Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach

This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of Riemannian metrics. We discuss the Riemannian metrics that can be defined thereon, and what is known about the properties of these metrics. We put particular emphasis on the induced geodesic distance, the geodesic equation and its well-posedness, geodesic and metric completeness and properties of the curvature.

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“…A careful analysis of the induced metric G H in the spirit of [91] should provide such a result for a fairly general Sobolev-type metric.…”

Section: Theorem 83mentioning

confidence: 99%

“…See [91] for a derivation of the geodesic equation for plane curves and [88] for the related geodesic equation on B e (M, R d ).…”

Section: Geodesic Equationmentioning

confidence: 99%

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

2014

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“…If we assume that the adjoint of the adjoint exists, which is not guaranteed in infinite dimensions [20], then we can then rewrite (30) as:…”

Section: The Adjoint Representation Of the Tangent-space Algebramentioning

confidence: 99%

Constructing an atlas for the diffeomorphism group of a compact manifold with boundary, with application to the analysis of image registrations

Twining

Marsland

2008

Journal of Computational and Applied Mathematics

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This paper considers the problem of defining a parameterization (chart) on the group of diffeomorphisms with compact support, motivated primarily by a problem in image registration, where diffeomorphic warps are used to align images. Constructing a chart on the diffeomorphism group will enable the quantitative analysis of these warps to discover the normal and abnormal variation of structures in a population.We construct a chart for particular choices of boundary conditions on the space on which the diffeomorphism acts, and for a particular class of metrics on the diffeomorphism group, which define a class of diffeomorphic interpolating splines. The geodesic equation is computed for this class of metrics, and we show how it can be solved in the spline representation. Furthermore, we demonstrate that the spline representation generates submanifolds of the diffeomorphism group, and we study this mapping.Explicit computational examples are included, showing how this chart can be constructed in practice, and that the use of the geodesic distance allows better classification of variation than those obtained using just a Euclidean metric on the space of warps.

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“…This result is proved (in a different form) in [16]. We make only a formal proof by identifying the equations satisfied by α and β.…”

Section: D(ϕφ) := Minmentioning

confidence: 99%

“…This is proved in [1], [16], and derives from an application of the general formula for the Levi-Civita connection on a Riemannian manifold, namely, (22) and on the identity (with notation from the proposition) …”

Section: Application: Parallel Translationmentioning

confidence: 99%

Jacobi fields in groups of diffeomorphisms and applications

Younes

2007

Quart. Appl. Math.

111

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Abstract. This paper presents a series of applications of the Jacobi evolution equations along geodesics in groups of diffeomorphisms. We describe, in particular, how they can be used to perform implementable gradient descent algorithms for image matching, in several situations, and illustrate this with 2D and 3D experiments. We also discuss parallel translation in the group, and its projection on shape manifolds, and focus in particular on an implementation of these equations using iterated Jacobi fields.

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Some geometric evolution equations arising as geodesic equations on groups of diffeomorphisms including the Hamiltonian approach

Cited by 39 publications

References 39 publications

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

Constructing an atlas for the diffeomorphism group of a compact manifold with boundary, with application to the analysis of image registrations

Jacobi fields in groups of diffeomorphisms and applications

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