Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

Martin Bauer; Martins Bruveris; Peter W. Michor

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Overview of the Geometries of Shape Spaces and Diffeomorphism Groups

Martin Bauer, Martins Bruveris, Peter W. Michor

arXiv:1305.1150v3May 6, 2013math.DGmath.AP

TL;DR

Shape spaces are nonlinear, making statistical analysis difficult, while their infinite-dimensional geometry complicates geodesic equations and numerical computation. This article surveys Riemannian structures on shape spaces and examines induced distances, geodesics, completeness, curvature, and related limitations.

Problem
Nonlinear shape spaces complicate statistics, and their infinite-dimensional geodesic equations may be partial or pseudodifferential equations rather than ordinary differential equations.
Method
The article surveys Riemannian structures on shape spaces and uses Riemannian submersions to study metrics on immersions and unparametrized shapes.
Results
The survey establishes that the L2 metric can induce identically vanishing geodesic distance, while a specified outer metric has distance bounded below by the Fréchet distance.
Takeaways & Limitations
Riemannian geometry provides geodesic methods for locally linearizing shape spaces and supports analysis through horizontal geodesics under Riemannian submersions.
Takeaways & Limitations
Numerical methods for computing exponential maps, geodesics, and distances remain an active research area, and the article does not comprehensively survey them.

Abstract

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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.

1 Introduction

This article surveys Riemannian structures on surface shape spaces and the relationships among parametrized shapes, unparametrized shapes, diffeomorphisms, and metrics. It emphasizes geodesic distance, geodesic equations, completeness, curvature, and unresolved analytical and numerical questions.

1 Introduction: Shape variability is modeled in nonlinear spaces, where Riemannian structures enable local linearization and geodesic-based statistics.The motivation includes applications in computational anatomy, mathematics, and computer vision.
1 Introduction: Shapes are represented as submanifolds diffeomorphic to a fixed compact manifold M, with parametrized and unparametrized versions distinguished by reparametrization.Immersions and embeddings model parametrized shapes; quotienting by Diff(M) produces shape spaces.
1 Introduction: Diff(M)-invariant metrics on immersion spaces induce quotient metrics through Riemannian submersions, while ambient diffeomorphism metrics provide an outer-metric construction.The outer construction measures deformation cost through right-invariant metrics on compactly supported ambient diffeomorphisms.
1 Introduction: The survey also connects diffeomorphism-group geometry to geodesic PDEs, including equations whose special cases include Camassa–Holm and Hunter–Saxton.For a metric defined by an operator L, the equation is written in terms of m = Lu.
1 Introduction: Key analytical questions concern geodesic distance, geodesic solvability, completeness, and curvature on infinite-dimensional, often weak Riemannian manifolds.Short-time solvability can be non-trivial, metric completeness is not generally expected, and several topics remain open.

2 Preliminaries

The preliminaries establish the geometric notation and explain how weak Riemannian metrics and Riemannian submersions transfer geometry from a top space to a quotient.

2 Preliminaries: The paper uses the Euclidean metric on R^d, induced pull-back metrics, volume forms, curvature tensors, and the Bochner–Laplacian as core geometric notation.For plane curves, derivatives and volume integration reduce to arclength expressions, and Gauß and mean curvature coincide as κ.
2 Preliminaries: Horizontal geodesics project to quotient geodesics, so solving the quotient geodesic equation can be replaced by solving the horizontal equation upstairs.Under the stated lifting condition, curves in the quotient correspond to horizontal curves up to the initial point.
2 Preliminaries: A weak Riemannian metric is injective as a tangent-to-cotangent map but need not be surjective, creating analytical complications and potentially obstructing Levi-Civita connections.When the connection exists, it is unique.
2 Preliminaries: For an invariant metric on a quotient construction, tangent vectors split into vertical and horizontal components, with the horizontal space defined by orthogonality to the vertical subbundle.The decomposition is written as X = X_ver + X_hor.
2 Preliminaries: A Riemannian submersion transfers the quotient metric from the top space when the induced fiberwise seminorm is a norm.The resulting quotient metric is weak, and the projection becomes a Riemannian submersion.

3 The spaces of interest

The paper defines parametrized and unparametrized shape spaces through immersions, embeddings, and diffeomorphism actions, while describing their manifold, quotient, and Lie-group structures.

3 The spaces of interest: Parametrized surfaces are modeled by immersions or embeddings q: M → R^d, and reparametrization acts by composition with Diff(M).Free immersions are those for which q ◦ ϕ = q implies ϕ = Id_M.
3 The spaces of interest: Imm(M, R^d), Imm_f(M, R^d), and Emb(M, R^d) are Fréchet manifolds.These spaces arise as open subsets under the relevant injectivity and embedding conditions.
3 The spaces of interest: The full immersed shape space is an orbifold with isolated singularities, so the paper uses free immersions to obtain a regular quotient manifold.The projection from free immersions has Diff(M) as structure group and is a smooth principal fibration.
3 The spaces of interest: Compactly supported diffeomorphisms form a regular Lie group with Lie algebra given by compactly supported vector fields and the negative usual Lie bracket.For noncompact Euclidean domains, the compactly supported subgroup is used because the full orientation-preserving group is not a usual smooth manifold in the stated topology.

4 The L2-metric on plane curves

The L2 metric on plane curves is simple and geometrically interpretable, but its induced geodesic distance vanishes; stronger metric classes are introduced to avoid this behavior.

4 The L2-metric on plane curves: The L2 metric on immersed plane curves induces a quotient metric whose horizontal tangent vectors are pointwise normal to the curve.Horizontal vectors have the form h(θ) = a(θ)n_c(θ).
4 The L2-metric on plane curves: The geodesic equation for horizontal curves reduces to a nonlinear hyperbolic PDE of second order rather than an ODE.Curvature depends implicitly on the scalar normal velocity, so eliminating curvature yields the PDE.
4.1 Properties of the L2-metric: The sectional curvature of the L2 metric on unparametrized plane curves is non-negative and unbounded.The curvature has an explicit Wronskian expression for orthonormal horizontal tangent vectors.
4.1 Properties of the L2-metric: The geodesic distance induced by G0 vanishes identically on both Imm(S1, R2) and Bi,f(S1, R2).Thus distinct curves can be connected by paths with arbitrarily small G0-length.
4.1 Properties of the L2-metric: Wild zig-zag paths explain the quotient-space collapse heuristically by making normal motion inversely proportional to curve length.Because the normal component is squared, the path length can be made arbitrarily small.
4 The L2-metric on plane curves: The paper therefore presents almost local, Sobolev-type, and ambient-diffeomorphism-induced metrics as stronger alternatives.The motivation is to prevent vanishing geodesic distance; an H1 metric also yields a locally well-posed gradient flow in the cited active-contour example.

5 Almost local metrics on shape space

Almost local metrics define shape-space geometries from local curvature and, mildly, global volume information. Under suitable conditions they yield point-separating distances on quotient shape spaces, while retaining vanishing distances along reparametrization fibers and unresolved well-posedness in key cases.

Metric construction: Diff(M)-equivariance of Ψ makes GΨ invariant and induces a Riemannian metric on the quotient shape space.The horizontal bundle consists of tangent vectors pointwise orthogonal to the immersion.
Metric construction: Almost local metrics combine local geometric quantities with a mild non-local dependence on total volume.If the metric factor depends only on volume, the metric is conformally equivalent to the L2-metric.
Geodesic distance: Under suitable conditions on Ψ, GΨ induces a point-separating geodesic distance on Bi,f(M, Rd).For distinct shapes C0 and C1, the resulting distance is positive.
Geodesic distance: The distance proof lower-bounds path length by swept area and uses Lipschitz continuity of an associated function.For planar curves with Ψ(c)=ℓc, the geodesic distance equals the infimum of swept area.
Geodesic distance: Almost local metrics may separate quotient shapes while assigning zero distance along reparametrization fibers and on the corresponding weighted L2-type Diff(M) orbit metric.Thus point separation on the quotient does not imply point separation on the immersion space.
Geodesic equations: Well-posedness of the almost-local geodesic equations remains unknown, including for the immersion and shape spaces in the planar case.This is also stated for the displayed curvature-weighted and conformal families.
Completeness and examples: For sphere shape spaces, the almost-local metrics considered can be geodesically incomplete, while the GA completion for planar curves lies among Lipschitz curves and contains all 1-BV rectifiable curves.Numerical geodesics can exhibit shrink-and-grow behavior when shapes are sufficiently far apart.

6 Sobolev type metrics on shape space

Sobolev-type metrics on immersions can descend to unparametrized shape spaces and support results on distance, geodesics, completeness, and curvature. The section develops operator-based constructions, plane-curve transforms, and corresponding analytical and geometric properties.

Metric construction: Sobolev-type inner metrics are defined through pseudo-differential operator fields L_q, typically symmetric, positive, and elliptic of order at least one.Reparametrization invariance makes the metric descend to the quotient shape space.
Metric construction: The horizontal bundle for Sobolev-type metrics generally cannot be described pointwise and instead requires inversion of the operator L_q.This contrasts with almost local metrics, whose horizontal vectors are pointwise normal to the surface.
Sobolev metrics on plane curves: For plane curves, the R-transform identifies the a = 1, b = 1 elastic metric with a flat L2 metric on an open subset, while the transformed closed-curve image has codimension two.The SRVT provides another computationally efficient representation of this metric and generalizes to arbitrary coefficients a and b.
Geodesic distance: High-enough-order Sobolev metrics induce point-separating geodesic distances on shape space, and the H1 metric admits path-length bounds using swept area or volume.The operator-field condition is expressed through a coercive comparison that ensures point separation.
Well-posedness of the geodesic equation: The geodesic equation for suitable invariant operator fields is locally well-posed, with smooth geodesic sprays, unique short-time solutions, and smooth dependence on initial data.The result extends from Sobolev completions to smooth immersions and implies well-posedness on the quotient shape space via horizontal geodesics.
Completeness and curvature: For higher-dimensional spheres, the immersion and shape spaces are not geodesically complete under the G_L metric in the stated order regime, while the H1 and H2 metric completions admit rectifiable-curve descriptions.For other model manifolds, completeness conditions remain unknown; the section also records non-negative curvature results for several plane-curve quotients.

7 Diffeomorphism groups

The section surveys diffeomorphism groups as deformation and reparametrization groups, then reviews geodesic equations, well-posedness, distance, completeness, and curvature for several metrics.

Diffeomorphism groups arise both as deformation groups acting on ambient spaces and as reparametrization groups for immersed shapes.
Metrics: Right-invariant metrics on diffeomorphism groups are defined from inner products on vector fields, including Sobolev-type and a-b-c metrics.
Well-posedness: For higher-order Sobolev metrics, geodesic sprays are smooth and geodesic equations have unique local solutions on suitable Sobolev completions.
Well-posedness: C1 Green’s functions yield global solutions in Lagrangian coordinates for measure-valued initial momenta, while the L2 spray on Diff(S1) is not smooth.
Geodesic distance: For Sobolev metrics, geodesic distance vanishes in established low-order cases, while non-degeneracy is conjectured above the threshold s > 1 2 in broader settings.
Completeness and curvature: High-enough-order Sobolev metrics are geodesically complete, and sectional curvature can be positive, negative, or constant positive depending on the metric and quotient.

8 Metrics on shape space induced by Diff(Rd)

The section constructs shape-space metrics from ambient diffeomorphism actions, using Riemannian submersions and reproducing-kernel Hilbert spaces, and summarizes distance results and limitations.

Group actions and quotients: Ambient diffeomorphisms act on embeddings and embedded submanifolds, whose orbits are open but need not be transitive.
Group actions and quotients: Embedded shape space can be represented both as Diff(Rd)/Diff(Rd)Q on an orbit and as Emb(M, Rd)/Diff(M) under reparametrization.
Metric construction: A right-invariant ambient metric induces a metric on embeddings through a Riemannian submersion, which then descends to unparametrized shapes.
Limitations: The image action is far from transitive, so a Riemannian metric on the full image space cannot be rigorously induced by ambient diffeomorphisms.
Metric construction: Reproducing-kernel Hilbert spaces provide a kernel-based representation of induced metrics and connect operator-based constructions with LDDMM formulations.
Geodesic distance: For plane curves with Sobolev ambient spaces, the induced geodesic distance is bounded below by the Fréchet distance, while stronger norms yield point-separating distances on embeddings.

9 The space of landmarks

Landmark space is a finite-dimensional space of distinct labeled points, with diffeomorphism-induced metrics that support explicit variational and geodesic formulations. For sufficiently regular kernels, geodesics exist globally, landmarks do not collide, and the induced distance is complete.

Landmark space consists of n distinct, labeled points in R^d and is an open subset of R^(nd).
Diffeomorphism-group metrics induce a Riemannian metric on landmark space through a reproducing kernel K.The metric is represented by a kernel matrix whose entries are K(q_i,q_j).
The landmark energy minimization problem is equivalent to the diffeomorphism-group problem, with trajectories satisfying the ODE q̇_i(t)=v(t,q_i(t)).The two minimization problems have equal energies and induced distances.
9.3 Completeness: For sufficiently high-order metrics, landmark space is geodesically complete and its geodesics prevent landmark collisions.Hopf–Rinow then yields metric completeness for the finite-dimensional C2 setting.
9.4 Curvature: The sectional curvature on L^2(R) depends only on the distance between two landmarks and can be expressed using the reproducing kernel.

10 Universal Teichm¨uller space as shape space

Universal Teichmüller space represents simple closed plane-curve shapes modulo translations and scalings through fingerprints. Its Weil–Petersson geometry is nonpositively curved and geodesically complete, while Teichons provide efficient finite-dimensional geodesic approximations with a concavity-related limitation.

The Weil–Petersson metric on universal Teichmüller space has nonpositive curvature and is geodesically complete.
Diff+(S^1)/PSL(2,R) parametrizes simple closed smooth plane curves modulo translations and scalings through the fingerprint of a shape.Shapes can be reconstructed from fingerprints by conformal welding.
The metric is a Sobolev metric of order 3/2 on the quotient by PSL(2,R), with sl(2,R) as the kernel of its seminorm.
Teichons are finite combinations of delta distributions whose parameters evolve through a finite-dimensional Hamiltonian system.They provide soliton-like solutions related to geodesics of shapes as landmarks relate to diffeomorphism-group geodesics.
Teichons efficiently approximate smooth shape geodesics, but crowd exponentially near concave parts of a shape.

11 The space of Riemannian metrics

The space of Riemannian metrics is studied primarily with the L^2 metric, whose geometry admits explicit distance, geodesic, completeness, and curvature results. Unlike several other shape spaces, its distance separates points, while its metric and geodesic completions have distinct structured descriptions.

11.2 Geodesic distance: The L^2 metric on Met(M) induces a point-separating geodesic distance, unlike the corresponding metrics on immersions, submanifolds, and diffeomorphism groups.The distance coincides with a distance obtained by integrating pointwise finite-dimensional metric distances.
11.2 Geodesic distance: Metrics stronger than L^2 retain point-separating distance under the stated uniform lower-bound condition, including many almost local and Sobolev-type metrics.
11.2 Geodesic distance: The set of metrics with total volume at most μ has finite diameter under the L^2 metric.The result follows from an upper bound for the geodesic distance depending on the dimension of M.
11.3 Geodesics: The L^2 geodesic equation decouples time and space, reducing to an ODE with an explicit solution formula.The geodesic is defined globally or only up to a finite time according to conditions involving H and Tr(H).
11.5 Completeness: The L^2 metric is incomplete both metrically and geodesically, while its metric completion is identified with measurable metric sections modulo an equivalence relation.
11.5 Completeness: The metric completion is a CAT(0) space, so it has minimizing geodesics between points and nonpositive Alexandrov curvature.
11.6 Curvature: The sectional curvature of the L^2 metric is non-positive.