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.
Metrics on shape spaces are used to describe deformations that take one shape to another, and to define a distance between shapes. We study a family of metrics on the space of curves, which includes several recently proposed metrics, for which the metrics are characterised by mappings into vector spaces where geodesics can be easily computed. This family consists of Sobolev-type Riemannian metrics of order one on the space Imm(S 1 , R 2 ) of parameterized plane curves and the quotient space Imm(S 1 , R 2 )/ Diff(S 1 ) of unparameterized curves. For the space of open parameterized curves we find an explicit formula for the geodesic distance and show that the sectional curvatures vanish on the space of parameterized open curves and are nonnegative on the space of unparameterized open curves. For one particular metric we provide a numerical algorithm that computes geodesics between unparameterized, closed curves, making use of a constrained formulation that is implemented numerically using the RATTLE algorithm. We illustrate the algorithm with some numerical tests between shapes.2010 Mathematics Subject Classification. 58B20, 58D15, 65D18.
Abstract. We study Sobolev-type metrics of fractional order s ≥ 0 on the group Diffc(M ) of compactly supported diffeomorphisms of a manifold M . We show that for the important special case M = S 1 the geodesic distance on Diffc(S 1 ) vanishes if and only if s ≤ 1 2 . For other manifolds we obtain a partial characterization: the geodesic distance on Diffc(M ) vanishes for M = R × N, s < 1 2 and for M = S 1 × N, s ≤ 1 2 , with N being a compact Riemannian manifold. On the other hand the geodesic distance on Diffc(M ) is positive for dim(M ) = 1, s > 1 2 and dim(M ) ≥ 2, s ≥ 1. For M = R n we discuss the geodesic equations for these metrics. For n = 1 we obtain some well known PDEs of hydrodynamics: Burgers' equation for s = 0, the modified Constantin-Lax-Majda equation for s = 1 2 and the Camassa-Holm equation for s = 1.
This paper discusses the mathematical framework for designing methods of large deformation matching (LDM) for image registration in computational anatomy. After reviewing the geometrical framework of LDM image registration methods, a theorem is proved showing that these methods may be designed by using the actions of diffeomorphisms on the image data structure to define their associated momentum representations as (cotangent lift) momentum maps. To illustrate its use, the momentum map theorem is shown to recover the known algorithms for matching landmarks, scalar images and vector fields. After briefly discussing the use of this approach for Diffusion Tensor (DT) images, we explain how to use momentum maps in the design of registration algorithms for more general data structures. For example, we extend our methods to determine the corresponding momentum map for registration using semidirect product groups, for the purpose of matching images at two different length scales. Finally, we discuss the use of momentum maps in the design of image registration algorithms when the image data is defined on manifolds instead of vector spaces.
We study completeness properties of the Sobolev diffeomorphism groups D s (M ) endowed with strong right-invariant Riemannian metrics when M is R d or a compact manifold without boundary. We prove that for s > dim M/2 + 1, the group D s (M ) is geodesically and metrically complete and any two diffeomorphisms in the same component can be joined by a minimal geodesic. We then present the connection between the Sobolev diffeomorphism group and the large deformation matching framework in order to apply our results to diffeomorphic image matching. IntroductionThe interest in Riemannian geometry of diffeomorphism groups started with [Arn66], where it was shown that Euler's equations, describing the motion of an ideal, incompressible fluid, can be regarded as geodesic equations on the group of volume-preserving diffeomorphisms. The corresponding Riemannian metric is the right-invariant L 2 -type metric. This was used in [EM70] to show the local well-posedness of Euler's equations in three and more dimensions. Also following [Arn66], the curvature of the Riemannian metric was connected in [Mis93; Pre04; Shk98] to stability properties of the fluid flow. The Fredholmness of the Riemannian exponential map was used in [MP10] to show that large parts of the diffeomorphism group is reachable from the identity via minimising geodesics. Mathematics Subject Classification (2010): Primary 58D05; Secondary 58B20 where L is a positive, invertible, elliptic differential operator of order 2r. For (possibly) non-integer orders, the most general family of inner products is given by pseudodifferential operators L ∈ OP S 2r of order 2r within a certain symbol class. The corresponding Riemannian metric isand it can be represented by the operatorOn Completeness of Groups of Diffeomorphisms 3 thus L ϕ is not a pseudodifferential operator with a smooth symbol any more. Pseudodifferential operators with symbols in Sobolev spaces were studied for example in [ARS86a; ARS86b; BR84; Lan06], but technical difficulties still remain. Strong Sobolev metricsHistorically most papers dealt with right-invariant Sobolev metrics on diffeomorphism groups in the weak setting, that is one considered H r -metrics on D s (M) with s > r; a typical assumption is s > 2r + d/2 + 1, in order to ensure that Lu is still C 1 -regular. The disconnect between the order of the metric and the regularity of the group arose, because one was mostly interested in L 2 or H 1 -metrics, but D s (M) is a Hilbert manifold only when s > d/2 + 1. It was however noted already in [EM70] and again in [MP10], that the H s -metric is well-defined and, more importantly, smooth on D s (M), for integer s when the inner product is defined in terms of a differential operator as in (1.1). The smoothness of the metric is not obvious, since it is defined viaH s and the definition uses the inversion, which is only a continuous, but not a smooth operation on D s (M). Higher order Sobolev metrics have been studied recently on diffeomorphism groups of the circle [CK03], of the torus [KLT08] and of gen...
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