5 anos). Mãe/responsável dos pacientes responderam a um questionário sobre perfil sóciodemográfico e história gestacional. Foi realizado exame oftalmológico de todos os pacientes. Eles foram agrupados em dois grupos de acordo com a exposição ao misoprostol durante a gestação e seus dados foram comparados. Resultados: Mães/responsáveis referiram gravidez indesejada em 36 (88%) dos casos. Destas, 19 (53%) fizeram uso de misoprostol no primeiro trimestre de gestação. Houve uma tendência do grupo de mães não expostas ao misoprostol de terem um estado civil estável (P=0,051). Duas (11%) mães do grupo de expostas ao misoprostol relataram primeiro grau incompleto e três (14%) do grupo de não expostas (P=0,538). A exposição das mães à cocaína, maconha, álcool e cigarro foi similar em ambos os grupos (P=0,297, P=0,297, P=0,428, P=0,444, respectivamente). Houve um caso (5%) de Rubéola no grupo de mães não expostas. As principais malformações associadas nos pacientes expostos e não expostos foram, respectivamente: estrabismo (72% e 77%), e diminuição da lágrima emocional (47% e 36%) e lagoftalmia (32% and 41%). Conclusão: Estado civil estável foi mais frequente em mães que não fizeram uso de misoprostol durante a gestação. Exposição à outros teratógenos e malformações oculares tiveram distribuição semelhante em ambos os grupos.Fundação Altino Ventura Department of OphthalmologyHospital de Olhos de Pernambuco Department of OphthalmologyUniversity of Illinois Department of Ophthalmology & Visual ScienceUniversidade Federal de São Paulo (UNIFESP) Department of OphthalmologyAssociação de Assistência à Criança DeficienteSanta Casa de Misericórdia de São Paulo Department of OphthalmologyUniversidade de São Paulo Faculdade de Medicina Departments of Neurology and PediatricsUniversidade de São Paulo Instituto de Biociências Department of BiologyUniversidade de São Paulo Department of OphthalmologyUniversidade Federal de Pernambuco Department of Pediatric SurgeryUniversidade de São Paulo Department of PsychiatryServices Group in Epileptic Child PsychiatryInstituto Cema Department of OphthalmologyUNIFESP, Department of OphthalmologySciEL