Overview of the Geometries of Shape Spaces and Diffeomorphism Groups
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.
Keywords:Shape Space Diffeomorphism Group Manifolds of mappings Landmark space Surface matching Riemannian geometry
- 1 Introduction
- 2 Preliminaries
- 3 The spaces of interest
- 4 The -metric on plane curves
- 5 Almost local metrics on shape space
- 6 Sobolev type metrics on shape space
- 7 Diffeomorphism groups
- 8 Metrics on shape space induced by
- 9 The space of landmarks
- 10 Universal Teichmüller space as shape space
- 11 The space of Riemannian metrics
The variability of a certain class of shapes is of interest in various fields of applied mathematics and it is of particular importance in the field of computational anatomy. In mathematics and computer vision, shapes have been represented in many different ways: point clouds, surfaces or images are only some examples. These shape spaces are inherently non-linear. As an example, consider the shape space of all surfaces of a certain dimension and genus. The nonlinearity makes it difficult to do statistics. One way to overcome this difficulty is to introduce a Riemannian structure on the space of shapes. This enables us to locally linearize the space and develop statistics based on geodesic methods. Another advantage of the Riemannian setting for shape analysis is its intuitive notion of similarity. Namely, two shapes that differ only by a small deformation are regarded as similar to each other.
In this article we will concentrate on shape spaces of surfaces and we will give an overview of the different Riemannian structures, that have been considered on these spaces.
1.1 Spaces of interest
We fix a compact manifold without boundary of dimension . In this paper a shape is a submanifold of that is diffeomorphic to and we denote by and the spaces of all immersed and embedded submanifolds.
One way to represent is to start with the space of immersions upon which the diffeomorphism group acts from the right via
The space is the space of parametrized type submanifolds of and the action of represents reparametrizations. The immersions and have the same image in and thus one can identify with the quotient
The space is not a manifold, but an orbifold with isolated singular points; see Sect. 3.3. To remove these we will work with the slightly smaller space of free immersions (see Sect. 3.1) and denote the quotient by
Similarly one obtains the manifold as the quotient of the space of embedded submanifolds with the group .
The spaces and are sometimes called pre-shape spaces, and is the reparametrization group. Their quotients and are called shape spaces.
We would like to note that our usage of the terms shape and pre-shape space differs from that employed in (68); (119); (42). In the terminology of (68) a pre-shape space is the space of labelled landmarks modulo translations and scalings and the shape space additionally quotients out rotations as well. For the purposes of this paper, the pre-shape space contains parametrized curves or surfaces and we pass to the shape space by quotienting out the parametrizations.
A Riemannian metric on that is invariant under the action of induces a metric on , such that the projection
is a Riemannian submersion. This will be our method of choice to study almost local and Sobolev-type metrics on and in Sect. 5 and 6. These are classes of metrics, that are defined via quantities measured directly on the submanifold. We might call them inner metrics to distinguish them from outer metrics, which we will describe next. This is however more a conceptual distinction rather than a rigorously mathematical one.
Most Riemannian metrics, that we consider in this article, will be weak, i.e., considered as a mapping from the tangent bundle to the cotangent bundle the metric is injective, but not surjective. Weak Riemannian metrics are a purely infinite-dimensional phenomenon and they are the source of most of the analytical complications, that we will encounter.
A way to define Riemannian metrics on the space of parametrized submanifolds is via the left action of on ,
Here denotes the Lie group of all compactly supported diffeomorphisms of with Lie algebra the space of all compactly supported vector fields, see Sect. 7. Given a right-invariant metric on , the left action induces a metric on , such that for each embedding the map
is a Riemannian submersion onto the image. This construction formalizes the idea of measuring the cost of deforming a shape as the minimal cost of deforming the ambient space, i.e.,
Here is an infinitesimal deformation of and the length squared , which measures its cost, is given as the infimum of , that is the cost of deforming the ambient space. The infimum is taken over all infinitesimal deformations of , that equal when restricted to . This motivates the name outer metrics, since they are defined in terms of deformations of the ambient space.
The natural space to define these metrics is the space of embeddings instead of immersions, because not all orbits of the action on are open. Defining a Riemannian metric on is now a two step process
One starts with a right-invariant Riemannian metric on , which descends via (1) to a metric on . This metric is invariant under the reparametrization group and thus descends to a metric on . These metrics are studied in Sect. 8.
Riemannian metrics on the diffeomorphism groups and are of interest, not only because these groups act as the deformation group of the ambient space and the reparametrization group respectively. They are related to the configuration spaces for hydrodynamics and various PDEs arising in physics can be interpreted as geodesic equations on the diffeomorphism group. While a geodesic on is a curve of diffeomorphisms, its right-logarithmic derivative is a curve of vector fields. If the metric on is given as with a differential operator , then the geodesic equation can be written in terms of as
PDEs that are special cases of this equation include the Camassa-Holm equation, the Hunter-Saxton equation and others. See Sect. 7 for details.
So far we encoded shape through the way it lies in the ambient space; i.e., either as a map or as its image . One can also look at how the map deforms the model space . Denote by the Euclidean metric on and consider the pull-back map
where is the space of all Riemannian metrics on and denotes the pull-back of to a metric on . Depending on the dimension of one can expect to capture more or less information about shape with this map. Elements of with are symmetric, positive definite tensor fields of type and thus have components. Immersions on the other hand are maps from into and thus have components. For , the case of surfaces in , the number of components coincide, while for we have . Thus we would expect the pull-back map to capture most aspects of shape. The pull-back is equivariant with respect to and thus we have the commutative diagram
The space in the lower right corner is not far away from , where denotes the connected component of the identity. This space, known as super space, is used in general relativity. Little is known about the properties of the pull-back map (2), but as a first step it is of interest to consider Riemannian metrics on the space . This is done in Sect. 11, with special emphasis on the - or Ebin-metric.
1.2 Questions discussed
After having explained the spaces, that will play the main roles in the paper and the relationships between them, what are the questions that we will be asking? The questions are motivated by applications to comparing shapes.
After equipping the space with a Riemannian metric, the simplest way to compare shapes is by looking at the matrix of pairwise distances, measured with the induced geodesic distance function. Thus an important question is, whether the geodesic distance function is point-separating, that is whether for two distinct shapes we have . In finite dimensions the answer to this question is always “yes”. Even more, a standard result of Riemannian geometry states that the topology induced by the geodesic distance coincides with the manifold topology. In infinite dimensions, when the manifold is equipped with a weak Riemannian metric, this is in general not true any more. The topology induced by the geodesic distance will also be weaker than the manifold topology. We will therefore survey what is known about the geodesic distance and the topology it induces.
The path realizing the distance between two shapes is, if it exists, a geodesic. So it is natural to look at the geodesic equation on the manifold. In finite dimensions the geodesic equation is an ODE, the initial value problem for geodesics can be solved, at least for short times, and the solution depends smoothly on the initial data. The manifolds of interest in this paper are naturally modeled mostly as Fréchet manifolds and in coordinates the geodesic equation is usually a partial differential equation or even involves pseudo differential operators. Only the regulat Lie group of diffeomorphisms with compact support on a noncompact manifold are modeled on -spaces. Thus even the short-time solvability of the initial-value problem is a non-trivial question. For some of the metrics, in particular for the class of almost local metrics, it is still open. For the diffeomorphism group the geodesic equations for various metrics are of interest in their own right. To reflect this we will discuss in Sect. 7 first the geodesic equations before proceeding with the properties of the geodesic distance.
It is desirable for applications that the Riemannian metric possesses some completeness properties. It can be either in form of geodesic completeness, i.e., that geodesics are extendable for all time, or metric completeness with respect to the induced geodesic distance. Since we are considering only weak Riemannian metrics on spaces of smooth shapes, we cannot expect the space to be metrically complete, but in some examples it is possible to at least describe the metric completion of shape space.
In order to perform statistics on shape space one can choose a reference shape, for example by computing the mean of a given set of shapes, and linearize the space around this shape via the Riemannian exponential map and normal coordinates. The curvature tensor contains information about the accuracy of this approximation. In general computing the curvature leads to long formulas that are hard to interpret, but in some cases curvature admits a simple expression. We collect the examples, where more is known about the curvature, usually the sectional curvature, than just the formula for it.
To summarize, we will deal with the following four properties of Riemannian metrics on shape spaces:
Geodesic equation and existence of geodesics
Geodesic and metric completeness
Properties of the curvature
Riemannian geometry on shape spaces is currently an active area of research. Therefore this paper is less an encyclopedic treatment of the subject but rather resembles an interim report highlighting what is known and more importantly, what is not.
1.3 Topics not discussed
There are many topics that lie outside the scope of this paper, among which are the following.
Changes in topology. In certain applications it may be of interest to consider deformations of a shape that allow for the development of holes or allow the shape to split into several components. In this paper we fix the model manifold and only consider submanifolds of diffeomorphic to . Thus by definition all deformations are topology-preserving. See (41); (131); (21) for topologically robust approaches to shape matching.
Non-geodesic distances. Many interesting distances can be defined on shape spaces, that are not induced by an underlying Riemannian metric; see for example (81); (103); (83). As we are looking at shape spaces through the lens of Riemannian geometry, these metrics will necessarily be left out of focus.
Subgroups of the diffeomorphism groups. The Riemannian geometry of the diffeomorphism group and its subgroups, especially the group of volume-preserving diffeomorphisms, has been studied extensively; see for example (120). It plays an important role in hydrodynamics, being the configuration space for incompressible fluid flow (44). While the full diffeomorphism group itself is indispensable for shape analysis, its subgroups have not been used much in this context.
Utmost generality. We did not strive to state the results in the most general setting. It is possible to consider shapes of higher codimension inside or curved ambient spaces; see (11). This would include examples like space curves or curves that lie on a sphere. It would also make the presentation more difficult to read.
Numerical methods. Since shape space is infinite-dimensional, computing the exponential map, the geodesic between two shapes or the geodesic distance are numerically non-trivial tasks. While we present some examples, we do not attempt to provide a comprehensive survey of the numerical methods that have been employed in the context of shape spaces. Finding stable, robust and fast numerical methods and proving their convergence is an area of active research for most of the metrics and spaces discussed in this paper. See (111); (113); (38); (39); (4); (57) for various approaches to discretizing shape space.
In this section we will introduce the basic notation that we will use throughout this article. On we consider the Euclidean metric, which we will denote by or . Unless stated otherwise we will assume that the parameter space is a compact, oriented manifold without boundary of dimension . Riemannian metrics on are usually denoted by . Tensor fields like and its variations are identified with their associated mappings . For a metric this yields the musical isomorphisms
Immersions from to – i.e., smooth mappings with everywhere injective derivatives – are denoted by and the corresponding unit normal field of an (orientable) immersion is denoted by . For every immersion we consider the induced pull-back metric on given by
for vector fields . Here denotes the tangent mapping of the map . We will denote the induced volume form of the metric as . In positively oriented coordinates it is given by
Using the volume form we can calculate the total volume of the immersion .
The Levi-Civita covariant derivative determined by a metric will be denoted by and we will consider the induced Bochner–Laplacian , which is defined for all vector fields via
Note that in the usual Laplacian is the negative of the Bochner–Laplacian of the Euclidean metric, i.e., .
Furthermore, we will need the second fundamental form Using it we can define the Gauß curvature and the mean curvature .
In the special case of plane curves ( and ) we use the letter for the immersed curve. The curve parameter will be the positively oriented parameter on , and differentiation will be denoted by the subscript , i.e., . We will use a similar notation for the time derivative of a time dependent family of curves, i.e., .
We denote the corresponding unit length tangent vector by
and is the unit length tangent vector. The covariant derivative of the pull-back metric reduces to arclength derivative, and the induced volume form to arclength integration:
Using this notation the length of a curve can be written as
In this case Gauß and mean curvature are the same and are denoted by .
2.2 Riemannian submersions
In this article we will repeatedly induce a Riemannian metric on a quotient space using a given metric on the top space. The concept of a Riemannian submersion will allow us to achieve this goal in an elegant manner. We will now explain in general terms what a Riemannian submersion is and how geodesics in the quotient space correspond to horizontal geodesics in the top space.
Let be a possibly infinite dimensional weak Riemannian manifold; weak means that is injective, but need not be surjective. A consequence is that the Levi-Civita connection (equivalently, the geodesic equation) need not exist; however, if the Levi-Civita connection does exist, it is unique. Let be a smooth possibly infinite dimensional regular Lie group; see (77) or ((76), Section 38) for the notion used here, or (106) for a more general notion of Lie group. Let be a smooth group action on and assume that is a manifold. Denote by the projection, which is then a submersion of smooth manifolds by which we means that it admits local smooth sections everywhere; in particular, is surjective. Then
is called the vertical subbundle. Assume that is in addition invariant under the action of . Then the expression
defines a semi-norm on . If it is a norm, it can be shown (by polarization pushed through the completion) that this norm comes from a weak Riemannian metric on ; then the projection is a Riemannian submersion.
Sometimes the the -orthogonal space is a fiber-linear complement in . In general, the orthogonal space is a complement (for the -closure of ) only if taken in the fiberwise -completion of . This leads to the notion of a robust Riemannian manifold: a Riemannian manifold is called robust, if is a smooth vector-bundle over and the Levi-Civita connection of exists and is smooth. See (88) for details. We will encounter examples, where the use of is necessary in Sect. 8.
The horizontal subbundle is the -orthogonal complement of in or in , respectively. Any vector can be decomposed uniquely in vertical and horizontal components as
Note that if we took the complement in , i.e., , then in general . The mapping
is a linear isometry of (pre-)Hilbert spaces for all . Here is the fiber-wise -completion of . We are not claiming that forms a smooth vector-bundle over although this will be true in the examples considered in Sect. 8.
Consider a Riemannian submersion between robust weak Riemannian manifolds, and let be a geodesic in .
If is horizontal at one , then it is horizontal at all .
If is horizontal then is a geodesic in .
If every curve in can be lifted to a horizontal curve in , then, up to the choice of an initial point, there is a one-to-one correspondence between curves in and horizontal curves in . This implies that instead of solving the geodesic equation in one can equivalently solve the equation for horizontal geodesics in .
3 The spaces of interest
3.1 Immersions and embeddings
Parametrized surfaces will be modeled as immersions or embeddings of the configuration manifold into . We call immersions and embeddings parametrized since a change in their parametrization (i.e., applying a diffeomorphism on the domain of the function) results in a different object. We will deal with the following sets of functions:
Here is the set of smooth functions from to , is the set of all immersions of into , i.e., all functions such that is injective for all . The set consists of all free immersions ; i.e., the diffeomorphism group of acts freely on , i.e., implies for all .
By ((26), Lem. 3.1), the isotropy group of any immersion is always a finite group which acts properly discontinuously on so that is a covering map. is the set of all embeddings of into , i.e., all immersions that are homeomorphisms onto their image.
The spaces and are Fréchet manifolds.
Since is compact by assumption it follows that is a Fréchet manifold by ((76), Sect. 42.3); see also (58), (90). All inclusions in (3) are inclusions of open subsets: first is open in since the condition that the differential is injective at every point is an open condition on the one-jet of ((90), Sect. 5.1). is open in by ((26), Thm. 1.5). is open in by ((76), Thm. 44.1). Thus all the spaces are Fréchet manifolds as well.
3.2 Shape space
Unparametrized surfaces are equivalence classes of parametrized surfaces under the action of the reparametrization group.
Theorem 3.2 (Thm. 1.5, (26))
The quotient space
is a smooth Hausdorff manifold and the projection
is a smooth principal fibration with as structure group.
For we can define a chart around by
with sufficiently small, where
is defined by and is the unit-length normal vector to .
The statement of Thm. 3.2 does not change, if we replace by and by .
3.3 Some words on orbifolds
is the prototype of a Riemannian submersion onto an infinite dimensional Riemannian orbifold. In the article ((122), Prop. 2.1) it is stated that the finite dimensional Riemannian orbifolds are exactly of the form for a Riemannian manifold and a compact group of isometries with finite isotropy groups. Curvature on Riemannian orbifolds is well defined, and it suffices to treat it on the dense regular subset. In our case is the regular stratum of the orbifold . For the behavior of geodesics on Riemannian orbit spaces see for example (1); the easiest way to carry these results over to infinite dimensions is by using Gauss’ lemma, which only holds if the Riemannian exponential mapping is a diffeomorphism on an -open neighborhood of 0 in each tangent space. This is rarely true.
Given a -invariant Riemannian metric on , one can define a metric distance on by taking as distance between two shapes the infimum of the lengths of all (equivalently, horizontal) smooth curves connecting the corresponding -orbits. There are the following questions:
Does separate points? In many cases this has been decided.
Is a geodesic metric space? In other words, does there exists a rectifiable curve connecting two shapes in the same connected component whose length is exactly the distance? This is widely open, but it is settled as soon as local minimality of geodesics in is established.
In this article we are discussing Riemannian metrics on , that are induced by Riemannian metrics on via Riemannian submersions. However all metrics on , that we consider, arise as restrictions of metrics on . Thus, when dealing with parametrized shapes, we will use the space and restrict ourselves to the open and dense subset , whenever we consider the space of unparametrized shapes.
3.4 Diffeomorphism group
Concerning the Lie group structure of the diffeomorphism group we have the following theorem.
Theorem 3.3 (Thm. 43.1, (76))
Let be a smooth manifold, not necessarily compact. The group
of all compactly supported diffeomorphisms is an open submanifold of (equipped wit a refinement of the Whitney -topology) and composition and inversion are smooth maps. It is a regular Lie group and the Lie algebra is the space of all compactly supported vector fields, whose bracket is the negative of the usual Lie bracket.
An infinite dimensional smooth Lie group with Lie algebra is called regular, if the following two conditions hold:
For each smooth curve there exists a unique smooth curve whose right logarithmic derivative is , i.e.,
Here denotes the right multiplication:
The map is smooth, where and is the unique solution of (4).
If is compact, then all diffeomorphisms have compact support and . For the group of all orientation preserving diffeomorphisms is not an open subset of endowed with the compact -topology and thus it is not a smooth manifold with charts in the usual sense. Therefore, it is necessary to work with the smaller space of compactly supported diffeomorphisms. In Sect. 7 we will also introduce the groups and with weaker decay conditions towards infinity. Like they are smooth regular Lie groups.
3.5 The space of Riemannian metrics
We denote by the space of all smooth Riemannian metrics. Each is a symmetric, positive definite tensor field on , or equivalently a pointwise positive definite section of the bundle .
Theorem 3.4 (Sect. 1.1, (52))
Let be a compact manifold without boundary. The space of all Riemannian metrics on is an open subset of the space of all symmetric tensor fields and thus itself a smooth Fréchet-manifold.
For each and we can regard as either a map
or as an invertible map
The latter interpretation allows us to compose to obtain a fiber-linear map .
4 The -metric on plane curves
4.1 Properties of the -metric
We first look at the simplest shape space, the space of plane curves. In order to induce a metric on the manifold of un-parametrized curves we need to define a metric on parametrized curves , that is invariant under reparametrizations, c.f. Sect. 2.2. The simplest such metric on the space of immersed plane curves is the -type metric
The horizontal bundle of this metric, when restricted to , consists of all tangent vectors, that are pointwise orthogonal to , i.e., for some scalar function . An expression for the metric on the quotient space , using the charts from Thm. 3.2, is given by
This metric was first studied in the context of shape analysis in (95). The geodesic equation for the -metric on is given by
Note that this is not an ODE for , because , being the curvature of , depends implicitly on . It is however possible to eliminate and arrive at (see ((95), Sect. 4.3))
a nonlinear hyperbolic PDE of second order.
Are the geodesic equations on either of the spaces or for the -metric (locally) well-posed?
The -metric is among the few for which the sectional curvature on has a simple expression. Let and choose such that it is parametrized by constant speed. Take two orthonormal horizontal tangent vectors at . Then the sectional curvature of the plane spanned by them is given by the Wronskian
In particular the sectional curvature is non-negative and unbounded.
This metric has a natural generalization to the space of immersions of an arbitrary compact manifold . This can be done by replacing the integration over arc-length with integration over the volume form of the induced pull-back metric. For the metric is defined by
For all its simplicity the main drawback of the -metric is that the induced geodesic distance vanishes on . If is a path, denote by
its length. The geodesic distance between two points is defined as the infimum of the pathlength over all paths connecting the two points,
For a finite dimensional Riemannian manifold this distance is always positive, due to the local invertibility of the exponential map. This does not need to be true for a weak Riemannian metric in infinite dimensions and the -metric on was the first known example, where this was indeed false. We have the following result.
The geodesic distance function induced by the metric vanishes identically on and .
For any two curves and there exists a smooth path with , and length .
For the space an explicit construction of the path with arbitrarily short length was given in (95). Heuristically, if the curve is made to zig-zag wildly, then the normal component of the motion will be inversely proportional to the length of the curve. Since the normal component is squared the length of the path can be made arbitrary small. This construction is visualized in Fig.2.
For vanishing of the geodesic distance is proven in (5); the proof makes use of the vanishing of the distance on and on .
In fact, this result holds more generally for the space . One can also replace by an arbitrary Riemannian manifold ; see (94).
The vanishing of the geodesic distance leads us to consider stronger metrics that prevent this behavior. In this article we will present three different classes of metrics:
Almost local metrics:
where is a suitable smooth function.
Sobolev type metrics:
where is a suitable differential operator.
Metrics that are induced by right invariant metrics on the diffeomorphism group of the ambient space.
4.2 Gradient flows on curves
The -metric is used in geometric active contour models to define gradient flows for various energies. For example the curve shortening flow
is the gradient flow of the energy with respect to the -metric.
The following example is taken from (85). The centroid based energy , with fixed and denoting the centroid, attains its minimum when . The -gradient of the energy is
We see from the second term that the gradient flow
tries to decrease the length of the curve for points with , but increase for . This latter part is ill-posed. However the ill-posedness of the gradient flow is not an intrinsic property of the energy, it is a consequence of the metric we chose to define the gradient. For example the gradient flow with respect to the -metric
is locally well-posed. See (125); (123); (124) for more details on Sobolev active contours and applications to segmentation and tracking. The same idea has been employed for gradient flows of surfaces in (140).
5 Almost local metrics on shape space
Almost local metrics are metrics of the form
where is a smooth function that is equivariant with respect to the action of , i.e.,
Equivariance of then implies the invariance of and thus induces a Riemannian metric on the quotient .
Examples of almost local metrics that have been considered are of the form
where is a function of the total volume , the mean curvature and the Gauß curvature . The name “almost local” is derived from the fact that while and are local quantities, the total volume induces a mild non-locality in the metric. If depends only on the total volume, the resulting metric is conformally equivalent to the -metric, the latter corresponding to .
For an almost local metric the horizontal bundle at consists of those tangent vectors that are pointwise orthogonal to ,
Using the charts from Thm. 3.2, the metric on is given by
Almost local metrics, that were studied in more detail include the curvature weighted -metrics
with in (95) and the conformal rescalings of the -metric
with in (135); (115), both on the space of plane curves. More general almost local metrics on the space of plane curves were considered in (96) and they have been generalized to hypersurfaces in higher dimensions in (3); (12); (13).
5.1 Geodesic distance
Under certain conditions on the function almost local metrics are strong enough to induce a point-separating geodesic distance function on the shape space.
If satisfies one of the following conditions
for some , then the metric induces a point-separating geodesic distance function on , i.e., for we have .
For planar curves the result under assumption 1 is proven in ((95), Sect. 3.4) and under assumption 2 in ((115), Thm. 3.1). The proof was generalized to the space of hypersurfaces in higher dimensions in ((12), Thm. 8.7).
The proof is based on the observation that under the above assumptions the -length of a path of immersions can be bounded from below by the area swept out by the path. A second ingredient in the proof is the Lipschitz-continuity of the function .
then the geodesic distance satisfies