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Geometry of Diffeomorphism Groups and Shape Matching

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Geometry of Diffeomorphism Groups and Shape Matching Imperial College London Department of Mathematics Geometry of Diffeomorphism Groups and Shape Matching Martins Bruveris May 31, 2012 Supervised by Prof. Darryl D. Holm Submitted in part fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics of Imperial College London and the Diploma of Imperial College London Declaration I herewith certify that all material in this dissertation which is not my own work has been duly acknowledged. Selected results from this dissertation have been disseminated in scientific publications as detailed in Section 1.4. Martins Bruveris 3 Abstract The large deformation matching (LDM) framework is a method for registra- tion of images and other data structures, used in computational anatomy. We show how to reformulate the large deformation matching framework for registration in a geometric way. The general framework also allows to gen- eralize the large deformation matching framework to include multiple scales by using the iterated semidirect product of groups. An important ingredient in the LDM framework is the choice of a suitable Riemannian metric on the space of diffeomorphisms. Since the space in question is infinite-dimensional, not every choice of the metric is suitable. In particular the geodesic distance, which is defined as the infimum over the length of all paths connecting two points, may vanish. For the family s of Sobolev-type H -metrics on the diffeomorphism groups of R and S1 we establish that the geodesic distance vanishes for metrics of order 0 s 1 . ≤ ≤ 2 The geodesic distance also vanishes for the L2-metric on the Virasoro-Bott group, which is a central extension of the diffeomorphism group of the circle. Vanishing of geodesic distance implies that the length-functional, which assigns to each curve in the manifold its length, has no global minima, when restricted to paths with fixed endpoints. We show that for the L2-metric on the diffeomorphism group of R and the Virasoro-Bott group doesn't have any local minima either. The large deformation matching framework is not the only approach to the registration and shape comparison. For curves and surfaces it is possible to define a Riemannian metric directly on the space of curves or surfaces and use geodesics with respect to this metric to measure differences in shape. We use the family of Sobolev-type metrics on surfaces from [7]. We show how to discretize the geodesic equations and solve the boundary value problem via a shooting method on the initial velocity. The discrete equations are implemented via the finite element method. 5 To my family. 7 Acknowledgments Over the last three years I had the fortune to meet many people, to whom I owe my gratitude and without whom this work would not have been possible. First I want to thank my adviser Darryl Holm, who gave me the opportunity to study in London and guided me on my first steps in the world of research. I also want to thank Peter Michor for his valuable advice, both technical and personal and for always making me feel welcome in Vienna. It is due to my colleagues in London, David Ellis, Laurent Risser, Sehun Chun, Fran¸cois- Xavier Vialard, David Meier, Chris Cantwell and Christopher Burnett, that I always felt like part of a team and never had to look far for someone to drink coffee with. It was a pleasure to collaborate with my friends in Vienna, Martin Bauer and Philipp Harms. My thanks also goes to Colin Cotter, Fran¸coisGay-Balmaz and Tudor Ratiu for interesting discussions and valuable advice. And finally I want to thank all the members of the ShapeFRG meetings, who made working in this field like being part of a large family. 9 Contents Abstract 5 Acknowledgments 9 1 Introduction 13 1.1 Diffeomorphism Group and Applications . 13 1.2 Content of this Work . 15 1.3 Contributions of this Work . 17 1.4 Publications . 18 2 Multiscale Registration 19 2.1 Geometry of Registration . 19 2.1.1 Motivation . 19 2.1.2 Abstract Framework . 24 2.2 Registration using Diff(Ω) . 31 2.2.1 The Setting . 31 2.2.2 Matching Problems . 34 2.2.3 Landmark Matching . 36 2.2.4 Image Matching . 37 2.2.5 Vector Field Matching . 39 2.3 Multiscale Registration . 41 2.3.1 Semidirect Products . 42 2.3.2 Semidirect Products of Diffeomorphism Groups . 45 2.3.3 Sums of Kernels . 49 2.3.4 The Order Reversed . 53 2.3.5 A Continuum of Scales . 56 2.3.6 Restriction to a Finite Number of Scales . 60 3 Geodesic Distance on Diffeomorphism and Related Groups 63 3.1 Overview of the Results . 63 11 3.2 Mathematical Background . 65 3.2.1 Diffeomorphism Groups . 65 3.2.2 Sobolev Spaces on Manifolds . 66 3.2.3 Sobolev Metrics on Diffc(M).............. 68 3.2.4 Virasoro-Bott Group . 69 3.3 Diffeomorphism Groups . 70 3.4 Virasoro-Bott Group . 82 3.5 Local Minima of the Length Functional . 89 3.6 Outlook . 100 4 Surface Matching 103 4.1 Background . 103 4.2 First Order Sobolev-type Metric . 104 4.3 Variation of the Metric . 105 4.4 The Helmholtz Operator and Duality . 110 4.5 Discretization . 113 4.5.1 The Geodesic Equation . 113 4.5.2 Computing the Gradient . 116 4.6 Numerical Experiments . 120 4.7 Outlook . 124 Bibliography 127 12 1 Introduction 1.1 Diffeomorphism Group and Applications The diffeomorphism group plays a central role in the field of computational anatomy. For the first time its use in biology was noted by D'Arcy Thomp- son in his book \On Growth and Form" [79]. Following the paradigm of pattern theory, introduced by Grenander [31, 33], we assume that the anatomical variety of an object of interest across the population can be explained by choosing a template, upon which a set of deformations acts, thus generating the entire population, potentially up to small-scale noise. In this setting the collection of anatomical objects forms a homogeneous space under the group of deformations [32]. The objects, which are stud- ied in computational anatomy are those that can be obtained by medical imaging procedures. They include volumetric gray-level images from MRI and CT scans, vector and tensor fields from diffusion tensor MRI, surfaces representing the outline of organs, curves in space representing white mat- ter fiber tracts and manually or automatically assigned feature points. The group of deformations is usually the diffeomorphism group of the ambient three-dimensional space. The aim of the pattern theory approach is to encode the differences be- tween two objects in the deformation that matches one object to the other. This allows one to ignore all the complexity inherent in anatomical struc- tures and only concentrate on the differences between them. Therefore, when given a template object I V and a target object I V from 0 2 targ 2 the population, denoted by V and assumed to be a vector space, as a first step, it is necessary to determine the deformation g G from the group G 2 of all deformations, mapping I0 to g:I0 = Itarg. This can be done using the large deformation diffeomorphic metric matching (LDM) framework of [40, 10, 61, 91, 93]. The objective of LDM is not just to determine a deformation g G such 1 2 13 that the group action g I of g G on the template I V approximates 1 0 1 2 0 2 the target I V to within a certain tolerance. Rather, the objective of targ 2 LDM is to find the optimal path gt G continuously parametrized by time 2 t R that smoothly deforms I0 through It = gtI0 to g1I0. The optimal path 2 gt G is defined as the path that costs the least in time-integrated kinetic 2 energy for a given tolerance. Hence, the deformable template method may be formulated as an optimization problem based on a trade-off between the following two properties: (1) the tolerance for inexact matching between the final deformed template g1I0 and the target template Itarg; and (2) the cost of time-integrated kinetic energy of the rate of deformation along the path gt. The former is defined by assigning a norm : V R to measure k · k ! the mismatch gtI I between the two images. The latter is obtained k 0 − targk by choosing a Riemannian metric : TG R that defines the kinetic j · j ! energy on the tangent space TG of the group G. In applications of LDM to the analysis of features in bio-medical images, the optimal path gt is naturally chosen from among the diffeomorphic trans- formations G = Diff(Ω) of an open, bounded domain Ω. The domain Ω will be taken to be the ambient space in which the anatomy is located. It can be shown that [64, 89] the optimal path of deformation satisfies an evolution equation, which is the geodesic equation for the metric, which was chosen to measure the kinetic energy. Therefore the whole path is encoded in its initial value at time t = 0. These geodesic equations on the diffeomorphism group naturally lead us the field of hydrodynamics [38]. The importance of the diffeomorphism group was also realized in a differ- ent context. In the seminal work of Arnold [3] it was realized that Euler's equations, which govern the motion of incompressible fluids, can be inter- preted as geodesic equations on the group of volume-preserving diffeomor- phisms with respect to the right-invariant L2 metric. This interpretation was used by Ebin and Marsden [23] to rewrite Euler's equation, a system of PDEs, as a second order ODE on a suitable Hilbert space and thus to prove the well-posedness of Euler's equations in three dimensions.
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